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Home , James Chadwick

Nuclear

Hans A. Bethe Floyd R. Newman, Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14853

[S0034-6861(99)04302-0]

I. HISTORICAL prising because the electric charges in the of the foil would deflect the ␣ particles by small angles and started in 1894 with the discovery of multiple deflections were expected. But to their surprise, the radioactivity of by A. H. Becquerel. Marie a few ␣ particles came back on the front side of the foil, and Pierre investigated this phenomenon in detail: and their number increased with increasing atomic to their astonishment they found that raw uranium ore weight of the material in the foil. Definitive experiments was far more radioactive than the refined uranium from with a gold foil were made by Geiger and Marsden in the chemist’s store. By chemical methods, they could 1909. separate (and name) several new elements from the ore Rutherford in 1911 concluded that this backward scat- which were intensely radioactive: (Zϭ88), tering could not come about by multiple small-angle (Zϭ84), a gas they called emanation (Z scatterings. Instead, there must also occasionally be ϭ86) (radon), and even a form of lead (Zϭ82). single deflections by a large angle. These could only be , at McGill University in Montreal, produced by a big charge concentrated somewhere in studied the radiation from these substances. He found a the . Thus he conceived the nuclear atom: each strongly ionizing component which he called ␣ rays, and atom has a nucleus with a positive charge equal to the a weakly ionizing one, ␤ rays, which were more pen- sum of the charges of all the electrons in the atom. The etrating than the ␣ rays. In a magnetic field, the ␣ rays nuclear charge Ze increases with the atomic weight. showed positive charge, and a charge-to- ratio cor- Rutherford had good experimental arguments for his 4 responding to He. The ␤ rays had negative charge and concept. But when in 1913 found the theory were apparently electrons. Later, a still more penetrat- of the hydrogen spectrum, Rutherford declared, ‘‘Now I ing, uncharged component was found, ␥ rays. finally believe my nuclear atom.’’ Rutherford and F. Soddy, in 1903, found that after The scattering of fast ␣ particles by He indicated also emission of an ␣ ray, an element of Z a stronger force than the electrostatic repulsion of the was transformed into another element, of atomic num- two He nuclei, the first indication of the strong nuclear ber ZϪ2. (They did not yet have the concept of atomic force. Rutherford and his collaborators decided that this number, but they knew from chemistry the place of an must be the force that holds ␣ particles inside the element in the periodic system.) After ␤-ray emission, Z nucleus and thus was attractive. From many scattering was transformed into Zϩ1, so the dream of alchemists experiments done over a decade they concluded that had become true. this attractive force was confined to a radius It was known that thorium (Zϭ90, Aϭ232) also was Rϭ1.2ϫ10Ϫ13A1/3 cm, (1) radioactive, also decayed into radium, radon, polonium and lead, but obviously had different radioactive behav- which may be considered to be the nuclear radius. This ior from the decay products of uranium (Zϭ92, A result is remarkably close to the modern value. The vol- ϭ238). Thus there existed two or more forms of the ume of the , according to Eq. (1), is propor- same chemical element having different atomic weights tional to the number of particles in it. and different radioactive properties (lifetimes) but the When ␣ particles were sent through material of low same chemical properties. Soddy called these . atomic weight, particles were emitted of range greater Rutherford continued his research at , and than the original ␣ particle. These were interpreted by many mature collaborators came to him. H. Geiger and Rutherford and James Chadwick as . They had J. M. Nuttall, in 1911, found that the energy of the emit- observed the disintegration of nuclei, from ted ␣ particles, measured by their range, was correlated up to potassium. with the lifetime of the parent substance: the lifetime gave the first theoretical expla- decreased very rapidly (exponentially) with increasing nation of natural radioactivity. In 1928 George Gamow, ␣-particle energy. and simultaneously K. W. Gurney and E. U. Condon, By an ingenious arrangement of two boxes inside each discovered that the potential barrier between a nucleus other, Rutherford proved that the ␣ particles really were and an ␣ particle could be penetrated by the ␣ particle He atoms: they gave the He spectrum in an electric dis- coming from the inside, and that the rate of penetration charge. depended exponentially on the height and width of the Rutherford in 1906 and Geiger in 1908 put thin solid barrier. This explained the Geiger-Nuttall law that the foils in the path of a beam of ␣ particles. On the far side lifetime of ␣-radioactive nuclei decreases enormously as of the foil, the beam was spread out in angle—not sur- the energy of the ␣ particle increases.

