Quantum World of Nuclear Physics

The Quantum World of Nuclear Physics

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Founder and Editor: Ardeshir Guran Co-Editors: M. Cloud & W. B. Zimmerman

About the Series Rapid developments in system dynamics and control, areas related to many other topics in applied mathematics, call for comprehensive presentations of current topics. This series contains textbooks, monographs, treatises, conference proceedings and a collection of thematically organized research or pedagogical articles addressing dynamical systems and control. The material is ideal for a general scientific and engineering readership, and is also mathematically precise enough to be a useful reference for research specialists in mechanics and control, nonlinear dynamics, and in applied mathematics and physics.

Selected Volumes in Series A Vol. 6

Mathematical Problems of Control Theory: An Introduction Author: G. A. Leonov

Vol. 7

Analytical and Numerical Methods for Wave Propagation in Fluid Media Author: K. Murawski

Vol. 8

Wave Processes in Solids with Microstructure Author: V. I. Erofeyev

Vol. 9

Amplification of Nonlinear Strain Waves in Solids Author: A. V. Porubov

Vol. 10 Spatial Control of Vibration: Theory and Experiments Authors: S. O. Fteza Moheimani, D. Halim, and A. J. Fleming Vol. 11 Selected Topics in Vibrational Mechanics Editor: I. Blekhman Vol. 12 The Calculus of Variations and Functional Analysis: With Optimal Control and Applications in Mechanics Authors: L P. Lebedev and M. J. Cloud Vol. 13 Multiparameter Stability Theory with Mechanical Applications Authors: A. P. Seyranian and A. A. Mailybaev Vol. 14 Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities Authors: V. A. Yakubovich, G. A. Leonov and A. Kh. Gelig Vol. 15 Process Modelling and Simulation with Finite Element Methods Author: W. B. J. Zimmerman Vol. 16 Design of Nonlinear Control Systems with the Highest Derivative in Feedback Author: V. D. Yurkevich

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Series A

^

B

k

Volume 17

Founder & Editor: Ardeshir Guran Co-Editors:

M. Cloud & W. B. Zimmerman

The Quantum World of Nuclear Physics

Yuri A. Berezhnoy Kharkov National University, Ukraine

\ ^ World Scientific NEW JERSEY • LONDON

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Preface

Quantum physics, which governs the motions and interactions of microobjects, is a firmly established part of our lives. It has provided working formulae for the design of nuclear reactors, electronic devices, and superconductive magnets, and has helped us understand processes occurring in solids, liquids, gases — even the stars. Many technologies are based directly on quantum laws, which differ in principle from the vastly more familiar laws of classical physics. Quantum mechanics was originally formulated to explain the structure and properties of atoms. Its further development showed that it can describe a huge variety of physical phenomena. It has served as a basis for the creation of atomic and nuclear physics, elementary particle physics, and solid-state physics. It underpins the operation of semiconductors and lasers, of nuclear reactors and weapons, etc. These directions within science and technology are fascinating, and the many engineering and military applications are of great importance in today's world. Various aspects of quantum mechanics, describing the motion and structure of molecules, atoms, atomic nuclei, and elementary particles, as well as the structure of substances, do receive attention in high school physics textbooks. The coverage given is, however, cursory at best; necessary concepts are typically treated in a way that is both incomplete and inconsistent, and the unfortunate reader is left without a solid understanding of this essential branch of physics. The creation and development of quantum mechanics have also led to deep alterations in certain philosophical views regarding the world in which we live. This alone warrants an integrated and sufficiently simple presentation of those quantum concepts that every educated person should know, regardless of his or her profession. V

vi

The Quantum World of Nuclear Physics

Quantum physics has shown that the main laws of Nature have a statistical rather than a dynamical character. This means that various physical processes follow probabilistic laws, and the strict determinism of classical mechanics is only revealed as a limiting case of this probabilistic description. Moreover, the probabilistic behavior is characteristic of not only large ensembles but of individual objects (molecules, atoms, atomic nuclei, and elementary particles) as well. Quantum mechanics may seem inaccessible to many people. The main barrier, however, is a simple hesitancy to abandon the habitual notions of classical physics that are subject to constant reinforcement during our everyday experience. Indeed modern physics, by its very character, lends itself to understanding by anyone willing to devote sufficient time to its study. The present book was written to facilitate this. No attempt has been made to expound quantum mechanics to its fullest extent; rather, the presentation is restricted to main ideas, concepts, and applications to the theory of atoms and subatomic structures. Despite the rather complicated mathematics that governs its behavior, the quantum world is a truly fascinating place to visit. The author invites anyone with an interest in modern science to "take the plunge" into this microscopic universe of atoms and nuclei. The book will have fulfilled its mission if it manages to spark some interest in atomic and nuclear physics. The author is indebted to Michael Cloud and William B.J. Zimmerman for the great work they have done editing the book, and to V.V. Pilipenko and Yu. P. Stepanovsky for reading the manuscript and offering useful comments. He is also grateful to Jaime Humberto Lozano Parada and V. Yu. Korda for production assistance. Yu. A. Berezhnoy

Contents

Preface

v

1. Quantum Mechanics

1

1.1 1.2 1.3 1.4 1.5 2.

3.

4.

Why Two Types of Mechanics? Main Ideas and Principles of Quantum Mechanics Measuring the Physical Characteristics of Microobjects . . . Structure of Atoms Structure of Matter

1 8 17 22 31

Fundamental Interactions

41

2.1 2.2 2.3 2.4 2.5

41 51 53 55 57

Gravitational Interaction Electromagnetic Interaction Weak Interaction Non-Conservation of Parity in Weak Interaction Strong Interaction

Structure of Atomic Nuclei

67

3.1 3.2 3.3 3.4

67 76 81 86

Composition and Properties of Nuclei Shell Model of Nuclei Collective Motions of Nucleons in Nuclei Superfluidity of Nuclear Matter

Radioactivity of Atomic Nuclei

89

4.1 The Law of Radioactive Decay 4.2 Alpha-Decay 4.3 Beta-Decay vii

89 98 102

viii

The Quantum World of Nuclear Physics

4.4 Gamma-Radiation of Nuclei 4.5 Exotic Types of Radioactivity 4.6 Application of Radioactive Isotopes 5. Nuclear Reactions 5.1 5.2 5.3 5.4 5.5

Conservation Laws in Nuclear Reactions Nuclear Reaction Mechanisms Nuclear Optics Accelerators Detectors of Particles

6. Fission of Atomic Nuclei 6.1 6.2 6.3 6.4

Nuclear Fission Mechanism Chain Fission Reactions Nuclear Reactors Man-Made Synthesized Elements

7. Nuclear Astrophysics and Controlled Nuclear Fusion 7.1 7.2 7.3 7.4 Index

Expanding Universe Creation of Atomic Nuclei Evolution of Stars Controlled Nuclear Fusion

106 107 109 115 115 119 127 138 145 147 147 155 159 162 167 167 171 176 181 187

Chapter 1

Quantum Mechanics

1.1

Why Two Types of Mechanics?

We live in a complicated world. Our sense organs provide a steady flow of information regarding the numerous phenomena that surround us. Powerful technological inventions also extend the reach of the human senses, giving us access to information more exact and complete than would otherwise be available. The world of our sense perception is the macroscopic one. Here physical phenomena are described by classical physics, which includes classical mechanics, continuum mechanics (hydrodynamics and the theory of elasticity), thermodynamics, and electrodynamics. Because classical physics deals with phenomena in which microscopic structure plays no significant role, it cannot yield a comprehensive theory of the structure of real substances. The laws of classical physics govern the motions of objects whose linear dimensions are sufficiently large: Rc\ > 10"6 m, say. Nothing more powerful than an optical microscope will be needed to observe such objects. Classical mechanics, in particular, describes the motions of planets, comets, stars, and galaxies. But there exists another world, inaccessible to direct observation through our sense organs. This is the amazing world of micro-objects, in which physical phenomena are subject to the laws of quantum mechanics. The dimensions of molecules, atoms, atomic nuclei, and elementary particles are very small and could be characterized as -Rqu < 1(T8 m. l

2

The Quantum World of Nuclear Physics

Thus we have two distinct physical theories. One describes macroscopic phenomena, the other microscopic phenomena. Why do two types of mechanics exist? The answer is far from simple. Let us pursue the question in more detail. Until the end of the 19th century, practically all physical phenomena were described using classical mechanics. This subject was originally expounded by Sir Isaac Newton in his "Philosophiae naturalis principia mathematica", a monumental work published in 1687. The appearance of each new experimental fact had required only modification to some old equation, or perhaps the introduction of a new one, but had not cast doubt on classical physics itself. Around the turn of the 20th century, however, existing approaches based on classical physics failed to describe new experimental data on atomic and subatomic phenomena. Two of these were black body radiation and the photoelectric effect. In physics, the term "absolutely black body" is used for an object that absorbs light (i.e., electromagnetic radiation) but does not reflect it. A model of a radiating absolutely black body is a closed box with impenetrable walls, in which a tiny hole is made so that we can observe the radiation. In 1860, Gustav Kirchhoff established that the intensity of black body radiation depends on temperature and frequency only, and not on the substance from which the walls are made. In 1896 Wilhelm Wien found that the radiation energy per unit volume and unit frequency (the density of black body radiation) decreases according to an exponential law /9(w,T)~w 3 exp(-aa;/r), where u> is the frequency, T is the temperature, and o is a constant. Wien's law holds for large frequencies (aui/T 3> 1). In 1900 Lord Rayleigh discovered that at low frequencies {aui/T ,T). The behavior of the radiation density in the intermediate range of frequencies was still an open question. At the end of the 19th century the German physicist Max Planck inves-

3

Quantum Mechanics

tigated the dependence of the electromagnetic radiation intensity of a black body on frequency. Naturally, Planck was an adherent of classical physics because there was no other science at that time. However, in 1900 he established that the available experimental data could be explained only under the assumption that black body radiation is emitted in discrete portions. The energy of such a portion is given by the formula E = hu), where h = A = 1.05457266 • 10~34 J • s.

(1.1)

The constant h (here the bar reminds us of the division by 2TT) is called Planck's constant, and the energy portions are called quanta (from the Latin word "quantum", meaning "how much"). Planck's constant has the dimension of action (or of angular momentum); i.e., h is a quantum of action. This result was really overwhelming, as a discrete character for energy is impossible in classical physics. Planck's law meant a qualitatively new approach which required new physical and conceptual levels of understanding. This was the first "brick" to be laid in the foundation of an emerging physical theory: quantum mechanics. For his work on the discovery of energy quanta, Planck was awarded the Nobel Prize in Physics for 1918. P / 21 I

0

co

Fig. 1.1 Blackbody radiation density p versus frequency u>. On the assumption that black body radiation is emitted in quanta, Planck obtained a formula for the radiation density, p(ui,T), which agrees with the known limiting cases for small and large frequencies and also stands in good agreement with the experimental data over the whole frequency

4

The Quantum World of Nuclear Physics

range (Fig. 1.1):

P^T) = ^{etlr_iy

(1-2)

where k is Boltzmann's constant. If the frequency is small (huj 2kT/(Tr2c3). For large frequencies (fko 3> kT) Planck's formula (1.2) turns into Wien's law p(u,T) = (w3h/TT2c3)exp{-fru/kT) See Fig. 1.1. The next major step was taken by Albert Einstein in 1905. He introduced the concept of a quantum of light and established that the photoelectric effect (the emission of electrons from a substance by light, a phenomenon discovered by Hertz in 1887) obeys the following law: Em = hw - A,

(1.3)

where Em is the maximum energy of an emitted electron, HUJ is the energy of the absorbed quantum of light, and A is the photoelectric work function (i.e., the energy required for emission of electrons from the substance). Einstein received the Nobel Prize in Physics for this discovery in 1921. Note that the term photon for a quantum of electromagnetic radiation was introduced by J. Lewis in 1926.l Today the photon is regarded as an elementary particle with zero rest mass and spin 1, possessing energy fko and momentum fuv/c. In 1911 Ernst Rutherford discovered the atomic nucleus, whose linear size turned out to be four orders of magnitude smaller than that of the atom. It became clear that an atom is a system in which electrons somehow move around a very small nucleus, and the planetary model of the atom arose. There was, however, an important contradiction with classical physics. An electron moving along a curved trajectory through an electric field must 1 Lewis regarded the quantum of light as an indivisible atom, an assumption which did not stand the test of time. Nonetheless, his term "photon" has become a standard term in physics and has even entered the popular lexicon. What would Star Trek be without photon torpedoes?

5

Quantum Mechanics

emit electromagnetic waves. The resulting loss of energy should be accompanied by a decay in the electron's orbit, this process continuing until the electron finally falls into the nucleus and the atom collapses. According to classical physics then, any atom must cease to exist after some finite time interval. But it was already known that atoms are actually stable and have no such constraint on their lifetimes. This contradiction was resolved by Niels Bohr in 1913 in his famous work "About the structure of atoms and molecules". Bohr theorized that an electron in an atom can remain for an infinitely long time only in certain states characterized by discrete values of the total energy. So the idea of discreteness, already known for electron energy, was used to explain atomic structure. Bohr obtained a correct formula for the electron energies in the hydrogen atom (the energy spectrum for this atom). For this purpose he used the equality of the centrifugal and electric forces acting on a charged particle (an electron) moving along a circular orbit in a Coulomb field:

(1-4)

^ = 4-

Here e is the electronic charge (traditionally, in formula (1.4) the CGSE system of units is used), m is the electron mass, and rn and vn are the orbital radius and velocity values. It is assumed that the mass of the nucleus is infinitely large in comparison to that of the electron. The number n {main quantum number} takes positive integer values n = 1, 2, 3 , . . . and serves to index the possible states of the atom (the electron orbits). Furthermore, Bohr supposed that the angular momentum of the electron can take only discrete values: (1-5)

mvnrn — nh.

While the relation (1.4) is usually referred to as "classical," the relation (1.5) is referred to as "quantum mechanical" as it involves the quantization of angular momentum (the requirement that angular momentum values be discrete). However, in today's established theory of quantum mechanics, the angular momentum cannot be written as a product of the momentum mvn and distance rn. So the left-hand side of (1.5) is a purely classical expression for the moment of momentum. From (1.4) and (1.5) one can readily obtain the quantities rn and vn: h2 me2

2

e2 hn

,

.

6

The Quantum World of Nuclear Physics

The total energy En of an electron in the hydrogen atom is a sum of the kinetic energy mv^/2 and the potential energy —e2/rn. Therefore, we find that _

me4

^—wj-

(L7)

The negative sign signifies that the electron exists in a bound state. The first five energy levels in the spectrum of the hydrogen atom are shown in Fig. 1.2, where one can see the levels getting closer as the binding energy of an electron decreases. Later, the energy expression (1.7) was obtained on the basis of quantum mechanics via solution of the Schrodinger equation.

S =

n=4

n=2

n=1 Fig. 1.2 The first five levels of the energy spectrum of the hydrogen atom. While deriving (1.7) Bohr postulated the concept of permissible orbits of the electron, having radii rn, as well as the possibility of emission and absorption of a certain portion (quantum) of electromagnetic radiation by the electron during its transition between orbits. This is the essential content of Bohr's two famous postulates. It is obvious that the energyfiu>of such a quantum must equal the difference of the electron energies for these orbits. If the electron passes from an orbit with energy En to one with energy Es (we assume that n > s), it emits a quantum (a photon) with energy tlaJ = En-Es.

(1.8)

Formula (1.8) is the Bohr frequency relation. Substituting the expres-

Quantum Mechanics

7

sion for the energy (1.7) into it, we obtain

» = £(?-;?)•

(")

By fixing the number s we obtain a series of frequencies (the Lyman series for s = 1, the Balmer series for s = 2, the Paschen series for s = 3) within which the number n defines a certain line. For each series we must keep n > s. Bohr noted that from (1.9) it is impossible to obtain other series of frequencies that were ascribed to hydrogen at the time. He was right to point out that these series belong to helium. Bohr's frequency relation (1.8) or (1.9) was important because the radiation frequency u> did not coincide with that of the electron orbit. For this reason the result differed in principle from, for example, those of Nicolson (who assumed these two frequencies must be equal). For his investigation of the structure of atoms and the radiation emanating from them, Bohr was awarded the Nobel Prize in Physics for 1922. Let us emphasize that in the works of Planck, Einstein, and Bohr, which formed the basis for the development of quantum mechanics, the corresponding final formulae were correct, but the methods by which they were found are subject to criticism. In fact, these scientific giants used their powerful intuition to guess the necessary results and causes of the corresponding physical phenomena. However, these results cannot be obtained in the framework of classical physics as it existed at that time. Quantum mechanics, as elaborated later, made it possible to derive these formulae by means of quite different and strict methods. In other words, the methodology for obtaining quantum results used by Planck, Einstein, and Bohr was not correct, and is now of mainly historical interest. Let us also note that the energy formula (1.7) obtained by Bohr is valid only for the hydrogen atom, but for more complex atoms Bohr's approach does not hold. Moreover, the approach is valid only for circular orbits, and cannot be applied to elliptical ones in this form. The explanation of radiation from an absolutely black body, the photoelectric effect, and the structure of the hydrogen atom required the introduction of new concepts that turned out to be incompatible with classical physics. This quandary was the impetus for the creation of a new theory: quantum mechanics. So it is important to ascertain whether there exist general properties of our world which require the existence of two types of mechanics. In physics, there is a relation between the fundamental symmetry prop-

8

The Quantum World of Nuclear Physics

erties of space and time and the laws of conservation of certain physical quantities. For example, the equivalence of all moments of time (temporal homogeneity) leads to the law of energy conservation for a closed (isolated) system. Similarly, the equivalence of all points of space (spatial homogeneity) leads to the law of conservation of momentum, and the equivalence of all directions in space (the isotropy of space) leads to the law of conservation of angular momentum. These symmetry properties are geometrical symmetries since they are not connected with concrete types of interaction. Space possesses one more geometrical symmetry, namely that of similarity (scale invariance), which is connected with changes in spatial scale. However, the laws of nature turn out not to be invariant with respect to a similarity transformation (scale change). In reality then, the principle of similarity is not valid. This can be formulated as follows: the absolute size of a body does matter; two systems that are geometrically similar but different in scale are fundamentally different in mechanical behavior. The failure of the similarity transformation was known to Galileo. He understood that if the sizes of animals and men were to be essentially increased, a significant increase in the firmness of their bones would become necessary; otherwise the bodies of giants would collapse under their own weight. Another example is a model of a building (e.g., a house) made of matchsticks. If such a model were scaled up to realistic dimensions, it would collapse similarly. So the principle of similarity is not valid at macroscopic scales. The existence of the smallest natural building blocks — molecules and atoms — which have finite sizes, the existence of the elementary electric charge, and the limiting speed of signal propagation (the speed of light), all imply a failure of the similarity principle and non-invariance of natural laws with respect to scale transformations. Hence follows the existence of two types of mechanics: (1) the classical version, valid at distances large compared with the linear sizes of molecules and atoms, and (2) the quantum version, whose laws describe physical processes at distances comparable with the linear sizes of molecules and atoms or smaller.

1.2

Main Ideas and Principles of Quantum Mechanics

In 1923 the French physicist Louis de Broglie suggested that a material particle having nonzero mass also possesses wave properties uniquely related to its mass and energy. He ascribed a wavelength to a free particle, which

Quantum Mechanics

9

is related to its momentum p by the formula

(10) The quantity X (or A) has been called the de Broglie wavelength for the particle. The concept of the wave nature of particles with nonzero rest mass was developed by de Broglie in a series of works in 1924. He received the Nobel Prize in Physics for the year 1929. No one in France could grasp the profundity of de Broglie's idea. His doctoral committee (consisting of the physicists Perrin and Langevin, the mathematician Cartan, and the crystallographer Mauguin) may not have even conferred his degree. But Langevin sent the thesis to Einstein for evaluation, and the answer was "He has lifted an edge of the great curtain." De Broglie's formula (1.10) deserves a place alongside Planck's formula E = hu and Einstein's formula E — me2. With de Broglie's discovery it became clear that material particles possess physical properties similar to those of quanta of electromagnetic radiation (photons): they have both wave properties and particle properties. The discovery of the wave nature of microparticles was a brilliant guess which led to a true upheaval in our understanding of the physics of microphenomena. In 1927 the Americans Davisson and Germer and, independently, the Englishman G. Thompson, discovered electron diffraction by crystals experimentally. This was a remarkable corroboration of the predicted wave properties of material particles. Davisson and Thompson received the Nobel Prize in Physics for 1937. We should stress that wave properties are inherent even in a single material microparticle. If one passes an electron beam of very low intensity through a crystal, so that individual electrons fly through the crystal independently of one another, then with sufficiently long exposure the same diffraction pattern will be observed as for a beam of high intensity. Since an individual electron causes the blackening of only one grain of the photoemulsion on the screen, electron diffraction means that one can only indicate a probability that an electron will reach some point of the screen. But it is impossible to calculate its trajectory. Thus, in 1924 two basic ideas of quantum mechanics were already known. The first was that of quantization: the possibility that a physical quantity will take a discrete series of values under certain conditions. The second was the wave-particle duality of quantum objects, i.e., that any

10

The Quantum World of Nuclear Physics

quantum object (photon, electron, atom, molecule, etc.) is both a corpuscle (particle) and a wave at the same time. Under different conditions the particulate or wave properties of a certain object can be manifested to a greater or lesser extent. Both quantum ideas were in absolute contradiction with classical physics, in which physical quantities take only continuous values. Moreover, in classical physics a particle and a wave are incompatible, whereas in quantum mechanics a particle and a wave cannot be separated. We stress that the wave-particle duality of a microobject should be understood as its potential ability to manifest its particulate or wave properties, depending on the conditions under which it is observed. Particles and waves are different forms of the same physical reality. In 1925 the German physicists Heisenberg, Born, and Jordan elaborated matrix mechanics, which was the first variant of quantum mechanics. In 1926 the Austrian physicist Erwin Schrodinger developed wave mechanics, based on an equation later named after him. To describe a microobject state, Schrodinger introduced the wave function (the i[>-function). In 19261927 the Englishman Paul Dirac made a great contribution to the elaboration of mathematical techniques for quantum mechanics, and in 1927 proposed the method of secondary quantization. For the creation of quantum mechanics, Werner Heisenberg was awarded the Nobel Prize in Physics for 1932. Schrodinger and Dirac were awarded the Nobel Prize in Physics for 1933 for the discovery of new productive forms of atomic theory. In 1926 Born gave a statistical interpretation of the wave function V>(r) as a probability amplitude such that the quantity |i/>(r)|2d3r yields the probability that the particle resides in volume d3r in the vicinity of the point r. We stress that the wave function can be a complex quantity. It can also depend on time. For his fundamental research in quantum mechanics, especially for his statistical interpretation of V'(i'), Born was awarded the Nobel Prize in Physics for 1954. Although the mathematical body of quantum mechanics was mainly elaborated in 1925-1926, it remained unclear why Heisenberg's matrix mechanics and Schrodinger's wave mechanics yield the same results. Development of mathematical techniques has made it clear that these are simply two equivalent variants of the same science, subsequently called quantum mechanics. The probabilistic character of the laws of quantum mechanics is due to the intrinsic randomness of the behavior of microobjects. Quantum mechanics can only predict the probability that a physical quantity will take a given value. For this reason, the probability concept plays a

Quantum Mechanics

11

fundamental role in quantum mechanics. However, probability in quantum mechanics differs substantially from this notion elsewhere in physics. The wave function is a probability amplitude and, generally speaking, can be a complex quantity. As distinct from other parts of physics, in quantum mechanics the wave functions (the probability amplitudes) are added to each other, but not the probabilities themselves. In this way, interference terms arise in the total probability. Let us illustrate the interference of probability amplitudes as an example of the superposition principle, which plays an important role in quantum mechanics. In the simplest variant it can be formulated as follows. If a given physical system can be in a state described by a wave function ip\ and the same system can be in another state described by a wave function fa, then it can also be in the state described by the function ^ = 01^1+^2^2,

(1-11)

where oi and a
M2.

The superposition principle leads to an important consequence: the equations of quantum mechanics must be linear. Indeed, the Schrodinger equation, which is the basic equation of nonrelativistic quantum mechanics, is linear. If the number of states in which the system can exist is greater than two, then the superposition principle appears as follows:

^ = X)a n ^ n ,

(1.12)

n

where the sum is extended over all possible states of the system. The number of system states can be infinite. Then the sum in (1.12) will contain an infinite number of addends. The probability that the system is in the state described by the wave

12

The Quantum World of Nuclear Physics

function (1.11) is defined by the expression

w = h/f = M V i l 2 + M2|V>2|2 + (aiaSVi^a + a\a^l^)-

(1.13)

The probability w consists of three terms. The first two are the probabilities of finding the system in the states 1 and 2, respectively, while the third represents an interference of the probability amplitudes. The occurrence of interference terms is characteristic of quantum mechanics. It is connected with the intrinsic wave properties of microobjects. The interference existing in quantum mechanics is the main conceptual difference between the classical and quantum descriptions of objects and processes. It can be illustrated by considering a particle motion along different paths from point 1 to point 2. In classical physics the probabilities of the particle motion along each path are summed. In quantum mechanics the amplitudes associated with these paths are summed, and then the squared modulus of the resulting amplitude is the probability of the particle having passed from point 1 to point 2. In 1927 Heisenberg established the uncertainty relation. Let us consider its simplest form. If we introduce the root-mean-square deviations (dispersions) for the coordinate and momentum <(Az)2} = (x2) - (x)\

<(APl)2> = {pi) -
(1.14)

then it turns out that their product for any microobject cannot be smaller than the value fh/2: ((Ax)2)((APx)2)

> ^.

(1.15)

The angle brackets in (1.14) and (1.15) denote the averaging of quantities in the microobject state under consideration. Formula (1.15) is the uncertainty relation for the position and momentum. It means that the more precisely we determine the position of a microobject, the greater the indeterminacy of its momentum component along the same axis will be, and vice versa. The uncertainty relation (1.15) shows that in quantum physics the simultaneous use of the notions of position and momentum is nonsense, although the very notions of position and momentum do have physical meaning. Uncertainty relations similar to (1.15) also exist for other pairs of quantities which are called conjugate pairs (for example, the energy of a microobject and the time of its interaction with a measuring device, the

Quantum Mechanics

13

projection of the angular momentum onto the axis of quantization and the azimuthal angle, etc.). Physical laws have, as a rule, the form of certain exclusions (restrictions) imposed upon physical quantities under consideration. However, only three exlusions hold throughout physics.2 The first is the uncertainty relation, which forbids the product of the dispersions of two conjugate physical quantities to be less than fr?/4. The second is the impossibility of existence of perpetual motion of the first and second kinds. The third does not allow a particle with nonzero rest mass to increase its velocity from a value v < c to a value v > c (the impossibility of crossing the light cone). The universal character of the uncertainty relation means that it is also valid in classical physics. However, owing to the smallness of the quantity h, it is never manifested in classical physics because in this case the uncertainties (dispersions) ((Ax)2) and ((Apx)2) are so small that they cannot be noticed. Let us emphasize that the existence of the uncertainty relation is a general law of quantum mechanics which is not connected with a method or accuracy of measurement of the corresponding quantities. The uncertainty relation (1.15) leads to an important consequence. Since it is impossible to determine simultaneously a position of a microobject and the corresponding component of momentum with arbitrary accuracy, then this implies the absence of any trajectory of the microobject, which is defined in classical physics as a function p(r). In other words, if we know the particle position at a given moment of time, we cannot know it at any subsequent moment. Rather, we can only determine the probability of finding the particle at a point in space at any subsequent moment. The absence of trajectories of microobjects turns out to be of great importance for systems of identical particles. If at a given moment of time we number identical particles, then at any following moment we cannot indicate where each of the numbered particles is. This property of systems of identical particles can be formulated in the following way: the physical properties of a quantum system of identical particles are invariant with respect to the permutation of any pair of particles in this system. In particular, the probability defined by the squared modulus of the wave function of a system of identical particles, \4>\2, is invariant with respect to such a permutation. However, it is clear that in this case the wave function ip itself may be not invariant with respect to the permutation of a pair of identical particles. 2 It is worth noting that the familiar conservation laws are not among these so-called global exclusions.

14

The Quantum World of Nuclear Physics

There are two possibilities: the wave function is symmetrical and does not change under the permutation of a pair of identical particles, or it is antisymmetrical and does change sign. It is proved in quantum mechanics that, if the wave function of a system of identical particles is symmetrical (antisymmetrical) at a given moment of time, then it will be symmetrical (antisymmetrical) for all time. Thus, a general problem arises to determine which systems of identical particles are described by symmetrical wave functions and which are described by antisymmetrical ones. It turns out that the symmetry properties of a wave function are completely denned by the spin (intrinsic angular momentum) of particles which, like the orbital moment, is measured in units of Planck's constant h. If the spin of a particle is an integer (possibly zero), then the wave function is symmetrical, and if the spin is half-integral, then the wave function is antisymmetrical. Spin is an inherent characteristic of a particle, like its mass and charge. It is a purely quantum characteristic, having no analogue in classical physics. In quantum mechanics the orbital moment of a particle, which is related to motion in space, takes only discrete values that are multiples of h. For this reason, in quantum mechanics it is convenient to measure angular momentum in units of Planck's constant h. The spins of different particles can have (in units of h) both integral and half-integral values. The spins of electrons, protons, neutrons, and neutrinos are equal to 1/2, the spins of 7r-mesons are equal to zero, and the spin of a photon equals 1. If identical particles possess an integral spin, then they obey the BoseEinstein statistics and are called bosons. If identical particles have a halfintegral spin, then they obey the Fermi-Dirac statistics and are called fermions. A system of identical bosons is described by a symmetrical wave function, and a system of identical fermions by an antisymmetrical wave function. In 1924-1925 Wolfgang Pauli investigated the properties of systems of identical particles, and established what was later to be called the Pauli exclusion principle: Two identical fermions cannot be in the same state. For this discovery Pauli was awarded the Nobel Prize in Physics for 1945. Although Pauli provided a proof of the exclusion principle, the latter had been used intuitively by Bohr in 1921-1922 when he classified atoms on the basis of their electron structure. The validity of the exclusion principle can be easily understood as fol-

Quantum Mechanics

15

lows. Let us assume that two identical fermions are in the same state. Then their permutation does not change the wave function of the system, since in this case the system turns into itself by virtue of the indistinguishability of the identical particles. On the other hand, the wave function of the system of two identical fermions must be antisymmetrical, i.e., it must change its sign under their permutation. For this reason, the wave function in the case under consideration equals zero. In other words, the probability of two identical fermions being in the same state is equal to zero. If one considers a system of identical particles at low temperature, then it turns out that a system of identical bosons behaves in a different way as compared with a system of identical fermions. At absolute zero all the identical bosons would be in their lowest energy state. In this case, Bose condensation arises. In the case of a system of identical fermions, no more than one particle can be in each state, even at absolute zero, owing to Pauli's exclusion principle. For this reason, a certain distribution of fermions over energies arises. Since atoms of different elements have different chemical properties, and since electrons in metals, even near absolute zero, have large energies, it is obvious that various physical characteristics of systems of identical bosons and fermions near absolute zero must be different. Let us consider another interesting quantum effect. The solution of the Schrodinger equation for the potential of a harmonic oscillator shows that a quantum oscillator with frequency u> has the lowest energy EQ = tvuj/2, which is called its zero-point vibrational energy. In other words, the quantum oscillator differs from the classical one in the fact that it does not rest in its lowest energy state, but performs these zero-point vibrations with energy E0- In quantum mechanics, it can be shown that the energy EQ is the minimum energy of the oscillator which is permitted by the uncertainty relation. This amazing property of quantum oscillators has been verified experimentally. In particular, it means that, when approaching absolute zero, atoms situated at points of the crystal lattice of a solid perform zero-point vibrations. One can investigate the scattering of electromagnetic radiation (photons) on atoms of the lattice. From the character of this scattering at temperatures close to absolute zero one can determine whether atoms at the points of the lattice rest or vibrate. These experiments have uniquely shown that atoms at the lattice points perform the zero-point vibrations with the energy Eo, because the zero-point vibrations substantially change the character of the photon scattering.

16

The Quantum World of Nuclear Physics

Zero-point vibrations are also performed by any physical field even in the absence of the particles that are quanta of this field. For example, zeropoint vibrations of an electromagnetic field lead to a small change of the energy levels of atoms as compared to those calculated without taking them into account. This phenomenon was for first time observed experimentally by the American physicist Willis Eugene Lamb in 1947. It has been called the Lamb shift, and has been explained by calculations carried out on the basis of quantum electrodynamics. For this discovery Lamb was awarded the Nobel Prize in Physics for 1955. The concept of a field plays a very important role in modern quantum physics. A field can exist even in the case when the corresponding particles are absent. This state of a field is called vacuum. However, there are zero-point vibrations that occur in a vacuum, which sometimes leads to a spontaneous creation of a pair consisting of particle and antiparticle. This pair is subsequently annihilated. For example, in an electromagnetic field electron-positron pairs can be spontaneously created and annihilated. It is for this reason that in quantum theory the distinction between a particle and field disappears. The successes of classical physics had led to a profound confidence in the possibility of unambiguous predictions of various physical events. The most prominent representative of this conception of necessity was the famous French scientist Pierre-Simon Laplace. In his words, "An intellect, which could know all the forces in the nature and the relative situation of the things composing it at a given instant and which could be powerful enough to subject them to an analysis, would be able to comprehend the motions of the greatest bodies in the Universe and of a smallest atom by a single formula, and nothing would remain unknown, both the past and the future would be open to him." So Laplace believed that a knowledge of the positions and velocities of all particles of a physical system made it possible to predict its behavior. This statement expresses the "Laplace determinism" that underlies classical physics. In quantum mechanics the situation is different: it is impossible to know the positions and velocities of particles simultaneously, by virtue of the uncertainty principle. One can only define the wave function at an initial moment of time, which is the most complete description of the state of a

Quantum Mechanics

17

physical system. In quantum mechanics, which uses the wave function, only a probabilistic description of events is possible. Laplace's classical causality does not hold for microobjects, whose behavior is inherently random. The probabilistic character of the quantum description of the behavior of microobjects is also rooted in the interaction of a microobject with the rest of the world. It is just the attempt to isolate a microobject or some physical system from the rest of the world that leads to the randomness in their behavior. In the words of Heisenberg, "It is necessary to draw one's attention to the fact that a system which should be considered according to the methods of quantum mechanics actually is a part of much larger system and, ultimately, of the whole world. It is in an interaction with this large system, and we have also to say in addition that microscopic properties of the large system are, at least to a great extent, unknown". That is why, by virtue of the probabilistic character of the quantum laws, the physical processes of microobjects are characterized by a probabilistic form of causality, of which Laplace's determinism is only a limiting case. However, one should not think that quantum mechanics is not a deterministic theory. It is deterministic in the sense that it defines the law of the wave function variation with time. Elements of unpredictability and randomness arise only with attempts to interpret a microobject on the basis of classical conceptions about its positions and velocities. 1.3

Measuring the Physical Characteristics of Microobjects

An important criterion for the validity of a physical theory is agreement between calculated predictions and experimental data. Another is the ability to predict results of future measurements from those already available. Hence the measurement of physical characteristics of microobjects is an issue of great importance in quantum mechanics. We are macroscopic objects living in the macroscopic world. This necessitates that any device (detector) used for quantum measurements also be a macroscopic object, so that we can perceive the information it obtains. Such a device must also, as a macroscopic system, exist in a rather unstable state that can be easily changed under the influence of a microobject. For example, in a bubble chamber a superheated transparent liquid instantly

18

The Quantum World of Nuclear Physics

boils when a charged particle flies through it (Sect. 5.5). The resulting bubbles of vapor allow us to see the particle trajectory. Here the thickness of the trajectory is large in comparison to the atomic scale, so the uncertainty relation is not violated. By such a rough approximation a microobject can be considered classically. The detector necessarily and significantly changes the state of the microobject with which it interacts. In classical physics it is assumed that the influence of a measurement device on an object can be made arbitrarily small. During a quantum measurement, one cannot in principle neglect the interaction between the detector and the microobject. Consequently, the uncontrollable character of the interaction between detector and microobject leads to the necessity of the probabilistic description of quantum processes, since the measurement destroys the initial quantum state of the microobject in an unpredictable way. The process of a quantum measurement is irreversible: as a result of the measurement, the wave function of the microobject changes abruptly; i.e., a reduction or "collapse" of the wave function occurs. Reconstruction of the initial state of the microobject is absolutely impossible after the measurement process has occurred. Thus the irreversibility of the process of measurement plays a fundamental role in quantum physics. By virtue of this irreversibility, an irreproducibility of a single measurement arises. Since in each act of measuring the interaction between the detector and microobject occurs in a different way, the measured results will be different. Only a sufficiently large number of measurements will give a certain stable pattern of the distribution of results. This pattern can also be obtained in another series including a sufficiently large number of measurements. The state of a microobject is not defined before a measurement. A series of measurements performed with the same detector over identical, as one would think, microobjects yields a set of different results. If a beam of electrons passes through a slit in a screen, then different electrons will reach different points of a photoplate and a certain diffraction pattern will appear. In this case one can only determine the probability that the electrons arrive at different points of the photoplate. In other words, a certain statistical distribution of electrons on the photoplate arises which is not chaotic. The task of quantum mechanics is to determine the probability distribution for various physical quantities characterizing microobjects. Let us now direct the electron beam onto a screen with two slits A and B (Fig. 1.3). If A is open and B is closed, then on the screen-detector we

19

Quantum Mechanics

Aw ....!?w

.....

x

Fig. 1.3 Distribution of electrons passing through a screen with two slits. observe the electron distribution I\ (x) that corresponds to a wave function ipi. If only B is open, the electron behavior will be described by a wave function ip2 and we will observe the electron distribution h{x). If A and B are open, then the wave function of the electrons is ip = V>i + 4>2- In this case each electron, owing to its wave nature, will pass through both slits simultaneously. The probability density of this process is denned by the expression w

= |^| 2 = i^i 2 + ^ 2 | 2 + (1/^2 + v.1^5).

(1.I6)

The third term in (1.16) describes the interference of waves passing through the two slits simultaneously. This case corresponds to the electron distribution I{x). If the electrons are allowed to pass through the slits opened in turn, then the probability distribution of this process will be defined by a sum of the probabilities of electron passage through each slit separately: w = \ipi\2 + \ip2\2-

(1-17)

In this experiment the interference disappears; i.e., controlling the electron passage through a certain slit in the screen destroys the interference. In other words, quantum measurement destroys interference and we observe

20

The Quantum World of Nuclear Physics

the electron distribution Iz{x). Quantum measurements are characterized by the union of a measuring device and the microobject over which the measurement is performed. It is this inseparable union of a macroscopic (classical) device and the microobject under analysis that leads to their uncontrollable interaction changing the microobject state. Quantum mechanics is a probabilistic theory, which makes it principally different from classical physics. However, it turns into classical mechanics in the limiting case when Planck's constant h becomes negligible. Formally, the transition to classical mechanics is performed when one lets this quantity decrease to zero: h —» 0. Quantum mechanics is grounded in, and therefore irrevocably linked to, classical mechanics. The limit h —> 0 can be understood as follows. If n ^> 1, then, according to (1.5), the angular momentum of an electron in an atom becomes very large relative to Planck's constant: mvnrn 3> h. In other words, in the case n ^> 1 the constant h can be neglected and the discreteness of the angular momentum disappears. Thus, quantum mechanics turns into classical mechanics when quantum numbers are large — this is the correspondence principle initially formulated by Bohr. In particular, this means that for large quantum numbers the frequency of radiation emitted by an atom at the transition from one state to another asymptotically coincides with the frequency predicted by the classical theory. If the atom passes from an excited state with energy En+i to a state with energy En, then the radiation frequency equals uin = (En+\ — En)/h, where En and En+\ are defined by (1.7). Then

_ roe4 [

( n

\2

For n » 1 we get 1 — (n/n + I) 2 « 2/n. Thus, we find that _ me4 _ vn h3n3

rn'

where the electron velocity vn and the "orbital radius" rn are defined by the formulae (1.6). In classical mechanics the frequency w of the electron revolution with velocity v along the orbit of radius r is equal t o u = v/r. We can see that for n » 1 the quantum result coincides with this. Moreover, the distance between neighboring energy levels of the hydrogen atom tends to zero for

Quantum Mechanics

21

large quantum numbers (n —• oo). In this limiting case, the discreteness of the energy spectrum becomes less significant and the atomic system behaves like a classical one. Because of its probabilistic approach to the explanation of microprocesses, quantum mechanics was rejected by some scientists during its development. One of the opponents of the quantum theory was Einstein, who helped sow the seeds for its creation. He proposed various arguments to prove that quantum mechanics is not valid. However, Bohr, also a giant of Physics, parried with cogent counterarguments in support of quantum mechanics. In 1935, a paper by Einstein, Podolsky, and Rosen appeared, in which they put forward their famous paradox. According to their initials, it became known as the EPR-paradox. We will not expound the essence of this or certain other paradoxes that were put forward during the development of quantum mechanics in order to prove its incompleteness. We shall only note that the outstanding intellect of Niels Bohr resolved all these paradoxes. Discussions regarding the completeness of the quantum-mechanical description of microobjects have led to the conjecture that the uncertainty in the behavior of a quantum object is explained by the existence of some "hidden" parameters, about which observers know nothing. Just the presence of these hidden parameters could lead to the probabilistic behavior of microobjects and to the uncertainty of the results of measurements. It followed from this approach that knowledge of the hidden parameters could allow one to predict exactly the microobject behavior, i.e., that the determinism of classical physics would triumph. The first proof of the nonexistence of hidden parameters was given by J. von Neumann. However, a formulation of the proposition was required which could experimentally corroborate the absence of hidden parameters. Finally, in 1965, Bell proposed a statement (the Bell theorem) that made it possible to ascertain experimentally the distinction between the predictions of quantum mechanics and the theory of hidden parameters. Experiments based on the Bell theorem were carried out by Clause and Freedman at the University of California in 1972, as well as by Aspect, Dalibard, and Roger at the Paris Institute of Optics in 1982. These and other experiments have proved the validity of quantum mechanics and the failure of the hidden parameter theory. Undoubtedly, experimental studies in this direction will continue. However, the theory of hidden parameters, at least in its present form, does not agree with experimental data.

22

1.4

The Quantum World of Nuclear Physics

Structure of Atoms

Bohr's 1913-1922 works, which investigated the structure of atoms, gave correct results which were, in fact, guessed, since the methods used to obtain them were incorrect. Later, these results were obtained on the basis of quantum mechanics. Bohr's formulas (1.6) and (1.7) are of value because they allow us to predict a characteristic linear size for the hydrogen atom, as well as a characteristic velocity and energy for the electron in the atom. For n = 1 (the ground state of hydrogen) we have n = - ^ w 0.53 • 1CT10 m, me1 e2 Vl = — « 2.2 • 106 m/s, TflP

4

£?i = - 2 ^ - « - 1 3 . 6 eV.

