Basdevant J.-L., Rich J., Spiro M. Fundamentals in Nuclear Physics: From Nuclear Structure to Cosmology

Подождите немного. Документ загружается.

XII Contents

6. Fission .................................................... 285

6.1 Nuclear energy......................................... 285

6.2 Fission products........................................ 287

6.3 Fission mechanism, ﬁssion barrier ........................ 290

6.4 Fissile materials and fertile materials...................... 295

6.5 Chain reactions ........................................ 297

6.6 Moderators, neutron thermalization ....................... 299

6.7 Neutron transport in matter ............................. 301

6.7.1 The transport equation in a simple uniform spherically

symmetric medium ............................... 302

6.7.2 The Lorentz equation ............................. 305

6.7.3 Divergence, critical mass .......................... 306

6.8 Nuclear reactors ....................................... 308

6.8.1 Thermal reactors................................. 309

6.8.2 Fast neutron reactors ............................ 316

6.8.3 Accelerator-coupled sub-critical reactors ............. 319

6.8.4 Treatmentand re-treatmentofnuclearfuel .......... 322

6.9 The Oklo prehistoric nuclear reactor ...................... 323

6.10 Bibliography........................................... 326

Exercises .................................................. 327

7. Fusion .................................................... 329

7.1 Fusionreactions........................................ 330

7.1.1 The Coulomb barrier ............................. 331

7.1.2 Reaction rate in amedium ........................ 335

7.1.3 Resonant reaction rates ........................... 338

7.2 Reactor performance criteria............................. 339

7.3 Magnetic conﬁnement ................................... 342

7.4 Inertial conﬁnement by lasers ............................ 346

7.5 Bibliography........................................... 349

Exercises .................................................. 349

8. Nuclear Astrophysics ..................................... 351

8.1 Stellar Structure ....................................... 351

8.1.1 Classical stars.................................... 352

8.1.2 Degenerate stars ................................. 359

8.2 Nuclear burning stages in stars........................... 363

8.2.1 Hydrogen burning ................................ 363

8.2.2 Helium burning .................................. 366

8.2.3 Advanced nuclear-burning stages ................... 369

8.2.4 Core-collapse .................................... 370

8.3 Stellar nucleosynthesis .................................. 373

8.3.1 Solar-system abundances .......................... 373

8.3.2 Production of A<60nuclei ....................... 376

8.3.3 A>60: the s-, r- and p-processes ................... 376

Contents XIII

8.4 Nuclear astronomy ..................................... 381

8.4.1 Solar Neutrinos .................................. 382

8.4.2 Supernova neutrinos .............................. 390

8.4.3 γ-astronomy ..................................... 392

Exercises .................................................. 394

9. Nuclear Cosmology ....................................... 397

9.1 The Universe today..................................... 399

9.1.1 The visible Universe .............................. 400

9.1.2 Baryons......................................... 401

9.1.3 Cold dark matter ................................. 401

9.1.4 Photons......................................... 402

9.1.5 Neutrinos ....................................... 403

9.1.6 The vacuum ..................................... 404

9.2 The expansion of theUniverse ........................... 405

9.2.1 The scale factor a(t) .............................. 407

9.3 Gravitation and the Friedmannequation .................. 410

9.4 High-redshift supernovae and the vacuum energy ........... 416

9.5 Reaction rates in theearlyUniverse ...................... 416

9.6 Electrons, positronsandneutrinos........................ 420

9.7 Cosmological nucleosynthesis............................. 424

9.8 Wimps................................................ 434

Exercises .................................................. 436

A. Relativistic kinematics .................................... 441

B. Accelerators .............................................. 445

C. Time-dependent perturbation theory ..................... 451

C.0.1 Transition rates between two states ................. 451

C.0.2 Limiting forms of the delta function ................ 453

D. Neutron transport ........................................ 455

D.0.3 TheBoltzmanntransportequation ................. 455

D.0.4 TheLorentzequation ............................. 456

E. Solutions and Hints for Selected Exercises ................ 461

F. Tables of numerical values ................................ 469

G. Table of Nuclei ........................................... 471

References .................................................... 507

Index ......................................................... 511

Introduction

Nuclear physics started by accident in 1896 with the discovery of radioactiv-

ity by Henri Becquerel who noticed that photographic plates were blackened

when placed next to uranium-sulﬁde crystals. He, like Poincar´eandmany

others, found the phenomenon of “Becquerel rays” fascinating, but he nev-

ertheless lost interest in the subject within the following six months. We can

forgive him for failing to anticipate the enormous amount of fundamental and

applied physics that would follow from his discovery.