S6 Reviews of , Vol. 71, No. 2, Centenary 1999 0034-6861/99/71(2)/6(10)/$17.00 ©1999 The American Physical Society Hans A. Bethe: Nuclear physics S7

On the basis of this theory, Gamow predicted that These paradoxes were resolved in 1932 when Chad- protons of relatively low energy, less than one million wick discovered the . Now one could assume electron volts, should be able to penetrate into light nu- that the nucleus consisted of Z protons and AϪZ neu- clei, such as Li, Be, and B, and disintegrate them. When trons. Thus a nucleus of mass A would have Bose Gamow visited , he encouraged the experi- (Fermi) statistics if A was even (odd) which cleared up menters at the to build accelera- the 14N paradox, provided that the neutron obeyed tors of relatively modest voltage, less than one million 1 Fermi statistics and had 2, as it was later shown to volts. Such accelerators were built by M. L. E. Oliphant have. on the one hand, and J. D. Cockcroft and E. T. S. Wal- Chadwick already showed experimentally that the ton on the other. mass of the neutron was close to that of the , so By 1930, when I spent a semester at the Cavendish, the minimum momentum of 1015 erg/c has to be com- the Rutherford group understood ␣ particles very well. pared with The penetrating ␥ rays, uncharged, were interpreted as Ϫ24 10 Ϫ14 high-frequency electromagnetic radiation, emitted by a Mncϭ1.7ϫ10 ϫ3ϫ10 ϭ5ϫ10 erg/c, (4) nucleus after an ␣ ray: the ␣ particle had left the nucleus Ϫ15 where Mn is the mass of the . Pminϭ10 is in an excited state, and the transition to the ground state small compared to this, so the wave function of neutron was accomplished by emission of the ␥ ray. and proton fits comfortably into the nucleus. The problem was with ␤ rays. Chadwick showed in The had been very dramatic. 1914 that they had a continuous spectrum, and this was and H. Becker found that Be, bom- repeatedly confirmed. Rutherford, Chadwick, and C. D. barded by ␣ particles, emitted very penetrating rays that Ellis, in their book on radioactivity in 1930, were baffled. they interpreted as ␥ rays. Curie and Joliot exposed par- Bohr was willing to give up in affin to these rays, and showed that protons of high en- this instance. Pauli violently objected to Bohr’s idea, and ergy were ejected from the paraffin. If the rays were suggested in 1931 and again in 1933 that together with actually ␥ rays, they needed to have extremely high en- the electron (␤-particle) a neutral particle was emitted, ergies, of order 30 MeV. Chadwick had dreamed about of such high penetrating power that it had never been for a decade, and got the idea that here at last observed. This particle was named the by was his beloved neutron. Fermi, ‘‘the small neutral one.’’ Chadwick systematically exposed various materials to the penetrating radiation, and measured the energy of II. THE NEUTRON AND THE DEUTERON the recoil atoms. Within the one month of February 1932 he found the answer: indeed the radiation consisted In 1930, when I first went to Cambridge, , of particles of the mass of a proton, they were neutral, nuclear physics was in a peculiar situation: a lot of ex- hence neutrons. A beautiful example of systematic ex- perimental evidence had been accumulated, but there perimentation. was essentially no theoretical understanding. The Chadwick wondered for over a year: was the neutron nucleus was supposed to be composed of protons and an elementary particle, like the proton, or was it an ex- Ϫ12 electrons, and its radius was supposed to be Ͻ10 cm. cessively strongly bound combination of proton and The corresponding momentum, according to quantum electron? In the latter case, he argued, its mass should mechanics, was be less than that of the hydrogen atom, because of the ប 10Ϫ27 binding energy. The answer was only obtained when PϾP ϭ ϭ ϭ10Ϫ15 erg/c, (2) Chadwick and Goldhaber disintegrated the deuteron by min R Ϫ12 10 ␥ rays (see below): the mass of the neutron was 0.8 MeV while from the mass me of the electron greater than that of the H atom. So, Chadwick decided, Ϫ17 the neutron must be an elementary particle of its own. mecϭ3ϫ10 erg/c. (3) 1 If the neutron was an elementary particle of spin 2, Thus the electrons had to be highly relativistic. How obeying Fermi statistics, the problem of spin and statis- could such an electron be retained in the nucleus, in- tics of 14N was solved. And one no longer needed to deed, how could an electron wave function be fitted into squeeze electrons into the too-small space of a nucleus. the nucleus? Accordingly, and Iwanenko inde- Further troubles arose with spin and statistics: a pendently in 1933 proposed that a nucleus consists of nucleus was supposed to contain A protons to make the neutrons and protons. These are two possible states of a correct atomic weight, and AϪZ electrons to give the more general particle, the nucleon. To emphasize this, net charge Z. The total number of particles was 2A Heisenberg introduced the concept of the isotopic spin 1 1 ϪZ, an odd number if Z was odd. Each proton and ␶z the proton having ␶zϭϩ 2 and the neutron ␶zϭϪ 2 . electron was known to obey Fermi statistics, hence a This concept has proved most useful. nucleus of odd Z should also obey Fermi statistics. But Before the discovery of the neutron, in 1931 Harold band spectra of , N2, showed that the N nucleus, Urey discovered heavy hydrogen, of atomic weight 2. Its of Zϭ7, obeyed Bose statistics. Similarly, proton and nucleus, the deuteron, obviously consists of one proton 1 electron had spin 2, so the nitrogen nucleus should have and one neutron, and is the simplest composite nucleus. half-integral spin, but experimentally its spin was 1. In 1933, Chadwick and Goldhaber succeeded in disinte-