(1.18)

The quantity r\ has been called the radius of the first Bohr orbit. It shows that the characteristic linear size of the atom, by order of magnitude, is Rat ~ 10~10 m. Using the constants m, e, H, one can compose the quantity rat = ?l3/(me4) ss 2.5 • 10~17 s, which is a characteristic atomic time. To understand to what extent the smallest particles of substance are small, let us consider a continuous chain composed of atoms lined up side by side. If every inhabitant of our planet had put one atom into this chain, its length would be equal to several centimeters. By virtue of the uncertainty relation (1.15), a microobject described by quantum mechanics has no trajectory. This means that there exist no electron orbits in atoms, only probabilities of different distances between the electron and nucleus. For example, for the ground state of the hydrogen atom (n = 1) this probability equals wi(r) = Cr2exp (~)

,

(1.19)

where C is a constant. The probability (1.19) has a maximum at r = r\, i.e., the radius of the first Bohr orbit is the most probable distance between the electron and nucleus in the ground state of the hydrogen atom. We stress that in reality the electron does not revolve around the atomic nucleus like a planet moving around the Sun. It simply exists in the atom and can be found at any point

23

Quantum Mechanics

of the atom with a strictly definite probability. According to quantum mechanics, a hydrogen atom in the ground state has spherical symmetry, whereas in classical mechanics a system consisting of two charged particles cannot be in a spherically symmetrical state. We note that the initial approach of Bohr, which is based on classical mechanics, gave the correct formula (1.7) for the energy of an electron in a hydrogen atom, but it led to an incorrect value of the electron orbital moment I. Indeed, for n = 1 (the ground state of the hydrogen atom), formula (1.5) yields the value I = 1 (in units of H); according to quantum mechanics, it should be I = 0. The structure of atoms is mainly denned by the electric interaction of atomic electrons with the nucleus and between the electrons themselves. If there are two or more electrons in the atom, then this is already a complex system of mutually interacting particles. However, it turns out that it is a good approximation for a complex atom to consider that each electron moves in an effective central field which is created by the nucleus and other electrons. Such a field is called the self-consistent field. A state of an electron in a central field (in a spherical atom) is characterized by the main quantum number n, the orbital moment I, and the projections of the orbital moment m and of the spin a on the quantization axis. The electron spin projection can take two values: a = ±^. The electron orbital moment projection (the magnetic quantum number) can take 21 + 1 values, since it takes integer values with the limits —l<m
0 s 2

1 2 p d 6 10

3 f 14

4 g 18

5 h 22

6 i 26

24

The Quantum World of Nuclear Physics

To denote the states of electrons with definite values of orbital moment, the literal notation presented in Table 1.1 is used. The lower line of Table 1.1 contains the total number of electrons in a closed shell with the given values of quantum numbers n and I. The state of the atomic electron with definite values of n and I is denoted by a figure indicating the value of n and a letter indicating the value of /. Sometimes, one refers to electrons with main quantum numbers n = 1; 2; 3; 4; 5; 6; ..., as being in the K-, L-, M-, N-, O-, P~, Q-, ..., shells, respectively.

1s

/

/ /

/

2s

/

2p

/

3s

/

3p

/

3d

/

/

/

/

4s 4p

'

/

4d 4f

/

5s 5p

/ /

/

5f

7s

/

6p

7p

/

5d

/

/

6s

/

6d

/

6f

7d

/

/

7f

Fig. 1.4 The order in which atomic shells arefilledis shown by arrows. The order of filling of the atomic shells is shown by the arrows in Fig. 1.4. However, one should remember that there are some exceptions to this rule. The distribution of electrons in shells is called the electron configuration of the atom. Let us consider this for some particular atoms. A hydrogen atom contains one electron in the state Is. In a helium atom there are two equivalent electrons in the same state, which differ by the values of their spin projections on the quantization axis. For this reason, an atom of helium has electron configuration Is 2 , where the superscript denotes the number of equivalent electrons in this state. A lithium atom has the electron configuration Is 2 2s, while a beryllium atom has the configuration Is 2 2s 2 . In the boron atom the first electron with I = 1 (the p-electron) appears. The electron configuration of this atom has the form Is22s22p. Further, the filling of the 2p-shell continues: a carbon atom has the electron configuration Is2 2s 2 2p 2 , a nitrogen atom has the configuration Is22s22p3, and so on. An atom of neon, which is an inert gas, has all shells filled: Is 22s22p6. In atoms of sodium and magnesium, the 3s-shell is filled. This is also true for the 3j>-shells in atoms of the elements from aluminum to argon. In atoms of potassium and calcium the 4s-shell is filled, and

Quantum Mechanics

25

in atoms of the elements from scandium to zinc the 3
26

The Quantum World of Nuclear Physics

spin of an atom. This was the beginning of quantum chemistry. According to their investigations, the valency is the doubled spin of an atom joining a compound. The same atom can manifest different valencies depending on the state in which it enters into a compound. Atoms enter into compounds in such a way that their spins should mutually compensate. Let us consider the valencies of elements of the basic groups. (1) Atoms of elements of the first group (Li, Na, K, Pb, Cs, and Fr, being alkaline metals) have spin 1/2 in the ground state and manifest the valency 1. Their first excited state lies far from the ground state. For this reason they usually enter into compounds in the ground states. (2) Atoms of elements of the second group (Be, Mg, Ca, Sr, Ba, and Ra, being the alkaline-earth metals) have zero spin in the ground state and do not enter into compounds in this state. However, they have an excited state whose last shell configuration is sp (instead of s2) with spin 1 and this is energetically close to the ground state. So atoms of these elements can enter into compounds in these excited states and manifest the valency 2, in this case. (3) Atoms of elements of the third group (B, Al, Ga, In, and Tl) have electron configuration of the last shell s2p and the spin 1/2 in the ground state. Close to the ground state they have an excited state with the configuration of the last shell sp2 and the spin 3/2. For this reason, the elements of this group manifest valencies 1 and 3. Exceptions are the first two elements B and Al, which are always trivalent. (4) Atoms of elements of the fourth group (C, Si, Ge, Sn, and Pb) have the electron configuration of the last shell s2p2 with spin 1 in the ground state, and close to it there is an excited state of configuration sp3 with spin 2. Hence they manifest valencies 2 and 4. The first two elements of this group, C and Si, mainly manifest valency 4 (an exception is, for instance, carbon monoxide CO). (5) Atoms of elements of the fifth group (N, P, As, Sb, and Bi) in the ground state have the configuration of the last shell s2p3 with spin 3/2. The closest excited state has configuration sp3s', where s' denotes the electron state which has main quantum number being greater by one than in the state s. The spin of this excited state is equal to 5/2. For this reason, the elements of the fifth group manifest valencies 3 and 5. (6) Atoms of elements of the sixth group (O, S, Se, Te, and Po) in the ground state have the configuration of the last shell s2p4 with spin 1. Besides, they have excited states with the configurations s2p3s' and

Quantum Mechanics

27

sp3s'p' with spins 2 and 3, respectively. For this reason, the elements of the sixth group manifest valencies 2, 4, and 6. An exception is oxygen, which is always bivalent because its atoms enter into compounds only in the ground state. (7) Atoms of the seventh group (F, Cl, Br, I, and At, being the halogens) in the ground state have the configuration of the last shell s2p5 with spin 1/2. They can also enter into compounds in the excited states with configurations s2p4s', s2p3s'p', sp3s'p'2, having spins 3/2, 5/2, 7/2, respectively. For this reason, they can manifest valencies 1, 3, 5, 7. An exception is fluorine, which is always univalent. (8) Atoms of elements of the eighth group (He, Ne, Ar, Kr, Xe, and Rn, being the inert gases) in the ground state have perfectly filled shells with zero spin. For this reason, they are practically always inert. Some can enter into compounds; this stems from the transition of electrons from the outer filled shell to the states of the unfilled d- and /-shell, which are close in energy. Quantum mechanics also makes it possible to explain the valencies of elements of the transition groups. In atoms of these elements the filling of the d- and /-shells occurs. Electrons in the d- and /-shells of atoms are situated deeper than the outer s- and p-electrons. For this reason, the interaction of these atoms with other atoms and molecules is usually weaker than for the atoms of basic group elements. In particular, among the compounds of transition group elements, molecules with nonzero spins are often encountered. Atoms of these elements can manifest both even and odd valencies, which are defined both by the interaction of outer electrons and by the possibility to enter into compounds in excited states when the deep-lying electrons pass over into the s- and p-shells. When atoms join a molecule, the electron densities in the filled shells change only slightly. However, in this case the electron densities in the unfilled shells change substantially. From this point of view, there are two extreme cases. The first is the heteropolar or ionic bond, in which all valence electrons pass over from some atoms to others and the molecules consist of charged ions with perfectly filled shells. For example, in the molecule NaCl, the sodium atom gives its s-electron to the chlorine atom, and the positively charged ion Na + together with the negatively charged ion Cl~ form a molecule of sodium salt. Another extreme case is the homopolar or covalent bond, in which atoms remain neutral and the valence electrons become "collective". Examples occur in the molecules H2, CI2, and so on.

28

The Quantum World of Nuclear Physics

Naturally, there also exist intermediate types of bond which are partially ionic and partially covalent in character. Let us now consider some aspects of the atom structure. The wave function of an electron in a central field, i.e., in a spherical atom, can be written in the form il>nlm{r,0,
(1.20)

where Rni(r) is the radial wave function, Yim{9,(p) is the spherical harmonic function whose definition can be found in mathematical reference books, and r, 9, tp are the spherical coordinates of the electron. We present the mathematical expressions for the first several normalized spherical harmonics: Ym{0,
Y1,±1(0,
Yw(9,
Ti\[fe±i'<'sme, V 47T

Y2,±1{0,
Y2,±2(9,v) =

-J]^e±2i*Sm2e.

The spherical function Yjm (9,
(1.21)

The squared modulus of the spherical function, \Yim(9, (p)\2, is a standard mathematical expression that contains no information about interaction of the electron with the atom field. Unfortunately, in some textbooks this quantity is called the "electron cloud" and pupils learn the pictures

29

Quantum Mechanics

representing \Yim(O,
(1.22)

ipni{r,O,ip) = Rnl{r) J2 CmYim(e,y), m=-l

where the cm are some constant (generally speaking, complex) coefficients. The quantity |c m | 2 defines the probability of the electron being found in an unobservable state with the definite value of m. Since the electron energy Eni in the state n, I does not depend on the quantum number m, it is usually said that this state is degenerate in m. The multiplicity of this degeneracy is equal to 21 + 1. Degeneracy of the state with the energy Eni with respect to the electron spin projection a is equal to 2 since
wnl(r,e,
£c m Y i m (0,cp)

2

.

(1.23)

m=-l

The angular dependence of the probability wni(r, #,
30

The Quantum World of Nuclear Physics

ulus of a superposition of spherical functions, but not of a single spherical function. The radial portion of probability \Rni(r)\2 contains all the information about interaction of the electron with thefieldof the atom and defines, in particular, the average distance between the electron and nucleus. If the quantity r2\Rni(r)\2 has one maximum, then this state corresponds to a circular orbit, as a classical analog. If it has two or more maxima, then the classical analogs for these cases are elliptic orbits. The electron energy in a hydrogen atom (1.7), as distinct from the electron energies in other atoms, depends only on the main quantum number n and not on the orbital moment I of the electron. Thus, an electron state in a hydrogen atom, having the energy En\, is degenerate in both m and I. For this reason, the wave function of an observable electron state in a hydrogen atom with energy En\ has the form n-l

I

(1.24)

iPn(r,0,cp) = Yl J2 cimRni{r)Ylm(e,v), 1=0

m=-l

where cim are some constant (generally speaking, complex) coefficients. The quantity |Q m | 2 defines the probability that the electron is in an unobservable state with definite values of the quantum numbers I and m. The probability that the electron in a hydrogen atom, having energy En, is at the point r,0,tp is equal to n-l

I

wn(r,e,cp) = \*ljn(r,e,v)\2= Y, Y, clmRni{r)Ylm{6,v)

2

(1-25)

/=0 m=-l

It is evident that, for the electron in a hydrogen atom which is in an observable state with the definite energy value En, it is impossible to separate a factor depending on the angles only. In other words, for a state of a hydrogen atom with the definite energy value En, the probability that the electron is located at the point r, 9, if is not separable into radial and angular factors. Thus an "electron cloud" is not a probability distribution describing the presence of an atomic electron at certain points in space for an observed state with definite energy, and does not correspond to modern concepts about atomic structure. This is simply an unsuccessful attempt to create a visual model of a microobject, which is impossible in principle. In serious textbooks on quantum mechanics, the notion of an "electron cloud" is, naturally, not used.

Quantum Mechanics

1.5

31

Structure of Matter

An explanation of the structure and various properties of solids, liquids, and gases turned out to be possible on the basis of quantum mechanics. Full consideration of these questions requires substantial analysis using complicated mathematical techniques that lie beyond the scope of this book. For this reason, we shall restrict ourselves only to the consideration of some important properties of the condensed media, which are connected with their microscopic structure. Many solids are crystals. A crystal is a form of substance in which a regular arrangement of atoms having a three-dimensional periodicity is observed. A crystal structure is defined by the geometry of the crystal lattice in whose points atoms, ions, and molecules are situated. In biological crystals, the lattice points are occupied by proteins, nucleic acids, or even viruses. Other solids are amorphous. These are characterized by an isotropy in their properties, i.e., by an absence of any strict periodicity in the arrangement of their atoms, ions, molecules, or groups thereof. Crystalline substances can be divided into three main groups with respect to their ability to conduct electric current: metals (conductors), dielectrics, and semiconductors. Metals have electrical conductivity in the range 108-106 (fim)"1. The corresponding range for dielectrics is 10~"810"10 (ffrn)"1. Semiconductors occupy an intermediate position between metals and dielectrics in terms of conductivity. We should note that the properties of substances with respect to the conduction of electric current can be substantially modified by external conditions such as temperature and pressure. Metals are widespread in Nature. It is enough to notice that under normal conditions 83 of the first 106 elements are metals. Metals exhibit a practically free conduction of electric current. Their conductivity decreases as temperature increases. A metal is a crystal whose lattice points contain ions, because some atomic electrons (the valence electrons) break away from their atoms. Collectively, these free electrons can be considered as a gas within the metal. At the metal's boundary, there exists a potential barrier that prevents electrons from flying away from the metal. If one applies a high-intensity electric field £ ~ 108 V/m to a metallic cathode, a cold emission current results. This is a purely quantum macroscopic phenomenon, which is connected with the sub-barrier passage of electrons known as the tunnel effect (Fig. 1.5). As a result, electrons

32

The Quantum World of Nuclear Physics

u(x)

barrier without electric field

U° *

metal

\

vacuum

u^eSx

x

Fig. 1.5 The tunnel effect. fly away from the metal and a current is observed. The dependence of this current /(£) on the electric field intensity £ is denned by an expression that can be obtained only on the basis of quantum theory: (1.26)

I(£)=Ioexp(-£o/£), where IQ and £Q are constants.

c

E>
o c

//////7T

15

free atom

crystal

Fig. 1.6 Energy levels of electrons within a crystal. Since electrons are fermions, they obey the Pauli principle. For this reason, electrons that have broken away from atoms and are practically free over the volume of the crystal occupy different energy levels (Fig. 1.6). Distinct from a free atom, in a crystal the permitted values of electron energies form bands as shown in Fig. 1.6. These bands are separated by forbidden regions. Each band contains a large number of energy levels that can be occupied by electrons. Widths of the bands increase as the electron binding energy decreases, i.e., the deepest atomic energy levels correspond

33

Quantum Mechanics

to very narrow bands in the crystal. Let us note that in Fig. 1.6 the simplest case is shown when the bands do not overlap. In reality, band overlapping is possible. Now consider the mechanism of electric current flow through a crystal. In an unexcited crystal all free electrons tend to occupy the lowest permitted energy levels. By the Pauli principle, electrons occupy all levels up to some highest occupied level, whose energy is called the Fermi energy Ef. Fig. 1.7 shows two cases of the highest occupied level situation in a crystal. From this viewpoint only two bands are of interest: the upper one, which is the conduction band, and the lower one — the valence band. Energy levels in all the bands lying below are completely occupied, and those in all the higher-energy bands are empty.

iHM

g <j / A W W A ? (D ^

XXXXXXXXX

MM /////////

TXTTAXTTTT

Fig. 1.7 The topmost energy level bands in a crystal. If the lowest empty level is situated far from the upper edge of the band (Fig. 1.7a), then under the action of an electric field applied to the crystal, the electrons situated near the empty levels increase their energy and pass over to the nearby empty levels. Their movement creates an electric current in the crystal. Such crystals are metals, and the electrons creating the electric current in them are called conduction electrons. Any metal is characterized by a certain electrical resistance. It is caused by scattering of the conduction electrons on thermal vibrations of the ions, forming the crystal lattice, and on structural inhomogeneities of the lattice (the impurity atoms and lattice defects). The resistance of a metal decreases with decreasing temperature and increases with increasing temperature. If the band is completely occupied (Fig. 1.7b), then the first empty level lying in the conduction band is separated from the last occupied level by a forbidden energy band of width e. The quantity e is usually equal to several eV. For example, for diamond it equals e = 5.5 eV. In this case,

34

The Quantum World of Nuclear Physics

the voltage applied to the crystal cannot transfer electrons from the valence band to the conduction band, i.e., such a crystal is a dielectric and does not conduct electric current. Only a very high voltage (the material's breakdown voltage) can transfer electrons to the conduction band. If a substance has a narrow forbidden region (s < 1 eV), then thermal motion can cause a small number of electrons in the crystal to pass from valence band to conduction band. The positions left empty in the valence band are called holes. These behave like positively charged particles with a mass equal to the electron mass and with the charge +e. The number of electrons having sufficient energy for the transition from valence band to conduction band is proportional to exp(—e/kT) for a wide range of temperatures T. It is obvious that the conductivity of the material will have a similar temperature dependence. Such substances are called semiconductors. Their electrical conductivity increases with temperature (in contrast to the behavior of metals as mentioned above), because of the increase in vibrational amplitude of the ions forming the crystal lattice. This mechanism for conductivity is characteristic of pure semiconductors. We should note that at sufficiently high temperatures some dielectrics become semiconductors. In a semiconductor, electrons are constantly making the transition from valence band to conduction band. The reverse transition occurs as well; here a conduction electron and a corresponding hole disappear as mobile charge carriers. This process is called recombination. At any given temperature an equilibrium arises between the numbers of electron transitions to and from the conduction band. In other words, the number of electrons in the conduction band and the number of holes in the valence band is conserved at any given temperature. Note that in semiconductors, electrical conductivity is due to both the motion of negatively charged electrons and the motion of holes in the opposite direction. Some substances, with regard to their electrical properties, occupy an intermediate position between metals and semiconductors. They have a comparatively small number of electrons in the conduction band even at very low temperatures, i.e., they can conduct only small currents. Such substances are called the semimetals (examples include Bi, Sb, and As). For practical aims one usually employs not pure semiconductors but deliberately impure ones. They contain a small amount of admixtures which substantially change their physical properties. The impurities are of two types: donors and acceptors. A donor impurity will donate surplus electrons to a semiconductor and create electron conductivity — the result is

35

Quantum Mechanics

an n-type semiconductor. An acceptor impurity will capture some electrons from the valence band and create hole conductivity, resulting in a p-type semiconductor. a

b

gf /////A , Si 5 03 =5

_,J

r y y

'//////

fc

iy V X X X X

Fig. 1.8 Creation of conductivity in doped semiconductors.

Fig. 1.8 shows how conductivity is created in impure or doped semiconductors. In the donor semiconductor (Fig. 1.8a) the electron to be released is on the energy level of a doped atom situated close to the conduction band. For this reason, even at a low temperature, such an electron can pass to one of the lowest empty levels of the conductivity band. Typical examples of donor semiconductors are Ge and Si containing sparse numbers of P, As, or Sb atoms that have replaced atoms of the main element at some lattice points. The ionization energy of the doping atoms is very small: e' « 0.01 eV for germanium and 0.04 eV for silicon. For this reason, even at 77 K nearly all doping atoms are ionized, and in this case the number of conduction electrons is determined by the donor concentration. In an acceptor semiconductor (Fig. 1.8b) the doping atom has an empty energy level close to the valence band. The capture of an electron from the valence band on this level of the acceptor leads to the rise of a hole in the valence band and to the creation of hole conductivity. Typical examples of acceptor semiconductors are Ge and Si containing sparse numbers of B, Al, or Ga atoms. It is possible to grow a semiconductor (e.g., germanium) crystal consisting of two individually doped regions: one n-type and the other p-type. Such a pn crystal, if it has a narrow boundary region between the n- and pregions, can be used as a rectifier {diode). If positive and negative voltages are applied to the p- and n-regions, respectively, electrons will pass from the n-region to the p-region and holes will move in the opposite direction. So a continuous current will flow through the crystal. For an applied voltage of opposite polarity, the current will be practically nil.

36

The Quantum World of Nuclear Physics

More complicated devices can be constructed from semiconductor crystals of the npn-type. Here a narrow p-type layer, a few hundredthousandths of a meter wide, is placed between two n-type regions. If, for example, a positive voltage is applied to the left end of such a semiconductor and a negative voltage to its right end, then current flows. The left portion (emitter) is similar to a triode filament — it emits electrons — while the middle (base) and right (collector) portions are analogous to the triode grid and anode, respectively. This npn-semiconductor (transistor) can function as an amplifier, increasing the voltage and power of a signal delivered to its emitter. A pnp-semiconductor in which hole conduction takes place will operate analogously. Let us consider the magnetic properties of solids. Those substances that can be magnetized in an external magnetic field are said to be magnetic. Inside a magnetic substance during magnetization, an intrinsic magnetic field arises. There exist weak-magnetic substances: diamagnets and paramagnets. Diamagnetic substances include He, Ar, Au, Zn, Cu, Ag, Hg, water, and glass. Paramagnetic substances include O, Al, Pt, the alkali metals, the alkaline-earth metals, and the rare-earth elements. In the absence of an external magnetic field, the atoms and molecules of diamagnets do not possess any intrinsic magnetic moments. If a diamagnet is placed in a magnetic field, then in each atom or molecule an additional current is introduced which creates a magnetic moment. According to Lenz's law, the current induced by an external magnetic field is always directed in such a way that it decreases the field inside the substance. For this reason, the induced magnetic moments are aligned opposite the direction of the external magnetic field, and the diamagnet is magnetized. When the external magnetic field is removed, the diamagnet is demagnetized. The atoms and molecules of a paramagnet have some intrinsic magnetic moments in the absence of an external magnetic field. In this case, however, random thermal motion leads to a chaotic orientation of their magnetic moments; i.e., the paramagnet does not possess an intrinsic magnetic field when an external magnetic field is absent. In an external magnetic field, the magnetic moments of atoms and molecules align along the field, and the paramagnet is magnetized. As already noted, when an external magnetic field is absent, the magnetic moments of atoms and molecules in a paramagnet are oriented chaotically. However, there exist substances which have an ordered structure in absence of an external magnetic field (Fig. 1.9). If in such a structure the mean magnetic moments of atoms are oriented in the same direction (Fig.

Quantum Mechanics

37

ttttttttttt ttUMtM

\/\A/\/\/ tmtmut -\/-\/-\/t+t+t*t*t*t ttitttittti Fig. 1.9 Orientation of magnetic moments in structured materials. 1.9a) so that a macroscopic volume magnetization arises, then the material is said to be ferromagnetic. But if the ordering is such that the overall magnetic moment equals zero (the magnetic moments of neighboring atoms are directed oppositely) and the macroscopic magnetization is absent, it is said to be antiferromagnetic (Fig. 1.9b). Finally, if the magnetic moments can have the opposite direction of ordering (Fig. 1.9c) but the magnetization is different from zero, the material is ferrimagnetic. The critical temperature above which the magnetic ordering disappears is called the Curie temperature or Curie point Tc in the ferro- and ferrimagnetic cases, and the Neel temperature or Neel point Tn in the antiferromagnetic case (after the French physicist Neel who, independently of Landau, explained the phenomenon of antiferromagnetism). Ferrimagnetism can be considered as the most general case of magnetic ordering. From this point of view, ferromagnetism and antiferromagnetism are particular cases of ferrimagnetism. Examples of ferromagnetics are Fe, Co, and Ni; antiferromagnetics include MgO, FeO, and NiO, while ferrimagnetics include Fe3O4, CoFe2O4, and NiFe2O4. Note that the critical temperature values for various magnetic substances lie in a very wide range.

38

The Quantum World of Nuclear Physics

For instance, the ferromagnetics Dy, Ni, and Fe have Curie temperatures of 37 K, 627 K, and 1043 K, respectively. Above the critical temperature, ferromagnetics and antiferromagnetics become paramagnets. At temperatures T < Tc, a ferromagnetic material consists of domains. A domain is a small volume of substance having linear sizes on the order of 10~4-10~5 m, inside which all magnetic moments of atoms are co-oriented, i.e., the domain has maximum possible magnetization. In the absence of an external magnetic field, the magnetic moments of domains inside a ferromagnetic sample are oriented chaotically so that the overall magnetic moment of sample is zero. Under the action of an external magnetic field, however, the magnetic moments of the domains align and the sample as a whole becomes magnetized. When the magnetic field is removed, some of the magnetic moments remain in alignment. For this reason, a residual magnetism can arise and permanent magnets can be created.

P

0

/

Tcr

T

Fig. 1.10 Superconductivity in mercury. Now let us consider phenomena that occur at very low temperatures. These are the superconductivity of some metals (e.g., Pb, Ta, Sn, Al, and Nb) and alloys (e.g., NbsSn), and the superfluidity of liquid helium. Superconductivity was discovered by the Dutch physicist Kamerlingh-Onnes in 1911, when he investigated the electrical resistance of mercury at low temperatures. It turned out that at the critical temperature Tcr =4.15 K, the resistance of mercury fell abruptly to zero (Fig. 1.10). Further, superconductivity of other substances was discovered. Each of them has its own critical temperature TCT for transition into the superconducting state. For his investigation of the properties of matter at low temperatures, which led to the production of liquid helium, Kamerlingh-Onnes was awarded the

Quantum Mechanics

39

Nobel Prize in Physics for 1913. If an electric current is induced in a superconducting metal ring, it will circulate in this ring without any damping, because its flow through the superconductor is not accompanied by any heat evolution. This property of semiconductors is used for the construction of superconducting magnets and other devices. It has also been established that a weak magnetic field does not penetrate a superconductor. The expulsion of magnetic field from a superconductor (the Meisner effect) means that in an external magnetic field a superconductor is a perfect diamagnet. The nature of superconductivity was explained by the American physicists Bardeen, Cooper, and Schrieffer in 1957. They were awarded the Nobel Prize in Physics for 1972 as a result. According to their theory, two conduction electrons in the crystal lattice of a metal create a bound state, the Cooper pair, which has zero total moment (spin). Let us consider the mechanism of creation of the Cooper pair in more detail. Two electrons experience Coulomb repulsion. However, an electron can excite or absorb a quantum of the lattice vibrations, which is called a phonon. If an electron excites a phonon and another one absorbs it, then such an exchange of a phonon leads to an attraction arising between the electrons; moreover, the attractive force in a superconductor turns out to be greater than the force of Coulomb repulsion. It is just this attractive force, arising from the presence of the crystal lattice, that leads to the pairing of electrons. Such a bound electron pair behaves like a particle. For this reason it is called a quasiparticle, which is a boson. At temperatures close to absolute zero, these quasiparticles experience Bose condensation. The Bose condensate, consisting of the electron pairs, moves freely (experiencing no resistance) through the crystal, which leads to superconductivity. The states of electrons in a superconductor are continuously changing and, for this reason, the compositions of electron pairs are continuously changing as well. Since the linear size of an electron pair is on the order of 10~6 m, this is an example of the long-range coupling of particles. At T = 0 K, all conduction electrons are paired. If T ^ 0, the probability of breaking an electron pair differs from zero. For this reason, at T ^ 0 the unpaired electrons form a normal electron "liquid" in the crystal, and the paired ones form a superconducting "liquid". Above T = Tcr the quasiparticles (Cooper pairs) become completely broken, and superconductivity disappears. Superfluidity is observed in liquid helium, which leaks through very narrow channels and capillaries without friction. This phenomenon was

40

The Quantum World of Nuclear Physics

discovered by the Soviet physicist Kapitsa in 1938. He established that at temperatures below T\ = 2.17 K (the lambda point), liquid 4He becomes superfluid. Kapitsa received the Nobel Prize in Physics for 1978 for this discovery and for his fundamental research on low temperature physics. For temperatures T > T\, liquid 4He is called He I; for T < T\, it is called He II. At the temperature T = T\ in liquid 4He, a phase transition of the second kind occurs, i.e., in this case its internal structure abruptly changes. Note that phase transitions of the second kind are undergone by ferromagnetics and antiferromagnetics at the Curie point, as well as by superconductors at the critical point. An explanation for superfluidity of liquid He II was given by Landau in 1941. According to his theory, there are two types of elementary excitations (quasiparticles) in He II: phonons and rotons. At finite temperatures, a part of He II behaves like a normal viscous liquid and another part behaves like a superfluid which possesses no viscosity. There is no friction when these liquids move through each other. The Soviet physicist Landau was awarded the Nobel Prize in Physics for 1962 for his pioneering theories regarding condensed matter, especially liquid helium. Consideration of the normal and superfluid components in He II is a convenient way to describe phenomena occurring in a quantum Bose liquid. This does not mean that the liquid is actually separated into two fractions. In reality, in He II two types of motion can occur simultaneously: one of them identical to a viscous liquid, and the other corresponding to a superfluid liquid. Both types of motion occur without transferring momentum from one to the other. In 1972-1974, superfluidity of 3He was discovered, which is observed at temperatures of several mK. In this case two atoms of 3He, which are fermions, create a pair which is a quasiparticle (boson) and the Bose condensation of these quasiparticles occurs, which leads to superfluidity of liquid 3He. Finally, we will note that quantum mechanics has made it possible to ascertain the physical laws of various processes and phenomena observed in condensed media, which in principle cannot be explained on the basis of classical physics. Studies of the microscopic structure of solids allowed people to create devices and mechanisms widely used in modern technology.

Chapter 2

Fundamental Interactions

2.1

Gravitational Interaction

Physics is concerned with matter: its structure and motion. The motion of matter is due to certain forces acting between bodies. The motion of galaxies and stars, of planets and comets, of electrons in TV sets and atoms, of nucleons and quarks in atomic nuclei, radioactive decays of atomic nuclei and elementary particles as well as all various processes in the Universe are caused by interactions between different physical objects. So it is not surprising that some of the most important questions in physics pertain to the study of these interactions. Over two millennia ago, the Greek philosopher Aristotle theorized that all substance in the Universe consisted of four elements — earth, air, fire, and water — and that these were subject to the action of two forces. The first was the force of gravity, which attracted earth and water downwards. The second was a "force of lightness", which served to attract fire and air upwards. Thus, Aristotle divided all of Nature into substances and forces. This approach has persisted in physics until the present day. Now, there are four known types of interactions: gravitational, electromagnetic, and the strong and weak nuclear forces. Let us consider each in turn. The gravitational interaction, by intensity, is the weakest of all the interactions known to us. The gravitational forces have a universal character. This means that all matter is subject to them; this is what the law of universal gravitation states. The range of gravitational forces is infinite. Gravitational forces are attractive. Gravitational interaction is mainly manifested between macroscopic bodies; it determines the motions of various cosmic objects: galaxies, stars, planets, etc. In the world of elementary particles, gravitational interaction is not directly apparent because of the very small 41

42

The Quantum World of Nuclear Physics

masses of the particles. Since the gravitational attraction of any two bodies is proportional to their masses, the notion of mass, which is one of the most important characteristics of a physical object, plays a fundamental role not only in the theory of gravitation, but in physics in general. The notion of mass has become more complicated after the creation of relativity theory. For this reason we shall consider it in detail.1 First, we shall consider the notion of mass in nonrelativistic (Newtonian) mechanics, which describes the motion of physical objects whose velocities are small relative to that of light. In this case, the momentum p of a physical body is related to its velocity v by the formula p = mv,

(2.1)

where the coefficient of proportionality m characterizing properties of the body is called the mass. The kinetic energy of the body is

(2.2) If a force F acts on the body, its momentum changes with time according to the law

(2.3) Since the acceleration a of the body is defined as the time derivative of its velocity,

(2.4) differentiation of (2.1) with respect to time leads to Newton's second law F = ma.

(2.5)

The mass m entering into (2.1) and (2.5) is called the inertia! mass. We stress that in nonrelativistic mechanics, (2.1) and (2.5) are equivalent. In these equalities, the mass plays the role of a coefficient of proportionality between the velocity and momentum, or between the acceleration and force. Now consider the gravitational attraction of bodies. The law of universal gravitation, discovered by Newton, states that the potential energy ^.B. Okun. The Concept of Mass. Uspekhi Fiz. Nauk, 1989, v. 158, N 3, p. 511.

Fundamental Interactions

43

of attraction of two bodies having masses M and m is determined by the formula Ug(r) = - G ^ ,

(2.6)

where the constant G = 6.67- 10~ u N-m2/kg2 and the negative sign signifies attraction. Differentiating equation (2.6) with respect to the vector r and taking into account the relation Fg = —dUg(r)/dr, we find the gravitational force

F. = - < £ £ , .

(2.7)

Equations (2.7) and (2.6) are equivalent, and the mass that appears in each is known as gravitational mass. It follows from (2.5) and (2.7) that the acceleration of a body moving in a gravitational field (e.g., a body falling in the gravitational field of the Earth) does not depend on its mass. If M is the mass of the Earth, we obtain

g = ^ = -<4r.

(2.8)

m r6 Since M « 6 • 1024 kg and the radius of the Earth RE « 6.4 • 106 m, we can substitute RE for r in (2.8) and find that g sa 9.8 m/s 2 . Galileo was the first to establish the universality of free-fall acceleration. Afterwards, the independence of the quantity g on the mass and substance of a falling body was confirmed by accurate experiments. The universality of g leads to the equality of the inertial and gravitational masses. In other words, the masses m of the same body, which enter into formulae (2.1), (2.5) and (2.6), (2.7), are the same. The equality of inertial and gravitational masses constitutes the principle of equivalence that plays an important role in relativity theory. According to this principle, when inside a system that moves with acceleration, one has no way to distinguish the accelerated motion from gravitation. In 1905 Einstein developed relativistic mechanics, which is also known as special relativity. This allows one to describe the motion of objects at velocities close to the speed of light c. The theory is based on the experimental fact that there exists a limiting speed of propagation for physical signals in Nature: c fa 3 • 108 m/s. General relativity, created by Einstein in 1915, accounts for gravitation. Many phenomena observed in the Universe and caused by the gravitational interaction of various cosmic objects

44

The Quantum World of Nuclear Physics

can be explained only on the basis of this theory. The special and general relativity theories are classical (non-quantum) theories. In relativistic mechanics, the relation between momentum and velocity of a freely moving body is defined by the formula

P = §v,

(2.9)

and the relation between energy and momentum is defined by E2=p2c2+m2c4.

(2.10)

The mass m and velocity v in (2.9) and (2.10) are the same quantities that enter into the formulae (2.1)-(2.7) of nonrelativistic mechanics. It follows from (2.9) and (2.10) that in relativistic mechanics the energy of a body with m ^ 0 does not become zero even when its velocity and momentum are equal to zero: v = p = 0. In other words, the body's rest energy EQ, according to relativity theory, is given by Eo = me2.

(2.11)

Relation (2.11) is the basis of conventional and atomic energetics, as well as of conventional and atomic military techniques. It was not known in nonrelativistic mechanics. It follows from (2.9) that E = pc if v = c. Substituting this into equation (2.10), we conclude that the mass of a particle moving with the speed of light is equal to zero. Conversely, if a particle has zero mass, it always moves with the speed of light. There is no reference frame for the massless particle in which it would rest; i.e., in any frame of reference such a particle moves with the speed of light. For this reason the photon, which is a massless particle, is doomed to be an "eternal wanderer", flying with speed c until it ceases to exist as a result of some electromagnetic process. It is convenient to express the energy and momentum of a particle with m ^ 0 through mass and velocity. For this purpose, we substitute (2.9) into (2.10):

Wl-J)=m 2 c 4 .

(2.12)

Denoting 7 = 1/^1 - v2/c2, we write (2.12) in the form E = mc2~f.

(2.13)

45

Fundamental Interactions

Substituting (2.13) into (2.9), we find that p = mvy.

(2.14)

The kinetic energy of a body in relativistic mechanics is (2.15)

Ek = E-E0=mc2{-1-l).

If u < c, expressions (2.14) and (2.15) turn into formulae (2.1) and (2.2) of nonrelativistic mechanics, respectively (for v -C c we have 7 « 1 + v2/2c2 + •••). This limiting case confirms the statement that the body mass m in nonrelativistic and relativistic mechanics is the same quantity. Now consider the relation between the force and acceleration in relativistic mechanics. For this purpose, let us differentiate equation (2.14) with respect to time. Then, taking account of definitions (2.3) and (2.4), we obtain F =ma7+^(a-v)73.

(2.16)

&•

It can be seen from (2.16) that generally, in relativity theory, the force has two different components: one directed along the acceleration and one directed along the velocity. This makes formula (2.16) differ in principle from formula (2.5) of nonrelativistic mechanics. Equation (2.16) correctly describes relativistic particle motion. It has been corroborated by numerous experiments. In particular, it formed the basis for the design of elementary particle accelerators that have operated around the world for years. Multiplying equation (2.16) by the velocity v, we find

a"V=m7(f+W)=^

(2'17)

Substituting (2.17) into (2.16), we have F-(F-v)^=ma7.

(2.18)

If the force is perpendicular to the velocity, F i v , then we obtain F = ma7.

(2.19)

However, if the force is parallel to the velocity, F || v, we find (2.20) F = ma 7 3 . We see from (2.18)-(2.20) that in special relativity theory, the ratio of the different forces to the acceleration is substantially dependent on the

46

The Quantum World of Nuclear Physics

mutual orientation of the force and velocity, that is, in relativistic mechanics the mass cannot be defined as the ratio of force to acceleration. In the nonrelativistic limiting case (v
v 2\

(r-v)vl

(2.21)

In the nonrelativistic limit u « c , the expression in square brackets reduces to r and E/c2 « m, i.e., we obtain (2.7). However, for v « c the quantity that could play the role of the gravitational mass of the relativistic particle depends not only on the particle energy, but also on the mutual orientation of the vectors r and v. If r || v, this quantity is equal to E/c2, but if r _L v, it is (E/c2)(l + v2/c2). For a photon we have m = 0 and v — c, i.e., when r || v, its "gravitational mass" equals E/c2, and for r J. v it equals 2E/c2. Hence, one can conclude that there is no sense in speaking about the gravitational mass of a photon or any other relativistic particle, since this quantity is different for objects moving at different angles with respect to the direction of the gravitational force. The mass of a body changes along with its internal energy. In particular, it changes during heating or cooling, as well as during changing the state of aggregation of the constitution of the body. For example, during the heating of iron by 200°C, the relative increase of its mass is Am/m = 10~12, while for the melting of ice into water we have Am/m = 3.7 • 10~12. This change of mass is very small and cannot be detected in experiments. Owing to the energy conservation law in chemical and nuclear reactions, the rest energies of bodies partially or completely transform into the kinetic energy of the reaction products if the overall mass of the particles entering into the reaction exceeds the overall mass of the reaction products. For example, during methane combustion the following chemical reaction occurs: CH4 + 2O2 —> CO2 + 2H2O.

(2.22)

Fundamental Interactions

47

As a result, the energy s = 35.6 MJ/m 3 is released. Since the density of methane is p = 0.89 kg/m3, we find that Am/m = e/pc2 = 4 • 10~10. The annihilation of an electron and a positron usually leads to the creation of two photons, and the whole rest mass of the initial particles transforms into the kinetic energy of the photons created. On the Sun and some stars, due to thermonuclear reactions, a fusion of four protons occurs that leads to the creation of an a-particle, two positrons, and two neutrinos: 4p—> a + 2e+ + 2ve.

(2.23)

The mass of four protons is 3755.08 MeV, the mass of a-particle is 3728.35 MeV, and the mass of two positrons is 1.02 MeV. In this process, the relative decrease of the mass is considerable: Am/m — 0.66 • 10~2. Thus, the relative change of mass Am/m in nuclear reactions is 78 orders higher than in chemical reactions. However, the mechanism of energy release in these reactions is the same: the rest energy of particles transforms into their kinetic energy (i.e., heat). Consider an atom of hydrogen, which consists of a proton and an electron. Experiments show that the mass of the hydrogen atom is less than the sum of the masses of a proton and an electron by several hundredthousandths of the electron mass. The explanation of this fact is connected with the circumstance that the energy of electric attraction of the proton and electron, binding these particles into the hydrogen atom, is considerable relative to the electron rest energy. More interesting examples are atomic nuclei. The simplest composite nucleus, the deuteron, consists of a neutron and a proton. The experimentally measured mass of deuteron is approximately 0.1% less than the sum of the masses of a neutron and a proton. The nucleus of uranium 238U contains 92 protons and 146 neutrons. The measured mass of this nucleus is approximately 5.2% less than the sum of the masses of the neutrons and protons entering into its composition. The examples presented show that mass is not an additive quantity; the mass of a composite object is not equal to the sum of the masses of the bodies from which it is composed. Now, let us discuss the role of the notion of mass in nonrelativistic mechanics and in relativity theory. The mass of an isolated physical object is conserved, and does not change under the transition from one frame of reference to another. In relativity theory, the mass determines the rest energy of the body. This property of the mass was not known in nonrelativistic mechanics. At the same time, in relativistic theory the mass of a physical

48

The Quantum World of Nuclear Physics

system is not a measure of the quantity of substance. In relativistic mechanics, there is no principal difference between substance (the objects with m =/= 0 — neutrons, electrons, atoms, molecules, etc.) and radiation (photons, which have m = 0). It is possible that photons are not the only particles having zero mass. Neutrinos and some other particles are difficult to observe experimentally because of their weak interaction with matter, but they are supposed to have zero mass as well. In relativistic theory, the mass of a system of particles (molecules, atoms, nuclei, etc.) does not coincide with the sum of the masses of the particles that compose it. In relativity theory, force has components directed along the acceleration and velocity. Therefore, the mass cannot be defined as the ratio of force to acceleration, that is, in this case the mass is not a "measure of inertia" of the moving body. We should stress that relativistic mechanics is a physical theory in a four-dimensional pseudo-Euclidean space (Minkowski space). For this reason, the three-dimensional laws of nonrelativistic mechanics, to which Newton's laws belong, are not realized in relativity theory in principle (aside from the limiting case v
F = mQ—1. at

(2.24)

Fundamental Interactions

49

Considering the force F to be constant, from (2.24) we determine the time interval t0 during which the body will reach the speed of light c:

j *!_=F_Jdtm

(2.25)

Performing the integration in (2.25), we find

*o = ^f.