In 1903, the third Nobel prize for Physics was awarded to Becquerel, and

to Pierre and Marie Curie. While Becquerel discovered radioactivity, it was

the Curies who elucidated many of its characteristics by chemically isolating

the diﬀerent radioactive elements produced in the decay of uranium. Ernest

Rutherford, became interested in 1899 and performed a series of brilliant

experiments leading up to his discovery in 1911 of the nucleus itself. Arguably

the founder of nuclear physics, he was, ironically, awarded the Nobel prize in

Chemistry in 1908.

It can be argued, however, that the ﬁrst scientists to observe and study

radioactive phenomena were Tycho Brahe and his student Johannes Kepler.

They had the luck in 1572 (Brahe) and in 1603 (Kepler) to observe bright

stellae novae, i.e. new stars. Such supernovae are now believed to be ex-

plosions of old stars at the end of their normal lives.

The post-explosion

energy source of supernovae is the decay of radioactive nickel (

Ni, half-life

6.077 days) and then cobalt (

Co, half-life 77.27 days). Brahe and Kepler

observed that the luminosity of their supernovae, shown in Fig. 0.1, decreased

with time at a rate that we now know is determined by the nuclear lifetimes.

Like Becquerel, Brahe and Kepler did not realize the importance of what

they had seen. In fact, the importance of supernovae dwarfs that of radioactiv-

ity because they are the culminating events of the process of nucleosynthesis.

This process starts in the cosmological “big bang” where protons and neu-

tron present in the primordial soup condense to form hydrogen and helium.

Later, when stars are formed the hydrogen and helium are processed through

Such events are extremely rare. In the last millennium, only ﬁve of them have

been seen in our galaxy, the Milky Way. The last supernova visible to the naked

eye was seen on February 23, 1987, in the Milky Way’s neighbor, the Large

Magellanic Cloud. The neutrinos and γ-rays emitted by this supernova were

observed on Earth, starting the subject of extra-solar nuclear astronomy.

2 Introduction

100

200

300

400

−4

−2

exp(−t / 111days)

log(flux)

t (days)

Fig. 0.1. The luminosity of Kepler’s supernova as a function of time, as recon-

structed in [4]. Open circles are European measurements and ﬁlled circles are Ko-

rean measurements. Astronomers at the time measured the evolution of the lumi-

nosity of the supernovae by comparing it to known stars and planets. It has been

possible to determine the positions of planets at the time when they were observed,

and, with the notebooks, to reconstruct the luminosity curves. The superimposed

curve shows the rate of

Co decay using the laboratory-measured half life. The

vertical scale gives the visual magnitude V of the star, proportional to the loga-

rithm of the photon ﬂux. V = 0 corresponds to a bright star, while V =5isthe

dimmest star that can be observed with the naked eye.

nuclear reactions into heavier elements. These elements are ejected into the

interstellar medium by supernovae. Later, some of this matter condenses to

form new stellar systems, now sometimes containing habitable planets made

of the products of stellar nucleosynthesis.

Nuclear physics has allowed us to understand in considerable quantitative

detail the process by which elements are formed and what determines their

relative abundances. The distribution of nuclear abundances in the Solar

System is shown in Fig. 8.9. Most ordinary matter

is in the form of hydrogen

(∼ 75% by mass) and helium (∼ 25%). About 2% of the solar system material

is in heavy elements, especially carbon, oxygen and iron. To the extent that

nuclear physics explains this distribution, it allows us to understand why we

We leave the question of the nature of the unknown cosmological “dark matter”

for Chap. 9.