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 S8 Hans A. Bethe: Nuclear physics grating the deuteron by ␥ rays of energy 2.62 MeV, and III. THE LIQUID DROP MODEL measuring the energy of the proton resulting from the disintegration. In this way, the binding energy of the A. Energy deuteron was determined to be 2.22 MeV. The most conspicuous feature of nuclei is that their This binding energy is very small compared with that binding energy is nearly proportional to A, the number of 4He, 28.5 MeV, which was interpreted as meaning of nucleons in the nucleus. Thus the binding per particle that the attraction between two nucleons has very short is nearly constant, as it is for condensed matter. This is range and great depth. The wave function of the deu- in contrast to electrons in an atom: the binding of a 1S teron outside the potential well is then determined sim- electron increases as Z2. ply by the binding energy ␧.Itis The volume of a nucleus, according to Eq. (1), is also ␺ϭexp͑Ϫ␣r͒/r, (5) proportional to A. This and the binding energy are the basis of the liquid drop model of the nucleus, used espe- ␣ϭ͑M␧͒1/2/ប, (6) cially by Niels Bohr: the nucleus is conceived as filling a with M the mass of a nucleon. compact volume, spherical or other shape, and its en- The scattering of neutrons by protons at moderate en- ergy is the sum of an attractive term proportional to the ergy can be similarly determined, but one has to take volume, a repulsive term proportional to the surface, into account that the spins of the two nucleons may be and another term due to the mutual electric repulsion of either parallel (total Sϭ1) or antiparallel (Sϭ0). The the positively charged protons. In the volume energy, spin of the deuteron is 1. The Sϭ0 state is not bound. there is also a positive term proportional to (NϪZ)2 The scattering, up to at least 10 MeV, can be described ϭ(AϪ2Z)2 because the attraction between proton and by two parameters for each value of S, the scattering neutron is stronger than between two like particles. Fi- length and the effective range r0 . The phase shift for nally, there is a pairing energy: two like particles tend to Lϭ0 is given by go into the same quantum state, thus decreasing the en- ergy of the nucleus. A combination of these terms leads 1 1 k cot ␦ϭϪ ϩ k2r , (7) to the Weizsa¨cker semi-empirical formula a 2 0 EϭϪa Aϩa A2/3ϩa Z2AϪ1/3 where k is the wave number in the center-of-mass sys- 1 2 3 2 Ϫ1 Ϫ3/4 tem, ␦ the phase shift, a the scattering length, and r0 the ϩa4͑AϪ2Z͒ A ϩ␭a5A . (9) effective range. Experiments on neutron-proton scatter- Over the years, the parameters a ,...,a have been ing result in 1 5 determined. Green (1954) gives these values (in MeV): atϭ5.39 fm, rotϭ1.72 fm, a1ϭ15.75, a2ϭ17.8, asϭϪ23.7 fm, rosϭ2.73 fm, (8) a3ϭ0.710, a4ϭ23.7, where t and s designate the triplet and singlet Lϭ0 3 1 states, S and S. The experiments at low energy, up to a5ϭ34. (10) about 10 MeV, cannot give any information on the The factor ␭ is ϩ1ifZand NϭAϪZ are both odd, ␭ shape of the potential. The contribution of LϾ0 is very ϭϪ1 if they are both even, and ␭ϭ0ifAis odd. Many small for EϽ10 MeV, because of the short range of more accurate expressions have been given. nuclear forces. For small A, the symmetry term (N Very accurate experiments were done in the 1930s on ϪZ)2 puts the most stable nucleus at NϭZ. For larger the scattering of protons by protons, especially by Tuve A Z and collaborators at the Carnegie Institution of Wash- , the Coulomb term shifts the energy minimum to ington, D.C., and by R. G. Herb et al. at the University ϽA/2. of Wisconsin. The theoretical interpretation was mostly Among very light nuclei, the energy is lowest for done by Breit and collaborators. The system of two pro- those which may be considered multiples of the ␣ par- 12 16 20 24 28 32 40 tons, at orbital momentum Lϭ0, can exist only in the ticle, such as C, O, Ne, Mg, Si, S, Ca. For 56 56 state of total spin Sϭ0. The phase shift is the shift rela- Aϭ56, Ni (Zϭ28) still has strong binding but Fe tive to a pure Coulomb field. The scattering length re- (Zϭ26) is more strongly bound. Beyond Aϭ56, the sulting from the analysis is close to that of the 1S state preference for multiples of the ␣ particle ceases. of the proton-neutron system. This is the most direct For nearly all nuclei, there is preference for even Z evidence for charge independence of nuclear forces. and even N. This is because a pair of neutrons (or pro- There is, however, a slight difference: the proton- tons) can go into the same orbital and can then have neutron force is slightly more attractive than the proton- maximum attraction. proton force. Many nuclei are spherical; this giving the lowest sur- Before World War II, the maximum particle energy face area for a given volume. But when there are many available was less than about 20 MeV. Therefore only nucleons in the same shell (see Sec. VII), ellipsoids, or the S-state interaction between two nucleons could be even more complicated shapes (Nielsen model), are of- investigated. ten preferred.