(2.26)

This erroneous result arises from the use of (2.5) instead of (2.16), which also contains the term with 7 3 . It follows from (2.16) that in reality a body with mo 7^ 0 can never reach the speed of light. A dogmatic acceptance of the wrong formula m = mo7, according to which the body mass increases with velocity, can turn out to be a serious obstacle for students who wish to acquire a deeper understanding of relativity theory in the future. Special relativity theory permits, in principle, the existence of particles that travel faster than the speed of light. These hypothetical particles, called tachyons (from the Greek word "tachis", meaning "fast"), must be created with super-light-speed velocities. Their velocities can never approach or drop below the speed of light if their masses are not equal to zero. However, such particles have never been observed. It follows from general relativity theory that space becomes curved under the action of masses of bodies. We can use the following analogy for this phenomenon. Imagine a sheet of thin rubber stretched between pegs in such a way that this sheet is flat. If we put a billiard ball upon this rubber, then it will create a pit, that is, a deformation of the rubber will occur and distortion of its surface will arise. Adding more balls will lead to an increase of the surface deformation and curvature. An analogous phenomenon is observed in space, if some matter is placed in it. However, the average density of matter in the Universe is small, p = 10~27 kg/m3, so space curvature is also very small. Nevertheless, there exist effects caused by this space curvature, which can be observed experimentally. The closest planet to the Sun is Mercury, which experiences the strongest gravitational force from our star. Its elliptic orbit is rather stretched out. According to general relativity, the major axis of the elliptic orbit of Mercury should turn around the Sun by approximately one degree each ten thousand years. This effect is called the turning of the

50

The Quantum World of Nuclear Physics

perihelion (the closest point to the Sun of the orbit) of Mercury. It was observed experimentally even before general relativity was put forth. Subsequently, even smaller changes in the orbits of the planets were measured. All these phenomena agree with the predictions of general relativity. According to general relativity, a ray of light in a gravitational field follows a curved, not straight, trajectory. This effect can be observed in the following way. The light from stars passing near the Sun is deflected from a straight line path. One can photograph the sky during a total solar eclipse and then, after a long time period (say, half a year), take another photograph of that part of the night sky which was close to the Solar disk during the eclipse. Since during this time the Sun will have moved into another region of the sky, the light from the stars we are interested in will not be influenced by the gravitational field of the Sun. A comparison of the photographs allows one to determine the deflection of light in the gravitational field of our star. Such observations were first carried out in 1919. They coincided with the predictions of general relativity theory. For the same reason, the wavelength of a radio signal transmitted from the Earth into space will be getting longer. In other words, for an observer in space the electromagnetic oscillations in the radiating antenna occur more slowly than for an observer on the Earth. For this reason, it turns out that time on the Earth's surface passes more slowly than it does at distant locations. The difference is approximately one second per fifty years, but modern atomic clocks can detect it. On the surface of the Sun the effect of time dilation is a thousandfold greater; it would be even greater on the surface of a neutron star. Thus, on the surface of a massive body, time passes more slowly than it does in space. On Earth, time passes more quickly on mountain peaks than it does in valleys. This effect has been observed experimentally by means of atomic clocks. The results of these measurements have exactly coincided with the predictions of general relativity. General relativity is a classical theory. A quantum theory of gravitation has not yet been created. However, it is clear from certain physical reasons that in such a theory a massless particle having spin 2 should figure, which is called the graviton (a quantum of the gravitational field). The gravitational force acting between two particles should be described in the quantum theory as an exchange of a graviton (or gravitons) between the interacting particles. The real gravitons must propagate in space in the form of waves, which are called gravitational waves in classical physics. However,

Fundamental Interactions

51

these waves are very weak, and it has not been possible to register them experimentally.

2.2

Electromagnetic Interaction

Electromagnetic forces act between bodies having electric charges. Electromagnetic interactions are also experienced by photons, which do not carry electric charge, and by neutral particles possessing magnetic moments (e.g., neutrons). In Nature, there are two kinds of electric charges: positive and negative. Between two positive or two negative charges there exist repulsive forces, and between positive and negative charges an attractive force acts. The electric force of the interaction between two charges is denned by Coulomb's law, which is similar in form to Newton's law of universal gravitation (2.7): Fe = k^-v.

(2.27)

Here e\ and e^ are the charges, and k is a constant whose value depends on the system of units used to measure force. The lack of a negative sign in (2.27) indicates that like charges repel and opposite charges attract, because an attractive force, as in the case of the gravitational interaction, is assumed to be negative. The SI unit of electric charge is the Coulomb (C). In order that the force may be expressed in Newtons, the constant in (2.27) should be equal to k = 8.987 • 109 N-m 2 /C 2 . There also exists a rarely used unit of electric charge called the electrostatic unit, defined in such a way that the constant would be k = 1. In this case, the force is measured in dynes (1 dyne = 1 g-cm2/s2), and the separation r in cm. Great progress in understanding electromagnetic interaction was made by the outstanding English physicist James Clerk Maxwell. It was Maxwell who introduced the notions of the electric and magnetic field intensities. The introduction of the field concept marked the rejection of the old "actionat-a-distance" concept — the idea of force as a direct action of one body upon another body located some distance away. The electric field can be considered as an external factor that does not depend on the properties of the given charged body, and that exerts a direct action upon it. The electric field is determined by the disposition of all charges in space. Analogously, the magnetic field is determined by all currents (moving electric charges) in space.

52

The Quantum World of Nuclear Physics

The field concept plays a fundamental role in modern physics. Besides electric and magnetic fields, which can be considered as a unified electromagnetic field, there exist other fields that constitute objective realities in the Universe. In other words, matter can exist in two forms: as particles, and as fields. Electromagnetism was discovered by the Danish physicist Hans Christian Oersted in 1820. This discovery had great repercussions on the development of science and technology (electric motors and generators, electric lighting, radio and television, information technology, particle accelerators, etc.). Later, Maxwell showed that the electromagnetic force is a sum of the electric and magnetic forces. At small velocities v -C c the magnetic force is small, but at large velocities it is of the same order of magnitude as the electric force. Thus, the scale that determines unification of the electric and magnetic forces is the scale of the velocities of charged particles. The greatest achievement of electromagnetism was the creation of the theory of the electromagnetic field by Maxwell in 1860-1865, which has been formulated as a set of equations named after him. Maxwell's equations unified the electric and magnetic fields into a single electromagnetic field. Maxwell established that visible light consists of electromagnetic waves with frequencies between 4 • 1014 and 7.5 • 1014 s" 1 . Since the frequency v is related to the wavelength A by the formula A = c/v, the wavelengths of visible light range from 7.5 • 10~7 to 4 • 10~7 m. Later it was shown that ultraviolet, infrared, and Roentgen rays, as well as gamma radiation and radio waves, are also electromagnetic waves. In large bodies (stars and planets), the amounts of positive and negative charges are almost the same, i.e., the electromagnetic forces between these objects are small. By contrast, electromagnetic interaction plays a definitive role in the world of atoms and molecules. Atoms, molecules, and solids owe their existence to electromagnetic forces. Electromagnetic interaction is also intrinsic to the chemical and Van der Waals forces that act between atoms at large distances, i.e., at distances large relative to the linear sizes of atoms. The range of action of electromagnetic interaction is infinite. Electromagnetic processes in the microworld are described by quantum electrodynamics. It allows us to explain numerous processes of an electromagnetic nature, in which elementary particles, atomic nuclei, atoms, and molecules take part. In quantum electrodynamics, the interaction of two charged particles is considered as an exchange of one or several photons which are quanta of the electromagnetic field. These photons are emitted by one charged particle and absorbed by another. Since these photons

Fundamental Interactions

53

cannot be registered experimentally, they are called virtual photons. The photon possesses, as we already stated, zero mass and spin 1. Note that all exchange particles (the quanta of different fields) always have integer spins, i.e., they are bosons.

2.3

Weak Interaction

A comparatively small number of fundamental particles among the several hundred currently known elementary particles are now considered as structureless. Leptons (from the Greek word "leptos", meaning "light") belong to this class. Now we know three pairs of leptons. These pairs are usually called the three generations of leptons. The first generation consists of the electron e~ and the electron neutrino ve. The second generation consists of the muon /J,~ and the muonic neutrino v^. The third generation consists of the tau-lepton T~~ and the neutrino vT corresponding to it. The masses of all neutrinos are apparently zero, or are so small that it has not been possible to measure them. The masses of other leptons are as follows: me = 0.511 MeV « 9.1 • 1CT31 kg, m^ = 106 MeV w 1.89 • lO""28 kg, and mT = 1784 MeV « 3.18 • 10~27 kg. All leptons have spin 1/2. The muon and tau-lepton are unstable. Their half-lives are £1/2(AO = 2-2 • 10~6 sand £ 1 / 2 (T) =3.15- 10" 13 s. How do the neutrino and antineutrino differ? Since in the theory it is usually assumed that these particles have zero rest mass, they must naturally move with the speed of light. In this case, it is possible to characterize a particle by the projection of its spin upon the direction of its momentum. This quantity is called helicity. The spin of a neutrino is directed against its momentum (left-handed helicity), and the spin of an antineutrino is directed along its momentum (right-handed helicity). Since these particles move at the speed of light, their helicities are conserved in any frame of reference. Leptons take part in weak interactions. In such a process, the numbers of leptons before and after the reaction are the same. Strictly speaking, the difference in the total number of leptons and the total number of antileptons is conserved. Moreover, the numbers of leptons of each generation are conserved separately. The conservation of the lepton number in weak interaction processes is called the leptonic charge conservation law. If one ascribes the leptonic charge +1 to any lepton and —1 to any antilepton, then the summary leptonic charge before and after any reaction is conserved

54

The Quantum World of Nuclear Physics

(all other particles have zero leptonic charge). An example of a leptonic process is muon decay: /z~ —> e~ + ve + v^ (here the overbar denotes an antiparticle). Hadrons also undergo weak interactions, however their main type of interaction is the strong one. For example, a free neutron decays according to the scheme n - t p + e" + ve. This process is the /3-decay of a neutron and is the basis of the /3-decay of atomic nuclei. The elementary reaction of the capture of an atomic electron by a neutron of the nucleus (usually from the if-shell), p + e~ —> n + ve, is a basis of the /('-capture which is a variety of the /3-decay. The pion (7r-meson) decays creating a muon and an antineutrino, n~ —> pT + PM. The particles taking part in a weak interaction exchange intermediate bosons. These possess, as the photon does, spin 1. There are three intermediate bosons: the charged W+- and W^-bosons, and the neutral Z°-boson. Exchange particles responsible for the weak interaction are very heavy. Their masses are mw = 80.22 ± 0.26 GeV « 1.43 • 10~25 kg and mz = 91.127 ± 0.007 GeV « 1.62 • 10~25 kg. Because of this, the range of action of the weak forces is very small, r\y = fr/m\yc « 10~18 m. The weak forces are short-range ones. In the theory of weak interaction, the process of neutron decay is considered as two successive steps. First, the neutron emits a W~-boson and transforms into a proton: n —> p + W~. Then the W~-boson decays into an electron and an antineutrino in a time r « 10~26 s: W~ —> e~ + ve. Analogously, the /3-decay of nuclei should be considered as a two-stage process in which, first, an intermediate boson is emitted which further decays into leptons. Intermediate bosons, predicted theoretically, were discovered in 1983. The Nobel Prize in Physics for 1984 was awarded to those who made decisive contributions to the large project that led to the discovery of the W- and Z-bosons. The engineer Van der Meer created the particle storage ring used in the experiment, and the physicist Rubbia led the team of 150 scientists. Similar to Maxwell's unification of electricity and magnetism, a theory has been developed that has unified the electromagnetic and weak interactions. For the creation of this theory, the Nobel Prize in Physics for 1979 was awarded to Glashow, Weinberg, and Salam. In the theory, the existence of the W±- and Z°-bosons has been predicted, and it has been established that three intermediate bosons and the photon must behave similarly at energies much greater than 100 GeV, i.e., they are just like

Fundamental Interactions

55

the same particle in this case. Nonetheless, at low energies these particles essentially differ. 2.4

Non-Conservation of Parity in Weak Interaction

Along with the properties of homogeneity and isotropy of space that lead to the laws of conservation of energy and momentum for an isolated physical system, there is one more symmetry property of space — the invariance of system properties with respect to mirror reflection, i.e., the simultaneous change in the signs of all spatial coordinates. In classical mechanics all processes are invariant with respect to the inversion transformation, since in this case the equations of motion do not change. In quantum mechanics this invariance is not always present. Before considering the invariance with respect to the mirror inversion in quantum processes, let us consider the left-right (mirror) symmetry in Nature. When a man looks in the mirror his face seems to be symmetrical. Leaves of trees, at first sight, also seem to be symmetrical. In fact, after scrutinizing these objects it turns out that small deviations from symmetry are always present. People and animals are nonsymmetrical creatures, since the heart and some unpaired organs (for example, the liver) are situated off to one side. It is surprising that Nature has not formed both possible types of people: those with hearts on the left side and those with hearts on the right side. All people (except for some anomalous cases) have their hearts on the left side. Thus, biology departs from complete bilateral symmetry. When studying microobjects, we also encounter asymmetry. The absence of bilateral symmetry in some organic compounds of biological structures was noticed by Paster as early as 1848. For example, DNA (deoxiribonucleic acid) molecules created by an organism look like right-handed screws. This rule is the same both for DNA created by humans and other mammals, and for DNA created by bacteria. The corresponding "left-handscrew" DNA molecules, which are the mirror reflections of the "right-handscrew" molecules, are never created by living organisms. How do organisms know that they only have to synthesize right-handed DNA molecules? Undoubtedly, this program is written in the genetic codes of all living organisms. But why did Nature choose just right-hand screws and not permit left- and right-handed DNA molecules with equal probabilities? It is clear that in the process of evolution of life on Earth, something had to occur which selected the right-handed sense of rotation in the structure of DNA molecules in biological systems. This phenomenon remains

56

The Quantum World of Nuclear Physics

unexplained. Now let us consider quantum objects. We have seen that the neutrino always has left-handed helicity, and that right-handed helicity can be possessed only by the antineutrino. Thus, in the world of leptons, the preference for the left-handed screw is also observed — moreover, it exists for all three types of neutrinos. In quantum mechanics the wave function for some cases (e.g., for a particle moving in a central field and having a definite value of the orbital moment) can remain unchanged or can change its sign under an inversion. Then the particle or quantum system state described by such a wave function is said to have a definite parity. If the wave function does not change under inversion, the parity is positive — otherwise, it is negative. Parity is a conserved quantity in electromagnetic and strong interactions. However, it turns out that parity is not conserved in weak interactions. Parity non-conservation was discovered in the decay of /('-mesons, which decay as particles with positive parity in some cases and negative parity in other cases. The American physicists Dao Lee and Chen Ning Yang established in 1956 that the proof of parity conservation only exists for electromagnetic and strong interactions, but in weak interactions parity is not conserved. For their fundamental investigations of parity laws, Lee and Yang were awarded the Nobel Prize in Physics for 1957. In 1957, Wu with her collaborators carried out a special experiment which confirmed parity non-conservation in weak interactions. The idea of Wu's experiment centered on the following. The /3-decay of cobalt nuclei was studied: 60 Co—> eoNi

+ e-+ve.

(2.28)

In this experiment, cobalt nuclei were polarized by means of a very strong magnetic field, H ~ 107 A/m, in such a way that their spins were oriented along a certain direction. The radioactive substance was cooled to ultralow temperatures T « 0.01 K in order to decrease the role of the thermal kinetic motion of the nuclei, which could destroy their polarization. To reach such ultralow temperatures, the method of adiabatic demagnetizing of paramagnet salts was used. In this case, the substance cooling is explained by the existence of the magnetocaloric effect. This occurs if heat inflow from outside is absent, then the work of the paramagnet demagnetizing is done solely due to its internal energy (the thermal kinetic energy). In this case, the preliminary magnetizing of the paramagnet should be performed isothermally. For technical reasons it was not easy to carry out this exper-

Fundamental Interactions

57

iment at that time. Nevertheless, it was accomplished. Electron emission was mainly observed in the direction opposite to the spin orientation of the nuclei, indicating the absence of mirror symmetry in the /3-decay process. In other words, during the /3-decay of nuclei due to weak interaction, parity is not conserved. Numerous experiments carried out afterwards have convincingly confirmed parity non-conservation in the /3-decays of different nuclei, i.e., the violation of the parity conservation law in weak interactions. Returning to neutrinos, we should note that these particles are always polarized since they always have left-handed helicity (their spins are always directed against their momenta). Thus, these leptons manifest parity non-conservation during the weak interactions. However, it remains unclear why the weak interaction prefers just left-handed helicity. 2.5

Strong Interaction

The existence of atomic nuclei was discovered by Rutherford in 1911, and he introduced the term "nucleus" in 1912. However, an understanding of the structure of atomic nuclei became possible only after the discovery of the neutron by Rutherford's disciple Chadwick in 1932, who observed neutron creation during the irradiation of beryllium by a-particles: a+

9Be

—•

12C

+ n.

(2.29)

For this discovery, Chadwick was awarded the Nobel Prize in Physics for 1935. At first, physicists supposed that the neutron is a composite particle consisting of a proton and an electron. However, in 1934 Chadwick and Goldhaber discovered that the neutron mass is larger than the sum of the masses of the proton and electron (now, it is known that the neutron mass is equal to 1.00083 of the sum of the masses of the proton and electron). For this reason, the neutron was acknowledged as a new elementary particle. After the discovery of neutrons, it became clear that atomic nuclei consisted of nucleons, i.e., of protons and neutrons. For the first time this notion regarding nuclei was developed by Heisenberg and, independently, by Ivanenko. The interaction between nucleons is called the nuclear interaction. For a long time it was presumed to be fundamental. The intensity of this interaction surpasses by far that of all other types of interaction. In order to compare the intensities of different interactions, let us con-

58

The Quantum World of Nuclear Physics

sider two protons separated by a distance r m h/(mpc) « 2 • 10" 16 m, where mp is the proton mass. We shall take the protons as point particles. In this case, the ratio of their energies due to the electromagnetic and strong (nuclear) interactions is a quantity of order 10~2, the ratio of their energies due to the weak and strong interactions is of order 10~5, and the ratio of their energies due to the gravitational and strong interactions is of order 10" 38 . The quantum theory of any interaction between two particles is based upon the exchange of some other particles. A theory of nuclear forces was first developed by the Japanese physicist Yukawa in 1935. According to this theory, the nuclear interaction between nucleons occurs because of the exchange of 7r-mesons. There are three such particles: two charged nmesons, TT+ and TT~, and one neutral 7r-meson, TT°. The spins of n-mesons are equal to zero. The masses of the charged pions (7r-mesons) are mn+ = mff- = 139.57 MeV « 2.48 • 1(T28 kg, and the mass of the neutral pion equals m / = 134.96 MeV sa 2.40 • 10~28 kg, i.e., the masses of pions are about 270-280 times greater than the electron mass. The pions were found experimentally in 1947. These particles are unstable. They decay according to the following schemes: TT+ —> fj,+ + v^ (fi+ is an antiparticle with respect to /x~), n~ —> ji~ + PM, TT° —> 7 + 7 (here 7 denotes a photon). The half-decay periods of the charged pions are 2.6 • 10~8 s, and that of the neutral pion is 0.83 • 10~16 s. For his prediction of the existence of mesons on the basis of theoretical work on nuclear forces, Yukawa was awarded the Nobel Prize in Physics for 1949. V(r) 1

0 —

\

1

\O5

'

to

'

1

1.5^

.

7

Fig. 2.1 Nuclear interaction potential. It is now known that the nuclear forces of the interaction between two nucleons cannot be explained on the basis of pion exchange only. In reality, the nuclear interaction potential is rather complicated (Figure 2.1). The nuclear forces of attraction between two nucleons at sufficiently large dis-

Fundamental Interactions

59

tances, r « 2 • 10~15 m, are due to the exchange of a ir-meson. At smaller distances, 0.4-10"15 m < r < 2-10~15 m, the nucleons apparently exchange two pions and, maybe, 77- or if-mesons which are, like pions, spinless particles (the mass of the neutral 77-meson is 549 MeV, that of the iC+-meson is 494 MeV, and that of the X0-meson is 498 MeV). At small distances r < 0.4 • 10~15 m, where the interaction between two nucleons has the character of repulsion, they exchange vector mesons (the p+-, p~-, and p°mesons have masses of 769 MeV « 1.4 • 10~27 kg, and the w°-mesons have masses of 783 MeV « 1.4 • 10" 27 kg) having spin 1. Since the exchange of a 7T-meson explains the nuclear forces at large distances, the range of influence of these forces can be determined as rn = h/m^c. Substituting the pion mass into this formula, we find rn ss 1.4 • 10~15 m. Thus, nuclear forces are short-range, which is similar to the weak interaction forces. The nuclear forces differ in essential ways from other known forces. These forces are mainly attractive, but at very small distances, r < 0.4 • 10~15 m, they become repulsive. The nuclear forces have a finite range of influence: at distances r > 2 • 10~15 m they are not practically observed. The nuclear forces possess the saturation property. For example, a nucleon does not interact with all other nucleons in the same nucleus, but only with several neighbors. The nuclear forces depend on the mutual orientation of spins of the interacting particles. They are noncentral forces and depend on the value of the total spin of the system of interacting particles. The nuclear forces do not depend on the electric charges of the interacting particles, i.e., the nuclear forces acting between two protons or between two neutrons are the same as the forces between a proton and a neutron. Nucleons are actually composite particles consisting of quarks, i.e., nuclear forces are secondary, and the strong interaction is defined by the forces between quarks. The particles undergoing the strong interaction are called the hadrons (from the Greek word "hadros", meaning "strong"). The concept of the quark structure of hadrons was independently formulated by Gell-Mann and Zweig in 1964. For his works on the classification of elementary particles and their interactions, Gell-Mann was awarded the Nobel Prize in Physics for 1969. Gell-Mann took the particle name "quark" from the 1939 novel Finnegans Wake by Irish writer James Joyce. This novel does not translate well into other languages because it toys with semantics and puns. Joyce's novel describes the life of Mr. Finn, who sometimes assumes an appearance of Mr. Mark. Three quarks are children of Mr. Finn, which often appear in the role of Mr. Finn (Mr. Mark) himself. From here, an association with elementary particle physics has arisen, since the

60

The Quantum World of Nuclear Physics

nucleons and some heavier particles consist of three quarks. The particles consisting of three quarks are called baryons (from the Greek word "barys", meaning "heavy"). The particles consisting of two quarks (strictly speaking, of a quark and an antiquark) are called mesons (from the Greek word "mesos", meaning "medium" or "intermediate"). We may also note that the word "quark" in the German means "curds". Let us consider the quark structure of some baryons in more detail. Protons have mass mp = 938.3 MeV w 1.67 • 10" 27 kg and spin 1/2. The neutron mass is equal to mn = 939.6 MeV sa 1.67 • 10~27 kg and its spin is 1/2. The masses of these particles differ only by 0.14% of the proton mass. Their main difference is that the proton has a positive elementary charge while the neutron is electrically neutral. To explain the structure of nucleons, it is sufficient to assume the existence of two types of quarks. These are denoted by the letters u and d (for "up" and "down", respectively). The up- and down-quarks form the first generation of quarks. The quarks possess spin 1/2, so that it is possible to construct particles with different spins out of them. The most amazing property of quarks is their fractional electric charge. The quark u has the charge 2e/3, and the quark d has the charge —e/3. Quarks are the only particles known to have fractional charges. Now, let us consider how it is possible to build up nucleons out of the quarks u and d. The proton consists of two quarks u and one quark d, and the neutron consists of one quark u and two quarks d. In order that the Pauli principle not be violated, the spins of the two quarks u that enter into a proton and of the two quarks d that enter into a neutron should be directed oppositely, i.e., every such pair of identical quarks inside a nucleon has zero spin, and the spin of the third quark defines the spin of the nucleon. An essential difficulty has arisen in the attempt to explain the structure of short-lived A-particles on the basis of the quark model. The A-particles have masses m A = 1232 MeV « 2.19 • lO"27 kg and spins 3/2. Four Aparticles are known, which differ by their charges: A + + , A + , A0, and A~. To construct the particle A + + , which has charge 2e, it is necessary to take three quarks u with parallel spins in order that their total spin would equal 3/2. However, in this configuration the Pauli principle, which prohibits any two identical fermions from being in the same state, would be violated. This difficulty can be removed if one supposes that quarks are characterized by an additional quantum number — the color — which can take three different values. It is assumed that any quark is always in one of three color states: blue, green, or red. For this reason, the particle A + +

Fundamental Interactions

61

consists of three quarks u — one blue, one green, and one red. Of course, the term "color" should not be understood literally; it is merely a label applied to a certain quantum state of a quark. The structure of hadrons is then determined by the following rule: the quarks entering into a hadron must have colors such that the hadron is a "white" object. Here, the optical rule is taken into consideration: by mixing equal amounts of the actual colors blue, green, and red, we get the color white. After the creation of the quark model, numerous searches for free quarks in Nature and in laboratory settings were carried out. However, there has been a singular failure to observe free quarks. Quarks can only be seen inside hadrons. For this, it is necessary to investigate the internal structure of hadrons (e.g., protons) using particles of very high energies. Since electrons take part only in electromagnetic interaction (their weak interaction practically plays no role against the background of the more intense electromagnetic interaction), from the character of their scattering on protons one can learn how electric charge is distributed inside a proton. It has been found from experiments that during interaction with protons, electrons are very often scattered to large angles, as if they collided with some electrically charged point objects when flying through a proton. Moreover, from the character of electron scattering, the electric charges of these point objects inside protons have been successfully determined. Their charges are equal to 2e/3 and —e/3, i.e., they coincide with the charges of quarks. For a homogeneous electric charge distribution inside protons, the electron scattering character would be quite different. Thus, quarks inside a proton have been successfully observed. The physical theory describing the strong interaction of quarks has been called quantum chromodynamics (from the Greek word "chromatos", meaning "color") because quarks are colored. It turns out that the interactions of quarks possess interesting peculiarities. At small distances, i.e., at distances much smaller than the linear size of the nucleon, r^ ~ 10~15 m, quarks behave practically like free particles. This phenomenon has been called asymptotic freedom. If the distance between two quarks increases, reaching the order of rjy, then the force of attraction of the quarks rapidly increases. In other words, it is impossible to tear quarks away from each other. For this reason, quarks are "locked" inside hadrons. This phenomenon is called confinement. It follows from quantum chromodynamics that a colored object (e.g., a quark) in a free state cannot be observed. Observable objects are only white objects, i.e., hadrons. During interaction, quarks exchange particles called gluons (from the

62

The Quantum World of Nuclear Physics

word "glue", since they glue together quarks inside hadrons). There exist eight different gluons. A gluon possesses spin 1 and its mass and electric charge are both zero. Gluons, like quarks, are colored objects. However, unlike a quark, each gluon bears two colors. For this reason, interacting quarks change their colors during gluon exchange. Since a gluon is a colored object, it, like a quark, cannot be observed as a free particle. However, several gluons can form a "white" object that can be observed. Such gluonic formations are called glueballs. Glueballs created in various processes do not live long, and decay into hadrons. Besides baryons, which are the hadrons that consist of three quarks, there also exist mesons — hadrons consisting of a quark and an antiquark. Let us consider the simplest family of these particles — pions — which includes three particles: TT+, TT~ , and n°. The meson TT+ consists of the quark u and the antiquark d (the charge of an antiquark is opposite to that of a quark). Moreover, the 7r+-meson contains the combination of the blue quark u and the antiquark d, whose color is complementary to blue, i.e., together they yield white. This combination is added to the same amounts of the configurations of the green quark u and the antiquark d, whose color is complementary to green, as well as of the red quark u and the antiquark d, whose color is complementary to red. The meson TT~ contains the combinations of corresponding colored quarks d and antiquarks u, and the meson TT° contains the combinations of u and u as well as of d and d. The spins of quarks and antiquarks that form the "white" pairs inside 7r-mesons are directed oppositely. For this reason, 7r-mesons have zero spins. There are also hadrons called vector mesons. Belonging to this class, in particular, are three p-mesons — p+, p~, and p° — whose spins equal unity. The /9-mesons are structurally similar to the 7r-mesons, but the spins of the quarks and antiquarks entering them are parallel in each "white" pair. This configuration of two quarks turns out to be "heavier" than the configuration with oppositely directed spins. For this reason, the masses of p-mesons are much greater than those of 7r-mesons. There are other known mesons that contain quarks and antiquarks, both of the first and other generations. Investigations of various processes observed at high energies have shown that, besides the first generation of quarks, there also exist two other generations. The second generation includes the charm quarks c and the strange quarks s. These carry electric charges 2e/3 and — e/3, respectively. Hadrons containing c-quarks are called charmed particles; those containing s-quarks are called strange particles.

Fundamental Table 2.1

63

Interactions

Symmetry of first generation leptons and quarks.

Electric charge (e)

-1

leptons antiquarks quarks antileptons

e

- |

- |

0

+|

+§

+1

i/e u

d d

u Pe

e

The third generation includes the top quarks t and bottom quarks b. Note that these designations parallel those of the first generation quarks u and d. The quark t carries charge 2e/3, while the quark b carries charge —e/3. So there are six known quarks; moreover, it has been established that there are no other quarks. One can notice a certain symmetry between quarks and leptons, which also have three generations. These generations of particles are usually displayed as

C-) (?) C-)

CD 0 0-

^

In Table 2.1, the observed symmetry of the leptons and quarks of the first generation is shown. All matter in the Universe consists only of first generation particles: the quarks u and d enter into nucleons forming atomic nuclei, and electrons, together with the nuclei, form atoms. While a supposition arises that it might be possible to build up matter using particles of the second and third generations, this has not yet been discovered in Nature. Hadrons containing quarks of the second and third generations have been obtained only under laboratory conditions (they have not been observed in cosmic rays). At very high energies, a collision of hadrons can create a quark-gluon plasma, i.e., the structure of hadrons is demolished and matter consisting of quarks and gluons is created. Such matter has interesting properties, and its investigation makes possible the extension of our knowledge about quarks and gluons. Let us summarize. It follows from what has been written above, that the interaction of two quantum objects is realized by an exchange of certain particles with integer spins. For the electromagnetic interaction these are photons; for the weak interaction they are the intermediate bosons W+, W~, and Z°; for the strong interaction they are gluons (eight of them); for the gravitational interaction they are gravitons that have not yet been observed experimentally. These particles, which are quanta of the corre-

64

The Quantum World of Nuclear Physics

sponding fields, have been called the gauge bosons. The spin of a particle which transfers interaction is of great importance: if its spin is even or zero, then two identical particles attract each other, but if its spin is odd, they mutually repel. The particles transferring interaction are bosons and, for this reason, they do not obey the Pauli principle. Consequently there is no restriction on the number of bosons that can be exchanged by the interacting particles in one act of the interaction, i.e., the force of interaction can be large. The particles transferring interaction are virtual particles and cannot be registered directly. However, the virtual particles do exist because one can measure magnitudes of the effects connected with the interaction of material particles. Under certain conditions, the gauge bosons also exist as real particles, i.e., in this case they can be registered by special detectors. So far, the question remains open whether leptons and quarks are true "elementary" particles or whether they consist of some other particles. To answer this question, it is necessary to "probe" their structure at distances much smaller than 10~18 m. To this end, particle accelerators must be constructed that could attain energies at least as high as 104 GeV. So far, such energies have not been reached, and the answer to the question regarding the structure of leptons and quarks is not known. In conclusion, let us again touch upon the nature of nuclear forces. First, let us consider the interaction of atoms at large distances. Atoms are electrically neutral objects. For this reason, the usual electric forces do not act between atoms at distances large compared to their linear sizes. However, atoms have finite sizes, and the electric charges in them are distributed in a certain way. Therefore, atoms can have, generally speaking, nonzero dipole, quadrupole, octupole, and other electric multipole moments, which are caused by their spatial distribution of charges. The presence of multipole moments leads to the fact that, as atoms approach, forces start to act. These have been named Van der Waals forces after the Dutch physicist who was first to study them in the last century. They are attractive, and their dependence on distance between atoms is denned by the formula Fw = -%, r1

(2.31)

where C is a positive constant. The nuclear forces between hadrons are similar to the Van der Waals forces in atomic physics. These forces decrease very rapidly with an increase in the distance between hadrons. Thus, nuclear forces are not fundamental;

Fundamental Interactions

65

they are a consequence of the superstrong quark-quark forces, which are the fundamental chromodynamical forces. It is these latter forces, caused by an exchange of gluons, that determine the strong interaction.

Chapter 3

Structure of Atomic Nuclei

3.1

Composition and Properties of Nuclei

In 1909 Geiger and Marsden, disciples of Rutherford, found that the most probable scattering angle of the 5.5 MeV.a-particles emitted by radioactive bismuth 214Bi (called RaC at that time), and passing through gold foil 4 • 10~7 m thick, was 0.87°. However, approximately one a-particle in 20,000 was scattered at an angle exceeding 90°, i.e., backward. Rutherford's genius helped him understand this deviation of a small number of cc-particles through large angles. He argued that a positively charged a-particle with sufficiently high energy could be backscattered by a collision with something extremely small, heavy, and charged within the atom. He introduced the notion of "atomic nucleus" by analogy with the cell nucleus in biology. Rutherford made a simple calculation and found that the linear size of nucleus was at least thousand times smaller than that of the atom. The formula he obtained (which was later named after him) made it possible to describe the scattering of one charged point particle by another. Rutherford was extremely lucky because his formula, obtained on the basis of classical notions, turned out to be correct in quantum mechanics, too, which was understood much later. Therefore, in 1911 Rutherford drew the conclusion regarding the existence of an atomic nucleus. An atomic nucleus possesses fascinating properties. It contains approximately 99.97% of the atomic mass, but occupies a volume ten thousand billion times smaller than that of an atom. This means that the atom, as is the case with all matter, consists mostly of empty space. In other words, the density of nuclear matter is ten thousand billion times greater than the density of Earth matter. The carrying capacity of a single truck is sufficient 67

68

The Quantum World of Nuclear Physics

to transport just 2 • 10~13 m3 worth of nuclear matter. In 1911, Geiger and Marsden began a research program with a series of precise measurements concerning the fraction of a-particles scattered at different angles by atoms. In 1913, they reported that the experimental data obtained were in good agreement with Rutherford's formula. This was the final confirmation of Rutherford's discovery of the atomic nucleus. The simplest atomic nucleus — that of the hydrogen atom — is a proton. However, two protons cannot form a bound state. In order to "glue" protons to form atomic nuclei, other particles should be added. After the discovery of a neutron by Chadwick in 1932, Heisenberg, and independently Ivanenko, proposed a realistic model of the atomic nucleus. According to that model, the atomic nucleus consists of protons and neutrons and, evidently, is a positively charged composite particle. The number Z of protons in a nucleus coincides with the atomic number, i.e., the number of electrons in a neutral atom. The sum of the number Z of protons and the number N of neutrons in a nucleus is called the mass number A = Z + N. Before 1932, some scientists believed that atomic nuclei consisted of protons and electrons. Then, the nucleus charge Z would be equal to the difference between the number of protons and the number of electrons. However, that assumption was incorrect. First, an electron cannot be confined in a region of space occupied by a nucleus because, according to the Heisenberg uncertainty principle, such an electron would have so much kinetic energy that it could not remain inside the nucleus. Indeed, Ap ~ h/ Ax, where Ap is the uncertainty in the electron's momentum. The uncertainty in the electron's coordinate is defined by the linear size of the nucleus: Arc ~ 10~15 m. Since the velocity of an electron inside the nucleus must be very high — on the order of c — its kinetic energy is of order AE ~ cAp ~ ch/Ax ~ 100 MeV. We see that AE is much greater than the experimentally measured binding energy of one particle inside a nucleus, which is of order 10 MeV. So an electron cannot be confined within a nucleus. Other experimental facts are consistent with the result that electrons cannot reside in nuclei. For instance, according to the electron-proton model, the 6Li nucleus should contain 6 protons and 3 electrons. Since proton and neutron spins are equal to 1/2 (in h units), the spin of 6Li (total spin of 6 protons and 3 electrons) will be a half-integer. However, experiments show that the 6Li nucleus possesses integer spin. The same holds for the nuclei of 2H, 10 B, and 14N. Nuclei that contain Z protons can have different numbers of neutrons. Nuclei having the same numbers of protons Z but different N (or A) are

Structure of Atomic Nuclei

69

called isotopes. This nomenclature for varieties of the same element was given by the British physicist Soddy in 1910, because in the table of chemical elements they occupy the same place (from the Greek "isos", meaning "same", and "topos", meaning "place"). For instance, hydrogen has 3 isotopes: light hydrogen (protium) 1 H, deuterium 2H, and tritium 3 H. Xenon (Xe) has the largest number of isotopes: 28 (118 < A < 145). For his contribution to our knowledge of the chemistry of radioactive substances, and his investigations into the origin and nature of isotopes, Soddy was awarded the Nobel Prize in Chemistry in 1921. Another British physicist, Aston, was awarded the Nobel Prize in Chemistry in 1922 for his discovery of isotopes in a large number of nonradioactive elements, and for his enunciation of the whole-number rule. Nuclei having the same A but different Z and N are called isobars or isobaric nuclei (from the Greek "baros", meaning "weight"). Nuclei with the same N but different Z (or A) are called isotones (from the Greek "tonos", meaning "strain"). A given nucleus with definite A and Z is called a nuclide. The mass of a nucleus depends on the total number of neutrons and protons it contains. Provided the nucleons in a nucleus do not interact with each other, making the nucleus a "gas" of free nucleons, its mass would be m(Z, A) = Zmp + (A — Z)mn, where mp and mn are the proton and neutron masses. However, allowing for the interaction between nucleons, the mass of a nucleus is smaller: m(Z, A) = Zmp + (A - Z)mn - AM,

(3.1)

where AM is called the mass defect. A stable nucleus has AM > 0. According to the theory of relativity, the nucleus rest energy EA = m(Z,A) • c2. The product of the nucleus mass defect and the squared velocity of light is the nucleus binding energy: W(Z, A) = AM • c2. Nucleus masses are measured in atomic mass units (a.m.u., or daltons). This unit is chosen to make the mass of the 12C carbon nucleus precisely equal to 12 a.m.u. Denoting u = 1 a.m.u., we introduce the magnitude Am = m(Z, A) - Au.

(3.2)

This quantity, called the mass excess, can be positive or negative. Conversion from a.m.u. to the units of energy, MeV, in which nuclear masses can

70

The Quantum World of Nuclear Physics

also be measured, is accomplished by the formula u = 931.5016 MeV.

(3.3)

Nuclear mass could also be expressed in kg through the conversion formula u = 1.66 • 10"27 kg.

(3.4)

Nuclear masses are measured experimentally with special devices called mass spectrometers. They can also be defined from measurements of the energies of different nuclear reactions, from the energies of a- and /3-decays, and from radiospectroscopic measurements of the frequencies of transitions between rotational levels of molecules. The latter measurements are able to define ratios of nuclear masses with high precision of the order of 10~5 —

-icr 6 . There are 287 nuclei in Nature. Most of these (168) are even-even nuclei having even numbers of protons and neutrons. There are only 4 stable oddodd nuclei: \R, |Li, 5°B, and y4N. There are 5 radioactive odd-odd nuclei: 19K, 2°V> sfLa, ^ 6 Lu, and | f T a . Elements that have Z = 43,61,85, and 87, and Z > 93, are obtained artificially because they do not occur terrestrially. There are no stable nuclei with A = 5,8,147, or A > 210. Tin has the largest number of stable isotopes: 10. Light nuclei (up to 20Ca) have approximately equal numbers of protons and neutrons. Further, with an increase in Z the number of neutrons in natural nuclei begins to exceed the number of protons. Thus, the heaviest natural nuclei have more than 1.5 times as many neutrons as protons. Nuclear binding energy can be denned on the basis of a hydrodynamical model, where a nucleus is treated as a drop of an incompressible charged liquid. Neglecting surface energy and the Coulomb interaction of protons, the nuclear binding energy would be proportional to the number A of nucleons in a nucleus (volume energy). The surface energy is proportional to the area of the nuclear surface, i.e., to the square of the nuclear radius R, and decreases the binding energy. Since the volume of a nucleus treated as a drop of an incompressible liquid is proportional to the number of nucleons A, the radius of a nucleus is proportional to A1/3: R = r0A1/3, where r0 can be considered as a constant value. The surface energy is proportional to A2/3. The electrostatic energy of repulsion of protons in a nucleus is proportional to the squared charge number Z2 and inversely proportional to the nuclear radius, i.e., to A" 1 / 3 . This also decreases the binding energy. Therefore, considering volume,

71

Structure of Atomic Nuclei

surface, and Coulomb energies in a hydrodynamical (drop) model, we are led to the following formula for the nucleus binding energy: W{Z,A)=aA-f3A2/3

(3.5)

--yZ2A~1/3,

where a, /?, and 7 are constants. The quantity f3 is connected to the coefficient of surface tension a of a drop of nuclear "liquid" by the correlation /3 = 47rrQ<7, while the constant 7 is denned by the squared elementary charge e: 7 = 3e 2 /(5ro). Formula (3.5) for the nuclear binding energy can be improved by the addition of two terms. Following the hypothesis of charge independence of nuclear forces, symmetry between protons and neutrons in a nucleus should exist. Thus the corresponding additional term (energy of symmetry) should depend only on the difference N — Z — A — 2Z and should be an even function of it, i.e., the energy of symmetry will be proportional to (A — 2Z)2. This additional term decreases the nuclear binding energy. Detailed analysis of the binding energies of nuclei shows that nucleons of the same sort in a nucleus prefer to join in pairs. Such "pairing" of nucleons leads to an increase in the nuclear binding energy. A corresponding additional term (the energy of "pairing") is denned by 5A~3/4, where

{

\d\ for even-even nuclei, 0 for nuclei with odd A,

(3.6)

— \S\ for odd-odd nuclei. So the final formula for nuclear binding energy is W(Z, A)=aA-

$A2'3 - 1Z2A~1'3

- C ^ " ^

+ 5A~3/A.