Introduction 3

live near a hydrogen burning star and are made primarily of elements like

hydrogen, carbon and oxygen.

A particularly fascinating result of the theory of nucleosynthesis is that

the observed mix of elements is due to a number of delicate inequalities of

nuclear and particle physics. Among these are

• The neutron is slightly heavier than the proton;

• The neutron–proton system has only one bound state while the neutron–

neutron and proton–proton systems have none;

• The

Be nucleus is slightly heavier than two

He nuclei and the second

excited state of

C is slightly heavier than three

He nuclei.

We will see in Chaps. 8 and 9 that modifying any of these conditions would

result in a radically diﬀerent distribution of elements. For instance, making

the proton heavier than the neutron would make ordinary hydrogen unstable

and none would survive the primordial epoch of the Universe.

The extreme sensitivity of nucleosynthesis to nuclear masses has gener-

ated a considerable amount of controversy about its interpretation. It hinges

upon whether nuclear masses are ﬁxed by the fundamental laws of physics

or are accidental, perhaps taking on diﬀerent values in inaccessible regions

of the Universe. Nuclear masses depend on the strengths of the forces be-

tween neutrons and protons, and we do not now know whether the strengths

are uniquely determined by fundamental physics. If they are not, we must

consider the possibility that the masses in “our part of the Universe” are as

observed because other masses give mixes of elements that are less likely to

provide environments leading to intelligent observers. Whether or not such

“weak-anthropic selection” had a role in determining the observed nuclear

and particle physics is a question that is appealing to some, infuriating to

others. Resolving the question will require better understanding of the origin

of observed physical laws.

Some history

The history of nuclear physics can be divided into three periods. The ﬁrst be-

gins with the discovery radioactivity of the nucleus and ends in 1939 with the

discovery of ﬁssion. During this period, the basic components (protons and

neutrons) of the nucleus were discovered as well as the quantum law governing

their behavior. The second period from 1947 to 1969 saw the development of

nuclear spectroscopy and of nuclear models. Finally, the emergence of a mi-

croscopic unifying theory starting in the 1960s allowed one to understand the

structure and behavior of protons and neutrons in terms of the fundamental

interactions of their constituent particles, quarks and gluons. This period also

saw the identiﬁcation of subtle non-classical mechanisms in nuclear structure.

Since the 1940s, nuclear physics has seen important developments, but

most practical applications and their simple theoretical explanations were

4 Introduction

in place by the mid 1950s. This book is mostly concerned with the simple

models from the early period of nuclear physics and to their application in

energy production, astrophysics and cosmology.

The main stages of this ﬁrst period of nuclear physics are the following

[5, 6].

• 1868 Mendeleev’s periodic classiﬁcation of the elements.

• 1895 Discovery of X-rays by Roentgen.

• 1896 Discovery of radioactivity by Becquerel.

• 1897 Identiﬁcation of the electron by J.J. Thomson.

• 1898 Separation of the elements polonium and radium by Pierre and Marie

Curie.

• 1908 Measurement of the charge +2 of the α particle by Geiger and

Rutherford.

• 1911 Discovery of the nucleus by Rutherford; “planetary” model of the

atom.

• 1913 Theory of atomic spectra by Niels Bohr.

• 1914 Measurement of the mass of the α particle by Robinson and Ruther-

ford.

• 1924–1928 Quantum theory (de Broglie, Schr¨odinger, Heisenberg, Born,

Dirac).

• 1928 Theory of barrier penetration by quantum tunneling, application to

α radioactivity, by Gamow, Gurney and Condon.

• 1929–1932 First nuclear reactions with the electrostatic accelerator of

Cockcroft and Walton and the cyclotron of Lawrence.

• 1930–1933 Neutrino proposed by Pauli and named by Fermi in his theory

of beta decay.

• 1932 Identiﬁcation of the neutron by Chadwick.

• 1934 Discovery of artiﬁcial radioactivity by F. and I. Joliot-Curie.

• 1934 Discovery of neutron capture by Fermi.

• 1935 Liquid-drop model and compound-nucleus model of N. Bohr.