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 Hans A. Bethe: Nuclear physics S9

B. Density distribution search in Berlin and found some sequences of radioac- tivities following each other. When Austria was annexed Electron scattering is a powerful way to measure the to Germany in Spring 1938, Meitner, an Austrian Jew, charge distribution in a nucleus. Roughly, the angular lost her job and had to leave Germany; she found refuge distribution of elastic scattering gives the Fourier trans- in . 2 form of the radial charge distribution. But since Ze /បc and F. Strassmann continued the research is quite large, explicit calculation with relativistic elec- and identified chemically one of the radioactive products tron wave functions is required. Experimentally, Hof- from uranium (Zϭ92). To their surprise they found the stadter at Stanford started the basic work. radioactive substance was , (Zϭ56). Hahn, in a In heavy nuclei, the charge is fairly uniformly distrib- letter to Meitner, asked for help. Meitner discussed it uted over the nuclear radius. At the surface, the density with her nephew, Otto Frisch, who was visiting her. Af- falls off approximately like a Fermi distribution, ter some discussion, they concluded that Hahn’s findings Ϫ1 were quite natural, from the standpoint of the liquid ␳/␳0Ϸ͓1ϩexp͑rϪR͒/a͔ , (11) drop model: the drop of uranium split in two. They with aϷ0.5 fm; the surface thickness, from 90% to 10% called the process ‘‘fission.’’ of the central density, is about 2.4 fm. Once this general idea was clear, comparison of the In more detailed studies, by the Saclay and Mainz atomic weight of uranium with the sum of the weights of groups, indications of individual proton shells can be dis- the fission products showed that a very large amount of cerned. Often, there is evidence for nonspherical shapes. energy would be set free in fission. Frisch immediately The neutron distribution is more difficult to determine proved this, and his experiment was confirmed by many experimentally; sometimes the scattering of ␲ mesons is laboratories. Further, the fraction of neutrons in the useful. Inelastic electron scattering often shows a maxi- nucleus, N/Aϭ(AϪZ)/A, was much larger in uranium mum at the energy where scattering of the electron by a than in the fission products hence neutrons would be set single free proton would lie. free in fission. This was proved experimentally by Joliot and Curie. Later experiments showed that the average number of neutrons per fission was ␯ϭ2.5. This opened C. ␣ radioactivity the prospect of a chain reaction. Equation (9) represents the energy of a nucleus rela- A general theory of fission was formulated by Niels tive to that of free nucleons, ϪE is the binding energy. Bohr and John Wheeler in 1939. They predicted that The mass excess of Z protons and (AϪZ) neutrons is only the rare of uranium, U-235, would be fis- sionable by slow neutrons. The reason was that U-235 ⌬Mϭ7.3Zϩ8.1͑AϪZ͒ MeV, (12) had an odd number of neutrons. After adding the neu- which complies with the requirement that the mass of tron from outside, both fission products could have an 12C is 12 amu. The mass excess of the nucleus is even number of neutrons, and hence extra binding en- ergy due to the formation of a neutron pair. Conversely, Eϩ⌬MϭEϩ7.3Zϩ8.1͑AϪZ͒ MeV. (13) in U-238 one starts from an even number of neutrons, so The mass excess of an ␣ particle is 2.4 MeV, or 0.6 MeV one of the fission products must have an odd number. per nucleon. So the excess of the mass of nucleus (Z,A) Nier then showed experimentally that indeed U-235 can over that of Z/2 ␣ particles plus AϪ2Z neutrons is be fissioned by slow neutrons while U-238 requires neu- trons of about 1 MeV. EЈϭEϩ⌬MϪ͑Z/2͒0.6 MeV