(3.7)

This formula, known as Weizsdcker's semi-empirical formula for the binding energy of the nucleus, was established in 1935 by the German physicist von Weizsacker. The quantities a, j3, 7, £, and 5 in (3.7) are derived through a comparison of calculations with experimental data. One widespread (but not unique) set of values is a — 15.75 MeV, /3 = 17.8 MeV, 7 = 0.71 MeV, C = 23.7 MeV, and |<5| = 34 MeV. Fig. 3.1 demonstrates the dependence of binding energy per nucleon W(Z, A)/A (MeV) as a function of the nucleus mass number A. It can be seen that the ratio W(Z,A)/A quickly increases in the region of the lightest nuclei, reaches a maximum at A = 56 (where W(Z, A)/A « 8.8 MeV), and then decreases smoothly to about 7.5 MeV for heavy nuclei. The approximate constancy of the binding energy per nucleon (except for

72

The Quantum World of Nuclear Physics

3 ,

I / I

^ ^ i

i

80

i

i

160

i

^

A

Fig. 3.1 Dependence of the binding energy per nucleon in a nucleus as a function of the mass number of the nucleus. the lightest nuclei) is connected with the property of saturation of nuclear forces: each nucleon in a nucleus effectively interacts only with its nearest neighbors and not with every other nucleon. Knowing the binding energy W(Z, A) of a nucleus, we can find its mass by the formula m(Z, A) = Zmp + (Aa

A

(3.8)

Z)mn

P ,.2/3

7 ry<2 , 1/3

C {A-2Z)2

5 ,_3/4

- -rA + ^A2/3 + -^Z2A~1/3 + \ —=A 3 / 4 . Formula (3.8) can help us derive the correlation between A and Z for the isotopes of nuclei being stable against /3-decay. Under /3-decay the number A is constant but Z changes. Assuming that (3.8) defines the dependence of nuclear mass on its charge number Z for a constant mass number A, it is possible to find the nucleus (isotope) with minimum mass (minimum rest energy) that will be stable against /3-decay. For this purpose, the derivative dm(Z, A)/dZ with constant A should be found and equated to zero. As a result we obtain 1.98 + 0.015A2/3' (3'9) The majority of nuclei have a spherical form. Some are deformed, i.e., their equilibrium form is not spherical. But this nonsphericity is small. Thus, in the first approximation these nuclei can be considered as spherical. Since the nucleus volume is proportional to the number of nucleons A, the nucleus radius is proportional to A1/3: R — TQA1^. Values of ro found Z =

Structure of Atomic Nuclei

73

by different experimental methods lie in the interval 1.1 • 10~15 m < r 0 < 1.5-10-15 m. One of the simplest methods of measuring the linear sizes of nuclei is the elastic scattering of fast neutrons by nuclei. This method is generally suitable for medium and heavy nuclei whose sizes substantially exceed the wavelength of scattered neutrons strongly absorbed by nuclei. In this case the neutron scattering is similar to the optical diffraction of light on a black sphere (Fraunhofer diffraction). The extremum positions in the angular distribution of scattered neutrons are completely denned by the nuclear radius. Experiments on the scattering of fast neutrons give ro = (1.3 — 1.4) • 1(T15 m. The distribution of charge (protons) in a nucleus is characterized by the scattering of electrons by nuclei because electrons only weakly interact with neutrons. Such experiments give values of nuclear (charge) radii smaller than those found from the analysis of neutron scattering: ro = (1.2 — 1.3) • 1CT15 m. The radii of a-radioactive nuclei can be derived from observing an adecay. The half-life of a-radioactive nuclei depends on the width of the potential well the a-particle is in. Thus, the measured half-life gives the well width and the nuclear radius as r0 = (1.45 - 1.50) • 10~15 m. Analysis of 7-spectra of muon atoms is an important method of finding nuclear sizes. Provided the negatively charged muon is close enough to the atomic nucleus, it can be captured by the latter and move in the electric field of the nucleus just like an electron. Such a system where one of the electrons is replaced by a muon is similar to a usual atom and is called a muon atom. Since the muon mass is more than 200 times greater than the electron mass (m^ « 207me), the mean distance between a muon and the nucleus in an atom is approximately 200 times smaller than the mean distance between electron and the nucleus in an atom, i.e., unlike an electron the muon is situated very close to the nucleus. The change in energy state of a muon in an atom is accompanied by emission of a photon, the energy of which depends substantially on the nuclear radius. Analysis of 7-spectra of muon atoms give values of nuclear radii r0 — (1.1 - 1.2) • 10~15 m. Studies of nuclear binding energies with the help of Weizsacker's formula, which contains the nuclear radius, give ro = (1.2 — 1.3) • 10~15 m. Experiments on the scattering of protons, a-particles, and different projectiles by nuclei give nuclear radii in the region ro = (1.2 — 1.5) • 10~15 m. An atomic nucleus should not be considered as a sphere with a sharp

74

The Quantum World of Nuclear Physics

P(r)

P(O)

1

0.5-

^ ^

1—\

Q

0

r

Fig. 3.2 Nuclear density p(r) as a function of the distance from the center of nucleus r. boundary because in the comparatively thin surface region the density of nuclear matter changes from a value close to that in the nucleus center to zero. The nuclear density as a function of distance from the nucleus center is well approximated by

Vc)'

P(r) =

( 3 - 10 )

1 + exp i-^-i where po is a normalization constant, C is the half-density radius, and d is the nuclear surface diffuseness value (Fig. 3.2). The density p(r) is characterized by the root-mean-square radius

( r 2 )= Id3rr2p(r).

(3.11)

One can introduce the radius R of an equivalent homogeneous density distribution. This radius corresponds to the density p(r) at d = 0 and a fixed atomic mass A. The radius R is related to (r2) by the formula ^

2

= (r2).

(3-12)

In this case the quantity R is exactly the nuclear radius. Note that nuclear matter in the nucleus center has an enormous density of 2.2 • 1017 kg/m3. In order to imagine how large this is, we can note that one cubic millimeter of nuclear substance would have a mass of 220000 tons! Like electrons, protons and neutrons have their own angular momenta — spins that equal 1/2 in h units. A nucleus consists of many protons and neutrons. Spins of separate nucleons of nuclei add according to the rule

Structure of Atomic Nuclei

75

for vector addition. Furthermore, nucleons inside a nucleus are in motion and have their own orbital moments. The latter are added with nucleon spins and give the total angular momentum of a nucleon at rest — its spin. The spin of a nucleus (in h units) can take integer and half-integer values: I = 0,1/2,1,3/2,.... Nuclei with odd A have half-integer spins, while nuclei with even A have integer spins. Ground state spins of stable even-even nuclei are equal to zero. Ground state spins of stable natural nuclei with odd A do not exceed 9/2. That means that nuclear spin is small compared to the sum of the absolute values of the spins and orbital momenta of all nucleons forming a nucleus. Hence it follows that the majority of nucleons in a nucleus move so that their spins and orbital moments compensate each other. Note that in Nature, there are nuclei 50V with the ground state spin / = 6, 138La with / = 5, 176Lu with 1 = 7, and 180Ta with 1 = 8. However, all these nuclei are radioactive (although their half-lives are very large). A deuteron is the simplest composite atomic nucleus, consisting of a neutron and a proton. A deuteron's spin is equal to unity, i.e., the spins of nucleons forming a deuteron are always codirected. In Nature there are no bound states of a neutron and proton having zero spin. A deuteron's binding energy is small, W& = 2.22 MeV, and this nucleus has no excited bound states. The study of the deuteron plays no less an important role in nuclear physics than the study of the hydrogen atom in atomic physics. For the discovery of heavy hydrogen (deuterium), composed of a neutron, a proton, and an electron, the American physicist Urey was awarded the Nobel Prize in Chemistry in 1934. Besides its own angular momentum (spin), an electron has also its own magnetic moment /ze = -\e\h/(2mec) = -9.274 • 10~24 J/T. The quantity MB = \e\h/(2mec) is called a Bohr magneton. The proton and neutron also have their own magnetic moments — \iv = 2.79/J.N and fin = — 1.91/nw — where fi^ = \e\h/(2mpc) = 5.05 • 10~27 J/T is a nuclear magneton. It can be seen that the proton magnetic moment is positive, i.e., is directed along the spin, while the neutron magnetic moment is negative and directed against the spin. Since a nucleus consists of many nucleons performing complicated motions, it also possesses a magnetic moment. This can be written as M/ = 9^NI, where g is called the gyromagnetic factor. If / = 0, then fii = 0; that is, magnetic moments of all stable even-even nuclei in ground states are equal to zero. One of the most important characteristics of an atomic nucleus is its

76

The Quantum World of Nuclear Physics

electric charge Ze, which defines the chemical properties of an element. However, charge does not give a complete picture of the electric characteristics of a nucleus, which is an extensional structure where knowledge of the spatial distribution of charge is important. That is why a nucleus should be characterized by multipole electric moments. The dipole moment of any nucleus in the ground state is always equal to zero. Thus, much attention is paid to quadrupole moments of nuclei. The simplest model of a quadrupole is a system of two equal and oppositelyoriented electric dipoles situated at some distance from each other. Electric quadrupole moments of nuclei with 7 = 0 and I — 1/2 are always equal to zero. Large quadrupole moments are usually observed for nuclei having nonspherical form.

3.2

Shell Model of Nuclei

In the model of nucleus as a liquid drop (hydrodynamical model) introduced before, it is assumed that nucleons in a nucleus are highly correlated (tightly bound) with each other. The opposite approximation is a model where nucleons in a nucleus are assumed to move independently of one another; in such a model, the nucleus is similar to gas contained in some volume. However, it is not a common gas but a quantum one. Since nucleons possess spin 1/2, they are fermions, and the gas of nucleons is a Fermi-gas under very low temperature. This gas is called a degenerate Fermi-gas, having properties substantially different from those of common classical gases. The Fermi-gas model, like the drop model, turned out to be too simplified to describe several properties of such a complicated system as an atomic nucleus. Experimental investigations have shown exceptional stability of nuclei having numbers of either protons Z or neutrons N equal to one of the numbers 2, 8, 20, 28, 50, 82, or 126. These numbers of nucleons are called "magic" numbers. First let us consider some experimental data reflecting the exceptional stability of magic nuclei with either Z or N equal to a magic number. If some particular nucleus has Z and N both equal to magic numbers, that nucleus is called double magic. Data on the relative abundance of different nuclei in Nature indicate the great stability of magic nuclei. Usually the relative abundance of isotopes with even numbers of nucleons among other isotopes of the element does not exceed 60% (except the lightest nuclei). However, the following exceptions

Structure of Atomic Nuclei

77

exist: 40Ca (N = Z = 20) at 96.9%; 52Cr (N = 28) at 83.8%; 88Sr (N = 50) at 82.6%; 138Ba {N = 82) at 71.7%; 170Ce (N = 82) at 88.5%. In Nature, there are usually 3-4 nuclei with a definite number of neutrons N and different Z. The number of such nuclei is greater if the number N of neutrons is magic: N = 20 (36S, 37C1, 38Ar, 39K, 40Ca); N = 28 (48Ca, 5 0 T i ) 5 1 V ) 5 2 C r ) 54p e ) .

N

=

5Q (86Kr)

8 7 R b ) 8 8 ^ 8 9 Y ) 9 0 ^ 9 2 M o ) . ^ = 82

(136Xe, 138Ba, 139La, 140Ce, 141Pr, 142Nd, 144Sm). The Z = 50 element (tin) has 10 stable isotopes — more than any other element. The probability of neutron capture by a nucleus having a magic number N of neutrons is substantially smaller than that for other nuclei. This is due to the anomalously small binding energy of a neutron in nuclei whose number of neutrons exceeds the magic number by unity. The energy of the first excited state of a magic nucleus is substantially greater than that for neighbor nuclei. The properties of radioactive decay also attest to the special stability of magic nuclei. All three natural radioactive series of nuclei experiencing a-decay finish on lead isotopes (Z = 82). Alpha-particles with the greatest energy are emitted by nuclei whose a-decay transforms them into a nucleus with Z = 82 and N = 126. Beta-decay that forms a magic nucleus is characterized by the greatest energies. Therefore, the existence of magic numbers of nucleons should be considered as experimentally confirmed. The problem of theoretical description of magic numbers then arises. This description is given by the shell model of the nucleus, which in many respects is similar to the shell model of the atom. The existence of magic numbers of nucleons in nuclei was first noticed by Bartlet in 1932 and Elsasser in 1933-1934. However, these works remained in obscurity because the theory of nuclear structure had just begun to develop. Much later, in 1949, the American physicist Maria Goeppert-Mayer and, independently, the German physicist Jensen, explained magic numbers by the concept of the nucleon shells in nuclei. Especially complicated was the explanation of the magic numbers 50, 82, and 126. It turned out that their existence could be explained only by adding purely quantum forces of spin-orbit interaction to the common central nucleon-nucleon forces. These forces are caused by the interaction of the spin of a nucleon with its angular momentum. They have no analogy in classical physics. Forces of spin-orbit interaction were well known in atomic physics, but their importance in the theory of nuclear structure appeared absolutely unexpected. These forces manifest themselves mainly in the surface region

78

The Quantum World of Nuclear Physics

of nuclei. The consideration of spin-orbit forces allowed Goeppert-Mayer and Jensen to develop the nuclear shell model and to explain various experimentally established properties of nuclei. Goeppert-Mayer and Jensen were awarded the Nobel Prize in Physics in 1963 for their discoveries concerning nuclear structure. At first sight, the nuclear shell model seems impossible to build. Indeed, a nucleus, unlike an atom, does not have any distinguished center of force, while nucleons, unlike electrons, strongly interact with each other. However, a more detailed analysis shows that several bases for the creation of a nuclear shell model do exist. Nucleons in a nucleus are in a state of rapid relative motion with respect to each other, and are separated by a distance on the order of 10~15 m. It turns out that due to the Pauli principle and repulsion at small distances, the collisions of nucleons with each other in a nucleus are comparatively rare events. Thus, it can be assumed that the motion of every nucleon takes place almost independently in the specific self-consistent field created by all nucleons of a nucleus. Provided a nucleus is spherically symmetric, the potential of the self-consistent field will depend only on the absolute value of the distance between the given point and the geometrical center of a spherical nucleus. Due to the short-range character of nuclear forces, the shape of this potential should be similar to the shape of the density distribution of nucleons in the nucleus, taking into account that the density is positive while the potential of attraction is negative: U(r) = -Cp(r),

(3.13)

where C is a positive constant value. According to quantum mechanics, nucleons that move in a selfconsistent field U(r) must be in different energy states, i.e., must fill several one-particle levels of energy. The nuclear ground state implies a total filling of all levels from the lowest level up to the level with some limiting energy EF {Fermi energy). When two nucleons collide, one of them should occupy a state with a lower energy. But this is not possible, because all the low lying energy levels are already occupied, and no nucleon can be added due to the Pauli principle. This fact, and the repulsive character of nucleon-nucleon forces at small distances, leads to the comparatively rare collisions of nucleons in a nucleus. Thus the existence of a self-consistent field of nucleons in a nucleus is the consequence of the fact that nucleons are fermions, and that nuclear forces are repulsive at small distances. As a first approximation, the potential of a self-consistent field U(r)

Structure of Atomic Nuclei

79

could be identical for protons and neutrons, since the Coulomb interaction of protons becomes essential only for quite heavy nuclei. This conclusion is supported by the coincidence of neutron and proton magic numbers. Since in the spherically symmetric field the angular momentum of a particle with respect to the field center is conserved, 2(21 + 1) nucleons of the same sort can occupy an energy level with the given I (here 21 + 1 is the number of possible values the projection of the nucleon's orbital momentum I onto the axis of quantization can acquire, 2 is the number of possible values of the projection of the nucleon's spin on that axis). During the formulation of the theory of the one-particle energy levels for nucleons in a nucleus, it was noticed that the separation between some neighboring levels appeared substantially greater than the typical mean distances between levels. The boundary between nucleon shells lies exactly in these places. If the given nucleus has all its nucleon shellsfilled,then it is especially stable and has zero spin, magnetic dipole and electric quadrupole moments. The nuclei ^He, g6O, fo^a, lo^a, a n d 828P° a r e doubly magic because their proton and neutron shells are totally filled. If a nucleus consists of a core with totally filled nucleon shells and one additional nucleon (the valence nucleon), then evidently the properties of such a nucleus will be defined by this nucleon. For instance, the nucleus 17 O has one neutron with the orbital moment I = 2 and with the total moment j = 5/2 above the closed shell. Therefore, the shell model predicts the spin of 17O equal to 5/2. This prediction agrees with experiment. The nucleus 17F has one proton in the state I = 2, j = 5/2 above the closed shell. Therefore, the spin of that nucleus is 5/2 as well. The shell model allows us to explain the many excited states of nuclei as the transitions of the valence nucleons to the excited states. Such states are called the one-particle excited states of nuclei. However, a nucleus can have excited states caused by the changes of states of the core of the nucleus. These states can be explained on the basis of a generalized shell model considering core excitations. As was mentioned above, nucleons of the same sort in a nucleus prefer to form pairs with zero spins. Particularly, any even-even nucleus has zero spin. Even if such a nucleus does not have all shells closed, it can be treated as a core in the shell model. The generalized shell model also allows one to consider nuclei having more than one nucleon above a core. If a nucleus lacks one nucleon to close a shell, then we say that such a shell has a "hole". Such one hole nuclei are also described by the shell model. It appears that the spin of such a nucleus is equal to the total

80

The Quantum World of Nuclear Physics

moment of the missing nucleon. For instance, the nucleus 3He could be treated as the 4He nucleus with one missing neutron in the state I = 0, j = 1/2. Thus the 3He spin is predicted to be 1/2, which is confirmed by experiment. Therefore, the shell model allows us to explain magic numbers, spins of different nuclei, some levels in the energy spectra of nuclei and, in several cases, the magnetic dipole and electric quadrupole moments of nuclei. Nevertheless, a number of unexplained levels remains. These levels are evidently caused by the core excitation when the excited state of the nucleus originates from the change of states of many nucleons. These levels have a "collective" character. In many cases, electric quadrupole moments of nuclei also appear much greater than the predicted ones. These deviations indicate the existence of collective degrees of freedom connected with the motion of many nucleons in a nucleus. The next paragraph is devoted to this issue. The model of nucleon associations, or the cluster model, is a variant of the nuclear shell model. In that model, nuclei (mainly light ones) are considered as consisting of several light nuclei that could exchange nucleons. For instance, the 6Li nucleus in such a model could be treated as composed of an a-particle and a deuteron, the 9Be nucleus could contain two aparticles and a neutron, and the 12C nucleus could contain three a-particles. The statement that some nuclei consist of light nuclei has a deep physical meaning. The matter of fact is that the correlations between nucleons caused by the character of nuclear interaction and the Pauli principle are probable in nuclei. These correlations are responsible for the formation of stable groups of nucleons (nucleon associations or clusters) in nuclei. The formation of clusters increases the binding energy of the nucleus. Extensive experiments testify to the existence of quite stable clusters in nuclei. The most stable is the a-cluster (a-particle). Two protons and two neutrons forming an a-cluster can occupy the same space and energy state. This results in the great binding energy of the a-particle: Wa = 28.3 MeV. The emission of one nucleon from an a-particle takes about 20 MeV, while for the majority of nuclei this value is about 8 MeV. The light nuclei with Z = N and total number of nucleons A divisible by four have a binding energy 90% of which is associated with a-clusters and only 10% with interaction between a-clusters. It is clear that the one-a-particle emission for such nuclei takes much less energy than the one-nucleon emission. The a-cluster nuclei are 12C, 16O, 20Ne, 24Mg, etc. (the 8Be nucleus is not stable).

Structure of Atomic Nuclei

81

The existence of a-particles inside heavy nuclei is confirmed, in particular, by their a-decay where an a-particle exists inside a nucleus in a ready form. Alpha-clusters are created with great probability in the surface region of a nucleus where the nuclear density is small relative to that in the interior. Note that a nucleon also is a cluster that consists of three quarks. The first model that considered the nucleus as consisting of clusters was the a-particle model. The model assumed that inside a nucleus, the a-particles were stable systems, i.e., a great probability existed to find a system of four nucleons localized in the nucleus and separated from other such systems. The possibility of nucleon exchange between separate apartides was neglected. The simplest a-particle model treats the 12C nucleus as a hard equilateral triangle with a-particles at the vertices. This model made it possible to explain a number of properties of 12C nuclei, and to calculate probabilities of scattering at small angles of electrons and nucleons by these nuclei. Similar results were obtained for the 16O nucleus treated as a hard equilateral tetrahedron with a-particles at the vertices. Further improvement of the a-particle model is connected with the possibility for a-particles to vibrate about their equilibrium locations. In the model of nucleon associations (cluster model), a-clusters could partially overlap and exchange nucleons with each other. Extensive calculations made on the basis of such a model explained various experimentally observed properties of nuclei, emphasizing the substantial role of nucleonnucleon correlations in nuclei.

3.3

Collective Motions of Nucleons in Nuclei

Numerous experiments have shown that many nuclei have electric quadrupole moments much greater than predicted by the shell model. Large quadrupole moments of nuclei are caused by the deviation of these nuclei from spherical symmetry. Indeed, if a nucleus has the form of an ellipsoid, then its electric quadrupole moment will be proportional to the amount of deformation. Nuclei with all nucleon shells filled are spherically symmetric. If a nucleus has one or a few nucleons above the closed shells, these nucleons can deform the nucleus via their interaction with the core. It was theoretically proved that one nucleon above the closed shell could have lower energy if its potential well was not spherical. Therefore, the nucleus consisting of a

82

The Quantum World of Nuclear Physics

core with closed shells and an additional nucleon (or nucleons) can achieve a state with lower energy if the core becomes nonspherical. In other words, the nuclear shape is defined by the competition between those nucleons comprising the closed shells that tend to give the nucleus a spherical shape, and those comprising the unclosed shells that tend to give the nucleus a nonspherical shape. As a result, the nucleus can acquire a nonspherical form that will be statically stable. Particularly, the nuclei of lantanoids and actinoids are nonspherical because their numbers of neutrons and protons lie between magic numbers. The possibility of the existence of nonspherical nuclei was first pointed out in 1950 by the American physicist Rainwater, who also explained the origin of the existence of large electric quadrupole moments. In 1950-52, the Danish physicists Aage Bohr (the son of Niels Bohr) and Mottelson developed the closed theory of collective motions of nucleons in nuclei. In 1975, for those works, A. Bohr, Mottelson, and Rainwater were awarded the Nobel Prize in Physics. In the first approximation, a nonspherical nucleus can be considered as an ellipsoid of rotation. If the deviation from sphericity is not large, the deformation can be defined by the magnitude (J = AR/R where AR = b — a and R = (o + 6)/2 (a and b are the semi-minor and semi-major axes of an ellipsoidal nucleus). In that case, the electric quadrupole moment of the nucleus is proportional to its deformation: Q ~ f3. Thus the values of deformations can be determined by the measured electric quadrupole moments of nuclei. Two types of collective motions of nucleons can be observed in nuclei. First, there are the vibrations of the nucleus, namely the vibration of its shape, which is not accompanied by changes in density because nuclear matter is in fact incompressible. Nonspherical nuclei can vibrate as well as spherical ones. The energy spectra of many nuclei have levels of a vibrational origin. The properties of such levels are predicted by the collective model of a nucleus. Another type of collective motion of nucleons is the rotation of nonspherical nuclei. If the rotation of a nonspherical nucleus is rather slow, as it is in many cases, then such rotation will not substantially influence the motions of separate nucleons inside the nucleus and the possible small vibrations of the nuclear surface. Under these conditions the rotational energy can be explicitly extracted from the total energy of a nucleus. In other words, the energy spectrum of a nonspherical nucleus comprises, besides the levels of a stationary nucleus, also the levels of its rotation. The

Structure of Atomic Nuclei

83

intervals between the rotational levels turn out to be small compared to those for the levels of a stationary nucleus (one-particle levels). If the nucleus takes the shape of a rotational ellipsoid, then such a quantum mechanical system can rotate only around the axis perpendicular to its axis of symmetry. Indeed, rotation of the nucleus around its axis of symmetry does not alter its orientation in space, thus its states are not distinguishable and the energy of a system does not change. Any rotation about an axis that is inclined at a definite angle to the axis of symmetry can always be represented as a result of rotations about the axis of symmetry and the axis perpendicular to the axis of symmetry. Since a change in the spatial orientation of a nucleus takes place only in the second case, a rotation of a nonspherical nucleus about an arbitrary axis becomes, in fact, a rotation about the axis perpendicular to the axis of symmetry. We emphasize that a spherical nucleus cannot rotate, so it has no rotational degrees of freedom. This statement also concerns any spherically symmetric quantum system whose states are indistinguishable upon spatial rotation. By means of quantum mechanics, it can be shown that the energy of rotation of a nonspherical nucleus with zero spin around the axis perpendicular to the axis of symmetry is

E/ = S J ( J+ 1)'

(3-14)

where / = 2,4,6,... are the moments (spins) of possible excited rotational states of the nucleus and J is the moment of inertia of the nucleus. Explanation of the moments of inertia of nuclei appears to be a rather complicated problem, since their values are between the magnitudes predicted by the models of a liquid drop and solid matter. Physical reasons leading to these values for the moments of inertia of nuclei are discussed below. Even-even nuclei have zero spins. All nucleons in such nuclei are paired. In order to break up one such pair, i.e., to transfer the ground state of the even-even nucleus into its one-particle excited state, an energy of about 2 MeV is needed. The energy of the first excited rotational level of a nonspherical nucleus is much smaller than the energy of pairing (almost ten times). Therefore, the rotational and one-particle levels are easy to distinguish in the energy spectra of nonspherical even-even nuclei. If the energy of the first excited rotational level of an even-even nucleus E2 is known, then its moment of inertia can be denned: J = 3h2/E2- Thus

84

The Quantum World of Nuclear Physics Table 3.1 Ratios E1/E2 for several nonspherical even-even nuclei whose theoretically predicted values are / ( / + l)/6. Nucleus

/ = 4

/ = 6

/ = 8

/ = 10

160 Dy

3.270 3.239 3.258 3.206 3.17 3.27 3.31 3.333

6.694 6.553 6.635 6.434 6.34 6.63 6.93 7

11.14 10.78 10.96 10.49 10.4 11.0 11.6 12

16.46 15.76 16.07 15.16 14.7 16.1 17.7 18.33

166 Yb 172 Hf

176W

186 Os

2 3 2 Th

238 U / ( / +1)/6

the energies of rotational levels will be determined by the formula

EI = ll(I + l)E2.

(3.15)

In this case the correlation between the energies of rotational levels, which follows from (3.15) and is called the rule of intervals, is valid: E2:E4:E6:E8:E10:...

= l: y : 7 : 12 : y ... .

(3.16)

Table 3.1 displays the experimentally measured ratios E1/E2 for several nonspherical even-even nuclei alongside the theoretically predicted values of J ( J + l ) / 6 . The contradictions between the predictions of the theory and the experimental data shown in Table 3.1 are explained first by the fact that the simplest model of nuclear shape — an ellipsoid of rotation — is chosen. The real shape of a nonspherical nucleus appears in fact more complicated. For instance, the shape of a nucleus could be a general ellipsoid, i.e., an ellipsoid having all three axes of unequal length. In some cases it could be even more complicated. Thus the rule of intervals (3.16) substantially changes. It is not always possible to separate rotations and vibrations of nuclei. Interaction between these two types of collective motions of nucleons also changes the rule of intervals (3.16). Therefore, the realistic rules of intervals considering different properties of nonspherical nuclei allow one to coordinate the calculated and measured energy spectra of rotational levels of nuclei. If the energy of a rotational level and the spin of an excited state are comparatively small, then the inner structure of the nucleus does not

Structure of Atomic Nuclei

85

change, i.e., the ties between coupled nucleons in a nucleus do not break. However, by bombarding the nucleus with heavy ions, an artificial superrotating nucleus can be created. In such a process, nuclei in high spin states can be formed. In these nuclei, almost all excitation energy is contained in the energy of rotation, i.e., the nucleus remains "cold" inside. Large rotational energy can lead to the breakdown of coupling of nucleons which, in its turn, can cause a change in nuclear shape. At comparatively small excitation energies (I < 15 — 20), a nonspherical even-even nucleus has the form of a prolate ellipsoid of rotation. With an increase of spin I, the centrifugal and Coriolis forces lead to the breakdown of coupling in nuclei, i.e., at I > 20 - 25 nuclear matter consists of uncoupled nucleons. In this case, the nucleus acquires the form of a three-axial ellipsoid, while its moment of inertia increases up to the solid matter value. If / > 60, then the nucleus acquires the form of an oblate ellipsoid of rotation. When / ss 85 - 100, the nuclei become unstable relative to fission. These are the maximum possible values of nuclear spins. The states of nuclei with high spins and excitation energies, when almost all excitation energy is the energy of rotation, are called yrast-states. This term is connected with the old Norman verb "hvirfla", meaning "to twirl". From this verb the Swedish adverb "yr" originates, the highest degree of which, "yrast", means "dizzy" or "stunning". However, this word should be comprehended as "bearing the most rotation". Let us briefly discuss some features of volume vibrations of nucleons in nuclei. If the nucleus absorbs a photon with high enough energy, then the electric field of the absorbed photon causes the coherent motion of all protons of the nucleus along the field direction. Evidently, this leads to the appearance of vibrations of all protons of the nucleus with respect to all its neutrons. In this case, the nucleus acquires some dipole moment. If the internal frequency of vibration of such a dipole moment coincides with the frequency of the absorbed photon, then the effect becomes especially powerful. This phenomenon is called giant dipole resonance. Usually the energy of its excitation for different nuclei is contained in the interval 15-25 MeV and can be determined from the empirical formula Eg = 78 • A~1/3 MeV.

(3.17)

More complicated types of giant resonances (quadrupole, octupole, etc.) can also be excited in nuclei. But such excitations require special consideration on the basis of quantum mechanics. That is why we do not discuss them here.

86

3.4

The Quantum World of Nuclear Physics

Superfluidity of Nuclear Matter

The hydrodynamic (liquid drop), shell, and collective models of the nucleus allow us to explain numerous properties of nuclei. However, important experimental data exist that cannot be explained by these models. The facts are as follows. (1) The existence of a gap in the energy spectra of even-even nuclei, and its absence in the spectra of odd and odd-odd nuclei. In other words, in odd and odd-odd nuclei the energy of the first excited level of nonrotational character is several tens of keV, while in even-even nuclei this energy is usually about 1 MeV. (2) The values of the moments of inertia of nonspherical nuclei are greater than those obtained from the hydrodynamic model, but are smaller than those found from the solid matter approach. Besides, the moments of inertia of odd nuclei are much greater than those for even-even nuclei. This difference cannot be explained on the basis of the collective model via the adding of one nucleon to the even-even core, because in that case the nucleus should have too much deformation. (3) The transition of the shape of the nucleus from spherical to ellipsoidal occurs when about 25% of all positions in the last unclosed shell of the nucleus are filled, while the independent particle model calculations show that all nuclei with nucleons in the unclosed shells should be nonspherical. (4) The differences in masses and binding energies of even-even, odd, and odd-odd nuclei require the introduction of an additional term in the formulae (3.7) and (3.8) considering the coupling of nucleons of the same kind. (5) The mean separation between the excited one-particle levels in odd nonspherical nuclei is approximately half that calculated in the model of a self-consistent field. (6) The measured probabilities (half-lives) of a- and /3-decay for some nuclei deviate from the values calculated in the independent particle model. Therefore the shell model, which assumes that nucleons move independently in a self-consistent field (the model of independent particles) and does not consider the nucleon-nucleon correlations leading to the coupling of nucleons of the same kind, cannot explain these experimental facts. These

Structure of Atomic Nuclei

87

facts are not explained in the collective model of the nucleus either. Thus we have to briefly discuss the physical consequences to which the nucleonnucleon pair correlations lead in nuclear matter. A nucleus is a system of strongly interacting nucleons having spin 1/2, i.e., fermions. Thus, a nucleus can be considered as a Fermi-liquid. If the forces of attraction act between the particles of a Fermi-liquid, then the phenomenon of creation of bound pairs of fermions occurs that appears to be extraordinary from the viewpoint of classical physics. In an infinite nuclear matter, the nucleons that are coupled are of the same kind, with equally valued and oppositely directed angular momenta and spins. Such a coupling is observed, for instance, for electrons in metals at low temperatures. It leads to the known phenomenon of superconductivity. As we have mentioned, the phenomenon of coupling of fermions is called the Cooper effect. In a finite-size nucleus, the nucleons that are coupled are of the same kind, occupying the states with the same main quantum numbers and orbital and total moments, but with opposite projections of total moments on the quantization axis. In other words, the coupled pairs are nucleons with quantum numbers n,l,j,m and n,l,j,—m (n is the main quantum number, I and j are the orbital and total moments of nucleon, and m is the projection of the total moment on the quantization axis). Cooper's effect takes place for any (even very weak) interaction between fermions, provided the interaction is an attraction. The usual self-consistent field in which nucleons move independently considers, in fact, only the major part of nucleon-nucleon interaction. The neglected small contribution is called the residual interaction. However, despite its smallness, the residual interaction can lead to the coupling of nucleons in nuclei because of its attractive character. Evidently, the effect of coupling of nucleons in nuclei leads to the superfluidity of nuclear matter. The superfluidity of a Fermi-liquid, as discussed earlier, is a special phase of matter existing at very low temperatures. Two coupled fermions form a system with zero spin, representing a stable formation similar to a particle. Therefore, such a group of two coupled fermions (Cooper's pair) is called a quasiparticle. It is a boson, and the Fermi-liquid in the superfluid state could be considered as a liquid of such quasiparticles. The theoretical description of the quasiparticle liquid is much easier than that for the nucleon liquid. The point is that nucleons strongly interact with each other, while the interaction between quasiparticles is very small.

88

The Quantum World of Nuclear Physics

gp^

0.40.2 -

°

0

0.2

^ ^ ^

0.4

p

Fig. 3.3 Ratios of the moments of inertia J of atomic nuclei calculated by the superfluid (solid line) and hydrodynamical (dots) models to their solid-body values Jo as functions of the nucleus deformation parameter (3. The circles are experimental data. Based on the model of superfluid nuclear matter, it is possible to explain many experimentally measured properties of nuclei. In particular, only the superfluid model describes the moments of inertia of nonspherical nuclei and the gaps in the energy spectra of even-even nuclei. Fig. 3.3 shows the experimentally measured and theoretically predicted moments of inertia of nonspherical nuclei. In conclusion, we note that all the nuclear models discussed here originate from rather different and sometimes opposing premises. Nevertheless, all the models describe different properties of nuclei quite well. This means that all models contain some unique underlying principle of nuclear structure. This principle is the concept of a self-consistent field in which nucleons move independently (the model of independent particles). Different nuclear models appear from different approaches to take account of the residual interaction. The existing nuclear models allow us to explain a great deal of experimental data and understand general regularities of nuclear structure.

Chapter 4

Radioactivity of Atomic Nuclei

4.1

The Law of Radioactive Decay

Human history records but a few great scientific discoveries made accidentally. The discovery of the radioactivity of atomic nuclei is one of them. Radioactivity is the process of spontaneous transmutation of an unstable nucleus into another one, which is accompanied by the emission of various particles and photons. Several elementary particles can undergo radioactive decay as well. In February 1896, the physics professor Becquerel was working at the Ecole Poly technique in Paris. He studied the abilities of various crystals under sunlight to emit a penetrating radiation similar to the X-rays that had been recently discovered by Roentgen. Becquerel supposed that crystals under the influence of light would emit rays that could register on the photographic plates covered by black paper. A screen made out of copper wires was placed between the crystal and the plate. Thus, after developing, the plate would be light-struck everywhere except the region covered by the copper wires. Among the crystals Becquerel was working with, by chance some uranium salts were stored, specifically uranium sodium bisulphate. Also by chance, the weather during the experiments was cloudy so that Becquerel put the plates into the box of the laboratory table together with the crystals of uranium salt. After a few days when the plates were developed, they appeared dark and demonstrated a very clear image of the screen, even though no sunlight had illuminated the uranium salt. Becquerel attributed the rays that registered on the plate to the uranium. Afterwards it was found that the same rays could be emitted by other elements as well. In 1898, Marie Sklodowska-Curie discovered that 89

90

The Quantum World of Nuclear Physics

the same rays were emitted by thorium; she and her husband Pierre Curie subsequently discovered radium. The term "radium" originates from the Latin word "radius", which means "ray". The Curies named this phenomenon radioactivity. In 1903, Becquerel and the Curies were awarded the Nobel Prize in Physics: Becquerel for his discovery of spontaneous (unmotivated) radioactivity, and the Curies for their joint research on the radiation phenomena discovered by Becquerel. Marie Sklodowska-Curie was also awarded the Nobel Price in Chemistry in 1911 for the discovery of the elements radium and polonium, for the isolation of radium, and for the study of the nature and compounds of this remarkable element. In 1899, Rutherford discovered a- and /?-rays. Afterwards he found that Q-rays are the ions of helium. The rays found in 1900 by French physicist Willard were different from a- and /3-rays. In 1903 Rutherford named these 7-rays and assumed they are electromagnetic radiation with quite a short wavelength similar to X-rays. In 1900 Rutherford introduced the notion of a half-life period. He was awarded the Nobel Prize in Chemistry for 1908 for his investigations into the disintegration of elements and the chemistry of radioactive substances. Only much later, after the discovery of the atomic nucleus by Rutherford in 1911, was it understood that radioactive radiation was emitted just by nuclei. Further investigation showed that many unstable nuclei with finite life-times exist in Nature. These nuclei change their states by themselves, and are called radioactive. Many of the known radioactive nuclei are artificial. Artificial radioactivity was discovered in 1934 by I. Joliot-Curie and F. Joliot-Curie. They were awarded the Nobel Prize in Chemistry in 1935. The Joliot-Curies bombarded aluminium foil with 5.3 MeV a-particles emitted by 210 Po, and among the reaction products they found not only nucleons but positrons. Moreover, the positrons were emitted during a certain period of time after the irradiation had stopped. Thus they observed the nuclear reaction 27Al

+ a —>

30 P

+ n,

(4.1)

in which the radioactive isotope of phosphorus 30 P was created to undergo the positron /3-decay (only the isotope 31 P is stable). Further, a great number of different radioactive isotopes were obtained, which have wide applications in nuclear physics and in other fields of science and technology. Various types of radioactive nuclei exist. As we mentioned, during adecay the nucleus itself emits an a-particle, i.e., the nucleus of the atom

Radioactivity of Atomic Nuclei

91

^He. If the radioactive nucleus emits an electron and an antineutrino, then such a process is called P~ -decay. If the radioactive nucleus emits a positron and a neutrino, then this is a /3+-decay. If the nucleus (the nuclear proton) captures an electron from the if-shell of the atom, the so-called Kcapture, then such a process is /3-decay as well. In this case the electron disappears while the electron antineutrino is emitted and the nucleus passes into an excited state. During 7-decay the excited nucleus emits one or more photons and passes into a lower lying or ground state. Some heavy nuclei can spontaneously decay into two or more parts. This phenomenon is called spontaneous fission and is also treated as a process of radioactive decay. The spontaneous fission of 235 U nuclei was discovered in 1940 by the Soviet physicists Flerov and Petrzhak. There also exist rare (exotic) types of radioactive decay of nuclei. The existence of proton and diproton radioactivity is possible close to the boundary of proton stability. The process of double /3-decay is possible. The radioactive decays in which the nuclei 12 C, 14 C, 16 O, 20Ne, and some others, are emitted have been well studied. The emission of delayed neutrons and protons is also considered as a radioactive decay. These decays are two-step cascade processes, because the emission of a delayed neutron or proton from a nucleus takes place after the preliminary emission of a positron or electron. Moreover, the emission of a nucleon is delayed by the time of the corresponding /3-decay, because the process of nucleon emission from a nucleus itself occurs almost instantaneously. The number of radioactively decaying nuclei decreases with time. The process has a statistical character, meaning that one can speak only about the probability per unit time for the given nucleus to decay. If the initial number of a certain type of radioactive nucleus is large enough, the general law of its decrease with time can be determined. Actually, the decrease of the number of nuclei dN(t) during the interval dt is proportional to both this interval dt and the current number of nuclei N(t): dN(t) = -XN(t)dt,

(4.2)

where A is the decay constant and the negative sign indicates the decrease in the number of nuclei during time dt. Formula (4.2) generates the law of radioactive decay N(t)=Noe-Xt,

(4.3)

92

The Quantum World of Nuclear Physics

where No = N(0) is the number of nuclei at the initial moment of time t = 0. Formula (4.3) reflects the random character of radioactive decay, and has been confirmed by numerous experiments. It is impossible to predict the moment of time when a given radioactive nucleus will decay. One can only evaluate the probability for it to decay at a given moment of time. For a given nucleus, the magnitude exp(—At) represents the probability that it will not decay during the time t. So 1— exp(—Xt) is the probability of decay during that same time. For instance, the decay constant for the 226 Ra nucleus is A — 1.37 • 1 0 ~ n s" 1 . Thus, during each second, on average, one nucleus of this isotope out of a hundred billion will decay. The decay constant has a simple physical meaning. It is equal to A = 1/i, where t is the mean lifetime of one radioactive nucleus. Evidently, during the time t the initial number of radioactive nuclei decreases by a factor of e K 2.718.... The principal problem of radioactive decay theory is finding the decay constant for the nuclei of a given isotope, which can also be measured experimentally. Formula (4.3) shows that the number of radioactive nuclei decreases with time according to the exponential law. The law is valid for all types of radioactive nuclei. However, it should be noted that this conclusion does not consider the dynamics of radioactive decay. Apparently the law (4.3) should be derived not from statistical considerations, but from the equations of quantum mechanics. Further, it should be kept in mind that the problem of radioactive decay is intrinsically nonstationary, because before the decay there are no products of decay outside the nucleus. Therefore, a decaying nucleus occupies one of the nonstationary states of its energy spectrum. The law of radioactive decay is caused by the character of this spectrum. The nonstationary character of the state leads, in fact, to a deviation from the exponential law. But that deviation takes place only at very small or very large times, and is not usually observable. In a-decay, for instance, it should be observed at t < 10~21f and t > 50£. The time ti/2 during which half of the initial number of identical radioactive nuclei decay is called the half-life. This quantity is related to the constant of decay by 0.6931 In 2 *1/a = - « —

. , (4.4)

Thus we find I = 1.443t1/2- After the time tk = fc*i/2, the number of

93

Radioactivity of Atomic Nuclei

1/2

-\

1/4

j.

-Nv

1/8

i

t -^^r^-^^

0

tm

2tm

3tm

t

Fig. 4.1 Number of radioactive nuclei N(t) as a function of time t. nuclei that have not yet decayed will be (Fig. 4.1) N(kt1/2)

= Noe~Xkt^

= 2~kN0,

(4.5)

where fc = 1,2,3,.... The values of the half-life periods £j/ 2 vary widely between radioactive nuclei. For a-decay, ti/2 varies from 3 • 10~ 7 s for 2 1 2 Po nuclei to 1.6 • 10 23 s (5 • 10 15 years) for 144 Nd nuclei. For /3-decay, the values of ti/2 range from 10~2 s to 2 • 10 15 years. The half-lives for 116 In and 115 In nuclei are 14 s and 1.9 • 10 22 s (6 • 10 14 years), respectively. Sometimes one type of nucleus will decay into another type that is also radioactive, and we see a chain of radioactive transmutations. In that case, we must replace (4.2) with a set of two linked equations in order to describe the overall process:

*m=.XlNl{t)i

«=AliVl(i)_W).

(4.6)

Here iVi and N2 are the numbers of the two types of nuclei, and X\ and A2 are the respective decay constants. The first equation in (4.6) describes the process of radioactive decay of the parent nuclei Ni. The second equation describes the decay of the daughter nuclei N2, and contains two terms on its right-hand side. The first represents the number of nuclei IV2 appearing under the decay of the Ni nuclei, while the second is the number of decaying

94

The Quantum World of Nuclear Physics

N2 nuclei. The solution of equations (4.6) has the form (4.7)

N^^N^e-^, N2(t) = N2(0)e-^ + M l M (e-Ai* _ e-^t\

?

( 4 8 )

A 2 — Ai

where Wi(0) and N2(0) are the numbers of nuclei Ni(t) and N2(t) at time t = 0. Equations (4.7) and (4.8) simplify substantially if Ai -C A2, i.e., if the half-life of the parent nuclei Ni is much greater than that of the daughter nuclei N2. Then we find JVi(t) « # ! ( ( ) ) ,

N 2 (t)wiV2(O)e- Aat + ^ 7 V i ( O ) ( l - e - A 2 t ) . A2 If we have no N2 nuclei at the beginning, ./V2(0) = 0, then

(4.9)

N2(t)*s^N1(p)(l-e-x>t). (4.10) A2 After a long period of time that substantially exceeds the half-life for the N2 nuclei (A2£ 3> 1), we obtain XiNiit) = X2N2(t).