• 1935 Semi-empirical mass formula of Bethe and Weizs¨acker.

• 1938 Discovery of ﬁssion by Hahn and Strassman.

• 1939 Theoretical interpretation of ﬁssion by Meitner, Bohr and Wheeler.

To these fundamental discoveries we should add the practical applications

of nuclear physics. Apart from nuclear energy production beginning with

Fermi’s construction of the ﬁrst ﬁssion reactor in 1942, the most important

are astrophysical and cosmological. Among them are

• 1938 Bethe and Weizs¨acker propose that stellar energy comes from ther-

monuclear fusion reactions.

• 1946 Gamow develops the theory of cosmological nucleosynthesis.

• 1953 Salpeter discovers the fundamental solar fusion reaction of two pro-

tons into deuteron.

Introduction 5

• 1957 Theory of stellar nucleosynthesis by Burbidge, Burbidge, Fowler and

Hoyle.

• 1960– Detection of solar neutrinos

• 1987 Detection of neutrinos and γ-rays from the supernova SN1987a.

The scope of nuclear physics

In one century, nuclear physics has found an incredible number of applica-

tions and connections with other ﬁelds. In the most narrow sense, it is only

concerned with bound systems of protons and neutrons. From the beginning

however, progress in the study of such systems was possible only because of

progress in the understanding of other particles: electrons, positrons, neutri-

nos and, eventually quarks and gluons. In fact, we now have a more complete

theory for the physics of these “elementary particles” than for nuclei as such.

A nuclear species is characterized by its number of protons Z and number

of neutrons N. There are thousands of combinations of N and Z that lead

to nuclei that are suﬃciently long-lived to be studied in the laboratory. They

are tabulated in Appendix G. The large number of possible combinations of

neutrons and protons is to be compared with the only 100 or so elements

characterized simply by Z.

A “map” of the world of nuclei is shown in Fig. 0.2. Most nuclei are

unstable, i.e. radioactive. Generally, for each A = N + Z thereisonlyoneor

two combinations of (N, Z) suﬃciently long-lived to be naturally present on

Earth in signiﬁcant quantities. These nuclei are the black squares in Fig. 0.2

and deﬁne the bottom of the valley of stability in the ﬁgure.

One important line of nuclear research is to create new nuclei, both high

up on the sides of the valley and, especially, super-heavy nuclei beyond the

heaviest now known with A = 292 and Z = 116. Phenomenological arguments

suggest that there exists an “island of stability” near Z = 114 and 126 with

nuclei that may be suﬃciently long-lived to have practical applications.

The physics of nuclei as such has been a very active domain of research in

the last twenty years owing to the construction of new machines, the heavy ion

accelerators of Berkeley, Caen (GANIL), Darmstadt and Dubna. The physics

of atomic nuclei is in itself a domain of fundamental research.Itconstitutes

a true many-body problem, where the number of constituents is too large

for exact computer calculations, but too small for applying the methods of

statistical physics. In heavy ion collisions, one discovers subtle eﬀects such as

local superﬂuidity in the head-on collision of two heavy ions.

This is of course a false paradox; the structure of DNA derives, in principle, com-

pletely from the Schr¨odinger equation and Quantum Electrodynamics. However

it is not studied it that spirit.

Diﬀerent isotopes of a same element have essentially the same chemical proper-

ties.

6 Introduction

Z=20

Z=28 Z=28

Z=50 Z=50

Z=82 Z=82

N=20

N=28N=28

N=50N=50

N=82N=82

N=126N=126

Z=20

nucleon emission

spontaneous fission

last neutron unbound

half−life> 10

proton

unbound

last

decay

A=100

A=200

Fig. 0.2. The nuclei. The black squares are long-lived nuclei present on Earth.

Unbound combinations of (N,Z) lie outside the lines marked “last proton/neutron

unbound.” Most other nuclei β-decay or α-decay to long-lived nuclei.