ϭEϩ7.0Zϩ8.1͑AϪZ͒. (14) E. The chain reaction The (smoothed) energy available for the emission of an Fission was discovered shortly before the outbreak of ␣ particle is then World War II. There was immediate interest in the EЉ͑Z,A͒ϭEЈ͑Z,A͒ϪEЈ͑ZϪ2,AϪ4͒. (15) chain reaction in many countries. This quantity is negative for small A, positive from To produce a chain reaction, on average at least one about the middle of the periodic table on. When it be- of the 2.5 neutrons from a U-235 fission must again be comes greater than about 5 MeV, emission of ␣ particles captured by a U-235 and cause fission. The first chain becomes observable. This happens when Aу208. It reaction was established by Fermi and collaborators on helps that Zϭ82, Aϭ208 is a doubly magic nucleus. 2 December 1942 at the University of Chicago. They used a ‘‘pile’’ of graphite bricks with a lattice of uranium metal inside. D. Fission The graphite atoms served to slow the fission neu- trons, originally emitted at about 1 MeV energy, down In the mid 1930s, Fermi’s group in Rome bombarded to thermal energies, less than 1 eV. At those low ener- samples of most elements with neutrons, both slow and gies, capture by the rare isotope U-235 competes favor- fast. In nearly all elements, radioactivity was produced. ably with U-238. The carbon nucleus absorbs very few Uranium yielded several distinct activities. , neutrons, but the graphite has to be very pure C. Heavy , and Otto Hahn, chemist, continued this re- water works even better.

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 S10 Hans A. Bethe: Nuclear physics

The chain reaction can either be controlled or explo- TABLE I. P and D phase shifts at 300 MeV, in degrees. sive. The Chicago pile was controlled by rods of boron 1 1 P Ϫ28 D2 ϩ25 absorber whose position could be controlled by the op- 3 3 erator. For production of power, the graphite is cooled P0 Ϫ10 D1 Ϫ24 3 3 by flowing water whose heat is then used to make steam. P1 Ϫ28 D2 ϩ25 3 3 In 1997, about 400 nuclear power plants were in opera- P2 ϩ17 D3 ϩ4 tion (see Till, 1999). In some experimental ‘‘reactors,’’ the production of heat is incidental. The reactor serves to produce neu- plete set of required measurements is given (Walecka, trons which in turn can be used to produce isotopes for 1995). The initial polarization, it turns out, is best use as tracers or in medicine. Or the neutrons them- achieved by scattering the protons from a target with selves may be used for experiments such as determining nuclei of zero spin, such as carbon. the structure of solids. Proton-proton scattering is relatively straightforward, Explosive chain reactions are used in nuclear weap- but in the analysis the effect of the Coulomb repulsion ons. In this case, the U-235 must be separated from the must, of course, be taken into account. It is relatively abundant U-238. The weapon must be assembled only small except near the forward direction. The nuclear immediately before its use. -239 may be used force is apt to be attractive, so there is usually an inter- instead of U-235 (see Drell, 1999). ference minimum near the forward direction. The scattering of neutrons by protons is more difficult to measure, because there is no source of neutrons of IV. THE TWO-NUCLEON INTERACTION definite energy. Fortunately, when fast protons are scat- tered by deuterons, the deuteron often splits up, and a A. Experimental neutron is projected in the forward direction with almost the full energy of the initial proton. A reasonable goal of nuclear physics is the determina- tion of the interaction of two nucleons as a function of their separation. Because of the uncertainty principle, B. Phase shift analysis this requires the study of nuclear collisions at high en- ergy. Before the second World War, the energy of accel- The measurements can be represented by phase shifts erators was limited. After the war, could be of the partial waves of various angular momenta. In built with energies upward of 100 MeV. This became proton-proton scattering, even orbital momenta occur possible by modulating the frequency, specifically, de- only together with zero total spin (singlet states), odd creasing it on a prescribed schedule as any given batch orbital momenta with total spin one (triplet states). of particles, e.g., protons, is accelerated. The frequency Phase shift analysis appeared quite early, e.g., by Stapp, of the accelerating electric field must be Ypsilantis, and Metropolis in 1957. But as long as only experiments at one energy were used, there were several B/m , ␻ϳ eff sets of phase shifts that fitted the data equally well. It in order to keep that field in synchronism with the or- was necessary to use experiments at many energies, de- bital motion of the particles. Here B is the local mag- rive the phase shifts and demand that they depend netic field which should decrease (slowly) with the dis- smoothly on energy. tance r from the center of the in order to keep A very careful phase shift analysis was carried out by 2 the protons focused; meffϭE/c is the relativistic mass of a group in Nijmegen, Netherlands, analyzing first the pp the protons which increases as the protons accelerate and the np (neutron-proton) scattering up to 350 MeV and r increases. Thus the frequency of the electric field (Bergervoet et al., 1990). They use np data from well between the dees of the cyclotron must decrease as the over 100 experiments from different laboratories and protons accelerate. energies. Positive phase shifts means attraction. Such frequency modulation (FM) had been developed As is well known, S waves are strongly attractive at in the radar projects during World War II. At the end of low energies, e.g., at 50 MeV, the 3S phase shift is 60°, that war, E. McMillan in the U.S. and Veksler in the 1S is 40°. 3S is more attractive than 1S, just as, at E Soviet Union independently suggested the use of FM in ϭ0, there is a bound 3S state but not of 1S. At high the cyclotron. It was introduced first at Berkeley and energy, above about 300 MeV, the S phase shifts be- was immediately successful. These FM cyclotrons were come repulsive, indicating a repulsive core in the poten- built at many universities, including Chicago, Pittsburgh, tial. Rochester, and Birmingham (England). The P and D phase shifts at 300 MeV are shown in The differential cross section for the scattering of pro- Table I (Bergervoet et al., 1990). The singlet states are tons by protons at energies of 100 to 300 MeV was soon attractive or repulsive, according to whether L is even or measured. But since the proton has spin, this is not odd. This is in accord with the idea prevalent in early enough: the scattering of polarized protons must be nuclear theory (1930s) that there should be exchange measured for two different directions of polarization, forces, and it helps nuclear forces to saturate. The triplet and as a function of scattering angle. Finally, the change states of JϭL have nearly the same phase shifts as the of polarization in scattering must be measured. A com- corresponding singlet states. The triplet states show a