(4.11)

Formula (4.11) is called the secular equation. This equation means that, if Ai -C A2, then after a long period of time the number of decays of the daughter substance A27V2(t) is equal to the number of decays of the parent substance \\Ni(t). The secular equation is widely used to obtain the halflives of long-lived radioactive isotopes. As an example we consider the following chain of radioactive decays of nuclei: 226 Ra nuclei undergo a-decay with half-life i1;/2(Ra) « 1600 year, and turn into radioactive 222 Rn nuclei; these undergo a-decay with halflife iiy2(Rn) w 3.8 days. Choosing times satisfying the condition £jy2(Ra) <§; t -C £]y2(Rn), we find the secular equation ARa%a

= *Rn%n-

( 4 - 12 )

In this case the number of appearing Rn nuclei coincides with the number of decaying Ra nuclei. The values of iVp^a and iVp^n can be determined by weighting the samples, while the value of Aj^n can be found by the comparatively small half-life period of Rn nuclei. Then the value of Aj^a for the Ra nuclei with the long half-life can be derived from the secular

95

Radioactivity of Atomic Nuclei

equation (4.12). Finally, using the correlation (4.4), it is easy to evaluate the long half-life of the Ra nuclei. The activity of a radioactive substance is denned by the number of decays per unit time. One decay per second is accepted as a unit of measure known as the Becquerel (Bk). This unit of measure is too small in practice, however, and 1 MBk = 106 Bk and 1 GBk = 109 Bk frequently appear instead. For instance, the radon 222Rn contained in 1 m3 of atmospheric air has an activity of about 4 Bk. One kg of uranium ore with 10% of pure uranium possesses an activity of 0.13 MBk. Cobalt-based radioactive samples used in medicine for radiotherapy have activities from 75 to 2 • 105 GBk. A nuclear bomb with an explosive power equivalent to that of 20 kilotons of trinitrotoluene creates an activity of about 7.4 • 1013 GBk upon explosion. An earlier unit of activity called the Curie (Ci) is now used only rarely; the conversion between these two units is given by 1 Ci = 3.7 • 1010 Bk. Sometimes a unit of activity called the Rutherford (Rd) is also used: 1 Rd = 106 Bk = 1 MBk. When the Earth was formed about 4.6 • 109 years ago, it was composed of the isotopes of various elements. Some of these were radioactive. Radioactive isotopes with half-lives substantially shorter than the Earth's age have long since decayed and are not observed in Nature. Therefore, the radioactivity of natural substances is determined by elements having half-lives greater than or equal to the Earth's age. 238U

a

-

234Th

— p

234Pa

-—~ p

234U

a

-

230Th

218 At

_^

- « p o ^

R n

oN , 21°Bi

y

214 Pb

"p

a

^

210TI

a

. "

P

226Ra

21 °Pb

P

21°Po

"

P

V

2"Bi

2 1 4 Po

a

206TI

206Pb

^

Fig. 4.2 The uranium series. Three series of radioactive nuclei exist in Nature. They begin from the

96

The Quantum World of Nuclear Physics , 235

U— a

—

2 3 1 T h - ^ 231Pa

p

219Rn

—

215Po

— a

—

227Ac

211Pb

1> \ >

-JT

227Th

.

« * 223Ra V ^Fr -p

211Bi i>

V

207TI

" "p

— a

207Pb

Fig. 4.3 The uranium-actinium series. very long-lived radioactive nuclei found on Earth. The uranium series (Fig. 4.2) starts from the 238U isotope (ti/2 = 4.5-109 years) undergoing the chain of transmutations (8 a-particles and 6 electrons are emitted) that lead to the formation of the stable 206 Pb isotope. The actinium-uranium series (Fig. 4.3) starts from the 235U isotopes (t1/2 = 7.1 • 108 years) undergoing a chain of transmutations (7 a-particles and 4 electrons are emitted) resulting in the stable 207 Pb isotope. The thorium series (Fig. 4.4) starts from the 232 Th isotope (£1/2 = 1.41-1010 years) undergoing a chain of transmutations (6 a-particles and 4 electrons are emitted) leading to the formation of the stable 208 Pb isotope. 232_.

Th — -

228D

Ra -r—

a

228.

Ac —r—

p

22I-,

224D

Th — -

(5

Ra — -

a

220,-,

Rn — -

a

a

X -Po

—

216

Po—

212Pb

—

212Bi

P

!>

208Pb

a

V -T,

/

Fig. 4.4 The thorium series. ^

233pg

N

a

233u _ _

2 2 ^ _ _

a

a

P

225Ra

225^

p

_ _

a

x2"Po

—

221

Fr— a

217At

— a

213Bi

V ^

>

a

209TI

s

209Pb

i

Fig. 4.5 The neptunium series.

^r

p

209Bi

Radioactivity of Atomic Nuclei

97

The neptunium series also exists (Fig. 4.5), starting from the 237Np isotope (£1/2 = 2.14 • 106 years), artificially created in nuclear reactors, undergoing a decay chain (7 a-particles and 4 electrons emitted) resulting in the stable 209Bi isotope. The series could have existed in the past for some time under natural conditions after the formation of the Earth. However, the half-lives of elements of the series are small compared to the Earth's age. Therefore, at present they are not observed in Nature. Many transuranium elements undergo a-decay and can also be attributed to these radioactive series. The 242 Pu nucleus (ij/ 2 = 5 • 105 years) transforms into the 238U nucleus. The 243Cm nucleus (£1/2 = 100 years) turns into the 239 Pu nucleus (£1/2 = 2.43 • 104 years), which in turn transforms into the 235U nucleus. The 244Cm nucleus (£1/2 = 19 years) turns into the 240 Pu nucleus (£1/2 = 6580 years), which then transforms into the 236U nucleus (t\/2 = 2.4 • 107 years), while the latter turns into the 232 Th nucleus. Finally, the 241Am nucleus (£1/2 = 470 years) transforms into the 237Np nucleus. Therefore all the known radioactive series can be continued to heavier transuranium elements. Table 4.1 Some radioactive nuclei found in the Earth's crust but that do not enter into the radioactive series Nucleus 40 K

87 Rb 113 Cd U5In

138 La 144 Nd 147 Sm 148 Sm 152 Gd 176 Lu 174 Hf 187 Re 1 9 0 Pt

Type of decay

t i ^ , yeas

K-capture p~ p~ P~ p~, K-capture a a a a p~ a p~ Q

1.26 • 109 4.8 • 1010 9-10 15 5.1 • 1014 1.1 • 1011 2.1 • 1015 1.06 • 1 0 u 8 • 1015 1.1 • 1014 3.6 • 1010 2.0 • 1015 4 • 1010 6 • 1011

0-,P+,

Radioactive elements that do not take part in these series of radioactive nuclei are also observed in Nature. Some of these with large half-lives are shown in Table 4.1. Note that the radioactive isotope 40K is widely used for dating minerals via the relation of percent contents of the isotopes 40K and 40Ar contained therein.

98

The Quantum World of Nuclear Physics

Despite the general character of the law of radioactive decay (4.3), the nature of the different types of radioactive decay is very different. An a-particle emitted during a-decay is in a nucleus in a ready form. It is formed in a nucleus, and exists there for quite some time before leaving the nucleus. This is not the case for the electron and antineutrino (or positron and neutrino) emitted during /3-decay — these are born just at the moment of decay. In fact, the existence of an electron inside a nucleus contradicts the data on values of spins and magnetic moments of nuclei. There is an even deeper physical reason for an electron not to exist in a nucleus: a conflict with the Heisenberg uncertainty principle (Section 3.1). Therefore, an electron and an antineutrino (or positron and neutrino) are born at the moment of /3-decay. A similar situation occurs with 7-decay, when a photon is born at the moment of its emission, because the nucleus does not contain a photon in a ready form. The interactions responsible for the different types of radioactive decays are very different. An a-decay is caused by the strong interaction, while (3-decs.y and 7-decay are caused by the weak forces and electromagnetic interaction, respectively.

4.2

Alpha-Decay

The mass of a nucleus that undergoes a-decay satisfies the inequality Tn(A,Z)>m{A-i,Z-2) + mcn

(4.13)

where m(A, Z) and m(A — 4, Z — 2) are the masses of the parent and daughter nuclei. An analysis of (4.13) indicates the possibility of a-decay for all the nuclei with A > 57, and this is confirmed by experimental data. Most of the kinetic energy released during a-decay is taken away by the a-particle, while the daughter nucleus takes only an insignificant part of it (about 2% for the heavy a-active nuclei). For instance, during the a-decay of the 212 Bi nucleus, the a-particle has energy Ea = 6.086 MeV while the daughter 208 Tl nucleus has energy Enuci = 0.117 MeV. The total energy of the a-decay of the 212 Bi nucleus is therefore E = Ea + EnucX = 6.203 MeV. The simplest theory of a-decay is based on the suggestion that before the decay, an a-particle exists in a nucleus in a ready form and performs its vibration motions. In other words, the probability for the four nucleons (two protons and two neutrons) to configure themselves into an a-particle in an a-radioactive parent nucleus is close to unity. The question arises how such an a-particle can leave the parent nucleus.

99

Radioactivity of Atomic Nuclei a

b

V(r)

V(r) B

£ 0

-K

-Sv^-----.

E R

~ r

0

R

r

Fig. 4.6 Potential wells of (a) a stable nucleus, and (b) an a-radioactive nucleus. E is the energy of an a-particle in the well. To answer the question, we consider the potential energy of an aparticle. If the a-particle were trapped inside the nucleus for an infinitely long time, its potential energy would be a potential well having width equal to the nuclear radius R and height infinity (Fig. 4.6a). The potential energy, in fact, has a finite height H (Fig. 4.6b). Moreover, its external part is the Coulomb energy of interaction between the free a-particle and the daughter nucleus, because the nuclear forces vanish at distances exceeding the nuclear radius. Consider as an example the process of a-decay of the 238U nucleus. Provided the nuclear radius and the amplitude of vibration of the a-particle are of the same order of magnitude, it appears that the a-particle in the nucleus moves with velocity va ~ 107 m/s. This particle strikes the internal wall of the potential well 1021 times per second, the radius of which for this nucleus is R ~ 10~14 m. Since the half-life of a 238U isotope is ti/2 « 4.5-109 years, before the a-particle leaves the nucleus it strikes the well wall, on average, 1038 times. The 238U nuclei emit a-particles with energy Ea = 4.21 MeV. The height of the potential barrier, when the angular momentum is equal to zero, is H = 2(Z - 2)e2/R. Since Z = 92 for uranium, H « 30 MeV. The barrier height H exceeds the a-particle energy Ea by almost 26 MeV. Therefore, from the viewpoint of classical mechanics, an a-particle cannot leave the nucleus, and a-decay cannot be explained in such a way. The theory of a-decay was developed in 1928 by Gamov and, independently, by Condon and Gurney. The theory is based on the quantum mechanical tunnel effect. In quantum mechanics, the probability that a particle will penetrate a barrier of finite height and width is nonzero. The

100

The Quantum World of Nuclear Physics

microparticle whose motion is described by the equations of quantum mechanics can undergo the tunnel transition from the potential well. This transition is also called the under barrier [tunnelling) passage of a particle. Calculations based on quantum principles allow us to obtain the probability of tunnelling under the potential barrier, and to derive the decay constant A. The uncertainty in the energy of an a-particle passing through the barrier should be of the same order of magnitude as the barrier height AE « H. Thus, for the uranium nucleus, AE « 30 MeV. According to the uncertainty principle for energy and time, we have At > H/2AE, where At is the time the a-particle takes to pass through the barrier. Therefore At > 10~23 s. The a-particle passes the barrier almost instantaneously, because the characteristic nuclear time is about 10~22 s. The study of radioactive nuclei shows that they possess an interesting feature: the smaller the half-life of the nucleus, the higher the energy of the a-particle emitted. For instance, the 212Po nucleus with half-life t\/2 = 3 • 10~7 s emits an a-particle with energy Ea — 8.95 MeV, while the 232 Th nucleus with tx/2 = 1010 years emits a-particle with energy Ea = 4.28 MeV. The energies of a-particles emitted by nuclei whose half-lives differ by a factor of 1030 differ by almost a factor of two. In 1911, data analysis allowed the German physicist Geiger and the English physicist Nuttall to establish the empirical law that now bears their names. This law relates the decay constant A to the mean free path of an a-particle in a substance (gas) under a given pressure and temperature: ( mQR?\ Ra In A—-—) =Aln—+B.

V.

n )

Ro

(4-14)

Here the constant A is the same for all the radioactive series existing in Nature, while the value of the constant B varies by approximately 5%. Also appearing in the formula are the mass ma of the a-particle, the nuclear radius R, the mean free path Ra of an a-particle in air, and a constant Ro having the dimension of length. The mean free path is the distance between the emitter and the point of complete stoppage (collision) measured along a straight line. The mean free path of an a-particle in a gas is related to its kinetic o Icy

energy by the correlation Ra = nEa with constant K. This formula allows one to connect the half-life of a nucleus with the energy of an a-particle

101

Radioactivity of Atomic Nuclei

emitted:

K^H-IH'

(4i5)

where the constants A' and B' are similar to A and B, while the constant Eo has the dimension of energy. The theory of a-decay based on quantum mechanics permits the derivation of a correct expression connecting the decay constant and the energy of the emitted a-particle:

In i^2^)

= C^fE~a + D,

(4.16)

where C and D are constants similar to A' and B'. Formula (4.16) gives the quantum mechanical formulation of the Geiger-Nuttall law. Formula (4.16) shows that a small change in a-particle energy Ea leads to a great change in the decay constant A. A 10% change in Ea changes the decay constant by about a thousand fold. If Ea < 2 MeV, the half-life becomes so long that a-decay becomes almost unobservable. That is why the energies of a-particles (except in some cases) for all the known a-active nuclei are contained in the interval 4 MeV < Ea < 9 MeV. As a rule, inactive nuclei have charge numbers Z > 82; moreover, Ea increases with Z. The exceptions are some nuclei of rare earth elements (gQ4Nd, 626Sm, etc.), 7|°Pt nuclei, and some artificially created nuclei. Table 4.2 The energy spectrum of a-particles created by the a-decay of the 212 Bi nuclei Group of .. , a-particles Ea, MeV Percent contents

ao 6.086 2? 2

ai

a2

6.047

5.765 ^ L7

6g g

az 5.622 ^

Q 15

as

04 5.603 ^

1A

5.481 ^

QMQ

Usually the same a-active nuclei emit a-particles with the same energy. But some nuclei emit a-particles with somewhat different values of kinetic energy. In this case we speak of a thin structure of the a-spectrum. As an example, Table 4.2 shows the energy spectrum of an a-particle emitted by 212Bi nuclei. Fig. 4.7 explains the thin structure of this spectrum. The a-decay of the parent 212Bi nucleus can take place with the formation of the daughter 208 Tl nucleus being not only in the ground state but also in different excited states. The transition from these excited states into the

102

The Quantum World of Nuclear Physics 2 1 2 Bi

\

V \ V \

\

6

-203

M e V

a\a\aW^aXaX

\\\V\\ zoa-pi ° -

4 7 3

^^\^

Ei MeV

— 0.327 ^ 0.040

°~~

Fig. 4.7 Thefinestructure of the spectrum of a-particles created by the a-decay of 212Bi nuclei.

ground state is accomplished by emitting photons. Nuclei also exist (e.g., 212 Po) that emit, in addition to the main group of a-particles with some energy, a few a-particles with somewhat greater energy (Table 4.3). Such a-particles are called long-passing. Their emission is explained by the adecay of the 212Po nucleus, which is formed in the excited state under the /3-decay of the 212Bi nucleus (Fig. 4.8). Table 4.3 The energy spectrum of a-particles created by the a-decay of the 212 Po nuclei Group of ,. .

a-particles Ea, MeV Percent contents

4.3

ao

8.780 ^ 1QQ

ai

oc2

9.492 ^

1Q_3

2

10.422 1Q_3

cez

10.543 1Q_2

Lg

Beta-Decay

Beta-decay is a process of spontaneous transmutation of an unstable nucleus into the isobar having its charge different from the initial one by unity.

Radioactivity of Atomic Nuclei ,,,

Bi 11.195 MeV „ < ^ ^ ^ < ^ .

^l^S—

103

E, MeV

212Po

^10.746

\ \ \ — 9*75

\ \ \ V~~8-949

0 \\\Vpb Fig. 4.8 Spectrum of long-range a-particles created by the a-decay of nuclei 212Po. Beta-decay with emission of an electron is energetically possible if the mass criterion is satisfied: m(A, Z) > m(A, Z + l) + me,

(4.17)

where me is the electron mass. If /3-decay is accompanied by the emission of a positron, this inequality should be satisfied: m(A,Z) >m(A,Z -l)+me.

(4.18)

For K-capture, the corresponding inequality has the form m(A, Z) + me > m{A, Z - 1).

(4.19)

Let us consider different types of/?-decay. The 10Be nucleus, being extra neutron-rich, undergoes electron /?-decay with a half-life of ti/2 = 2.5 • 106 years and turns into the stable 10B nucleus. The 13N nucleus, having fewer neutrons than protons, undergoes positron /?-decay with t1/2 = 10 min and turns into the stable 13C nucleus. The 37Ar nucleus undergoes if-capture, absorbing an electron from the if-shell of the atom, and turns into the 37C1 nucleus (ti/2 = 35 days). As formerly discussed, a free neutron also undergoes /3-decay, turning into a proton and emitting an electron and an electron antineutrino. The half-life of the free neutron is about ten minutes. Note that neutrons in an atomic nucleus are usually stable and do not decay, provided the nucleus is not radioactive.

104

The Quantum World of Nuclear Physics

Some nuclei can undergo two types of /3-decay. For instance, the 48V nuclei undergo /3+-decay (positron emission) in 58% of cases and ^-capture in 42% of cases, turning into 48Ti nuclei (£1/2 = 16 day) in either case. The 74As nuclei undergo /3~-decay (electron emission) in 53% of cases and /3 + decay in 47% of cases, turning into 74Se nuclei in the first case and 74Ge nuclei in the second case (ti/2 = 17.5 day either way). In addition, there are nuclei that undergo all three types of /3-decay. An example is the 80Br nucleus undergoing /3~-decay in 92% of cases, /?+-decay in 3% of cases, and if-capture in 5% of cases (£ly/2 = 18 min in any case). The 80Br nucleus turns into the 80Kr nucleus by /3~-decay and into the 80Se nucleus by (3+decay. If m(A, Z) > m(A, Z + 2) + 2me then, in principle, the possibility of double /?-decay exists. Here the initial nucleus (^4, Z) simultaneously emits two electrons and two antineutrinos and turns into the nucleus {A, Z + 2). Although the probability of such a process is very small, it has been observed experimentally. For instance, double /3-decay has been observed for 48Ca nuclei turning into 48 Ti nuclei, 76Ge nuclei turning into 76Se nuclei, etc. Provided the (3~- or /3+-radioactive nucleus is neutron (proton) rich, the final nucleus can be formed in a state having energy that exceeds the energy of neutron (proton) separation. In this case, the daughter nucleus will subsequently emit a delayed neutron (proton). Since the neutron or proton is emitted by the excited nucleus almost instantly, the delay is denned by the time of /3-decay. The nature of /3-decay has presented a number of conceptual difficulties. Actually, before a-decay, the a-particle is already inside the nucleus in a ready form. But an electron or a positron cannot exist inside the nucleus, for this contradicts the uncertainty principle. Therefore we cannot say that an electron or a positron leaves the nucleus. On the other hand, during a-decay the emitted a-particle has a clearly defined energy or discrete spectrum of energies if the daughter nucleus is formed in an excited state. Experiments show that electrons or positrons appearing during /3-decay do not have a discrete spectrum but are characterized by a continuous spectrum with a definite upper boundary. At first sight, this important feature of /3-decay seemed to contradict the conservation law of energy and angular momentum. At the beginning of the investigation into /3-decay, the idea was even suggested that those laws were not universal and could be violated, and that the phenomenon of (3decay was just the process where the laws were violated. Even Niels Bohr

Radioactivity of Atomic Nuclei

105

supported this viewpoint at first. The principal solution of the /?-decay problem is associated with the name of Pauli, who suggested that one more particle, unobservable in experiments, is emitted besides the electron or positron in /3-decay. This means that the particle is chargeless, as charge is conserved during /3-decay. Moreover, the particle should have either zero or very small mass in order to avoid violation of (4.17) and (4.18). The particle acquired the name "neutrino", meaning "small neutral particle", because the name "neutron", meaning "neutral particle", was already reserved for the nuclear particle. The spins of neutrinos and antineutrinos are equal to 1/2 in accordance with the conservation laws for energy and angular momentum during /3-decay. Using this idea, Fermi constructed the first theory of /3-decay in which the process was considered as a simultaneous generation of two particles — electron and antineutrino. Actually, following the conservation law for the number of leptons, the generation of an electron with lepton charge +1 should be accompanied by the generation of an antineutrino with lepton charge -1. During (3+-decay, the particles formed are a positron with lepton charge -1 and a neutrino with lepton charge +1. The theory should be constructed in such a way that those particles do not exist in a nucleus, but appear at the moment of /?-decay. Fortunately, at that time the theory of quantum electrodynamics already existed. Here the emission of a photon by some quantum mechanical system was considered as a process of its generation but not its escape from the system. Such an approach required the development of a notion of quantization for the electromagnetic field. An electron or any other charged particle or system of charged particles (an atom, a nucleus) can interact with this field. Moreover, photons, the quanta of the electromagnetic field, can be generated via such an interaction. The approach became the starting point for the creation of the theory of /3-decay. By analogy with quantum electrodynamics, two quantum fields are introduced in this theory — an electron-positron field, and a neutrino field with electrons (positrons) and neutrinos (antineutrinos) as the quanta. Then /3-decay is presented as a process of generation of these particles by the nucleus, similar to the process of generation of a photon by an atom.

106

4.4

The Quantum World of Nuclear Physics

Gamma-Radiation of Nuclei

If a nucleus is in an excited state, it can spontaneously emit a photon and pass to a state with lower energy. Such a radiation transition can occur once, if the nucleus emits one photon and passes into the ground state, or many times, forming a cascade of successively emitted photons. Note that photons are always born at the moment of 7-decay of a nucleus, because there are no photons inside a nucleus. Photons emitted by a nucleus can have different angular momenta. If the photon takes the angular momentum I = 1, the radiation is called dipole radiation. If Z = 2, it is called quadrupole radiation; if I = 3, it is called octupole radition, and so on. The electromagnetic transitions of the electric and magnetic types and the corresponding photons are also distinguished. The electric transitions are caused by the redistribution of electric charges in a nucleus, while the magnetic transitions are caused by the redistribution of magnetic moments and currents. Another interesting phenomenon is called nuclear isomery. An isomer is an excited state of a nucleus with energy close to that of the ground state but with a very different spin. Usually the difference is A/ > 4. The nature of nuclear isomery was explained by von Weizsacker in 1936. The probability of the transition of a nucleus from the excited isomer state to the ground state via the emission of a photon appears to be very small, while the lifetime of the state is large — it can be hours, days, or even weeks. Such an excited state is metastable. The evolution of the excited (metastable) state of a nucleus can occur in two ways. The first is /3-decay of an isomer. The second consists of the emission of a photon, with the transition of an isomer into another excited state with lower energy, and the consequent emission of the electron of internal conversion. Internal conversion is a process of direct emission of an atomic electron without preliminary radiation of a nuclear photon. In such a process, the mono-energy electrons are emitted whose energies are determined by the energy of the excited state of the nucleus and by the state of the electron in the atom. Usually the .^-electrons are emitted with the greatest probability. Nuclear isomery is not a rare phenomenon. There exist more than a hundred long-lived nuclear isomers. Some examples are as follows: the 107Ag nucleus in the excited state with energy 0.093 MeV ( i ^ = 44.3 s); the 113In nucleus in the excited state with energy 0.393 MeV (ti/2 = 104 min); the u 9 Sn nucleus in the excited state with energy 0.089 MeV (i 1/2 « 250 days).

107

Radioactivity of Atomic Nuclei

The Mossbauer effect proves useful in the investigation of nuclear structure. Mossbauer, a German physicist, discovered the effect in 1958 and was awarded a Nobel Prize in 1961. The effect concerns the resonant absorption of a photon by a nucleus "frozen" into a crystal. In this case, the effect of nuclear recoil during photon absorption transforms into the effect of the recoil of the whole mass of crystal lattice and, consequently, the resonance width coincides with the natural width of the spectral line. In order to observe the Mossbauer effect, the crystal is cooled to very low temperatures. The effect plays an important role in applied physics. It is used for the direct measurement of the superthin splitting of nuclear levels caused by the nuclear spin and magnetic moment, and for the determination of the radii of excited nuclei. It allows one to study the superthin fields in metals and alloys, and is implemented for determining the phases of diffracted waves in monocrystals, for gathering information on crystal structure, and for investigating ordered magnetic states. There are many other fields of physics, chemistry, and biology where the application of the Mossbauer effect helps us study the structure and properties of different objects.

4.5

Exotic Types of Radioactivity

Many heavy nuclei can undergo spontaneous fission. This usually competes with a-decay, the probability of which is much greater. Evidently, the half-life of a nucleus due to a-decay is much smaller than that due to spontaneous fission (Table 4.4). Thus, spontaneous fission is a rather rare type of radioactive decay. Table 4.4 Nucleus 232Th 235U 238U 238Pu 239Pu 240Pu 242Pu 241

Am

Half-lives for some heavy nuclei

ti/2> spont. fission (years)

*i/2> a-decay (years)

1.3 • 10 1 8 1.9 • 10 1 7 5.9 • 10 1 5 4.9 • 10 1 0 5.5 • 10 15 1.3 • 1 0 u 71010 2.3-1014

1.41 • 10 1 0 7.1 • 10 8 4.5 • 10 9 89.6 2.43 • 10 4 6.58 • 10 3 3.5-10 5 470

108

The Quantum World of Nuclear Physics

If a nucleus is proton rich it can decay by proton emission. In 1970, the isomer states of cobalt nuclei were observed to decay (ti/2 = 0.25 s) with the emission of 1.57 MeV protons (in 1.5% of cases) and positrons (in 98.5% of cases). The competition with /3+-decay makes the process of proton emission hardly probable. Proton decay was later observed for some other nuclei. For instance, in 1982, the artificially obtained ^ L u nuclei were shown to be unstable in the ground state. They decay and emit 1.217 MeV protons (£^2 = 85 ms). Some nuclei cannot emit a proton, but decay with the simultaneous emission of two protons. In 1983, the following chain of decays was observed: ?§A1 —» flMg + e-+i> e ,

22Mg

_ ^ 20 Ne +

2p

(420)

Some nuclei first undergo /3-decay and then emit protons and neutrons. Since the emission of a nucleon from the nucleus occurs almost instantly, the emission of the delayed proton or neutron is defined by the time for /?-decay of the nucleus. In such a process the nucleus (A, Z +1) undergoes /3+-decay or if-capture and turns into the nucleus (A, Z) in the excited state, then emits a proton and turns into the nucleus (A — 1,Z — 1). As examples, note §C nuclei emitting 8.24 MeV and 10.92 MeV protons (t1/2 = 0.126 s), 83O emitting four groups of protons with energies from 1.44 MeV to 7.0 MeV (£1/2 = 0.0089 s), ioNe emitting five groups of protons with energies from 1.68 MeV to 7.04 MeV (t1/2 = 0.108 s), and some others ( ^ T e , s^Hg). Sometimes the fission fragments of nuclei form in excited states with excitation energies greater than the neutron separation energy. They can undergo /3~-decay and then emit the delayed neutron. Particularly, these include the 29Na, 30Na, and 31Na nuclei. Double neutron radioactivity is also possible when the nucleus emits two neutrons simultaneously. Also well known are the exotic radioactive decays with emission of the 12 C, 14 C, 16 O, 20Ne, 24Ne, 28Mg, and 32Si nuclei. However, the half-lives of these are very large and this hampered their experimental observation for a long time. The competition with /3-decay makes these exotic decays comparatively rare. This circumstance also considerably complicates their observation and discovery. Due to progress in the manufacture of semiconductor detectors, these types of decays were successfully investigated in the 1980s. For instance, in 1984 the decays of 222Ra, 223Ra, 224Ra, and 226Ra nuclei, emitting 14C nuclei, were observed. In 1985, the decays of 230 Th, 231 Pa, 232U, 233U, and 234U nuclei emitting 24Ne nuclei, and the decays of 234U, 236 Pu, and

Radioactivity of Atomic Nuclei

109

238pu n u c j e i ; emitting 28Mg nuclei, were studied. In 1989, the radioactive decays of 238 Pu nuclei were investigated in detail. These nuclides can emit a-particles with ti/2 = 88 years, 28Mg nuclei with i j / 2 = 2 • 1018 years, and 32Si nuclei with ij/2 = 6 • 1017 years, and also can spontaneously fission with ti/2 = 5 • 1010 years. The discovery of other types of radioactive decay is quite possible.

4.6

Application of Radioactive Isotopes

Radioactive isotopes have been widely used in medicine for a long time. Cardiovascular diseases are often accompanied by serious alterations of vessels and aberrations in heart function. Knowing the total amount of blood and the speed of blood flow is very important for correct diagnosis of these diseases. The measurements of these parameters should be made "in vivo". To determine the parameters of bloodflow,a small amount (0.25 cm3) of physiological solution containing about 1 MBk of 24Na radioactive isotope is injected into a vein in the elbow. This isotope undergoes electron /?decay with t1/2 « 15 hours. The nuclide transforms into a 24Mg nucleus in an excited state, which relaxes into the ground state with emission of 1.38 MeV and 2.76 MeV photons. These photons easily pass through human tissue and are detected by a special detector that allows one to observe blood flow until the radioactive preparation is uniformly distributed over the total volume of blood. Numerous studies have shown that in a healthy human, blood flows from one hand to another in about 15 s, while it flows from hand to foot in about 20 s. Passing through the heart, the radioactive preparation is diluted by quite a large volume of blood. The greater the volume of blood flow, the faster the dilution of the preparation, the degree of which can be estimated by the change in concentration of radioactive nuclei after the blood leaves the heart. So by measuring the radioactivity of the injected preparation, one can determine the volume of blood flow. It is about 5.5-6 1/min for a healthy human, but can be 2-3 times smaller with heart disease. Diagnosis of the diseases of different organs by radioactive isotopes is based on the fact that organisms will concentrate definite chemical elements in these organs. For instance, a thyroid gland is able to accumulate iodine in its tissues, while phosphorus, calcium, and strontium precipitate in bones. To diagnose the diseases of a thyroid gland, the radioactive isotope 131 I, undergoing electron /3-decay with ti/2 ~ 8 days, is injected into the

110

The Quantum World of Nuclear Physics

human body. The 131I nucleus decays with the creation of the 131Xe nucleus in an excited state. The 131Xe nuclei transit into the ground state with emission of photons. Detecting this 7-radiation, one can evaluate the speed of concentration of iodine in the thyroid gland and thereby assess the disease of that organ. The radioactive isotope 131I can also be used in diagnosing liver function if we inject into the human body a special organic dye. This dye concentrates in the liver. The speed at which it does this, as well as the speed at which it subsequently leaves the liver, allow one to assess liver function. Analogous methods are applied to study the functioning of the kidneys, stomach, duodenum, and intestines. Radioactive isotopes allow one to reveal malignant tumors. Besides the 131I isotope, the radioactive phosphorus 32 P isotope, which undergoes electron /3-decay with i j / 2 « 14.3 days, is used to achieve that goal. The tumor cells intensely accumulate radioactive phosphorus unlike the cells of a healthy tissue. This phosphorus isotope emits electrons with a mean free path in the tissues of the human body of about 0.03-0.08 m. Hence the 32 P isotope helps to diagnose tumors situated close to the body surface, and also in the cavity of the mouth and larynx, and in the gullet. In this case, the emitted electrons can be detected by a special detector. The radioactive isotope 198Au undergoes electron /3-decay with ti/2 ~ 2.7 days, creating the 198Hg isotope in an excited state, the transition from which into the ground state is accompanied by the emission of 0.41 MeV photons. That is why the 198Au and 131I isotopes are injected into a vein with the physiological solution (198Au), or as a part of a special preparation (131I) used as a diagnostic for the internal organs (brain, liver, thyroid gland, etc). Besides diagnostics, radioactive isotopes are widely used for medical purposes. Radioactive therapeutics are applied for the irradiation of a malignant tumor itself, with the aim of its destruction, or for the irradiation of the whole body in order to influence the nervous system or immunity. Radioactive therapeutics are usually carried out with the help of a cobaltic gun containing a radioactive isotope of cobalt that emits photons. Application of radioactive isotopes for medical purposes is not restricted to diagnostics and medical treatment. With the help of radioactive isotopes, one can observe the dispersion of drugs in the body and see how harmful substances reach different organs. One can also sterilize medical instruments and materials through the use of 7-rays, use portable 7-ray sources instead of X-rays apparatus, and even obtain marked microflora.

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111

Sterilization by 7-radiation made it possible to replace expensive repeated-use glass syringes by cheap plastic disposable syringes. Packed along with needles into the individual packets, the plastic disposable syringes are placed in cardboard boxes and irradiated by gamma-rays (photons) having energy greater than 1 MeV and emitted by the radioactive isotope 60Co (ti/2 ~ 5.3 years). The 7-rays penetrate the packets and sterilize the syringes and needles, which remain sterilized as long as the integrity of the packets is not compromised. Natural radioactive 14C nuclei undergoing electron /3-decay with ti/2 ~ 5730 years are of great practical interest in archaeology and paleontology. The 14C isotope is created in the atmosphere under the influence of neutrons born in processes initiated by cosmic rays: 14N(n,p)14C. The quantity of 14C nuclei in the atmosphere is 1012 times smaller than that for 12C nuclei. Since the mean intensity of cosmic rays is in fact stable over time, the concentration of 14C isotope in the atmosphere is actually constant. Living organisms absorb 14C and 12C isotopes from the atmosphere. Plants absorb 14C as a part of the carbonic acid gas CO2; herbivorous animals eat plants, and predators feed on herbivorous animals. Thus the ratio of the quantity of 14C isotope to the total amount of carbon in a living plant or animal remains constant. If the plants or organisms die, the quantity of radioactive 14C isotope in them decreases according to the law of radioactive decay (4.3). Hence the measurement of the relative quantity of 14C isotope in fossil plants or the remains of animals allows one, with the help of that law, to estimate the time that has passed since the moment of their death. Such a method of dating organic substances is called carbon dating. It was proposed in 1946 by the American scientist Libby. For his method of using carbon-14 for age determination in archaeology, geology, geophysics, and other branches of science, Libby was awarded the Nobel Prize in Chemistry in 1960. As an example, let us estimate the age of a tree in which the ratio of the quantity of 14C nuclei to the quantity of 12C nuclei is equal to 2/3 of this ratio for modern trees. Evidently, during the time that has passed since the moment of death, the quantity of 14C nuclei in the tree is 2/3 of its initial value, i.e., N(t) = 2iVo/3. Equation (4.3) yields t = t1/2 ln(3/2)/ln2 « 3350 years. The first tests of carbon dating were made on organic archaeological monuments of known ages, and gave good results. Further, the method made it possible to disentangle many chronological mysteries not investigated in other ways. At present, the carbon dating method is deemed one

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of the most reliable for dating finds of organic origin. Forensic Science often uses an activation analysis. Its essential idea is in obtaining the artificial radioactive isotopes after irradiation of a sample by charged or neutral particles. For example, the irradiation by neutrons leads to their capture by nuclei and the creation of j3~ -radioactive isotopes. The character of decay of such isotopes allows one to identify tiny samples whose masses sometimes are of the order of 10~12 kg! For instance, although the qualitative composition of human hair is stable, the quantity of different substances in the hair of different people differ. Therefore, hair is as individual a feature of a human as fingerprints are. It is known that the hair of the emperor Napoleon I contained arsenic, the dose of which exceeded the norm by about 10 times. Napoleon was most probably poisoned by substances containing arsenic and released into the air by the wallpaper of his room. In the USA, a method has been patented for detecting explosives in the luggage of air passengers, based on the fact that explosives contain the nitrogen isotopes 14N and 15N. Irradiation by neutrons transforms them into the radioactive 16N isotope with ti/2 = 7 s. This isotope emits 6 MeV photons. The discovery of such radiation after the irradiation of luggage by neutrons signals that luggage contains substances with nitrogen, probably explosive. Since the half-life of 16N isotope is small enough, the application of this method does not take much time. Luggage can be simply moved by conveyor past the neutron source and the X-ray detector. 4

Fig. 4.9 The scheme of radioisotope energy generator: 1 — radioisotope, 2 — converter, 3 — isolator, 4 — thermopairs. There exist energy sources of comparatively small power that use radioactive isotopes as "fuel". The charged particles emitted by radioactive nuclei in such a power supply are absorbed by a substance, and the heat

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113

obtained this way is converted into electricity. Fig. 4.9 shows the scheme of such a device. Charged particles emitted by the radioactive isotope 1 are absorbed by the converter 2. Thermoelements (thermopairs) 4 are placed in the insulated cover 3. Their joints share the temperature of the converter, while the outside ends exist at a lower temperature. In such energy generators the /3-radioactive 90Sr (i x / 2 = 27.7 years) and the a-radioactive 238 Pu (ti/2 = 87.5 years) are usually used. These sources of electric energy are successfully used on artificial satellites, shining buoys, isolated meteorological stations situated in inaccessible places, and in pacemaker devices that stimulate heart function.

Chapter 5

Nuclear Reactions

5.1

Conservation Laws in Nuclear Reactions

The various processes of interaction between nuclei and nuclei or nuclei and other particles are called nuclear reactions. They can result from the strong (nuclear) interaction, or from the electromagnetic and weak interactions. The strong interaction can cause a nuclear reaction if the distance between particles is on the order of 10~15 m, because only at such distances can strong forces act. The electromagnetic interaction is responsible for the nuclear reactions between nuclei and photons or charged leptons. The weak interaction causes nuclear reactions between nuclei and neutrinos. Nuclear reactions can change the internal states of colliding particles, and can lead to the creation of new ones. The first artificial nuclear reaction was conducted by Rutherford in 1919. For this purpose he used a-particles emitted by a radioactive bismuth isotope 214Bi, then known as RaC. Alpha-particles emitted by that isotope had an energy of about 5.5 MeV. Passing through a tube filled with gaseous nitrogen, a-particles caused the appearance of new particles whose free path substantially exceeded that of the a-particles. These long-free-path particles were detected by a scintillation screen coated with sulphurated zinc. Rutherford determined that they were protons. Rutherford observed the nuclear reaction in which a-particle entered the nitrogen nucleus, adhered, and emitted a proton: 14N

+ a —-> 17O+p.

(5.1)

Previously unknown, the oxygen isotope 17O was the first element to be created artificially. These experiments were difficult to perform because the transformation of nitrogen into oxygen occurs very rarely. The twenty 115

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The Quantum World of Nuclear Physics

registered nuclear reactions (5.1) required a million a-particles to be emitted by bismuth. Rutherford, the consummate scientist, solved the age-old riddle that had puzzled the medieval alchemists who, for many centuries, tried to transform one element into another. In subsequent years, Rutherford accomplished nuclear reactions with 17 light nuclei including boron, fluorine, sodium, aluminium, lithium, and phosphorus. Further study of nuclear reactions required particles with higher energies and beams of particles with greater intensity than the radioactive sources could give. Thus, in 1932 at Cambridge University, Rutherford's two disciples Cockroft and Walton created a high voltage generator — the first accelerator of elementary particles. Using that device, they observed the reaction 7U+p

—> 2a

(5.2)

induced by the 0.125 MeV accelerated protons. Thus began the era of the accelerators that have substantially widened the possibilities for conducting various experiments in nuclear physics. In 1951, Cockroft and Walton were awarded the Nobel Prize in Physics for their pioneering work on the transmutation of atomic nuclei by artificially accelerated atomic particles. An interaction between a particle a with a nucleus A can occur in several different ways: a + A, a + A*, b + B, a + A —> < ... d + g + D, z + Z+...

(5.3)

.

Particles that react are called an initial channel, while particles that result from a reaction are called a final channel. A reaction can have a number of final channels. The first two final channels in the scheme (5.3) represent elastic scattering, when initial particles are identical to the final ones, and inelastic scattering (A* denotes nucleus A being in an excited state). A • reaction channel is characterized by energy, spin, etc. A reaction a + A —> b + B is sometimes denoted as A(a, b)B. If in the experiment all of the reaction products are completely determined, then this process is called an exclusive reaction. If in the final channel only one particular particle is registered, then this process is called an

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117

inclusive reaction. The study of exclusive reactions gives the most complete information about a nuclear process. Let us consider some of the conservation laws that hold during nuclear reactions. In nuclear reactions electric charge is always conserved. This means that the sum of the charges of all particles in the initial channel is equal to that sum in the final channel. The conservation of a baryon charge in a nuclear reaction means the following. A baryon is a particle consisting of three quarks. If any baryon is attributed the baryon charge equal to +1 and any antibaryon is attributed the baryon charge equal to -1, then the sum of the baryon charges of particles in the initial channel will coincide with that sum in the final channel. Baryon charges of photons, leptons, and mesons are equal to zero. For instance, the reaction p + 7 —* p + n is forbidden because the baryon charges in the initial and final channels are equal to 1 and 2, respectively. Also forbidden is the process of annihilation of an electronpositron pair into one neutron, e~~ + e+ —* n, because the baryon charge in the initial channel is equal to zero while that charge in the final one is equal to unity. Possible, however, is the annihilation of the electron-positron pair into the nucleon-antinucleon one, e~ + e+ —> n + h, e~ + e+ —> p + p, where the line above a letter denotes an antiparticle. It is very important that the conservation of baryon charges forbids the annihilation of a hydrogen atom as p + e~ —> 27; to this we owe the stability of our Universe. The laws of conservation of electric and baryon charges are also valid in the radioactive decay of nuclei. The law of conservation of energy in a nuclear reaction a + A —> b + B means that the following equality is valid: Ea + EA = Eb + EB,

(5.4)

where the energy of the ith particle according to (2.10) is equal to Ei = i/p?c 2 + TTI?C4. Here pj and rrn are the momentum and mass of the ith particle. The energy of the ith particle can also be written as Ei — rriiC2 + Ti, where the first term represents the rest energy of the ith particle while Ti is its kinetic energy. Thus, equation (5.4) can be given in the form mac2 + Ta + mAc2 + TA = mbc2 + Tb + mBc2 + TB.

(5.5)

The magnitude Q = Tb + TB - Ta - TA = (mo + mA - mb - mB)c2

(5.6)

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The Quantum World of Nuclear Physics

is called the energy or heat of reaction. It can be either positive or negative. If Q = 0, the nuclear reaction is a process of elastic scattering. If Q > 0, the reaction is called an exoergic or exothermic reaction. Energy is released in such a reaction. For instance, the reaction 2H

+ 3 H —>4 He + n

(5.7)

is exoergic with Q = 17.6 MeV. If Q < 0, the reaction is said to be endoergic or endothermic. In this case energy is absorbed. An endoergic reaction always has a threshold. The threshold energy is the minimum energy needed for the proceeding of that reaction; it is given by

r, = ( 1 + ^ )

W

|.

(5.8)

We see that Ti > \Q\ because part of the absorbed energy is turned into kinetic energy of the particles that fly away (their center of mass always moves in the laboratory system). As an example of an endoergic reaction, we consider the charge-exchange nuclear reaction in which the initial proton is absorbed and the final neutron flies out: p+7Li—yn+7Be,

Q = -1.643MeV.