Introduction 7

Nuclear physics has had an important by-product in elementary particle

physics and the discovery of the elementary constituents of matter, quarks

and leptons, and their interactions. Nuclear physics is essential to the under-

standing of the structure and the origin of the world in which we live. The

birth of nuclear astrophysics is a decisive step forward in astronomy and in

cosmology. In addition, nuclear technologies play an important role in mod-

ern society. We will see several examples. This book is intended to be a ﬁrst

introduction to a large variety of scientiﬁc and technological ﬁelds. It can be

a ﬁrst step in the study of the vast ﬁeld of nuclear physics.

Bibliography

On the history of nuclear and particle physics:

1. Abraham Pais Inward Bound

, Oxford University Press, Oxford, 1986.

2. Emilio Segr´e,FromXraystoQuarks

, Freeman, San Francisco, 1980.

Introductory textbooks on nuclear physics

1. B. Povh, K. Rith, C. Scholz and F. Zetsche, Particles and Nuclei

, Spring-

er, Berlin, 2000.

2. W.N. Cottingham and D.A.Greenwood, Nuclear Physics

, Cambridge Uni-

versity Press, Cambridge, 2002.

3. P.E. Hodgson, E. Gadioli and E. Gadioli Erba, Introductory Nuclear

Physics, Clarendon Press, Oxford, 1997.

4. Harald Enge, Introduction to Nuclear Physics

, Addison-Wesley, Read-

ing, 1966.

5. J. S. Lilley, Nuclear Physics

, Wiley, Chichester, 2001.

Advanced textbooks on nuclear physics

1. Nuclear Structure A. Bohr and B. Mottelson, Benjamin, New York, 1969.

2. M.A. Preston and R.K. Bhaduri, Structure of the Nucleus

, Addison-

Wesley, Reading, 1975.

3. S.M. Wong, Nuclear Physics

, John Wiley, New York, 1998.

4. J.D. Walecka,Theoretical Nuclear and Subnuclear Physics

, Oxford Uni-

versity Press, Oxford, 1995.

5. A. de Shalit and H. Feshbach,Theoretical Nuclear Physics

, Wiley, New

York, 1974.

6. D.M. Brink, Nuclear Forces

, Pergamon Press, Oxford, 1965.

7. J.M. Blatt and V.F. Weisskopf, Theoretical Nuclear Physics

, John Wiley

and Sons, New-York, 1963.

1. Basic concepts in nuclear physics

In this chapter, we will discuss the basic ingredients of nuclear physics. Section

1.1 introduces the elementary particles that form nuclei and participate in

nuclear reactions. Sections 1.2 shows how two of these particles, protons and

neutrons, combine to form nuclei. The essential results will be that nuclei

have volumes roughly proportional to the number of nucleons, ∼ 7fm

per

nucleon and that they have binding energies that are of order 8 MeV per

nucleon. In Sect. 1.3 we show how nuclei are described as quantum states. The

forces responsible for binding nucleons are described in Sect. 1.4. Section 1.5

discusses how nuclei can be transformed through nuclear reactions while Sect.

1.6 discusses the important conservation laws that constrain these reactions

and how these laws arise in quantum mechanics. Section 1.7 describes the

isospin symmetry of these forces. Finally, Sect. 1.8 discusses nuclear shapes.

1.1 Nucleons and leptons

Atomic nuclei are quantum bound states of particles called nucleons of which

there are two types, the positively charged proton and the uncharged neutron.

The two nucleons have similar masses:

= 939.56 MeV m

= 938.27 MeV , (1.1)

i.e. a mass diﬀerence of order one part per thousand

− m

=1.29 MeV . (1.2)

For nuclear physics, the mass diﬀerence is much more important than the

masses themselves which in many applications are considered to be “inﬁ-

nite.” Also of great phenomenological importance is the fact that this mass

diﬀerence is of the same order as the electron mass

=0.511 MeV . (1.3)

Nucleons and electrons are spin 1/2 fermions meaning that their intrinsic

angular momentum projected on an arbitrary direction can take on only

thevaluesof±¯h/2. Having spin 1/2, they must satisfy the Pauli exclusion

principle that prevents two identical particles (protons, neutrons or electrons)