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 Hans A. Bethe: Nuclear physics S11 tendency toward a spin-orbit force, the higher J being interaction based on the transfer of a meson from more attractive than the lower J. nucleon i to j, and a second meson from j to k. The main term in this interaction is C. Potential VijkϭAY͑mrij͒Y͑mrjk͒␴i•␴j␴j•␴k␶i•␶j␶j•␶k , (19) In the 1970s, potentials were constructed by the Bonn where Y is the Yukawa function, and the Paris groups. Very accurate potentials, using the exp͑Ϫmcr/ប͒ Nijmegen data base were constructed by the Nijmegen Y͑mr͒ϭ . (20) mcr/ប and Argonne groups. We summarize some of the latter results, which in- The cyclic interchanges have to be added to V123 . There clude the contributions of vacuum polarization, the mag- is also a tensor force which has to be suitably cut off at netic moment interaction, and finite size of the neutron small distances. It is useful to also add a repulsive cen- and proton. The longer range nuclear interaction is one- tral force at small r. pion exchange (OPE). The shorter-range potential is a The mass m is the average of the three ␲ mesons, m 2 1 2 sum of central, L , tensor, spin-orbit and quadratic spin- ϭ 3 m␲0ϩ 3 m␲Ϯ. The coefficient A is adjusted to give the 3 orbit terms. A short range core of r0ϭ0.5 fm is included correct H binding energy and the correct density of in each. The potential fits the experimental data very . When this is done, the binding energy of well: excluding the energy interval 290–350 MeV, and 4He automatically comes out correctly, a very gratifying counting both pp and np data, their ␹2ϭ3519 for 3359 result. So no four-body forces are needed. data. The theoretical group at Argonne then proceed to cal- No attempt is made to compare the potential to any culate nuclei of atomic weight 6 to 8. They used a meson theory. A small charge dependent term is found. Green’s function Monte Carlo method to obtain a suit- The central potential is repulsive for rϽ0.8 fm; its mini- able wave function and obtained the binding energy of mum is Ϫ55 MeV. The maximum tensor potential is the ground state to within about 2 MeV. For very un- about 50 MeV, the spin-orbit potential at 0.7 fm is about usual nuclei like 7He or 8Li, the error may be 3–4 MeV. 130 MeV. Excited states have similar accuracy, and are arranged in the correct order. D. Inclusion of pion production