(5.9)

Assuming the nucleon (proton and neutron) masses to be the same and the masses of the 7Li and 7Be nuclei to be equal to seven nucleon masses, we can use (5.8) to find that Tt « 1.88 MeV. Here are some further examples. The process of annihilation of a hydrogen atom is further forbidden by the law of energy conservation because the sum of the rest energies of a proton and an electron is smaller than the neutron rest energy (mn-mp — me)c2 = 0.78 MeV, while the kinetic energy of an electron belonging to a hydrogen atom has an order of magnitude of about 10~5 MeV. In other words, the process of annihilation of a hydrogen atom would be energetically allowed if the electron in this atom would have energy exceeding 0.78 MeV. Let us now ask why a neutron does not decay in deuterium and in the 3He nuclei following the scheme n — > p + e~ + i>, while such a process takes place in the 3H nuclei (tritium) and causes their /?-radioactivity. Since there are no bound states for the systems comprising two and three protons, the decay of a neutron in deuterium or in the 3He nuclei is forbidden by the law of energy conservation due to the inequalities (mn — mp — me)c2 < £<$, (mn — mp — me)(? < Eh where (mn — mp — me)c2 = 0.78 MeV and the

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119

deuterium and 3He nuclei binding energies are equal to Sd = 2.22 MeV, Eh = 7.72 MeV. However, for the tritium nuclei, the inequality (mn — mp — me)c2 > et—£h holds where the tritium binding energy is equal to e* = 8.48 MeV. Therefore, tritium nuclei appear /3-radioactive: 3H —> 3He + e~ + v ( b + B, the momentum conservation law can be written in the form Pa + PA=Pb + PB,

(5-10)

where pj is the momentum of the ith particle. If the target nucleus A does not move, then pA — 0 and p a = pfc + p B . The angular momentum conservation law for the same reaction has the form Ia + IA + laA = lb + IB+hB,

(5.H)

where Ij is the spin of the zth particle, and laA and Its are the orbital moments of relative motion of the particles in the initial and final channels. Conservation laws impose certain restrictions on the physical characteristics of particles emitted due to a nuclear reaction. 5.2

Nuclear Reaction Mechanisms

Let us consider nuclear collisions with slow particles whose energies are substantially smaller than the binding energies of atomic nuclei. Most important is the fact that the interaction between the projectile (hadron) and the nucleus is large. The energy of this interaction is of the same order of magnitude as the energy of interaction between nucleons entering into a nucleus. Owing to that, the nuclear collision problem appears essentially as a many-body problem. Note that this statement has nothing to do with nuclear collisions of particles, the energies of which are substantially greater than the energy of nuclear interaction. The strong interaction between a projectile and a nucleus leads to the fact that soon after the collision and fusion with the nucleus, the projectile loses a substantial part of its energy, which is transferred to a nucleon entering into the nucleus. This redistribution of energy has a statistical character. Thus, none of the particles of the nucleus will have enough energy to overcome the nuclear forces of attraction and immediately leave the system consisting of the initial nucleus and the projectile. Only after

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a long time has elapsed, when due to fluctuations some particle acquires enough energy to overcome the forces of attraction acting on it from the rest of particles, will it leave the nucleus. Therefore we come to a very important conclusion: the projectile and the target nucleus can be treated as a unified quantum-mechanical system that exists and does not decay during a long period of time substantially exceeding the typical nuclear time. In other words, the lifetime of such a composite system is essentially greater than the time the projectile takes to fly through the nucleus. Assuming the velocity of the projectile ~ 108 m/s and the linear size of the nucleus ~ 10~14 m, the typical nuclear time is estimated as rnuci ~ 10~22 s. During the time this system exists, its properties do not differ from those of ordinary nuclei in highly excited states. Therefore the system formed by the fusion of the initial nucleus and the projectile is usually called a compound nucleus. We emphasize that the compound nucleus is in a state with positive energy, i.e., its energy exceeds the energy needed to separate at least some nuclear particles. After a long period of time the excitation energy of the compound nucleus can accidentally concentrate on a single particle, which in this case gets an opportunity to leave the nucleus. It is not necessary that the particle that leaves the compound nucleus be the same as the projectile that formed the nucleus. On the contrary, it is unlikely that the nature of the initial and final particles would be the same, because there are various possibilities for compound nuclear decay. Still more unlikely is that, even provided the nature of both particles is the same, the internal state of the nucleus will not change. More probable is that after the particle's emission, the remaining nucleus will be in an excited state. If the projectile and the emitted particle are identical but the energy of the final nucleus differs from that of the initial nucleus, then this is an inelastic scattering. If the projectile and the emitted particle are different by their nature, then this is a nuclear reaction. During nuclear collisions, in some cases, radiation processes play a substantial role because the emission of a photon requires a smaller concentration of energy than the emission of a particle with nonzero rest mass. A photon can take away less energy than a nucleon emitted from a nucleus. Thus, at small excitation energies the lifetime of the compound nucleus is mainly determined by the interaction of nuclear particles with radiation, although this interaction is rather small. Hence, the lifetime of the compound nucleus with comparatively small excitation energy, which only slightly ex-

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Nuclear Reactions

ceeds the nuclear binding energy in the nucleus, is very long as compared to the typical nuclear time. For instance, the typical lifetime of the excited 52Cr nucleus that emits photons with energy ~ 1 MeV is 0.65 • 10~13 s, which exceeds the typical nuclear time by many orders of magnitude. The role of radiation processes in nuclear collisions in comparison with atomic collisions is enhanced due to the wandering of the projectile in the nucleus, owing to which the time the projectile spends in the nucleus appears much longer than the typical nuclear time. Therefore, in nuclear collisions with comparatively small projectile energies, two stages should be distinguished: the formation of the quasistationary long-lived compound nucleus, and its decay. These features of nuclear collisions were first explained by Bohr in 1936. In other words, a nuclear collision with the formation of a compound nucleus and its decay occurs according to the scheme a + A —> C* —> b + B,

(5.12)

where C* denotes a compound nucleus being in excited state. The excitation energy of a compound nucleus is equal to Ec = (m a + mA -mc)c2+

mATa

,

(5.13)

ma + mA where ma, rriA, and me are masses of the particles a and A and the compound nucleus C in its ground state, while Ta is the kinetic energy of the particle a. The lifetime of a compound nucleus is very long relative to the typical nuclear time. During this lifetime, the compound nucleus forgets about the way it was formed, because the energy of the projectile is statistically distributed between the nucleons of the nucleus. Thus the decay of a compound nucleus in the final channel / should be treated independently from the mechanism of the formation of a compound nucleus in the initial channel i. Hence the probability of the nuclear reaction under consideration Wif is the product of the probability of creation of the compound nucleus Wi and the probability of its decay wf. Wif = WiWf.

(5-14)

The final result of the nuclear reaction is denned by the competition between the different possible processes of decay of the compound nucleus, which are compatible with the general conservation laws.

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The Quantum World of Nuclear Physics

In order to prove the independence of the processes of creation and decay of a compound nucleus, special investigations of several nuclear reactions were performed. For the first time this was done using the nuclear reactions r 63 Zn + n, 60Ni + a - ^ I 62Zn + n + n, i 6 2 Cu + n + p,

(5.15)

studied along with the reactions 63Cu

( 63Zn + n, + p —» } 62Zn + n + n, i 6 2 Cu + n + p.

(5.16)

In reactions (5.15) and (5.16), the same 64Zn compound nucleus was formed. Within experimental error, the angular distributions of the same particles created in these reactions coincided independently of the ways the compound nuclei were formed. Among the other experiments conducted, one should mention 12C + 63Cu and 16O + 59Co with the compound nucleus 75Br created. The angular distributions of a-particles (or protons) emitted as a result of the decay of the compound nuclei were identical. Many other nuclear reactions were experimentally studied, which confirmed the hypothesis of Bohr. The concept of a compound nucleus as a quasistationary system makes sense if the number of nucleons in the compound nucleus, among which the projectile's energy is distributed, is large enough. On the other hand, if the energy of the projectile is too high, a nucleus becomes transparent. Thus, the concept of compound nucleus can be used if the projectile (nucleon) mean free path in the nuclear matter is small compared to the linear nuclear size. Only under this condition will the projectile be absorbed by the nucleus with large probability. Moreover, the nucleon separation energy from the compound nucleus e must be large compared to the excitation energy per nucleon (E + e)/(A+ 1), where E is the projectile energy and A is the number of nucleons in the target nucleus. Hence we get the condition E 10 and E < 50 MeV. The probability of a nuclear reaction that occurs via a compound nucleus usually has a sharp maximum at a certain excitation energy (the projectile energy). This energy is the most favorable one for the decay of a compound nucleus through a certain final channel. With the increase

Nuclear Reactions

123

of excitation energy, the probability of a particular nuclear reaction will decrease because a new final channel opens. Like usual stable nuclei, a compound nucleus is characterized by a certain spectrum of energy levels. Since a compound nucleus is a quasistationary system, each of these levels has a certain width. The width of a level F is connected with its lifetime r (more precisely, with the lifetime of the compound nucleus in this state of its energy spectrum) by the relation r = - = hw,

(5.17)

where w is the probability of transition of a compound nucleus from the given state into all other possible states per unit time. If the decay of a compound nucleus can occur in various ways, then

w = J2wf, /

( 5 - 18 )

where Wf is the probability of decay of the compound nucleus through the / t h final channel. Then the total width of a level F is equal to r = ^r/. /

(5.19)

The quantity F/ is called the partial width of a level, corresponding to the decay of the compound nucleus through the /-channel. Evidently, the relation F/ = hwj holds. One can talk, for instance, about the neutron width F n , the radiation width F 7 , and the widths Tp and Ta corresponding to the proton and aparticle escape, understanding these quantities as the probabilities of decay of a compound nucleus with the emission of neutrons, photons, protons, and a-particles, expressed through the energy units. The lifetime of a compound nucleus is long relative to the typical nuclear time, so the widths of levels of a compound nucleus appear to be small compared to the binding energy of a nucleon in a nucleus. However, this does not mean that the level widths are also small relative to the neighbor levels separation D. Two cases are possible: when the level widths F are small relative to the level separation D, and when they are of the same order of magnitude or even greater than it. The first case occurs in the region of small energies, while the second occurs at high energies. If a compound nucleus is formed through the capture of a slow particle by the initial nucleus, then the excited state of

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The Quantum World of Nuclear Physics

this compound nucleus is characterized by a rather small width. But if the projectile is fast, the compound nucleus is formed with overlapping levels. Only in the first case F -C D can one say that the separated levels of the compound nucleus comprise a region of a quasidiscrete spectrum of levels. In the second case F > D the overlapped levels form a region of a quasicontinuous spectrum of levels. An important feature of the first case is the strong manifestation of the dependence of the compound nucleus formation probability on the energy of the projectile. This dependence has a resonance character: at certain values of projectile energy, the probability of compound nucleus formation and, therefore, the probability for some nuclear reactions to occur, becomes very large. In quantum mechanics it is shown that the total probability of a resonance reaction is defined by the simple formula CY F

(5.20)

where C is a constant, F e is the elastic scattering level width, F r is the reaction level width, F = F e + F r is the total width of a level, Er is the resonance value of energy, and E is the projectile energy. Formula (5.20) is called the dispersion formula or the Brett- Wigner formula because it was derived in 1936 by American physicists Breit and Wigner. In the second case (overlapping levels) there is no clearly pronounced dependence between the probability of nuclear reaction and the energy. Instead, the probability of a certain nuclear process is denned by the joint action of a large number of levels, and originates from the overlapping of the single level probabilities over an energy interval containing a large number of levels. The possibility of such overlap substantially simplifies consideration of this case. If the projectile energy is large, E > 50 MeV, a compound nucleus might not form because the corresponding probability is too small. In this case, a projectile hitting the nucleus close to its surface can suffer one or several collisions with nucleons entering the nucleus and located near the surface. Having lost some of its energy, the particle can fly away without creating a compound nucleus. In such a process the projectile can even change its nature (e.g., a neutron can turn into a proton, a deuteron can turn into a neutron, etc.). Nuclear processes that occur without creating a compound nucleus are called direct nuclear reactions. Note that these can also take place in a range of comparatively small energies where a compound nucleus is formed. Evidently, direct nuclear reactions are surface processes

Nuclear Reactions

125

in which a small number of nucleons contained in the near-surface region of the target-nucleus participate. Sometimes processes are observed in which the composite system created due to the fusion of a projectile with a nucleus decays before the energy transferred to the nucleus by the projectile has been statistically redistributed among all nucleons of that system. Those processes are called preequilibrium nuclear reactions. They are intermediate between compound nuclear reactions and direct ones. They are sometimes also called multistep nuclear reactions. Let us consider the process of absorption of a particle by a nucleus in more detail. The energy of a slow particle (e.g., neutron) colliding with a nucleus is first transferred to one or two nucleons comprising a nucleus, which then transfer it to other nucleons. Hence, the process of interaction of a particle with a nucleus can be treated as a sequence of stages, reflecting the consequent nucleon-nucleon interactions. These stages are characterized by the number of particle (p) - hole (h) pairs called the exciton number. A hole is a vacant nucleon state formed due to a one-particle excitation of a nucleus, which is the transition of a nucleon from a state with energy lying lower than the maximum energy Ep (Fermi energy) of nucleon states into a state with energy exceeding Ep. At every stage, the emission of pre-equilibrium particles is possible. The probability of emission of such particles decreases from stage to stage, because the excitation energy is distributed more uniformly among the nucleons with every new stage. Eventually the compound nucleus attains the state of statistical equilibrium. Then it can emit particles, if possible, until it reaches the ground state. Note that the smaller the projectile's energy, the smaller the probability of emission of pre-equilibrium particles. The excited state of a nucleus formed after the first stage is a state in which two particles (the captured projectile and the nuclear nucleon in the excited state) have energy greater than EF and one hole (a vacant nucleon state) has energy smaller than Ep (Fig. 20). Such a state is called the door-way state and is denoted by 2plh (two particles and one hole). The next stage could be the emission of an initial particle with such energy and angular momentum that the nucleus will return to the ground state; this constitutes an elastic scattering of the projectile as a preequilibrium process. Such an elastic scattering process is characterized by the final width F^. On the other hand, particles having energies greater than EF can interact with one of the remaining nucleons of the nucleus. The composite system so formed will have three particles with energies

126

The Quantum World of Nuclear Physics a

a

LJ

U ^ 3

Fig. 5.1 Different stages of the nucleon-nucleus interaction: 1 — the nucleus before the interaction; 2 — the door-way state 2plh; 3 — the state 3p2h; a — the projectile. greater than Ep and two holes 3p2h (Fig. 5.1). The width of such a state is called the damping width and is denoted by F^. The consequent stages of the formation of a composite system can be considered by analogy with the first two. Thus, the long-lived state of a compound nucleus is formed after a large number of two-particle nucleon-nucleon interactions, passing through a series of intermediate states that get more and more complex with the number of stages. In other words, during the formation of a compound nucleus, a certain sequence of states arises, which is called the hierarchy of configurations. As already discussed, the simplest state of a compound system is the initial state 2plh. By contrast, the states of a compound nucleus, in which many nucleons participate, are very complicated. They are usually described statistically, while the process of proceeding through the compound nucleus formation is described as "the noise". Let us consider one more intermediate type of nuclear reaction called reactions of deep inelastic transfer. These reactions are observed when two colliding nuclei have energies that exceed the height of their Coulomb barrier B = ZiZze21'{R\ + R2), where Ri and R2 are the radii of colliding nuclei. The mechanism of these reactions contains features characteristic of the direct nuclear reactions and reactions proceeding via a compound nucleus formation. In deep inelastic transfer reactions, strong coupling occurs

Nuclear Reactions

127

between the initial and final channels, i.e., the products of decay "remember" the process of collision of initial particles, which is also a feature of direct nuclear reactions. However, the angular distributions of the products of these reactions attest to the statistical equilibrium with respect to some degrees of freedom. In this feature, deep inelastic transfer reactions are similar to reactions held via compound nucleus formation. The mechanism of a deep inelastic transfer reaction assumes that during the collision of two nuclei a complex called a double nuclear system is formed. It can be assumed that during the collision, nuclei interact as two liquid drops that roll along one another and stick together for a short time due to the large viscosity of nuclear matter, making nucleons able to penetrate the surface of contact. Despite the partial interpenetration of nuclei during the collision, their shell structures provide conservation of individuality of nuclei under the strong interaction. It appears that even the frontal collision of heavy nuclei, whose energies exceed the Coulomb barrier, does not lead to their total fusion and the formation of a compound nucleus. Note that the double nuclear system changes its state very quickly, which is different from the case of a compound nucleus with its quasistationarity. Therefore, the deep inelastic transfer reaction occurs much faster than the reaction that takes place through compound nucleus formation. Sometimes the deep inelastic transfer reaction is called quasifission.

5.3

Nuclear Optics

The phenomena and processes in which atomic nuclei participate are studied by nuclear physics and described by the equations of quantum mechanics. However, despite the fact that the quantum laws that control the behavior of nuclear objects differ substantially from those that govern the macroscopic world, there exists a wide analogy between some nuclear and classical processes. Detailed consideration shows that various nuclear processes have analogues in optical phenomena. These originate from the existence of wave properties of quantum objects. Such nuclear processes comprise the special branch of nuclear physics known as nuclear optics. Substantial information on the structure of molecules, atoms, atomic nuclei, and elementary particles is gained from the study of their interaction with other quantum objects. For this purpose, elastic and inelastic scattering of particles by nuclei, first of all, and then various nuclear reac-

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The Quantum World of Nuclear Physics

tions are studied in nuclear physics. If the energy of the scattered particle is large enough, the elastic scattering of such a projectile by a nucleus is similar to the scattering of light (an electromagnetic wave) by a spherical liquid drop having a definite refractive index and absorption factor. In classical physics (optics), such a scattering process can be described by its complex refractive index. Let us consider the scattering of a nucleon (nucleon wave) by a nucleus. Outside the nucleus, a nucleon wave is characterized by the wave vector k = y/2mE/h, where m and E are the nucleon mass and energy. Inside the nucleus, the nucleon wave vector is K = \j2m{E — U)/h where U is the effective potential of the nucleon-nucleus interaction. The refractive index v of nuclear matter is defined by

" = f = f~l-

(5-21)

It can be seen that if the refractive index is a complex quantity v — v\ + ivi {i>\ and V2 are real values), the potential of interaction of a projectile with the target-nucleus should be a complex quantity as well: f — V — iW

r < R

(5.22)

where R is the nuclear radius, and V and W are real and positive. The negative sign of the potential indicates the attraction and absorption of the projectile by the nucleus. If the energy of the scattered nucleons is large enough to satisfy the condition E = h2k2/2m ^> \U\, the expression for the refractive index can be approximated as 1,-1

U _

mV

imW

(5.23)

Hence we have mV

"1 = 1 + *w I

mW U2 =

W

(5-24)

Thus, the specific complex nuclear potential U(r) has a simple physical meaning: its real part describes the refraction of a scattered wave by nuclear matter, while its imaginary part causes the absorption of scattered particles by nuclei. The complex nucleon-nucleus potential is called the optical potential, while the model that uses complex potentials to describe the interaction of

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129

hadrons with nuclei is called the optical model. The essence of the optical model of nuclear scattering should be found in the fact that the many-body interaction of the projectile with the individual nucleons of the nucleus (or other particles which comprise the nucleus) is replaced by the effective twobody complex nucleon-nucleon potential. In other words, in the optical model a very complicated many-body problem is reduced to a simple twobody problem. In the optical model a nucleus is considered as a drop of nuclear liquid characterized by definite refractive and absorption properties. The basics of the optical model of nuclear scattering were first formulated by American physicists Feshbach, Porter, and Weisskopf in 1953. The model that considers a nucleus as a liquid drop is a very simplified approach, because such a model does not consider a great number of properties of atomic nuclei (e.g., a shell structure). Despite that, however, the optical model of nuclear scattering has turned out to be a powerful theoretical tool for the description of a great many interaction processes between hadrons and atomic nuclei. As an example, Fig. 5.2 shows the cross sections for the 17 MeV protons elastically scattered by iron, cobalt, nickel, copper, and zinc nuclei as functions of the scattering angle: points are experimental data, and curves are calculated via the optical model. The cross section, da/dfl, is the number of particles ejected into the solid angle element, divided by the density of the projectile flux. This quantity is measured in mb/sr, where mb is a millibarn (1 mb = 10~31 m2) and sr is a steradian. It can be seen that the optical model allows a good description of the experimental data on the elastic scattering of protons by various nuclei. Data analysis based on the optical model gives parameters of the complex optical potentials that characterize properties of particular nuclei. If the energy of a projectile (hadron) is large enough to fulfill the condition A
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The Quantum World of Nuclear Physics

Fig. 5.2 Angular distributions of 17 MeV protons elastically scattered by different nuclei. The curves have been calculated by the optical model; the points are experimental data. screen are parallel. The parallel rays also travel from the near and far sides of the scatterer to the observation point. Therefore, Praunhofer diffraction alters the directions of the light rays that undergo diffraction near the edge of a screen. Diffraction scattering is an interference phenomenon. In optical Praunhofer diffraction from a black disk, the ray 1 scattered by the near side of the scatterer interferes with the ray 2 scattered by the far side (Fig. 5.3). An interference pattern results because the phases of near- and far-side rays after scattering differ: the path length of the far-side ray exceeds that of the near-side ray by L « 2R9 (the scattering angle 6 of Fraunhofer diffraction

131

Nuclear Reactions

1

Fig. 5.3 Rays of light undergoing interference during Fraunhofer diffraction from a black sphere.

is small). Small scattering angles 9 < X/R « 1 (^ = A/2TT) dominate in the Fraunhofer diffraction pattern. About 84% of neutrons are scattered by the strongly absorbing nuclei into that region of angles. The Fraunhofer diffraction pattern is characterized by the alternation of maxima and minima of intensity of scattered particles, while the angular distribution of elastically scattered particles is similar to the ratio of intensities of scattered and incident light in optics. Real properties of atomic nuclei differ from those of the black disk in optics. The nucleus has a thin diffused surface layer in which the density of nuclear matter changes from the value typical for the nucleus center to zero. The nucleus surface diffuseness consideration leads to a more rapid (exponential) decrease in the envelope of maxima of the Fraunhofer pattern for nuclear scattering, unlike optics where the maxima envelope decreases as 0~3. The existence of the nuclear surface diffuseness is similar to the socalled apodization (the change of the function of the pupil) in optics. The scattering of particles by nuclei is also influenced by the refraction of scattered nucleon waves in the semitransparent surface layer of the nucleus and the Coulomb interaction of charged particles with nuclei (diffraction of "charged" rays). These two effects cause the partial filling of minima of the calculated angular distribution of scattered particles, which is confirmed by the available experimental data. Diffraction scattering of charged particles by nuclei was studied for the first time by Soviet physicists Akhiezer and Pomeranchuk in 1945. In nuclear scattering, inelastic processes are also possible in which the projectiles cause the transition of nuclei into excited states. Under the

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The Quantum World of Nuclear Physics

Fig. 5.4 Angular distributions (mb/sr) of 1 GeV protons elastically (1) and inelastically (2) scattered by 208 Pb nuclei calculated by the diffraction theory; the points are experimental data.

conditions of diffraction such processes have a diffractive character. Note, however, that the processes of inelastic scattering have no analogy in optics. Fig. 5.4 depicts the angular distributions (mb/sr) of elastic and inelastic scattering of protons by the lead nuclei. The typical Fraunhofer diffraction pattern (the alternation of maxima and minima) is clearly seen. Diffraction scattering is also inherent for the composite nuclei such as deuterons, 3H nuclei, a-particles, and other light nuclei. Besides elastic and inelastic diffraction scattering, the composite nuclei can participate in the processes of partial or total dissociation of a projectile in the field of a nucleus. For instance, a deuteron can break up into a neutron and a proton (diffractive dissociation of the deuteron). The process is possible in which one of a deuteron's nucleons is absorbed by a nucleus while the other one is released. Such a process is called the stripping reaction. The dissociation

133

Nuclear Reactions

processes for complex nuclei have a diffractive character but, like inelastic scattering, have no analogy in optics. Scattered particles and nuclei can have spins. Collisions of such particles with nuclei cause the polarization of particles and nuclei, meaning that after collision the majority of the spins of particles and nuclei acquire certain directions. Polarization phenomena in diffraction scattering, as well as spins themselves, also lack a classical analogy. Therefore, diffraction processes in nuclear physics are more diverse than in optics.

p

Fig. 5.5 Rays of light interfering during the Fresnel diffraction from the edge of a black sphere. Besides Praunhofer diffraction, Fresnel diffraction is also observed in optics. In this case either the light source and the observation point are both placed at finite distances from the black screen, or only one of these points is. The Fresnel diffraction pattern arises from interference between the direct ray of light 1, which travels from the source to the observation point, and the ray 2 scattered by the near side of the scatterer. Possible variants of scattering that lead to the Fresnel diffraction pattern in optics are schematically displayed in Fig. 5.5. Let us clarify why the Fresnel diffraction pattern is observed in various experiments on the scattering of charged particles by strongly absorbing nuclei. At first sight it seems that the conditions for Fresnel diffraction cannot be satisfied in nuclear scattering. Indeed, to observe Fresnel diffraction the distance between the nucleus and the particle source or the particle detec-

134

The Quantum World of Nuclear Physics

tor should be finite. In the case of neutron scattering this is impossible, because one would need to place the neutron source at a distance from the nucleus that should be of the same order of magnitude as the nuclear disk. However, the situation is essentially changed when we turn from neutrons to charged particles. Because of Coulomb interaction, particles are scattered by the nucleus as if they came from a virtual point source located at a finite distance a from the nucleus. The Coulomb interaction must be strong enough to distort the scattered waves so that the wavefront has a significant curvature near the nucleus. If the nucleus strongly absorbs the projectiles and their wavelength A is small compared to the nuclear radius R, then the scattering pattern will be analogous to the Fresnel diffraction pattern from a black disk in optics. The distance o is defined by the expression

(5.25) where n = Z\Zi&2 fhv is the Sommerfeld parameter, Z\ and Zi are the charge numbers of colliding particles, and v is the projectile's velocity. Formula (5.25) shows that the Fresnel diffraction conditions are fulfilled when a strong Coulomb interaction is present: n 3> 1. In this case the strong electric field near the nuclear surface acts on the incident waves like a diverging lens if the sign of the projectile's charge coincides with that of the nucleus, and like a converging lens if these signs are opposite. The focal distance of this lens is determined by

/ = «-£•

( 5 - 26 )

Thus the Fresnel diffraction pattern in nuclear scattering is formed by the interference between the strong Coulomb and nuclear interactions. If we consider the ratio of the intensity of scattered particles to the intensity of pure Coulomb scattering of a point particle having the projectile's mass and charge on a point particle having the target-nucleus mass and charge, then, under the conditions of Fresnel diffraction, this ratio will be similar to the ratio of intensity of the light scattering by the edge of a halfplane to the intensity of the falling light in optics. The diffraction of the Fresnel type in nuclear collisions was explained for the first time by Frahn in 1966. As an example, Fig. 5.6 shows this ratio of intensities for the elastic scattering of oxygen nuclei by lead nuclei. One can formulate general conditions that allow us to clarify when in nuclear scattering the Fraunhofer or Fresnel diffraction pattern is observed.

135

Nuclear Reactions

0(0) «R(8)

0.5 -

0

\ |

i

i

20

i

'

i~~—^-**<

40

i

.

60 9°

Fig. 5.6 Fresnel diffraction pattern for the ratio of the intensity of 170.1 MeV 16O nuclei elastically scattered on 208 Pb nuclei to the intensity of the pure Coulomb scattering of these particles: the solid line and dashes are the calculation results by the diffraction model, taking into account the surface diffuseness and semitransparency of nuclei and without taking these into account, respectively; the points are experimental data. For this purpose one should introduce the parameter p=(kR-2n)-. (5.27) a It appears that when p « l the Fraunhofer scattering diffraction pattern is observed, while at p > 1 the Fresnel pattern is observed. Thus the quantity p totally defines the type of diffraction scattering pattern. Different scattering processes having the same values of p give the same diffraction pattern, and the angular distributions of scattering particles coincide. This is the essence of the scaling law for nuclear diffraction, and the quantity p is called the scaling parameter. Another interesting type of nuclear scattering also has an analogy in optics. This is rainbow scattering. The physical explanation of this phenomenon was achieved after the discovery of the laws of refraction and total internal reflection of light on the boundary of two media, and also the laws of propagation of electromagnetic waves. The possibility of rainbow scattering in nuclear and atomic collisions was predicted by Ford and Wheeler in 1959. Recall the essence of the rainbow phenomenon in optics. There we consider the refraction and internal reflection of light rays in a liquid drop when the linear size of the drop is large relative to the wavelength of light. Fig.

136

The Quantum World of Nuclear Physics

Fig. 5.7 Passage of rays of light through a drop of transparent liquid.

5.7 shows the simplest case of ray propagation through a semitransparent drop. It is seen that the parallel beam of light rays falls on the spherical drop. The beam undergoes refraction when entering the drop, propagates inside the drop, undergoes total internal reflection from the drop surface, and finally leaves the drop while experiencing refraction once again. Fig. 5.7 shows that there exists some maximum angle 9max between the directions of falling and exit rays (rays leaving the drop at angles greater than 9max do not exist). In the vicinity of the angle 9max the typical rainbow thickening of rays is observed. The limiting angle for the water drop is 9max « 42°, which corresponds to the scattering angle 9 = 180° — 9max « 138°. If the drop radius is much larger than the light wavelength, the limiting angle 9max does not depend on the drop size. The colored rainbow in the atmosphere is caused by different values of refraction coefficients for rays with different wavelengths, for which the thickening of rays takes place at different values of 8maxA rainbow scattering phenomenon exists in quantum mechanics. The analogy between the scattering of light and particles consists in the existence of the limiting angle (the rainbow angle 9r) around which the classic trajectories (rays) are thickened, which means an increase in the intensity of scattered particles. The envelope of rays which thicken around the rainbow angle is called a caustic. In nuclear physics rainbow scattering also exists. In this case the following behavior of the angular distribution of particles elastically scattered by

Nuclear Reactions

137

nuclei is observed: in a region of comparatively small scattering angles, the angular distribution oscillates — but these oscillations are quickly damped. Then the angular distribution contains a large and wide maximum (rainbow maximum), after which the intensity of scattered particles decreases rapidly and smoothly. The rainbow angle separates two regions. The region with 6 < 9r is the illuminated rainbow side, while the region with 9 > 6r is the dark (shadow) side from the viewpoint of the optics analogy. Note that in classical mechanics the angular distribution has a singularity if the scattering angle 6 approaches 9r in the illuminated region, and turns into zero in the shadow region.

Fig. 5.8 Angular distribution of 140 MeV a-particles elastically scaterred by 50Ti nuclei: the curve is the result of theoretical calculations, the points are experimental data, and the arrow shows the rainbow angle. A nuclear rainbow is mainly observed in the scattering of light nuclei etc., with energies E > 25 — 30 MeV/nucleon, by medium and heavy nuclei. In this case, a little transparency and large refraction of nuclear matter with respect to the scattered particles are essential. Fig. 5.8 depicts the ratio of the intensity of the elastically scattered 140 MeV a-particles on 50 Ti nuclei to the intensity of the pure Coulomb scattering (as in Fresnel diffraction): points are experimental data, the curve is the result of theoretical calculations, and the arrow marks the rainbow angle 9r. Fig. 5.8 demonstrates the typical pattern of nuclear rainbow scattering. The scattering of heavy ions by nuclei is characterized by much larger

3He, 4He, 6Li,

138

The Quantum World of Nuclear Physics

absorption and much smaller transparency relative to the case of light ions. So the nuclear rainbow is not manifested in those processes. However, sometimes a very small transparency (10-15 times smaller than for light ions) exists. Thus, a hint of the formation of rainbow maxima called the rainbow ghost is observed.

Note that, when the rainbow scattering effect is observed in the angular distributions of elastically scattered nuclei, analogous effects of refraction of scattered waves can be revealed in the angular distributions of inelastic scattering and other quasielastic processes at 6 > 6r. Rainbow scattering is an example of the non-diffractive process of interference of rays propagating through the drop. The angular distribution of scattering particles calculated with the formulae of classical mechanics can also have singularities at 6 = 0° and 6 = 180°. In optics and quantum mechanics such singularities do not exist, but there is an increase in the intensity of scattered rays and particles in the forward and backward directions. This phenomenon is called the glory. An example of the glory in optics is the halo around the head of a man standing on a small hill when his head shields the Sun. The different processes of interaction between particles and nuclei at high enough energies that have analogies in optics (the optical model of elastic scattering, the Fraunhofer and Fresnel types of diffraction scattering, rainbow scattering, and the glory) and also those that have no direct optical analogies (inelastic diffraction scattering, diffractive break-up of composite particles, and some nuclear reactions) considered above are unified in nuclear optics — the peculiar bridge between classical physics (electrodynamics) and quantum mechanics.

5.4

Accelerators

Every epoch in human history is evidenced by some original monuments. The Egyptian pyramids and the towers of Babylon, the sculptures of Ancient Greece and the aqueducts of Ancient Rome, the Mayan temples and the medieval cathedrals of Western Europe — these are silent witnesses to past times. What monuments of our epoch will find people in a few centuries? It is hard to say, but among the most interesting may be the giant accelerators that have been built in different countries all over the world. What are these grandiose and expensive constructions for? Let us try to answer that question.

Nuclear Reactions

139

Accelerators allow us to obtain beams of charged particles having enormous energies and intensities. The accelerated particles are mainly electrons and protons, and, for some special purposes, the nuclei of various elements (usually light nuclei). Accelerators exist that produce protons of 500 GeV. Beam intensities in accelerators can be very large — up to 1016 particles per second. Moreover, these beams can be focused on a target of several mm2. Accelerators were first created to probe molecules, atoms, and atomic nuclei in order to study their structure and the mechanisms of various microprocesses. In fact, the nucleons in nuclei are bound much more tightly than the atoms in molecules. The energy cost of separating atoms in the most tightly bound molecule of carbon oxide, CO, is 11.1 eV. If we want to separate the neutron and proton in the most weakly bound nucleus of deuterium (deuteron), we need to spend 2.22 MeV. Therefore, the energies needed to split atomic nuclei and to study mechanisms of various nuclear reactions appear much greater than the energies used to study chemical processes. In nuclear reactions with positively charged baryons (proton-nucleus and nucleus-nucleus collisions), electric forces of repulsion are involved: F = Zj^e 2 /'(Ri + -R2)2, where i?j and R2 are the radii of colliding nuclei. The nuclear forces of attraction between these particles start to act at distances r ~ 10" 15 m. In order that these particles come together at such distances, energy exceeding the Coulomb barrier B = ZiZ^e2/{R\ + R2) should be spent. For instance, to collide protons with a 16O nucleus, an energy of 4 MeV is needed. To place a proton inside the 238U nucleus, an energy of 15 MeV should be used. The creation of a new particle with mass m in a nuclear collision takes a minimum energy of me2. An energy of more than 140 MeV is needed to create one charged ?r-meson via the collision of nuclear particles. The creation of an nucleon-antinucleon pair p — p or n — n requires an energy of 1.9 GeV. High energies are not only needed to carry out nuclear reactions and study the creation of new particles. They are also necessary to study the detailed structure of known microobjects. In order to discern the details of a microobject having linear size D, it is necessary to use accelerated particles with the de Broglie wavelength X < D. If nonrelativistic particles (e.g., protons) are used, their minimum kinetic energy T should be equal

140

The Quantum World of Nuclear Physics

to

(5.28) Thus, in order to study the details of an object having a linear size of 10~15 m, the energy of the protons should be about 20 MeV. Formula (5.28) shows that a decrease in linear size of the objects under study requires a substantial increase in the energy of the particles. Since D ~ 1/y/T, to decrease D by a factor of 10, the energy of the particles should be increased a factor of 100. The study of smaller and smaller distances requires a substantial increase in the energy of the accelerated particles. A charged particle can be accelerated if it passes through some potential difference: a particle with a charge q passing through a potential difference V acquires the energy E = qV. To keep this energy, the particle should move in vacuum, otherwise all the energy will be transferred to air molecules. Besides, a powerful source of charged particles is needed to make the beam intense enough. Thus, the main parts of any accelerator are the particle source (injector), the accelerating device, and the vacuum creation system (vacuum pumps). The accelerating system described above cannot accelerate particles to high energies. In fact, under several kilovolts the probability of electric breakdown becomes large. Other technical problems are hard to avoid. Therefore, different types of accelerators were constructed. The most widespread of these are considered below. One of the first accelerators was the electrostatic generator or Van de Graaff generator (Fig. 5.9), devised by the American physicist Van de Graaff in 1931. The generator has a large hollow conductor standing on columns of insulator. The strip on which the special device generates positive charges transfers them to the conductor. Positively charged particles (e.g., protons) move from the source (conductor) into the vacuum tube where they are accelerated through the potential difference and directed towards the target by the deviating magnet. Usually electrostatic generators accelerate protons to an energy of about 10 MeV. Tandem electrostatic accelerators where the charge exchange of hydrogen ions takes place (the ion H~ having two electrons loses them and turns into H + ) allows one to obtain particles with energies twice as large. The intensity of a beam of protons accelerated by an electrostatic generator is very large — it reaches 100 /iA. The maximum energy of the protons accelerated by a modern tandem electrostatic generators can approach 30-

Nuclear Reactions

Fig. 5.9

141

The scheme of the Van de Graaff generator.

40 MeV. Van de Graaff generators are mainly used to study the structure of atomic nuclei. In order to reach high energies, indirect methods of acceleration were developed. In 1929 the American physicist Lawrence proposed the first cyclotron, which started to work in 1931. In 1939 Lawrence was awarded the Nobel Prize in Physics for the invention and development of the cyclotron and for results obtained with it, especially with regard to artificial radioactive elements. The working principles of a cyclotron are as follows. A particle with charge q and mass m, moving in a magnetic field of intensity H and with speed v in the plane perpendicular to the direction of the magnetic field, has a circular trajectory (Fig. 5.10). A complete revolution of the nonrelativistic particle around the circle takes a time T = 2TTm/(no\q\H), where /i 0 = 4?r • 10~7 H/m. If v «C c, then the rotation time T does not depend on v. Particles are accelerated in a vacuum chamber situated between the poles of an electromagnet. The semicircle electrodes (duants) are isolated metal disks to which a potential difference

142

The Quantum World of Nuclear Physics

Fig. 5.10

The scheme of a cyclotron.

V is applied. Passing through slot 1, the particle acquires an additional energy \q\V. When it approaches 2, the sign of the electric field is inverted, which means that the particle passing through this slot acquires additional energy \q\V. The sign of the electric field is inverted every half period T/2. Therefore, the particle passing through any slot is accelerated each time, acquiring an additional energy \q\V. After n revolutions, the particle acquires energy 2n|g|V. The higher the velocity of the particle, the larger the radius of its trajectory R = mv/{no\q\H), meaning that the trajectory is a spiral. The particle must move all the time in a homogeneous magnetic field. There is a limit for the construction of such a field on a plane. The energy of the particle accelerated in a cyclotron is limited by the fact that when the velocity is comparable to the velocity of light, the period of rotation starts to increase and it is impossible to retain synchronism between the revolutions of the particles and the sign inversions of the voltage. This limiting energy is about 20 MeV for protons and just 0.01 MeV for electrons. Thus, in order to reach higher energies of accelerated particles, it is necessary to change the working principles of a cyclic accelerator. In 1945, the American physicist McMillan and the Soviet physicist Veksler independently proposed the idea of a synchrotron. In this device particles are injected into the accelerating circle with initial energy EQ. Special magnets keep particles on the circular trajectory with the radius R. Another system of magnets preserves the beam from divergence in space (the beam is collimated). Particles are accelerated inside special resonance cavities operating at frequency w. When the accelerating devices are turned off, the particle with energy J5o and momentum poi after being placed in the accelerating circle, moves with velocity VQ = poc2/Eo (the particle is considered as relativistic) and

Nuclear Reactions

143

makes a complete rotation during the time

T = ^f°.

(5.29)

The frequency of revolution per unit time in this case is

(5.30) In order to hold particles on the circle trajectory with radius R, a magnetic field should be created with intensity Ho\q\R

(5.31)

If the accelerating device is turned on, then its frequency should be larger than the angular frequency fi by a factor of k times (where k is an integer) in order to add energy to the particle at the necessary times. The frequency cj should increase with the energy of the accelerated particle until the latter becomes ultrarelativistic, i.e., when the relation pc = E is valid. The intensity of the magnetic field should also increase. Thus, in order to accelerate the particle, we should satisfy both of the following relations:

»- M -*!§-*!• "=ism-

(5-32)

The acceleration process takes place as follows. First, a bunch of particles with energy Eo is injected into the ring. Then the frequency u> and the magnetic field intensity H are continuously increased in such a way that both conditions (5.32) are satisfied at any time. During acceleration, the energy of the particles increases from the initial value Eo to a final value E. The time of acceleration is different for different accelerators. For the largest and most modern accelerators, it is about 1 s. It follows from (5.32) that in order to shorten the ranges of alteration of the frequency w and the intensity of the magnetic field, it is worthwhile to inject particles with maximum possible energy EQ. Therefore, for instance, on the synchrotron in Batavia (USA), where protons are accelerated to 300 GeV, the injector consists of an electrostatic accelerator accelerating protons to 0.75 MeV and a linear accelerator increasing the proton energy up to 200 MeV. Then the protons enter the booster synchrotron, which they leave with an energy of 8 GeV, and reach the accelerating ring of a large synchrotron having a diameter in excess of 2000 m. Proton synchrotrons are also called synchrophasotrons.

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The Quantum World of Nuclear Physics

Fig. 5.11 The scheme of a collider.

Further increase in the energy of acceleration appears to be a very complicated task from both the technical and economical points of view. Therefore the relative energy of colliding relativistic particles should be increased in another way, e.g., by colliding intersecting beams of high energy. If two proton beams with relativistic energies 21.6 GeV collide with each other, their relative energy is 1000 GeV. A few accelerators with intersecting beams, also called colliders (Fig. 5.11), already exist. Besides cyclic accelerators of particles, there also exist linear accelerators. In these, accelerated particles are driven along a straight line. However, linear acceleration is not free from several difficulties. For instance, the Stanford linear accelerator (20 GeV electrons) has a length of 3000 m. Further increase in energy requires lengthening of the accelerator, which is a difficult technical and economical problem.