Nucleon-nucleon scattering ceases to be elastic once VI. NUCLEAR MATTER pions can be produced. Then all phase shifts become ϩ Ϫ complex. The average of the of ␲ , ␲0, and ␲ ‘‘Nuclear matter’’ is a model for large nuclei. It as- is 138 MeV. Suppose a pion is made in the collision of sumes an assembly of very many nucleons, protons, and two nucleons, one at rest (mass M) and one having en- neutrons, but disregards the Coulomb force. The aim is ergy EϾM in the laboratory. Then the square of the to calculate the density and binding energy per nucleon. invariant mass is initially In first approximation, each nucleon moves indepen- ͑EϩM͒2ϪP2ϭ2M2ϩ2EM. (16) dently, and because we have assumed a very large size, its wave function is an exponential, exp(ik•r). Nucleons Suppose in the final state the two nucleons are at rest interact, however, with their usual two-body forces; relative to each other, and in their rest system a pion is therefore, the wave functions are modified wherever two produced with energy ␧, momentum ␲, and mass ␮. nucleons are close together. Due to its interactions, each Then the invariant mass is nucleon has a potential energy, so a nucleon of wave 2 2 ͑2Mϩ␧͒2Ϫ␲2ϭ4M2ϩ4M␧ϩ␮2. (17) vector k has an energy E(k)Þ(ប /2m)k . Consider two particles of momenta k1 and k2 ; their Setting the two invariant masses equal, unperturbed energy is EϪMϭ2␧ϩ␮2/2M, (18) WϭE͑k1͒ϩE͑k2͒, (21) a remarkably simple formula for the initial kinetic en- and their unperturbed wave function is ergy in the laboratory. The absolute minimum for meson production is 286 MeV. The analysts have very reason- ␾ϭexp͓iP•͑r1ϩr2͔͒ϫexp͓ik0•͑r1Ϫr2͔͒, (22) ably chosen EϪMϭ350 MeV for the maximum energy where Pϭ(k1ϩk2)/2 and k0ϭ1/2(k1Ϫk2). We disregard at which nucleon-nucleon collision may be regarded as the center-of-mass motion and consider essentially elastic. ␾ϭeik0•r, (23) V. THREE-BODY INTERACTION as the unperturbed wave function. Under the influence of the potential v this is modified to The observed binding energy of the triton, 3H, is 8.48 ␺ϭ␾Ϫ͑Q/e͒v␺. (24) MeV. Calculation with the best two-body potential gives 7.8 MeV. The difference is attributed to an interaction Here v␺ is considered to be expanded in plane wave between all three nucleons. Meson theory yields such an states k1Ј , k2Ј , and

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with deep and attractive potential wells, are the qualita- eϭE͑k1Ј͒ϩE͑k2Ј͒ϪW, (25) tive explanation. The next proton or neutron must be in Qϭ1 if states k1Ј and k2Ј are both unoccupied, a relative p state, so it cannot come close, and, in addi- Qϭ0 otherwise. (26) tion, by the exchange character of the forces (see Sec. IV.C), the interaction with the ␣ particle is mainly repul- Equation (26) states the Pauli principle and ensures that sive: thus there is no bound nucleus of Aϭ5, neither eϾ0 always. It is assumed that the occupied states fill a 5He nor 5Li. The ␣ particle is a closed unit, and the most Fermi sphere of radius kF . We set stable light nuclei are those which may be considered to be multiples of the ␣ particles, 12C, 16O, 20Ne, 24Mg, etc. v␺ϭG␾, (27) But even among these ␣-particle nuclei, 16O is special: and thus define the reaction matrix G, which satisfies the the binding energy of ␣ to 12C, to form 16O, is consid- equation erably larger than the binding of ␣ to 16O. Likewise, 40Ca is special: it is the last nucleus ‘‘consisting’’ of ␣ Ϫ3 3 particles only which is stable against ␤ decay. ͗k͉G͉k0 ;P,W͘ϭ͗k͉v͉k0͘Ϫ͑2␲͒ ͵ d kЈ The binding energies can be understood by consider- Q͑P,kЈ͒ . ͗k͉v͉kЈ͘ ͗kЈ͉G͉k;P,W͘ ing nuclei built up of individual nucleons. The nucleons E͑PϩkЈ͒ϩE͑PϪkЈ͒ϪW ͮ may be considered moving in a square well potential (28) with rounded edges, or more conveniently, an oscillator This is an integral equation for the matrix ͗k͉G͉k0͘. P potential of frequency ␻. The lowest state for a particle and W are merely parameters in this equation. in that potential is a 1s state of energy ␧0 . There are two The diagonal elements ͗k͉G͉k0 ,P͘ can be transcribed places in the 1s shell, spin up and down; when they are into the k1 , k2 of the interacting nucleons. The one- filled with both neutrons and protons, we have the ␣ particle energies are then particle. The next higher one-particle state is 1p, with energy W͑k ͒ϭ k k ͉G͉k k ϩ͑ប2/2M͒k2. (29) 1 ͚ ͗ 1 2 1 2͘ 1 ␧0ϩប␻. The successive eigenstates are k2 With modern computers, the matrix Eq. (28) can be ͑1s͒, ͑1p͒, ͑1d2s͒, ͑1f2p͒, ͑1g2d3s͒ solved for any given potential v. In the 1960s, approxi- with energies mations were used. First it was noted that for states out- side the Fermi sphere, G was small; then E(PϮkЈ)in ͑␧0͒, ͑␧0ϩប␻͒, ͑␧0ϩ2ប␻͒, ͑␧0ϩ3ប␻͒. the denominator of Eq. (28) was replaced by the kinetic The principal quantum number is chosen to be equal to energy. Second, for the occupied states, the potential the number of radial nodes plus one. The number of energy was approximated by a quadratic function, independent eigenfunctions in each shell are W͑k͒ϭ͑ប2/2M*͒k2, (30) ͑2͒, ͑6͒, ͑12͒, ͑20͒, ͑30͒, M* being an effective mass. so the total number up to any given shell are It was then possible to obtain the energy of nuclear 2 , 8 , 20 , 40 , 70 , ... . matter as a function of its density. But the result was not ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ satisfactory. The minimum energy was found at too high The first three of these numbers predict closed shells at a density, about 0.21 fmϪ3 instead of the observed 0.16 4He, 10O, and 40Ca, all correct. But Zϭ40 or Nϭ40 are fmϪ3. The binding energy was only 11 MeV instead of not particularly strongly bound nuclei. the observed 16 MeV. The solution to this problem was found independently Modern theory has an additional freedom, the three- by Maria Goeppert-Mayer and H. Jensen: nucleons are body interaction. Its strength can be adjusted to give the subject to a strong spin-orbit force which gives added correct density. But the binding energy, according to the attraction to states with jϭl ϩ1/2, repulsion to jϭl Argonne-Urbana group, is still only 12 MeV. They be- Ϫ1/2. This becomes stronger with increasing j. The lieve they can improve this by using a more sophisti- strongly bound nucleons beyond the 1d2s shell, are cated wave function. ͑1f ͒, ͑2p1f 1g ͒, ͑2d3s1g 1h ͒, In spite of its quantitative deficiencies nuclear matter 7/2 5/2 9/2 7/2 11/2 theory gives a good general approach to the interaction ͑2 f3p1h9/21i13/2͒. of nucleons in a nucleus. This has been used especially by Brown and Kuo (1966) in their theory of interaction The number of independent eigenfunctions in these of nucleons in a shell. shells are, respectively, ͑8͒, ͑22͒, ͑32͒, ͑44͒. VII. SHELL MODEL So the number of eigenstates up to 1f7/2 is 28, up to 1g9/2 A. Closed shells is 50, up to 1h11/2 is 82, and up to 1i13/2 is 126. Indeed, nuclei around Zϭ28 or Nϭ28 are particularly strongly The strong binding of the ␣ particle is easily under- bound. For example, the last ␣ particle in 56Ni (ZϭN stood; a pair of neutrons and protons of opposite spin, ϭ28) is bound with 8.0 MeV, while the next ␣ particle,