Nuclear Reactions

5.5

145

Detectors of Particles

Particles accelerated to high energies interact with targets, after which the results of reactions should be detected. Charged particles passing through a substance collide with its atoms. Many collisions lead to atomic ionizations, i.e., to the emission of electrons from atoms and to the transformations of the latter into positive ions. A fast particle passing through a substance creates many ions that can be observed by various techniques. In 1908 Geiger, in association with Rutherford, devised a particle counter which consisted of a metal thread stretched along the axis of a metal cylinder and isolated from the latter. The cylinder was filled with a gas under a pressure of about 0.1 atmosphere. The thread was given a slight positive potential. A fast particle passing through the cylinder ionized the gas, which led to an electric discharge between the thread and the cylinder, and the emerging impulse of current could be detected. In 1928 the counter was improved by Geiger, in association with Muller, and was called the Geiger-Miiller counter. Another type of particle detector is the scintillation counter. The first such device, called the spinthariscope, was invented by the Englishman Grookes in 1903. In this device the charged particles hit a screen coated with zinc sulfide (ZnS) and caused light flashes on it. Rutherford, along with his collaborators and other nuclear physicists at the turn of the 20th century, observed these flashes in their experiments and counted them with an unaided eye. This was very tiresome and did not give the needed precision. In 1944 the human eye was replaced by the photomultiplier, which breathed a second life into these devices. Instead of the screen coated with zinc sulfide, crystals of sodium iodide (Nal) with the addition of some quantity of thallium, or special plastics, came into use. The Wilson chamber, constructed in 1912, has played an important role in the development of the particle detection technique. This chamber is filled with a gas consisting of the supersaturated vapor of a liquid. If charged particles pass through the vapor, the ions thus created become condensation centers: a track consisting of the resulting liquid drops can be observed and photographed. For his method of making the paths of electrically charged particles visible by condensation of vapor, the Englishman Wilson was awarded the Nobel Prize in Physics for 1927. The Englishman Blackett was awarded the Nobel Prize in Physics for 1948 for his development of the Wilson cloud chamber method. In semiconducting detectors, the charged particle creates ions in a solid

146

The Quantum World of Nuclear Physics

semiconductor made of germanium and silicon. Such detectors have high resolution, but they usually require cooling to very low temperatures in order to reduce the level of noise caused by the thermal excitation of atoms. In 1952, the American physicist Glaser constructed a bubble chamber. For this invention he received the Nobel Prize in Physics for 1960. The bubble chamber is filled with a superheated transparent liquid. Passing through the liquid, a charged particle causes the liquid to "boil up" at the spots where it collides with atoms; i.e., bubbles of vapour arise around the ions. These bubbles form the particle track, which can be photographed. Giant bubble chambers exist to register high energy particles. For instance, the bubble chamber at the Argonne National Laboratory (USA) contains 20 m3 of hydrogen. This is a very complex and expensive device, because liquid hydrogen boils at -246° C and the necessary pressure in the chamber is several atmospheres. In the early 1970s in CERN (Geneva, Switzerland), a giant (for those times) bubble chamber was built. The chamber was filled with freon. In this chamber, the results of rather rare interaction events between neutrinos and protons were observed; this enabled the observation of quarks inside a proton. The spark chamber was also created to detect particles. This device is a multilayer capacitor with a large number of plates to which high voltage is applied. Ions are formed during the propagation of a charged particle through the device, and electrical breakdown takes place in the form of a spark. The track of sparks can be photographed or registered by other means (using special electronics). In order to register high energy particles, volume nuclear photoemulsions are sometimes used. A charged particle regenerates silver passing through this photoemulsion. After development and fixing, the photoemulsion shows tracks of particles and "stars" formed as a result of the dissociation of photoemulsion nuclei caused by particle passage. Many discoveries in nuclear physics (in particular, that of the 7r-meson) have been made in this way. Modern technologies for the registration of the products of interaction between elementary particles and nuclei possess a powerful storehouse of detecting devices, which allow us to investigate events with high precision. This technology is very complex. Detectors of high energy particles are giant structures, the sizes of which sometimes approach that of a three-story building. They are equipped with complex and expensive electronics, by means of which the nuclear and subnuclear processes occurring in detection devices are automatically registered and analyzed.

Chapter 6

Fission of Atomic Nuclei

6.1

Nuclear Fission Mechanism

When heavy nuclei capture neutrons, it is possible for a nucleus to split into two or more parts. Nuclear fission was discovered by the German physicists Hahn and Strassmann in 1939. Hahn was awarded the Nobel Prize in Chemistry in 1944 for that. The discovery of fission was preceded by the fundamental works of Fermi on the irradiation of uranium nuclei by neutrons. The Nobel Prize in Physics was awarded to Fermi for his demonstrations of the existence of new radioactive elements produced by neutron irradiation in 1938. The phenomenon of nuclear fission was explained by the Austrian physicist Meitner and the English physicist Frisch in 1939. They called this new kind of nuclear reaction "nuclear fission due to the seeming similarity with the process of cell fission leading to the reproduction of bacterium." Then, using the analogy between a nucleus and a drop of liquid, in 1939 N. Bohr and the American physicist Wheeler developed the theory of nuclear fission. That year, an analogous theory was independently proposed by the Soviet physicist Frenkel. Neutron induced fission is observed in goTh, giPa, and 92U, and also in the transuranium elements with Z > 93. If the nucleus with (Z, A) breaks into parts, then two nuclei with (Z\,A{) and (Z2, A2) are formed, provided that Z\ + Zi = Z. Besides the fission fragments, fission is accompanied by neutrons (fission neutrons), the number v of which per fission event varies between 2 and 5 for different nuclei (Table 6.1). In uranium fission, the mean value of this magnitude is u = 2.3 ±0.3. Fission neutrons do not have the same energy, but are characterized by a certain energy spectrum. Fig. 147

148

The Quantum World of Nuclear Physics

F

i

103

\

"I I

0

1

I

4

I

I

I

8

_ En

Fig. 6.1 Energy spectrum of fission neutrons for 239 Pu nuclei (arbitrary units). 6.1 shows the typical spectrum of fission neutrons. This is well reproduced by the formula F(En) = Csmh^/2E^exp(-En),

(6.1)

where En is the energy of a fission neutron (MeV) and C is a constant. If nuclear fission is induced by neutrons, then the conservation law for the baryonic number gives A + 1 = A\ + A^ + v. The mass number values of the fission fragments A\ and A2 are not exactly definite. In uranium fission, their mean values are in the ratio 2:3. Fig. 6.2 depicts the distribution of fission probability versus the masses of the uranium fission fragments formed under the influence of thermal (very slow) and fast neutrons. Fission fragmentation is clearly an asymmetrical process in the distribution of "daughter masses". As the excitation energy of a nucleus increases, the asymmetry of this distribution decreases. An enormous amount of energy is released in nuclear fission. In actuality, at the beginning of fission two fragments are separated by a distance po = Ri + R2 (where Ri and R2 are the radii of the fragments). Thus, their relative electrostatic potential energy is Z\Z2&2/p®. After fission this

149

Fission of Atomic Nuclei

N

102 - |

^

\

10" 3 - I A

rt-4

I

I |

I

80

I

I

I

I

110

1

I

140

ll

„

A

Fig. 6.2 Yield of fission fragments (arbitrary units) for 235 U nuclei as a function of nucleus mass number: 1 — fission by thermal neutrons, 2 — fission by 14 MeV neutrons. Table 6.1 Average number of neutrons v emitted in one act of fission of some nuclei (at left — fission by thermal neutrons, at right — spontaneous fission) Nucleus

v

Nucleus

v

2 2 9 Th

2.08 ± 0 . 3 2.407 ±0.007 2.87 ±0.009 2.874 ±0.015 3.07 ±0.04 3.25 ±0.10 3.430 ±0.047 4.56 ±0.21

2 3 2 Th

2.13 ±0.14 1.99 ±0.07 2.150 ±0.015 2.141 ±0.011 2.51 ±0.011 2.69 ± 0.032 3.756 ± 0.010 3.98 ±0.14

235U 2 3 9 Pu 2 4 1 Pu

Am 2 4 2 Am 2 4 3 Cm 249 Cf 241

238U 2 4 0 Pu 2 4 2 Pu 2 4 2 Cm 2 4 4 Cm 252 Cf 2 5 4 Fm

energy is converted into kinetic energy of the fission fragments. Simply putting Z\ = Zi = 46 and R\ = R
150

The Quantum World of Nuclear Physics

about 4-9 MeV, while the typical energy of a chemical reaction between two molecules is on the order of several eV. The energy released in uranium fission induced by neutrons is three million times greater than the heat energy released in the burning of coal, and twenty million times greater than the trinitrotoluene blast energy, if these substances are taken in comparable amounts. More precise calculations allowing for various effects of nuclear fission give approximately the same value of the released energy: 200 MeV. Prom calculations of fission energy an important condition can be obtained, which shows when fission is energetically possible: Z2/A > 17. This condition is fulfilled for all nuclei starting from 4°8Ag, for which Z2/A w 20. Following these conditions, we find that all nuclei with A > 110 should not be stable against fission. However, fission is observed only for the heaviest nuclei — Th, Pa, U — and the transuranium ones. This means that simply the energy feasibility of fission is insufficient to induce fission. As we will see, fission is prevented by the existence of a potential barrier called the fission barrier.

o o i

CO OO 2

3 4 Fig. 6.3 Stages of nuclear fission.

To understand the appearance of the fission barrier, we should consider the different stages of nuclear fission shown schematically in Fig. 6.3. Having absorbed the neutron, the nucleus gets excited. Then the excited nucleus becomes deformed, and the nuclear shape begins to vibrate. Subfigure 1 marks the shape of the initially spherical nucleus of radius R. Then the nucleus is deformed into an ellipsoid with semiaxes a and b (subfigure 2). If the energy of excitation is high enough, the vibrational amplitude becomes appreciable. The surface tension forces tend to return the nucleus to its initial spherical shape. The Coulomb force of repulsion between two positive charges centered at the foci of an ellipsoid tend to enhance deformation and separate the nucleus into two fragments. If the Coulomb forces

151

Fission of Atomic Nuclei

of repulsion exceed the forces of surface tension, a contraction is formed in the central part of the nucleus (subfigure 3). Each of the fragments generated tends to acquire a spherical shape, which makes the contraction more narrow. Further Coulomb repulsion leads to the total separation of fragments (subfigure 4). As calculations show, the nucleus energy increases with the increase of deformation j3, provided the latter is small, while at rather high deformations the energy of the fragments becomes smaller than that of the initial nucleus. In other words, the nucleus energy depends on the deformation /3 in the manner shown in Fig. 6.4.

P Fig. 6.4 The nucleus energy E(/3) as a function of the deformation parameter /3; Ef is the height of the fission barrier. We see that the states of the initial nucleus and fission fragments are separated by a barrier of height Ef. This means that the possibility of fission is not defined only by the energetic preference (i.e., that the energy of fragments is smaller than the initial energy of the nucleus). A nucleus undergoing the fission process must pass through the potential barrier. In spontaneous nuclear fission, where no particles are captured by the nucleus, a tunneling through the potential barrier occurs. Due to the substantial height of the barrier, the probability of such tunneling is small; consequently, so is the probability of spontaneous fission. However, for appreciable fission, to be able to break into two fragments, the nucleus must gain additional energy (for example, by capturing a neutron or other particle) exceeding the barrier height. Therefore, the theory of nuclear fission states the fundamental problem of determining the height and form of the fission barrier.

152

The Quantum World of Nuclear Physics

0.0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fig. 6.5 The shape of a nucleus in the saddle point for different values of the fissionability parameter x.

In the theory of nuclear fission, a very important role is played by a quantity called the parameter of fission x — (Z2/A)/(Z2/A)o, where (Z2/A)o is estimated to be about 45-50. This parameter characterizes the stability of a spherical nucleus against fission. If x > 1, the nucleus is unstable. If a; < 1, the nucleus is stable against small deformations; however, sufficiently large deformations can make it unstable. This critical deformation which causes fission corresponds to the saddle point state of the nucleus. The critical deformation must be greater as x becomes smaller. Fig. 6.5 shows the shapes of nuclei in the saddle point for various values of x. The fragments created during fission are usually /3-radioactive, because they are neutron-rich. If the fragment excitation energy is greater than the energy of neutron separation, the fragment can emit a neutron. Such neutrons are called secondary. If the neutron is emitted by a fragment after /3-decay, then the time of emission is determined by the half-life of the proceeding /3-decay, because the neutron is emitted almost instantly. Thus, such secondary neutrons are also said to be "delayed". Fission fragments possess high ionizing ability and, therefore, have short free paths in matter. This phenomenon is caused by the high velocity of fragments, due to which they lose some of their electrons. Thus, the fragments acquire large effective charge, which results in high ionizing ability of fragments and their short free paths in matter.

Fission of Atomic Nuclei

153

Nuclei of some transuranium elements can capture electrons and create long-lived excited states. The nuclei in these states can undergo fission with the half-life periods equal to several minutes. These fission processes are said to be delayed. The delayed fission can also take place after the /3-decay of some transuranium nuclei. Nuclear fission into two fragments is the most frequent and probable kind of fission. But fission into three or more fragments is also possible, although rare. For instance, the probability of fission into three or four approximately equal fragments taking place due to the capture of slow neutrons by 235U nuclei is no more than one event per 105 events of binary fission. The emission of a-particles along with two massive fragments of the usual type is a firmly established type of ternary fission. It is observed that an a-particle is emitted directly from the nucleus under fission and, thus, is a fission fragment just like the other two heavy fragments are. The a-particle emitted possesses high energy 10 MeV < Ea < 40 MeV. The probability of such a process is about 400 times smaller than the probability of common fission into two fragments. Also possible is ternary fission in which the 3H nucleus is emitted along with two heavy fragments. This process takes place 1-2 times per 104 binary fission events. The transition of a fission fragment from its excited state to its ground state is accompanied by the emission of photons (instantaneous photons). The /3-decay of fragments can be accompanied by photon emission. The time of emission is determined by the half-life of the fragment, meaning that such photons are emitted over a long period of time. Nuclear fission can be initiated by slow and fast neutrons. The likelihood of fission under the influence of neutrons is different for different isotopes of nuclei, and is substantially dependent on the neutron energy. Let us consider the isotopes 235U and 238U. The isotope 235U, upon capturing neutrons, forms the isotope 236U. The fission barrier for the latter is Ef (236) = 5.8 MeV. The same value for 239U, originating from 238U, is JB/(239) = 6.2 MeV. The energy dowry of the neutron is en+Tn, where en is the energy of neutron separation and Tn is the neutron kinetic energy. The neutron separation energy for 236U is en(236) = 6.4 MeV, while for 239U it is en(239) = 4.76 MeV. Thus, the barrier height for 236U is smaller than the neutron separation energy, while for 239U it is the reverse. Therefore, 235 U nuclei can undergo fission induced by neutrons with arbitrary small kinetic energies, while 238U nuclei can only undergofissionby fast neutrons whose kinetic energy is greater than Ef (238) - £n(239) « 1.44 MeV.

154

The Quantum World of Nuclear Physics

The calculations of the fission barrier, based on the model of a homogeneous charged incompressible liquid nuclear drop vibrating due to the competition and compromise between the surface and Coulomb forces, allow one to derive an estimate of the fission barrier and its shape. However, the macroscopic model of a nucleus as a charged liquid drop does not account for a very important property of a nucleus, namely its shell structure. Thus, the fission barrier calculated for the liquid drop should be corrected by adding a term obtained from considering the nuclear shell structure. In this case it is assumed that the shell structure is valid for deformed nuclei, but the energies of one-particle levels depend on deformation. E'

\

^ \

/i

PW \^== / \ ~/_

ground state

I P.

\ — /

\ , \ isomeric

\y \ \ fission' isomeric state \ \ spontaneous \fission

t P2

t P3

t P4

^ P

Fig. 6.6 Schematic plot of thefissionbarrier, taking into account the shell correction (solid line) as a function of the deformation parameter j3. The dashed line shows thefissionbarrier calculated in the liquid-drop model. In 1967 the Soviet physicist Strutinsky showed that the shell correction modifies the barrier, which can give rise to additional extrema. Fig. 6.6 shows the simplest modified double-humped barrier. It has two potential wells and two maxima, situated according to the extrema of the shell correction. The energy levels corresponding to the states of the nucleus before fission are situated in the potential wells. The existence of the second local minimum leads to the possibility of a state in which the nucleus has deformed differently from the deformation in the first well. The second well situated between two maxima has a depth of about 3 MeV for the nuclei around plutonium. In this well, we have several metastable states called the isomers of shape, the lifetimes of which are contained in the interval from 10~17 s to 10~3 s. We emphasize that the isomers of shape

155

Fission of Atomic Nuclei

have nothing in common with the usual isomers discussed above, which differ from the ground states by high values of spin. The half-life periods of spontaneous and isomeric fission of several nuclei are presented in Table 6.2. Note that nuclear fission can be initiated also by charged particles (protons, deuterons, a-particles, etc.) and photons (photofission). Table 6.2 Nucleus 235U

U 238U 239Pu 240Pu 242Pu 2 4 1 Am 2 4 2 Am 244Cm 246Cm 236

6.2

Half-life periods of spontaneous and isomeric fission for some nuclei tj/2 (years), spontaneous fision

t\/2 (10~ 9 s), isomeric fision

1.9 -10 1 7 2-1016 0.8 • 10 16 5.5 • 10 15 1.4 • 10 11 7 • 10 10 2-1014 9.5-1010 1.35 • 10 7 1.7 • 10 7

19.7 ± 4 . 9 70 ± 2 0 195 ± 30 (8 ± 1) • 10 3 4.4 ± 0 . 8 28 (1.5±0.6)-103 (13.5 ± 1.2) • 10 6 100 10

Chain Fission Reactions

The neutron-induced fission of heavy nuclei results not only in the formation of fragment nuclei, but also produces secondary neutrons which, in turn, can initiate fission of the nuclei. These neutrons have continuous distribution of energy and the majority have energies around 1-2 MeV, while their maximum energy is about 10 MeV. Therefore, the possibility of a nuclear chain reaction arises. The notion of chain reaction has been known in chemistry for a long time. For instance, during coal combustion, carbon atoms react with oxygen atoms and produce carbon dioxide. This reaction is exothermic, because an energy of about 4.2 eV is liberated for each CO2 molecule created. However, to start burning, coal should acquire some initial energy, i.e., it should be set on fire. The energy liberated during the creation of only one CO2 molecule turns out to be enough for the burning of neighboring carbon atoms to begin. Therefore, coal burning is an example of a self-maintained chemical chain reaction. Under favorable conditions, neutron-induced nuclear fission can become a self-maintained nuclear fission chain reaction. Neutron physics exploits the following (rather conventional) classifica-

156

The Quantum World of Nuclear Physics

tion of neutrons with respect to their energies. Neutrons whose energies are contained in the interval from 0.025 eV to 0.5 eV are called slow. If the order of magnitude of the neutron energy is the same as that for the thermal motion of particles, then such neutrons are called thermal. The thermal energy for the temperature of 300 K is about 0.025 eV, while the temperature of 1000 K corresponds to a thermal energy of 0.086 eV. Neutrons whose energy is smaller than thermal are called cold. If the neutron energy is smaller than 10~4 eV, the neutrons are ultracold. Neutrons with energies lying in the interval from 0.5 eV to 103 eV are called overthermal, while intermediate neutrons are in the energy interval from 103 eV to 0.5 MeV. Neutrons are fast if their energy exceeds 0.5 MeV. Let us consider the conditions for a nuclear chain fission reaction to occur. We proceed from the simplest case of an infinite multiplicative system. Such a system produces neutrons continuously, and the matter whose nuclei can undergo fission occupies the whole space. In such a system, we may take no account of the neutrons that leave a system having limited size. We shall assume that the system contains only nuclei capable of neutron induced fission. Since nuclear fission is accompanied by fast neutrons, the processes of fission and inelastic neutron scattering are highly likely. At the same time, radiative capture of neutrons whose probability is rather small can be, nevertheless, significant. If the probability of inelastic scattering is small, then in order for the chain reaction to proceed, the condition v > 1 must hold. This condition must hold also in the case where the probability of inelastic neutron scattering is not small, but the inelastically scattered neutrons are able to induce fission. Fission is possible if the kinetic energy of the majority of inelastically scattered neutrons exceeds the difference between the barrier height Ef and the neutron binding energy in the nucleus en, because only in this case is the probability of fission induced by the inelastically scattered neutrons sufficient. If the inelastically scattered neutrons are not able to induce fission, the possibility of chain reaction is determined, in addition to the condition on v, by the probability of inelastic scattering and the energy spectrum of inelastically scattered neutrons. Conditions can be created that favor chain reactions induced by slow (thermal) neutrons. Suppose, due to the high probability of inelastic scattering, that neutrons can leave the energy interval in which they can split nuclei. Then the chain reaction cannot be induced by fast neutrons, yet radiative capture is possible and its role increases with decrease of neutron energy. For this purpose, the system should be filled with a moderator,

Fission of Atomic Nuclei

157

i.e., a light element whose nuclei effectively moderate neutrons by means of elastic scattering but weakly absorb them. The chain reaction induced by slow neutrons therefore occurs, e.g., in a system comprising uranium and light moderator (graphite, heavy water). Fission of the basic uranium isotope 2381J is induced only by fast neutrons with energies exceeding 1.44 MeV. Thus, the high probability of elastic neutron scattering prevents chain reaction by fast neutrons from taking place for the basic uranium isotope as a material for fission. Nonetheless, the uranium isotope 235U, the concentration of which in the natural mixture of uranium isotopes is about 0.72%, is capable of slow neutron fission and the fission probability appears especially high exactly in the region of thermal energies. By the introduction of a moderator whose nuclei are incapable of radiative neutron capture at the energies considered, it is possible to decrease the role of the radiative neutron capture by the nuclei of the basic uranium isotope, which transforms into plutonium in this way. This means that when neutron energy is in the thermal range, favorable conditions are formed for the chain reaction with the 235U isotope as a fission material. In the natural mixture of uranium isotopes, unfavorable conditions for chain reaction occur down to the lowest energies of neutrons due to the high probability of radiative capture of neutrons by 238U nuclei and the small probability of fission of 235U nuclei because of their small concentration. Fortunately, the situation changes dramatically in the vicinity of thermal energies where neutron energies are smaller than the energy of the lowest resonance level of the 238U nucleus, which is around a few eV. Now we can formulate the conditions required for chain reaction induced by slow neutrons. Let us denote by P the probability that an initial fast neutron is captured by an uranium nucleus, which then undergoes fission. Then the mean number of secondary neutrons resulting from fission induced by the initial neutrons is k = Pv. This quantity is called the multiplication coefficient of the system. If the system is infinite, meaning that neutrons do not leave it, then a self-developing chain reaction can take place when k > 1. If k < 1, it cannot. Until now, we have considered an infinite multiplicative system. In order for a finite system to support a self-maintained chain reaction, its size must exceed a certain critical size. The existence of a critical size has a simple physical meaning. Indeed, neutron production in a system is a volume effect, while neutron escape from a system with finite size is a surface effect. If the size of a system is small, the surface escape of

158

The Quantum World of Nuclear Physics

neutrons is more important than their volume production. With increasing size, the role of surface effects decreases, while volume effects become more important. Surface area increases as the square of the linear size of the system, while the volume increases as its cube. The critical size of a system is denned by the condition that the number of neutrons created inside the system coincides with the number of neutrons leaving the system through its surface. The condition k > 1 can be not met for a homogeneous mixture of moderator and fission material in an infinite system. At the same time, this condition will hold for a heterogeneous system containing separate blocks of fissionable material placed in the moderator. For example, such a system can be designed as uranium bars placed in a certain order in the moderator. The advantage of a heterogeneous system follows from the decrease of resonance absorption in the case when the fissionable material is placed in separated blocks. The following consideration makes this statement clear. In a homogeneous system, the neutrons are in the vicinity of the 238U nuclei which can absorb them all time during the process of slowing down. In the block-built system, the neutron can move trough the "danger" zone near the level of resonance absorption, when it is far from uranium blocks. In other words, a neutron can be moderated to the thermal energy with a greater probability than in the case of a homogeneous system. Therefore, the radiative capture of neutrons in a system with multiple block domains can, be substantially lowered relative to a homogeneous system. Resonance absorption is also decreased in a block heterogeneous system because at neutron energies in the region of strong resonance absorption, the interiors of blocks are screened by their outside layers and therefore are not effectively used. This screening leads to a substantial decreasing of the radiative absorption of neutrons with energies close to resonance by uranium nuclei situated in the block, relative to the radiative absorption of neutrons by single nuclei. The chain reaction of fission can be harnessed as a tremendous source of energy. This aim is achieved in special devices, nuclear reactors, and also during the explosions of atomic bombs.

Fission of Atomic Nuclei

6.3

159

Nuclear Reactors

The reserves of conventional fuel on Earth are limited. According to expert forecasts, oil and gas reserves will be exhausted in approximately 100 years, while coal reserves will vanish in 300-500 years. So it is obvious that the problem of energy cannot be solved without nuclear energy. In order to use the chain reaction of nuclear fission to obtain energy, this reaction must be made controllable. That is why nuclear reactors are complex devices in which chain reactions are totally controlled. If the chain reaction in a reactor gets out of control, a catastrophe becomes possible — an explosion of the reactor, accompanied by dramatic consequences for humanity and the environment. Just such a situation led to the explosion of the reactor at the Chernobyl atomic power station (USSR) in April 1986. The chain reaction in the reactor is regulated by an absorber of neutrons. Boron and cadmium are usually used as absorbers. If a decrease in the energy released by the reactor is needed, then additional absorbers should be introduced into the reactor and the chain reaction of fission will be damped. It is clear that the absorbers must be put into the active zone of the reactor very quickly, otherwise a delay of even part of a second will lead to a catastrophe. That is why regulation of the chain reaction in the reactor is carried out by means of computers. In a reactor, the neutron generations continuously replace each other. The active zone always contains a large amount of neutrons: any cubic centimeter of the reactor contains about a half billion neutrons. If at the beginning we have no thermal neutrons in the reactor, then some (undoubtedly not all) will cause nuclear fission and, therefore, fast secondary neutrons will appear. These neutrons then become thermal owing to the moderating elements. New thermal neutrons, in turn, induce new nuclear reactions and the generation of new neutrons. This process lasts until a chain reaction of nuclear fission occurs. Let us denote the number of secondary neutrons by n-i and introduce the mean lifetime of neutrons of the same generation. This lifetime is the time during which one generation of neutrons is replaced by another. For reactors with thermal neutrons, this time has order of magnitude 10~3 s - 10~4 s, while for the reactors with fast neutrons it is 10~6 s. The ratio of the number of neutrons in a generation to their number in the previous generation, taken on the same stage of their temporal evolution, is called the effective multiplication coefficient of neutrons: keff = ni/no. The quantity p — (keff — l)/keff is called the reactivity of

160

The Quantum World of Nuclear Physics

the reactor. Depending on the values of keff or p, three regimes of reactor functioning are distinguished. If keff > 1 and p > 0, the regime is supercritical. This occurs, e.g., at the start when the reactor is sped up to the required power. If keff = 1 and p = 0, the regime is critical and the reactor works with constant power. If keff < 1 and p < 0, the regime is subcritical. This occurs, for instance, when the reactor is stopped and its power is gradually decreased. Therefore, the functioning of the reactor of a nuclear power station is efficient when it works in the critical regime. To stay within this regime requires a rather hard technological task: continuous control of all the parameters of such a complicated device as a nuclear reactor. An important problem in constructing reactors is the sink of energy (heat) out of the active zone. Special coolants are employed for this purpose: water, carbon dioxide CO2, heavy water, or liquid (molten) sodium. The coolant is heated in the active zone of the reactor and transfers its heat energy to the external device or secondary circuit. In the active zone, the coolant is strongly irradiated and its nuclei acquire an induced radioactivity. If the coolant transfers the energy directly to the turbine that generates electricity, then we can restrict ourselves to a one-circuit scheme to channel energy out of the active zone. If we have two or even three circuits, then thermal energy will be transferred to the user by a pure (not radioactive) coolant. The efficiency of a nuclear reactor is very high. Gigantic energies are released in nuclear fission: the total fission of 1 kg of uranium produces the same energy as that generated during the total combustion of 2000-3000 tons of coal. Therefore, countries that lack sufficient reserves of natural coal, gas, and oil are forced to build nuclear stations. The first nuclear reactor was built in Chicago in December 1942 under the guidance of Enrico Fermi. The first nuclear reactor in the USSR was developed under the direction of Kurchatov and launched in December 1946 in Moscow. Since that time, many reactors have been built to obtain energy and to perform scientific investigations. Nuclear reactors are also widely used as power sources for the engines on ships and submarines. A modern nuclear power station is a very complicated structure having the height of a 10-floor building. However, despite its technological complexity, a nuclear power plant possesses substantial advantages in comparison to a conventional power plant. The cost of electricity produced by a nuclear plant is smaller than the cost of electricity generated at a conventional power station. A 1 GW nuclear plant spends about 1 kg of 235U

Fission of Atomic Nuclei

161

isotope per day, while a thermal power station of the same power burns a whole trainload of coal or oil during the same time period. In the region of a nuclear power station, ecological impact is kept to a minimum, largely taking the form of thermal pollution. Under normal operating conditions, the probability of lethal irradiation in this region is smaller than the probability of being struck by lightning. At the same time, each ton of coal contains about 80 grams of uranium. Therefore, the smoke from a thermal power station contains much more radioactive material than the gaseous emissions from a nuclear plant. Moreover, the smoke from thermal power plants contains many additional harmful chemical substances (e.g., sulphurous gas). More than 400 nuclear reactors in 26 countries produce over 300 GW of electrical energy, which is about 16% of all the energy generated in the world. In France, for instance, nuclear power stations produce more than 80% of the electricity supply. Humanity has no current viable alternatives to atomic energy. In principle, a nuclear reactor could exist in natural conditions. This can be confirmed by the following considerations. At present, natural uranium contains 0.72% of 235U isotope. This amount is not enough for the existence of a reactor with water as a moderator of neutrons. But the half-lives of 235U and 238 U are 7.13 • 108 and 4.51 • 109 years, respectively. This means that 2 • 109 years ago, the concentration of 235U isotope in the uranium was about 3%, i.e., approximately the same as in modern nuclear reactors working on enriched 235U fuel. With this concentration, a nuclear chain reaction could occur with groundwater as a moderator. Analysis of uranium ore mined in Oklo (West Africa) has shown that it contains 0.64% of 235U isotope. It also contains plenty of rare-earth elements created during uraniumfission,and a small amount of 239 Pu which appears as a result of neutron capture by 238U nuclei and the subsequent f3~ -decay. Geological study of the uranium deposit in Oklo has shown that it is situated in an old river delta. Taking into account the amount of plutonium created, the power of this natural reactor was estimated as 25 kW and the duration of its work estimated as 600 million years. The quantity of the "consumed" 235U isotope gives the age of the reactor as about 1.8 billion years. If several pieces of fissionable material, the total amount of which is greater than the critical mass, are rapidly united, an explosion of tremendous power occurs. This idea forms the basis for nuclear weapons. To make the mutual approach of these pieces rapid, an ordinary explosive is used.

162

The Quantum World of Nuclear Physics

The pieces can be placed close to each other, but separated by a layer of a substance that strongly absorbs neutrons. To induce explosion in this case, it is enough to rapidly remove the neutron absorber or to introduce a neutron source to counteract the absorber. The first nuclear bomb was detonated on a testing ground in the United States on 16 July 1945. 6.4

Man-Made Synthesized Elements

The list of elements that exist in Nature ends with plutonium (Z = 94). However, plutonium and neptunium (Z = 93) exist in Nature in extremely small amounts in uranium ores. For instance, the ratio of the number of 239pu n u c i e j t o t n e number of uranium nuclei in pitchblende deposits in Colorado, which contain 50% uranium, is 7.7 • 10~12. This plutonium has been created as a result of the interaction between uranium nuclei and the neutrons of the secondary component of cosmic rays. Therefore, we can state that uranium (Z = 92) is the last of the elements existing in Nature. Elements with Z > 92 are called the transuranium elements. Their existence was predicted by Rutherford in 1903, and in 1935 the possibility of their synthesis was proposed by Enrico Fermi. These elements can be produced artificially using different nuclear reactions. At present, the transuranium elements with atomic numbers 93 < Z < 109 have been discovered (Table 6.3). Let us briefly discuss the production of transuranium elements. Neptunium was obtained in 1940 in the reaction of radiative capture of a neutron by a uranium nucleus with subsequent /3-decay: 238IJ

239TJ

+

> 239 N p

_

n

+

g

-

+

+

239 u

^

+

( g_ 2)

7 )

^

=

33 5 m

i n

(g 3)

This element was named after the planet Neptune. The 239Np nucleus is radioactive: it undergoes /3~-decay with £1/2 = 2.35 days. This neptunium isotope was first identified by its half-life, and then its properties were studied. At present, 11 neptunium isotopes are known to have mass numbers 231 < A < 241. The longest-lived isotope of neptunium is 237Np, which experiences a-decay with £1/2 = 2.14 • 106 years. Plutonium was obtained in 1940 from the reactions 238U+ 2

H

^

238 N p

+

2 n )

(g 4 )

163

Fission of Atomic Nuclei Table 6.3

Transuranium elements

„ Z

.. Name

_ , , Symbol

Year ,. . discovered

First discovered . isotope

93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109

Neptunium Plutonium Americium Curium Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium Rutherfordium Dubnium Seaborgium Bohrium Hassium Meitnerium

Np Pu Am Cm Bk Cf Es Fm Md No Lr Rf Db Sg Bh Hs Mt

1940 1940 1944 1944 1949 1950 1952 1952 1955 1957 1961 1968 1968 1974 1981 1984 1982

2 3 9 Np

238 Np

—>

238 Pu

2 3 8 Pu 2 4 1 Am 2 4 2 Cm 2 4 3 Bk 245 Cf 2 5 3 Es 2 5 5 Fm 2 5 6 Md 254 No 2 5 7 Lr 257 Rf 2 6 0 Db 259 Sg 2 6 2 Bh 265 Hs 2 6 6 Mt

+ e~ + v, t1/2 = 2.1 days.

ti/2 ' 2.35 days 86.4 years 458 years 162.5 days 4.5 hours 44 min 20 days 22 hours 1.5 hours 3s 8s 4.8 s 1.6 s several ms 4 ms 2.4 ms 5 ms

(6.5)

It was named after the planet Pluto. The 238 Pu isotope is a-radioactive with ti/2 = 86.4 years. Another plutonium isotope, 239 Pu, was obtained in 1941 as a product of/3^-decay of 239Np. This isotope is also a-radioactive. The large half-life ti/2 = 2.43 • 104 years of the 239 Pu nuclei created in nuclear reactors in large quantities allows one to use this isotope as a fuel for special nuclear reactors and as an explosive in nuclear bombs. At present, 15 plutonium isotopes are known to have mass number 232 < A < 246. The longest-lived isotope of plutonium is 242Pu, which has ti/2 = 3.8 • 105 years and is a-radioactive. For the discovery of plutonium, American physicists McMillan and Seaborg were awarded the Nobel Prize in Chemistry in 1951. Americium was discovered in 1944 from the reactions

241 Pu

239pu +

n

^

24OPu

+ 7,

(6.6)

240Pu +

n

_^

241 Pu

+ 7)

(6.7)

—>

241Am

+ e~ + D, t 1 / 2 « 14 years.

(6.8)

This element received its name (from the word America) because it was homologous to the element Z = 63 europium (named in honor of Europe).

164

The Quantum World of Nuclear Physics

The 241Am nuclei are a-radioactive with £1/2 = 470 years. At present, we know 11 americium isotopes with mass numbers 237 < A < 247. The longest-lived isotope is a-radioactive with t\/2 = 7800 years. Americium is famous because the 242Am isotope has an isomer of shape with the half-life of spontaneous fission ti/2 = 1-4 • 10~2 s. This isomer was discovered in Dubna (Russia) in 1962 and was the first observed isomer of shape. Curium was produced in 1944 by the reaction 2 3 9 Pu+ 4

He-^

242Crn

+ n.

(6.9)

This element, with Z = 96, was named in honor of Pierre and Marie Curie, who discovered actinium — the first element from the actinide series. In the symbol Cm, the first letter comes from the surname Curie, while the second is taken from the Christian name of Marie Curie. Now we know 13 curium isotopes with mass numbers 238 < A < 250. The longest-lived isotope, 248Cm, experiences a-decay as its main decay mode with t^ji = 4.7 • 105 years. In 1949 berkelium was obtained in the reaction 241 Am

+ 4He—>

243Bk

+ 2n.

(6.10)

This element was so named because it was produced at the laboratory of the University of California at Berkeley near San Francisco. At present, 9 berkelium isotopes with mass numbers 243 < A < 251 have been produced. The longest-lived isotope, 247Bk, is a-radioactive with t^/i = 1380 years. Californium was produced in 1950 in the same laboratory as berkelium, and was named for the university. The element was created in the reaction 242 Crn+ 4

H e - ^ 2 4 5 Cf+n.

(6.11)

Sixteen californium isotopes are known, with mass numbers 240 < A < 255. The longest-lived isotope, 251Cf, is a-radioactive with t^^ — 1600 years. Einsteinium (Z = 99) and fermium (Z = 100) were first found in the several hundred kilograms of corals that remained after the explosion of a thermonuclear bomb on the Bikini atoll in 1952. Their names were given in honor of the two outstanding physicists of the 20th century. Fourteen isotopes of einsteinium with 243 < A < 256, and fifteen isotopes of fermium with 243 < A < 258, are known. The longest-lived isotopes of these elements are a-radioactive: 254Es with ty/% — 480 days, and 257Fm with £]y2 = 79 days.

165

Fission of Atomic Nuclei

Mendelevium (Z ~ 101) was obtained in 1955 in the reaction 2 5 3 Es+ 4 H e - ^

256Md

+ n.

(6.12)

Named in the honor of Mendeleev, who discovered the periodic table, this element illuminated the validity of the periodic law for the elements with atomic numbers greater than 100. Eleven mendelevium isotopes are known to have 248 < A < 258. Nobelium (Z = 102) was discovered in nuclear reactions of irradiation of 244Cm nuclei by 13 C, studied using the cyclotron of the Nobel Institute in Stockholm from where it acquired the name. Lawrencium (Z = 103) was produced in 1961 at the University of California. It was named in honor of the American physicist Lawrence, who was one of the creators of the first accelerator. The element with Z = 104 was obtained in Dubna in 1964, where it was proposed to be named kurchatovium, and at the University of California in 1968, where it was named rutherfordium. Since doubts still exist whether this element was really discovered in 1964, the final name for the element is rutherfordium. The element with Z = 105 was obtained in Dubna in 1968, and its name dubnium is devoted to the town where it was discovered. The elements with Z > 106 are also proposed to receive the names of outstanding scientists: Z = 106 (1974, Dubna and independently the University of California) — seaborgium; Z — 107 (1981, the accelerator of heavy ions in Darmstadt, Germany) — bohrium; Z = 108 (1984, Darmstadt) — hassium; Z = 109 (1982, Darmstadt) — meitnerium. In 1996-1999 the elements with Z = 112 and A = 283 and also with Z = 114, A = 287-289 were found. In 2000-2003 the elements with Z = 116,118 and Z = 115, A = 287 and A = 288 were synthesized. However, these results must be confirmed in other laboratories of the world. Apparently, in coming years, the creation of superheavy elements will continue. Let us now discuss the discovery of radioactive elements with Z = 43,61,85,87. These nuclei are not observed in Nature, except for the element with Z = 43 which is found in several young stars and in uranium ores in insignificant amounts. Technetium (Z = 43) was discovered in 1939. Its name comes from the Greek "technetos", meaning "artificial". The reaction was Mo+ 2 H—> Tc + n.

(6.13)

Now we know 16 technetium isotopes with 92 > A > 107. The longest-lived isotope, 97 Tc, experiences the K-capture with tx/2 = 2.6 • 106 years.

166

The Quantum World of Nuclear Physics

Promethium (Z = 61) was found in 1946 in the fission products of 235U in nuclear reactors. It was named for the mythical hero Prometheus, who stole fire from Zeus and gave it to man. The name emphasizes the method of creation of the element with use of the energy of nuclear fission. The longest-lived isotope, 145Pm, undergoes K-capture with t1/2 = 18 years. Astatine (Z = 85) was produced in 1940 from the reaction 209 Bi+ 4

He^

211At

+ 2n.

(6.14)

The name originates from the Greek "astatos", meaning "unstable". Astatine was predicted by Mendeleev under the preliminary name "eka-iodium". The longest-lived isotope of this element, 210At, has half-life ti/2 — 8.3 hours (mainly K-capture). Prancium (Z = 87) was discovered in 1938 in the decay products of the natural radioactive element 227Ac. It was named for France, where it was discovered. The element was predicted by Mendeleev under the preliminary name "eka-caesium". Its longest-lived isotope, 223Fr, undergoes the /3~decay with £1/2 = 22 min. Note that all the artificially obtained elements are radioactive with halflives much smaller than the age of the Earth: 4.6 • 109 years. That is why these elements are not present on the Earth (except for small amounts in the uranium ores, where they are produced as a result of radioactive decay of nuclei).

Chapter 7

Nuclear Astrophysics and Controlled Nuclear Fusion 7.1

Expanding Universe

There are 287 different isotopes of nuclei known in Nature. They include both stable nuclei and nuclei whose half-lives exceed the age of the Earth. All these nuclei have been found on Earth. There is only one element, technetium, which has not been found on Earth but is observed in young stars. In fact, the half-life of the longest-lived isotope of technetium is on the order of 105 years, i.e., the nuclei of technetium on Earth have already completely decayed, but they still exist in sufficiently young stars. The quantities of different isotopes in Nature differ substantially (Fig. 7.1). Light nuclei prevail in Nature; heavy nuclei are the most sparse. Hydrogen and helium by far surpass all other elements in their abundance. Together they form 98% of the substance of our galaxy. The curve in Fig. 7.1 shows the relative abundance of the isotopes of nuclei in the solar system and in the main sequence stars which are close to the Sun by their mass and age. This is called the solar curve. Questions about its shape are closely related to the general problem of the origin and evolution of the Universe. At the present time, the generally accepted theory is that of the Big Bang, as a result of which each particle of matter rushed away from every other particle. At the moment of the Big Bang, temperature, and density were infinite. Immediately after the Big Bang, the temperature of matter was huge. The concept of the Big Bang and the picture of the hot Universe at the early stages of its development were proposed by George Gamov in 1946 in the famous paper he published together with his postgraduate student Alpher. Gamov also persuaded the outstanding physicist Bethe to be a coauthor of this paper — that way, the author list would read "Alpher, 167

168

The Quantum World of Nuclear Physics

H 1Q I

hydrogen combustion

1 He £ Si

8 - 1 helium combustion \ . carbon and oxygen combustion silicon combustion fJf \ /

8 ®

1

CD

\ V4

\ \ l\ —

4- \ \

iron g r o u p

/I

I 2T\M wT ^\ N=/ 82 Q.

£

\

/

\

W 0 - Li-Be-B i

-21

0

Z = 5 0

\ /, \JW\ i

100

i

z=82 N=126 I i

200

A

Fig. 7.1 Dependence of the logarithm of the relative abundance of elements (arbitrary units) on nucleus mass number. Bethe, Gamov" and would correspond to the three first letters of the Greek alphabet: alpha, beta, gamma. Gamov, who possessed a keen sense of humor, thought this would befit a paper about the origin of the Universe. Because of its huge energy, the initial Universe started to expand and gradually became cooler. Gamov called the initial substance ylem which, in translation from the ancient English, can be roughly understood as "an initial substance from which the creation of elements occurred". Of course, under the huge temperatures during the first moments after the Big Bang, no bound matter could exist. At that time not only could there be no molecules and atoms, but also no nuclei or nuclear constituents: protons and neutrons. There only existed photons, electrons, positrons, neutrinos, antineutrinos, and other leptons and, besides these, the constituents of nucleons — quarks, antiquarks, and gluons — which were in the state of a quark-gluon plasma. Immediately after the Big Bang, owing to the huge temperatures (in 10~43 s after the Big Bang the temperature was approximately 1032 K) and

Nuclear Astrophysics

169

the correspondingly huge particle energies, there was in fact no difference between the various types of interaction: the strong, electromagnetic, weak and, probably, gravitational. Rather, there existed a single fundamental interaction. In that case, photons, leptons, quarks, and other particles moved like free objects and together formed a heat radiation whose temperature, however, was not constant and changed over time. At the moment of the Big Bang, it is probable that matter possessed all possible symmetry properties: the number of electrons was the same as the number of positrons, and the number of quarks was the same as the number of antiquarks. Further, the expansion of the Universe was accompanied by cooling of matter. Similar to condensation forming water drops during water vapor cooling, during the cooling of the quark-gluon plasma the bound states of these particles, i.e., nucleons and antinucleons, were created as well. The evolution of the Universe after the Big Bang can be imagined as a succession of four eras (or epochs) leading to its present state with an average density of matter p ~ 10~27 kg/m3 and average temperature T « 2 . 7 K . Some 10~10 s after the Big Bang, the density of matter far exceeded that of nuclear matter and equaled p ~ 1018 kg/m3. The temperature was also huge: T ~ 1013 K. During this epoch the strongly interacting particles, hadrons, were created. This hadron epoch lasted about 10~4 s, until the temperature decreased to the rest energy of the lightest hadron, the 7r-meson (T « 1012 K). By the end of the hadron epoch, the density of matter was comparable to that of nuclear matter. The next period in the evolution of the Universe can be called the "lepton" era. Its duration was approximately equal to 10 s, until the temperature of matter decreased to the threshold of the photoproduction of an electron-positron pair (T « 0.7 • 1010 K). By the end of the lepton era, the density of matter was p ~ 107 kg/m3. At that time, leptons could not already be created spontaneously, and radiation consisted mainly of photons. Thus the "radiation" era began, which ended when photons could exist separately from matter. This era ended approximately 106 years after the Big Bang. The temperature at its end was T ~ 107 K. Further expansion and cooling of the Universe led to the "star" era, which lasts until the present age. In 1965, the American scientists Pensias and Wilson discovered the existence of an isotropic photon flux in the Universe, having no "seasonal" oscillations, which is called relict radiation. This radiation is uniformly distributed over the celestial sphere, and its energy corresponds to the heat

170

The Quantum World of Nuclear Physics

radiation of an absolutely black body at temperature T « 2.7 K. In 10~2 s after the Big Bang, relict radiation had temperature 1011 K, and in 106 years — 3 • 103 K. Now, relict radiation is isotropically distributed in the Universe with density 5 • 108 photons/m3. Earlier, it was predicted theoretically by Gamov on the basis of the expanding Universe model. Relict radiation is a witness to the Big Bang and to the three subsequent epochs in the development of the Universe. Its existence allows us to conclude that the Universe was hot during the early stages of its expansion. For the discovery of relict radiation, Pensias and Wilson were awarded the Nobel Prize in Physics for 1978. Moments after the Big Bang, the number of particles apparently coincided with the number of the corresponding antiparticles, but thereafter the former became somewhat larger than the latter, i.e., the so-called baryon asymmetry arose. Just from astrophysical observations it is known that the ratio of the number of particles (electrons and nucleons) in the Universe to the number of photons is on the order of 10~9. The theory of baryon asymmetry was first developed by the Soviet physicist Sakharov in 1967. There also exists another viewpoint on the fact that the electron, proton, and neutron — rather than the positron, antiproton, and antineutron — are the main elementary particles in Nature. In fact, the problem is to indicate the cause that has led to the separation of matter and antimatter, but not to their annihilation. In other words, in this approach it is supposed that matter and antimatter, after their separation, have disseminated to different parts of the Universe. Those parts of the Universe observed by us do not contain antimatter. It appears that one act of nucleon-antinucleon annihilation leads to the emission of several 7r-mesons (on the average, six 7r-mesons per annihilation act). During contact of a cluster of matter with a cluster of antimatter, the annihilation of particles and antiparticles would occur at the contact surface, and the 7r-mesons emitted from both sides of the surface would create a macroscopic pressure called the annihilation pressure. This would lead to separation of the media. These processes could proceed in the hadron era when the density of substance by far exceeded that of nuclear matter. However, this model suffers from various difficulties yet to be resolved.

Nuclear Astrophysics

7.2

171

Creation of Atomic Nuclei

After the creation of protons and neutrons, nucleons — not quarks — became the fundamental "bricks" of the Universe. Atoms also include electrons, and there are also fluxes of photons and neutrinos in the Universe. Gradually, during the expansion and cooling of the Universe, complex nuclei began to be formed from the nucleons, and an increasingly important role was played by different nuclear reactions. These not only led to the creation of new nuclei, but also became an energy source. The origin of stellar and solar energy can be explained exactly in this way. An important role in the creation of light and medium nuclei (up to the elements of the iron group, inclusively) was played by nuclear reactions with charged particles, and in the creation of heavier nuclei — by neutron capture and /3-decay. Thus, a question of a paramount importance for studying the creation of atomic nuclei is the relation between the quantities of protons and neutrons at the early stage of evolution of the Universe. At sufficiently high temperatures nucleons interacted with leptons, i.e., their mutual transformations took place: p + e~ <—> n + ue,

(7.1)

p + Pe <—> n + e+.

(7.2)

Reactions (7.1) and (7.2) defined the equilibrium between proton and neutron concentrations. At T > 1011 K the proton and neutron concentrations were approximately the same. As a result of the expansion and cooling of the Universe, there were more and more protons and fewer and fewer neutrons since the neutron mass is somewhat greater than the proton mass, and proton creation in the reactions (7.1) and (7.2) is energetically more favorable than neutron creation. The velocities of reactions (7.1) and (7.2) rapidly decrease as temperature decreases, so these reactions almost completely ceased seconds after the Big Bang. Then, the relative concentration of neutrons in nucleon matter was around 15%, which denned the future ratio between the concentrations of hydrogen and helium in Nature. Let us now consider the processes for creating complex nuclei. The simplest complex nucleus is a deuteron, which consists of a proton and a

172

The Quantum World of Nuclear Physics

neutron. This could be created via the fusion of a proton and a neutron: p + n—> d + 7,

Q = 2.225 MeV.

(7.3)

It could also be created from the fusion of two protons, with emission of a positron and a neutrino: p + p—>d + e+ + v,

Q = 1.19 MeV.

(7.4)

Because of the Coulomb barrier, the latter reaction can proceed only at higher temperatures than the reaction (7.3). During the further capture of neutrons by deuterons, 3H nuclei were created. Under the /?-decay of 3 H nuclei, 3He nuclei were created, which captured one more neutron and transformed into 4He nuclei. However, it is more probable that the 3He nucleus, capturing a neutron, would decay into two deuterons than create an a-particle. Nevertheless, a fraction of the 3He nuclei was transformed into a-particles via neutron capture. 4He nuclei could also be created in another way, namely, through capture of a proton by a deuteron with creation of a 3 He nucleus and further fusion of two 3He nuclei: d + p—> 3He + 7, 3 He+ 3 He—+

4He

+ 2p,

Q = 5.49 MeV, Q = 12.85 MeV.

(7.5) (7.6)

This possibility plays a fundamental role in the origin of the energy of stars. Together with the fusion reaction of two protons into a deuteron, it forms the proton-proton cycle. Here the transformation of protons into 4He nuclei occurs as follows: six initial protons are transformed into an a-particle, two free protons, two positrons, and two neutrinos, i.e., hydrogen "burns out" creating helium. In each cycle, 26.21 MeV of energy is released. The creation of helium was finished approximately 100 s after the Big Bang. At that time, matter in the Universe consisted of 70% protons and 30% 4He nuclei. This composition of the Universe remained unchanged until the processes of synthesis of heavier nuclei started once stars formed. The ratio between the concentrations of hydrogen and helium nuclei in Nature defined the further synthesis of various nuclei. If this ratio were different, the relative concentrations of various elements in the Universe would be different as well. The whole world would be quite different. The correct prediction of the relative abundance of helium nuclei in the Universe

Nuclear Astrophysics

173

is the second (after relict radiation) great achievement of the model of the Big Bang. The further creation of nuclei by neutron capture is impossible, as the 5He nucleus does not exist. Indeed, the 5He nucleus decays into two parts immediately after its formation, which creates a insurmountable barrier for the synthesis of heavier nuclei by the successive capture of neutrons by nuclei. The 5Li nucleus, another candidate for a nucleus with A = 5, is also unstable. The same difficulty is also encountered further, since there is no stable nucleus with mass number A = 8. The abundances of Li, Be, and B nuclei in Nature are rather low (Fig. 7.1); nevertheless, they are too high to be consistent with the synthesis of these nuclei directly inside stars. Indeed, these elements have rather small binding energy. For this reason they would be destroyed faster within stars, the higher the temperature of the star. These nuclei, apparently, were created partially from stellar explosions and partially in the cold matter from fission of different nuclei under the action of cosmic rays. Now we have to explain how the carbon nuclei 12C could be synthesized. If the density of 4He nuclei is very large, then fusion of three a-particles into a 12C nucleus can occur. The probability of this process is much higher than that for triple collisions of a-particles, since it usually proceeds through the fusion of a-particles which forms an excited 8Be* nucleus via the resonance level of the latter. The reaction a + a —> 8Be is impossible, since 8Be in its ground state is unstable. However, at the energy E ~ 0.1 MeV there exists an excited state of 8Be*, whose lifetime is T ~ 10~16 s. Although this time seems small, it is large on the nuclear "scale" because the characteristic nuclear time is Tnuc ~ 10~22 s. In other words, in an excited 8Be* nucleus two a-particles have time to perform about 106 oscillations relative to each other before they fly apart. This is time enough for a third a-particle to approach them to form a 12C nucleus. The 4He nucleus plays an important role in the synthesis of heavier nuclei, as well as in the synthesis of the 4He nuclei themselves. Along with the proton-proton cycle described above, the carbon-nitrogen cycle is also essential; it consists of the following reactions: p+

1 2

C^

13N

+ 7,

1 3 N _ ^ i3 C + e + +

l/)

Q = 1.95 MeV,

(7.7)

Q = 1.50 MeV,

(7.8)

174

The Quantum World of Nuclear Physics

p+

13C

—-+

14N

+ 7,

Q = 7.54 MeV,

(7.9)

p+

" N __» i5 O + 7 ,

Q = 7.35 MeV,

(7.10)

Q = 1.73 MeV,

(7.11)

15O—> is N + e + + p

+

15N

_»

12C

I/)

+ 4He,

g = 4.96 MeV.

(7.12)

In this cycle, 14N nuclei are created which were absent otherwise and are the most abundant odd-odd nuclei in Nature. As a result of the carbonnitrogen cycle, four protons transform into an a-particle, two positrons, and two neutrinos. In this case the number of 12C nuclei is unchanged; i.e., in this cycle, carbon is a catalyst and is not consumed. In the carbonnitrogen cycle 25.03 MeV of energy is released. This cycle is also a source of stellar energy. Since the probabilities of all reactions entering into the proton-proton and carbon-nitrogen cycles are essentially dependent on the stellar temperature, the total energy yield due to these nuclear processes essentially depends on the stellar temperature as well. At temperatures of 15-20 million °C, both cycles lead to approximately the same energy release. At lower temperatures, the dominant cycle is the proton-proton cycle, and at higher temperatures the carbon-nitrogen cycle dominates. These cycles of stellar energy production were established by Bethe in 1938, and in 1967 he was awarded the Nobel Prize in Physics for this discovery and for his contribution to the theory of nuclear reactions. After nitrogen, the oxygen nucleus 16O follows, which is formed under the capture of an a-particle by a 12C nucleus. Within the Sun and main sequence stars, the most abundant elements among those heavier than helium are carbon (0.39%) and oxygen (0.85%). The human body consists of 18% of carbon and 65% of oxygen by mass (the remainder is largely hydrogen). For this reason, the determination of the ratio of abundances of 12C and 16O nuclei created during the consumption of helium is a very important task. This ratio is defined, first of all, by the relative rates of the reactions 3a —> 12C and 12C + a —> 16O + 7. If the first reaction will proceed much more rapidly than the second, then very few oxygen nuclei will be formed by helium consumption. In the opposite case, there will be very few carbon nuclei. The ratio of rates of the reactions under consideration and, as a consequence, the ratio of the abundances of 12C:16O nuclei in Nature, are

175

Nuclear Astrophysics

defined by the energy level structure of these nuclei. It is only the 12C nucleus, which has a level (excited state) with energy E = 7.656 MeV and spin 7 = 0, which is situated somewhat higher than the sum of the rest energies of 8Be and 4He nuclei (E = 7.370 MeV) and than the total rest energy of three a-particles (E = 7.277 MeV). This means that a 12C nucleus can be created in the course of a resonance reaction. If the 12C nucleus had no favorably situated resonance level, then the rate of creation of carbon would be much less than in the presence of such a level. Thus, the existence of favorably situated resonance levels of 8Be* and 12C* nuclei has led to the sufficiently high concentration of carbon nuclei in Nature. Now let us consider the 16O nucleus. If this, like the 12C nucleus, had a favorably situated resonance level, then the synthesis of 16O nuclei by the capture of a-particles by 12C nuclei would proceed so rapidly that there would be very little carbon in Nature. In reality, the 16O nucleus has levels with energies E = 6.05 MeV (I = 0) and E = 6.92 MeV (I = 2) which are close to the sum of the rest energies of the 4He and 12C nuclei (E = 7.16 MeV). However, these levels are situated lower than the sum of the rest energies of the 4He and 12C nuclei, so that in reality the resonance capture of a-particles by carbon nuclei does not occur. Thus the evolution of stars, which leads to the synthesis of heavier nuclei and the existing ratio between the quantities of carbon and oxygen nuclei in Nature, is denned by three circumstances: (1) the instability of the 8Be nucleus in its ground state and the presence of the resonance level of the 8Be* nucleus, (2) the existence of the favorably situated resonance level in the 12C nucleus, and (3) the absence of such a favorably situated resonance level in the 16O nucleus. In the further evolution process, the absorption of a-particles by 16O nuclei and by heavier nuclei makes it possible to explain the creation of 20Ne, 24Mg, 28Si, and other nuclei. Nuclei with the mass numbers lying between the mass numbers of 16O, 20Ne, 24Mg and so on could be formed in the neutron and proton capture reactions. However, along with these processes, the creation of nuclei with A > 20 could also occur as a result of the nuclear fusion reactions. At temperatures T « 2 - 1 0 9 K, carbon fusion could take place:

f 20Ne + a, i2C+

i 2 C — • ) 23 N a

+ P )

(7.13)

i23Mg + n. This cycle of reactions makes it possible to explain the abundances of nuclei

176

The Quantum World of Nuclear Physics

with mass numbers 20 < A < 32. The creation and abundances of nuclei with mass numbers 32 < A < 42 can be explained by oxygen fusion, which could occur at temperatures of about 3.6 • 109 K: 1 6 O+ 1 6

C 28Si + a, O ^ I 31 P + p, [ 31 S + n.

(7.14)

The fusion of neon and silicon allows one to explain the creation of nuclei up to nickel. In this way, one can explain the synthesis processes for the nuclei up to the iron group elements A < 65. However, the binding energy per nucleon reaches its maximum for nuclei of this group, and then begins a gradual decrease. For this reason, the iron group elements cannot play the role of a fuel; that is, fusion with energy release ceases as soon as elements of this group have been created. This explains the sufficiently large relative abundance in Nature of the nuclei having charge numbers close to that of the iron nucleus. In other words, the elements of the iron group are the "nuclear ashes" that have been formed during the fusion of stellar matter consisting of lighter nuclei. However, in Nature there exist stable nuclei with mass numbers up to A — 209, and natural radioactive nuclei up to A = 238. It should be noted that the relative abundances of nuclei of the heavy elements are very small. A typical abundance for the heavy nuclei is 1010 times less than the abundance of hydrogen! The extremely low abundances of heavy nuclei attest to the accessory nature of their synthesis processes. These processes could be reactions of neutron or proton capture by nuclei of the iron group as well as by the heavier nuclei being formed. These nucleon capture reactions could alternate with the /3-decays of nuclei. The nucleon capture processes go on until neutrons and protons are created in the stars due to burning or explosion. As soon as the reactions in which energy is released cease, the synthesis of heavy nuclei also ceases. The mechanisms of nuclear synthesis described above allow one to calculate the relative abundances of isotopes and to explain the main features of the curve shown in Fig. 7.1. 7.3

Evolution of Stars

All available data on the observation of galaxies and stars show that the Universe on a sufficiently large scale is homogeneous. That is, on the average

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177

it looks the same regardless of where and in which direction it is observed. In other words, on the scale of hundreds of millions of light years, the Universe looks as if it is homogeneously filled with billions of galaxies, the distances between which are, on the average, equal to several million light years (1 light year is « 0.95 • 1016 m — the distance traveled by light during one year). The large-scale homogeneity of the Universe leads to the cosmological principle.

In 1929, the American astronomer Hubble discovered that any two galaxies in the Universe are moving apart with velocities proportional to the distance between them (the Hubble law). For example, a galaxy 108 light years away is moving away from us with a velocity of about 2 • 106 m/s. A galaxy twice as far away is receding with twice the velocity, etc. It follows that the Universe is expanding, i.e., it is not stationary. Hubble's discovery has finally demolished the notions about the static and stable Universe that were accepted wisdom from the time of Aristotle. Considering the reverse process, we can conclude that approximately 15-20 billion years ago all galaxies (all the substance in the Universe) were gathered at one point: "the singularity". This is exactly the time of creation of the Universe by the Big Bang. Before the creation of the Universe, i.e., before the moment of the Big Bang, the concept of time was meaningless. The Big Bang is the origin of time; earlier time moments are simply not defined, because the concept of time is connected with the motion of matter. This circumstance, that the concept of time is connected with the creation of the Universe, was first pointed out by St. Augustin (354-430), the bishop of the city of Gippon in North Africa, in his work "About God's town". At present, space is filled with galaxies united into gigantic clusters comprising hundreds and thousands of galaxies. The formation of galaxies started approximately a billion years after the Big Bang, out of the mixture of hydrogen and helium that filled all space. Our galaxy consists of about 1011 stars, and the Sun is an unremarkable example of a peripheral star whose age is about 4.6 • 109 years. The synthesis of atomic nuclei takes place in stars via nuclear reactions proceeding at very high temperatures. For this reason, these processes and the explanation of the abundances of nuclei in Nature on the basis of them are closely connected with the problem of the structure and evolution of stars. The gravitational forces inside a star tend to decrease its volume, and the gas pressure inside the star counteracts this. Since these pressure and

178

The Quantum World of Nuclear Physics

temperature values are huge, atoms in stars are completely ionized; the matter inside stars, which consists of free electrons and nuclei, is a plasma (for example, at the center of the Sun the pressure is around 2 • 1010 atm, and the temperature is equal to 1.4 • 107 K). The huge pressure inside a star is sustained by the energy released during nuclear reactions. A star is in the state of equilibrium where gravitational and internal pressures are mutually balanced while nuclear reactions of hydrogen fusion occur (the proton-proton and carbon-nitrogen cycles). However, at some point, the hydrogen "fuel" will be used up. Then the internal gas pressure will start to decrease and gravitational compression will begin; this will again lead to an increase of pressure and temperature inside the star. At certain sufficiently high temperatures, new nuclear reactions initiate (e.g., helium fusion) and equilibrium will be reached again. In this case, the synthesis of new nuclei will occur. In other words, in the process of stellar evolution, the stages of burning and compression alternate. The burning process can proceed calmly, as with the Sun, or it can be accompanied by an explosion (the explosions of supernovas). The energy radiated by the Sun is approximately 3.8 • 1026 J/s. For this, about 630 tons of hydrogen must burn every second. Owing to the synthesis of helium from hydrogen, the Sun will continuously radiate this energy during next 57 billion years. Then, due to the exhaustion of its hydrogen stocks, helium burning will start up. The Sun will quickly transfer into the red giant stage, and in only ten thousand years after this it will transform into a white dwarf. After the end of burning of light nuclei leading to the creation of nuclei up to the iron group, stellar evolution enters a new stage. Due to gravitational compression, the star can transform into a black hole, white dwarf, or neutron star, or it can decay totally. The final result in this case will depend on the initial mass of the star, which is the most essential parameter characterizing its dynamics. With a stellar mass twice as large its luminosity is almost thirty-fold greater, and with a stellar mass half as large, its luminosity is thirty-fold lower. There exist stars whose masses are more than ten-fold greater that of the Sun. Such a star shines like a million Suns, but loses its energy relatively soon because the nuclear fuel in its central portion burns down rapidly. If the initial mass of a star does not exceed four solar masses (the mass of the Sun is 2 • 1030 kg), then the star is slowly compressed until the density at its center reaches 10 10 -10 U kg/m3 and its surface temperature becomes 104 K. Such a star is called a white dwarf. Its size will be comparable to

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179

that of the Earth if its mass is comparable to that of the Sun. A white dwarf is a "cold" star, since its temperature is insufficient for nuclear synthesis reactions to occur. If the initial mass of a star exceeds four solar masses, then it explodes via gravitational compression (the supernova outburst). Supernovas are stars that suddenly explode, become very bright for several months, and then gradually die away. Supernovas are explosions classified by released power of 1034 J/s. This huge energy is over two orders of magnitude greater than the released power of the next category of exploding object — the novas — and is approximately 2.5 • 107 times greater than the power released from the Sun. Supernova outbursts are the most powerful star explosions in Nature. Investigation of supernovas is important for the elucidation of the evolution of stars, of the origin of cosmic rays, and of the formation of atomic nuclei of different elements. According to numerous estimates, approximately one supernova occurs every 30 years in a giant galaxy. The giant galaxy closest to1 us is the Andromeda nebula situated 2.2 million light years away. This enormous distance complicates the observation of supernovas outbursts in this galaxy. In our galaxy a supernova happens, on average, once every 28 years. In recorded history, three very bright supernovas have been observed in our galaxy. Their appearance in 1006 and 1054 was described in Chinese chronicles, and in 1604 the Kepler star was observed. After 1604, more than 600 supernovas have been discovered, but all were distant and weak. The event of the century for astronomers and physicists was the bright outburst of a supernova in 1987 in the Big Magellanic Cloud, a young galaxy where the active creation of stars is ongoing. This supernova is relatively close to us, only 150000 light years away (the radius of our Galaxy is about 50000 light years). The supernova discovered in 1987 resulted from the explosion of a star whose radius was equal to 30-60 solar radii. Its mass was 10-30 times the solar mass. After an explosion of a supernova, an inactive stellar nucleus remains; this can become a neutron star, decay completely, or transform into a black hole. Here the deciding factor is the mass of the stellar nucleus remaining after the explosion of the supernova. If its mass is less than the critical mass, approximately 1.7 solar masses of the Sun (known as the Chandrasekhar mass), then the stellar nucleus will reach its equilibrium state when its radius will be about 104 m, and the density at its center will be about the same as the nuclear density, 1018 kg/m3, or even greater. This is a neutron star or pulsar. If the mass of the stellar nucleus is less than the critical

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mass, gravitational compression leads to the formation of a black hole. Only one of five supernovas typically gives rise to a neutron star. A neutron star is formed when, due to gravitational compression, a star of normal mass reaches such density that electrons acquiring very large energies begin to be captured by protons: p + e~ —> n + ue. A neutron star consists of the neutrons so created, which are bound together by gravitational forces. The existence of neutron stars was predicted in the mid-1930s. However, there was little hope of observing one. The discovery of neutron stars became possible after the development of radioastrophysics. In 1967, scientists from Cambridge University in England discovered a new class of celestial objects, situated beyond the solar system, that radiate periodic radio signals. These objects were called pulsars, even though they do not actually pulse but rather rotate very rapidly. For the discovery of pulsars, the English radio astronomer Hewish was awarded the Nobel Prize in Physics for 1974. Now, more than 100 pulsars are known. Each has its own radiation period in the radio frequency range. Periods of all the known pulsars lie within the range from 0.03 s to 4 s, and over time they gradually increase. There is a pulsar with the very short period of 0.033 s in the Crab nebula within the constellation Taurus, where in 1054 Chinese astronomers observed an extraordinarily bright supernova. The supposition is that this pulsar arose as a consequence of the observed supernova. At present, the Crab nebula is an object consisting mainly of gas and occupying space with a diameter of about 7.5 • 1013 m. It is expanding with a velocity of around 1.1 • 106 m/s. Pulsars are identified as neutron stars, and their periods are the periods of rotation of the neutron stars. The increase in the period of a pulsar is caused by an energy loss. The loss of rotational energy of the pulsar observed in the Crab nebula is a quantity of about the same size as the total energy radiated by this nebula. So the pulsar is the main source of the energy radiated by the huge Crab nebula. Note that pulsars were observed not only in the radio frequency range: there exist objects that periodically radiate visible light. Pulsar radiation is apparently caused by the spiral motion of relativistic charged particles in the strong magnetic field of the star, which are thrown away from the rapidly rotating equatorial regions of the pulsar. The asymmetry of the magnetic field relative to the rotation axis leads to radiation emission from certain regions of the pulsar. On Earth, the directed fluxes of radiation rotate together with the star. This phenomenon is similar to

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the rotating beam of a searchlight. The huge mass of the pulsar leads to its nearly constant period of radiation registered on Earth. To calculate time variation of the rotation period of a pulsar, one needs to know its moment of inertia. For this, one requires a sufficient model of the structure of a neutron star. It is assumed that such an object has an atmosphere several meters thick. Since the pressure in the external layer is close to zero, the most stable atoms under these conditions are those of iron, 56Fe. Their state having least energy is a crystal lattice. For this reason, the external layer of a pulsar is a "hard crust" consisting of a thin layer of iron atoms. When moving to the center of a neutron star, the density of matter continuously increases. Under the crust of iron a plasma lies — a fluid consisting of electrons and nuclei. The density of this layer is about 107 kg/m3, and its thickness is about 103 m. If the density of matter of a pulsar reaches 109-1010 kg/m3, then neutron-enriched nuclei are formed. Due to the specific conditions inside neutron stars, these nuclei cannot experience /3-decay and are, for this reason, stable. When the density exceeds 4 • 1014 kg/m3, a neutron "liquid" begins to form inside the star. With further increase in density, p > 2.5-1017 kg/m3, a continuous liquid consisting of neutrons, protons, and electrons is created, with protons forming only 4% of this liquid. Thus, inside a pulsar the major fraction of particles consists of neutrons, hence the name "neutron star". If the density of matter within a pulsar exceeds the nuclear density, then at the center of the star the creation of mesons and of quite a number of other elementary particles becomes possible. Finally, let us note that the theory of the Big Bang and the expanding Universe is based on classical physics augmented by relativity theory. Taking account of quantum effects can essentially alter this elaborate picture of the evolution of the Universe. However, the quantum theory of gravitation is not developed yet and it is difficult to say how this picture will change.

7.4

Controlled Nuclear Fusion

The stocks of subterranean coal, oil, and gas are limited. Their combustion leads to environmental pollution. For this reason, mankind urgently requires other energy sources. Typical nuclear reactors in which energy is a product of fission through a chain reaction of heavy nuclei have two essential shortcomings: first, they yield large amounts of radioactive waste whose disposal is a difficult issue; second, they lead to thermal pollution. Conse-

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quently, the idea of using controlled nuclear fusion for energy production through fusion of light nuclei appears so attractive. Nuclear fusion proceeds inside stars at very high temperatures (energies) reaching hundreds of millions of degrees. During the collision of two nuclei, fusion can occur if the approach distance is around 10~15 m, where the nuclear forces of attraction already act. For this they must surmount a Coulomb barrier of height B = Z\Z2e2 / (R\ + R2), where Z\, Z2 and R1: R2 are the charge numbers and radii of the colliding nuclei. In order for the nuclei to fuse, their energy should exceed the height of the Coulomb barrier: E > B. Since for the collision of light nuclei, the Coulomb barrier height is about B ~ 0.1 MeV, fusion reactions can proceed at temperatures T ~ 109 K. Such reactions proceeding at very high temperatures are called thermonuclear reactions. At the temperatures necessary for thermonuclear synthesis, matter is in a state of completely ionized plasma which is a mixture of nuclei and electrons. Plasma is the fourth state of matter; it exists at very high temperatures T > 104 K. Plasma becomes completely ionized at T > 107 K. The particles that compose the plasma have very large kinetic energies and tend to fly apart from one another. For this reason, the problem of plasma confinement in a certain restricted volume arises. Since the temperature is so high that no material can remain in the solid state, the problem of plasma confinement is complicated by the necessity to isolate it from the walls of the enclosure (the fusion reactor) that confines the plasma. To obtain controlled, self-sustained fusion reactions of light nuclei, termed controlled nuclear fusion (CNF), it is necessary that the plasma density and temperature are sufficient during a long interval. Then, a greater amount of energy could be produced from CNF than will be needed for the heating and confinement of the plasma. Furthermore, it is necessary to take into account the energy losses due to heat leakage from the walls of the fusion reactor, brake radiation of electrons, and other effects. One should also remember the huge plasma pressure upon the walls of the reactor. It must not exceed more than a few hundred atmospheres, which can be resisted by the construction materials. The nuclear fusion reaction intensity is defined by the confinement parameter nr (n is the plasma density, r is the time of plasma confinement). A fusion reactor will yield an energy gain, i.e., it will have a positive efficiency, upon fulfillment of a criterion established by the English physicist Lawson in 1957. For a deuterium-tritium plasma in which the following

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reaction proceeds 2 H + 3H—-> 4He

+ n,

Q = 17.6 MeV,

(7.15)

the Lawson criterion reads: TIT > 1020 s/m3,

T ~ 108 K.

(7.16)

Huge stocks of deuterium 2H exist on our planet. These can be obtained from the electrolysis of water, since the content of deuterium in water is 0.015%. Tritium 3H is unstable. It experiences the /3~-decay with ti/2 = 12.2 years, and is therefore absent in Nature. It is usually obtained from the reaction n+

6 Li—> 4 He+ 3 H,

Q = 4.8 MeV.

(7.17)

For energies E < 0.2 MeV (T < 2 • 109 K), the dependence of the probability of the reaction (7.15) for synthesizing 4He nuclei on energy is determined by the Gamov formula

wG(E) = -expl-]J-£\.

(7.18)

Here C is a constant, EQ = 2TT 2 Zi^e 4 \ij'h 2 , and fi = m-\rn.2/(jn\ + m?) where mi and m^ are the masses of deuterium and tritium. At high temperatures T, the energies of nuclei in plasma are not identical, but are distributed statistically according to the Maxwell distribution

f(E) = ^expf—j.

(7.19)

For this reason, the total probability w(E) of reaction (7.15) for the synthesis of 4He nuclei is defined by the product of the quantities WQ [E) and f{E):

v,(E) = ZexP(-yp§-if\.

(7.20)

The probability w(E) reaches its maximum at the energy Em = (£ 0 T 2 /4) 1 / 3 . The function w(E) for the reaction (7.15) is shown in Fig. 7.2. Since Eo « 0.02 MeV and T « 108 K, the quantity w(E) has its maximum at the value Em = 0.064 MeV. At E < Em the probability w(E) decreases

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w(E) |

0

-

E

F

Fig. 7.2 Probability of the reaction of synthesis of 4He nuclei as a function of energy.

with decreasing energy, and at E > Em it decreases with increasing E. At T « 108 K the probability of reaction (7.15) is maximum. To realize CNF, it is necessary to solve two problems: heating the plasma to T ~ 108 K, and confining the plasma at this temperature long enough for an appreciable part of the deuterium and tritium nuclei to merge. For plasma confinement, strong magnetic fields of special configuration (magnetic traps) are employed, and for heating the plasma one can use, for example, radio frequency methods. These are based on the interaction of different types of electromagnetic waves with plasma. Wave energy, under certain conditions, is efficiently absorbed by charged particles; this leads to heating of the plasma. A magnetic trap usually takes the form of a torus in order to prevent plasma from escaping along the magnetic field lines. Then the magnetic field lines are circles, and the charged particles must move along spirals wound around these circles. In reality, it is impossible to create a completely uniform toroidal magnetic field. For this reason, particles moving along the field lines will gradually shift (drift) perpendicular to the magnetic field, and this will result in the plasma rejection upon the outer wall of the chamber. This shortcoming can be removed in special systems called tokamaks and stellators. These devices create magnetic field configurations that suppress charged particle drift. Hence, plasmas in tokamaks and stellators can be confined for a sufficiently long time. The theory of toroidal magnetic confinement of plasmas has been developed by the Soviet physicists Tamm and Sakharov, who proposed a tokamak in 1951. In the same year,

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the American physicist Spitzer proposed a stellator. Besides tokamaks and stellators, there also exist traps of other types for plasma confinement (the adiabatic, ambipolar, and gas dynamic traps, etc.). In the early 1970s, tokamaks were built in countries carrying out investigations on CNF. These studies have shown that the time of plasma confinement in these devices increases with their sizes. Construction of very large tokamaks is complicated and expensive. For this reason, it is necessary to join the efforts of many countries. Note that along with the methodology for CNF in toroidal systems, there also exist other possibilities (the pulse laser heating of an ice grain consisting of a mixture of deuterium and tritium and having a diameter of several millimeters, plasma heating by a high-current beam of relativistic electrons, etc.). If the energy of the fusion of light nuclei is released very rapidly, then it will not be CNF — but an explosion. The hydrogen bomb is based on this principle. During a fusion explosion, the reactions (7.15) and (7.17) occur. These mutually maintain each other, keeping unchanged the numbers of neutrons and 3H nuclei. During the explosion of a hydrogen bomb, the energy of the fusion of light nuclei is released for a very short time interval of about 10~6 s. In this case, to prevent plasma from a premature escape, it is enough to use a hard shell casing for the bomb. The preliminary heating of plasma up to the ionization temperature of 107 K is accomplished by explosion of a nuclear fission bomb. The explosion of a hydrogen bomb, in contrast to CNF, is a thermonuclear reaction of nonstationary character.

Index

accelerators, 138 acceptor impurities, 34 actinium, 164 actinoids, 25 alpha-decay, 98 americium, 163 amorphous solids, 31 Andromeda nebula, 179 annihilation pressure, 170 antiferromagnetic material, 37 antimatter, 170 Aristotle, 41 astatine, 166 asymptotic freedom, 61 atomic mass unit, 69 atomic structure, 22 Balmer series, 7 baryon asymmetry, 170 baryons, 60 basic groups, 25 beam collimation, 142 berkelium, 164 beta-decay, 102 Big Bang theory, 167 black body radiation, 2 black hole, 179 Bohr magneton, 75 Bohr orbit, 22 Bohr, Niels, 5 Bose condensation, 15 Bose-Einstein statistics, 14

bosons, 14 bottom quarks, 63 breakdown voltage, 34 Breit-Wigner formula, 124 bubble chamber, 146 calcium, 109 californium, 164 carbon, 174 carbon dating, 111 carbon-nitrogen cycle, 173 caustic, 136 chain reaction, 155 Chandrasekhar mass, 179 charm quarks, 62 charmed particles, 62 Chernobyl, 159 classical physics, 1 closed shell, 23 cold emission, 31 cold neutron, 156 collider, 144 color, 60 compound nucleus, 120 conduction band, 33 conduction electron, 33 conductivity, 31 confinement, 61 confinement parameter, 182 conjugate pair, 12 conservation laws, 115 controlled fusion, 181 187

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controlled nuclear fusion, 182 Cooper pair, 39 correspondence principle, 20 cosmological principle, 177 Coulomb's law, 51 covalent bond, 27 critical regime, 160 critical size, 157 crystal, 31 Curie temperature, 37 Curie, M., 89, 164 Curie, P., 90, 164 curium, 164 damping width, 126 daughter nuclei, 93 de Broglie wavelength, 9 de Broglie, Louis, 8 decay constant, 91 deep inelastic transfer, 126 degeneracy, 29 degenerate Fermi-gas, 76 delayed fission, 153 delayed neutron, 152 detector, 17 deuterium, 183 deuteron, 47, 75, 171 diamagnet, 36 dielectrics, 31 diffraction scattering, 129 diode, 35 dipole radiation, 106 Dirac, Paul, 10 direct nuclear reaction, 124 dispersion, 12 dispersion formula, 124 domain, 38 donor impurity, 34 door-way state, 125 double nuclear system, 127 down-quarks, 60 drift, 184 duants, 141 dubnium, 165 effective multiplication coefficient,

159 Einstein, 43 Einstein, Albert, 4 einsteinium, 164 electromagnetic interaction, 51 electron configuration, 24 electron shell, 23 electrostatic unit, 51 endoergic reaction, 118 endothermic reaction, 118 energy of reaction, 118 energy spectrum, 5 energy-momentum tensor, 46 equivalent electrons, 23 europium, 163 exciton number, 125 exclusive reaction, 116 exoergic reaction, 118 exothermic reaction, 118 expanding universe, 167 fast neutron, 156 Fermi energy, 33, 78 Fermi, E., 160, 162 Fermi-Dirac statistics, 14 fermions, 14 fermium, 164 ferrimagnetic material, 37 ferromagnetic material, 37 ferrum series, 25 field concept, 51

final final channel, 116 fission,147

fission barrier,150 fission neutron, 147 fission parameter, 152 forbidden region, 32 francium, 166 Fraunhofer diffraction, 129 free electrons, 31 Fresnel diffraction, 133 fundamental interactions, 41 fusion, 181 Galileo, 8 gamma-radiation, 106

189

Index gauge bosons, 64 Geiger-Muller counter, 145 Geiger-Nuttall law, 101 general relativity, 43 generations, 53 giant dipole resonance, 85 global exclusions, 13 glory, 138 glueballs, 62 gluons, 61 gravitational interaction, 41 gravitational mass, 43 gravitational waves, 50 graviton, 50 gyromagnetic factor, 75

K-capture, 91 Kirchhoff, Gustav, 2

hadrons, 59 half-life, 92 heat of reaction, 118 Heisenberg, Werner, 10 helicity, 53 heteropolar bond, 27 hierarchy of configurations, 126 hole, 125 holes, 34 homopolar bond, 27 Hubble law, 177

Mossbauer effect, 107 magnetic substance, 36 magnetic trap, 184 mass, 42 mass defect, 69 mass excess, 69 mass number, 68 mass spectrometer, 70 Maxwell distribution, 183 Maxwell, James Clerk, 51 mean free path, 100 Meisner effect, 39 Mendeleev, 25 mendelevium, 165 mesons, 60, 62 metals, 31

inclusive reaction, 117 inert gases, 25 inertial mass, 42 infinite multiplicative system, 156 initial channel, 116 interference, 19 intermediate bosons, 54 intermediate neutron, 156 internal conversion, 106 iodine, 109 ionic bond, 27 isobaric nuclei, 69 isobars, 69 isomer, 106 isomers of shape, 154 isotones, 69 isotopes, 69, 109

Lamb shift, 16 Lamb, Willis Eugene, 16 lambda point, 40 lanthanoids, 25 Laplace determinism, 16 Laplace, Pierre-Simon, 16 lawrencium, 165 Lenz's law, 36 leptons, 53 light year, 177 linearity, 11 Lyman series, 7

Minkowski space, 48 moderator, 156 multiplication coefficient, 157 multistep nuclear reaction, 125 muon atom, 73 Neel temperature, 37 neon, 176 neptunium, 162 neutron absorber, 159 neutron star, 179 Newton, Sir Isaac, 2 nickel, 176 nitrogen, 174

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The Quantum World of Nuclear Physics

nobelium, 165 nuclear fission, 147 nuclear interaction, 57 nuclear isomery, 106 nuclear magneton, 75 nuclear optics, 127 nuclear reaction, 115 nuclear reaction mechanisms, 119 nuclear reactor, 159 nuclei, 67 nucleons, 57 nuclide, 69 octupole radiation, 106 Oersted, Hans Christian, 52 optical model, 129 optical potential, 128 overthermal neutron, 156 oxygen, 174 palladium series, 25 paramagnet, 36 parent nuclei, 93 parity, 56 partial width, 123 particle detectors, 145 Paschen series, 7 Pauli exclusion principle, 14 Pauli, Wolfgang, 14 permissible orbits, 6 phonon, 39 phosphorus, 109 photoelectric effect, 2, 4 photoemulsion, 146 photofission, 155 photon, 4 pion, 54 Planck's constant, 3 Planck, Max, 2 planetary model, 4 plasma, 182 platinum series, 25 plutonium, 162 polarization, 133 polonium, 90 pre-equilibrium nuclear reaction, 125

principle of equivalence, 43 promethium, 166 proton-proton cycle, 172 pulsar, 179 quadrupole radiation, 106 quanta, 3 quantization, 9 quantum chromodynamics, 61 quantum electrodynamics, 52 quantum measurement irreversiblity of, 18 quark, 59 quasifission, 127 quasiparticle, 39, 87 radio frequency methods, 184 radioactivity, 89, 90 radium, 90 rainbow ghost, 138 rainbow maximum, 137 rainbow scattering, 135 rare-earth elements, 25 Rayleigh, Lord, 2 Rayleigh-Jeans law, 2 reactivity, 159 recombination, 34 rectifier, 35 relict radiation, 169 residual interaction, 87 residual magnetism, 38 rest energy, 44 rule of intervals, 84 Rutherford, Ernst, 4 rutherfordium, 165 saddle point state, 152 saturation, 59 scale invariance, 8 scaling parameter, 135 scattering, 67 Schrodinger equation, 6, 11, 15 Schrodinger, Erwin, 10 scintillation counter, 145 secondary neutron, 152 secular equation, 94

191

Index

self-consistent field, 23 semiconductor, 34 semiconductors, 31 semimetals, 34 shell model, 76 silicon, 176 similarity, 8 slow neutron, 156 solar curve, 167 Sommerfeld parameter, 134 spark chamber, 146 spatial homogeneity, 8 spatial isotropy, 8 special relativity, 43 spherical harmonic, 28 spin, 14 spinthariscope, 145 spontaneous fission, 91 stars, 176 stellator, 184 strange particles, 62 strange quarks, 62 stripping reaction, 132 strong interaction, 57 strontium, 109 subcritical regime, 160 superconductivity, 38 supercritical regime, 160 superfluidity, 38, 86 supernova outburst, 179 superposition principle, 11 synchrophasotrons, 143 synchrotron, 142 synthesized elements, 162 tachyons, 49 technetium, 165, 167 temporal homogeneity, 8 thermal neutron, 156 thermonuclear reactions, 182 thin structure, 101 threshold energy, 118 tokamak, 184 top quarks, 63 trajectories, absence of, 13 transistor, 36

transition groups, 25 transuranium elements, 162 tritium, 183 tunnel effect, 31 tunneling passage, 100 ultracold neutron, 156 ultraviolet catastrophe, 2 uncertainty relation, 12 under barrier passage, 100 up-quarks, 60 uranium, 89, 147-149, 157, 162 valence band, 33 Van de Graaff generator, 140 Van der Waals forces, 64 vector mesons, 59, 62 virtual photons, 53 wave function, 10 symmetry of, 14 wave-particle duality, 9 weak interaction, 53 Weizsacker's formula, 71 white dwarf, 178 Wien's law, 2 Wien, Wilhelm, 2 Wilson chamber, 145 ylem, 168 yrast-states, 85 zero-point vibrational energy, 15