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60 ¯ in Zn (ZϭNϭ30) has a binding energy of only 2.7 creates an electron ␾e and an antineutrino ␾␯ , and con- 90 MeV. Similarly, Zr (Nϭ50) is very strongly bound verts a neutron ␺n into a proton ␺p . The electron and and Sn, with Zϭ50, has the largest number of stable the neutrino are not in the nucleus, but are created in isotopes. 208Pb (Zϭ82,Nϭ126) has closed shells for pro- the ␤ process. All operators are taken at the same point tons as well as neutrons, and nuclei beyond Pb are un- in space-time. stable with respect to ␣ decay. The disintegration Fermi assumed a vector interaction in his first ␤-decay 212Po 208Pbϩ␣ yields ␣ particles of 8.95 MeV while paper. 208Pb→204Hgϩ␣ would release only 0.52 MeV, and an ␣ The Fermi theory proved to be essentially correct, but particle→ of such low energy could not penetrate the po- Gamov and Teller later introduced other covariant com- tential barrier in 1010 years. So there is good evidence binations allowed by Dirac theory. Gamov and Teller for closed nucleon shells. said there could be a product of two 4-vectors, or ten- Nuclei with one nucleon beyond a closed shell, or one sors, or axial vectors, or pseudoscalars. Experiment nucleon missing, generally have spins as predicted by the showed later on that the actual interaction is shell model. Vector minus Axial vector, (32) and this could also be justified theoretically. B. Open shells The ␤-process, Eq. (31), can only happen if there is a vacancy in the proton state ␺ . If there is in the nucleus The energy levels of nuclei with partly filled shells are p a neutron of the same orbital momentum, we have an usually quite complicated. Consider a nucleus with the allowed transition, as in 13N 13C. If neutron and proton 44-shell about half filled: there will be of the order of differ by units in angular momentum,→ so must the lep- 244Ϸ1013 different configurations possible. It is obvi- tons. The wave number of the leptons is small, then the ously a monumental task to find the energy eigenvalues. product (kR)L is very small if L is large: such ␤ transi- Some help is the idea of combining a pair of orbitals tions are highly forbidden. An example is 40K which has of the same j and m values of opposite sign. Such pairs angular momentum Lϭ4 while the daughter 40Ca has have generally low energy, and the pair acts as a . Lϭ0. The radioactive 40K has a half-life of 1.3 Iachello and others have built up states of the nucleus ϫ109 years. from such . This theory was satisfactory to explain observed ␤ de- cay, but it was theoretically unsatisfactory to have a pro- VIII. COLLECTIVE MOTIONS cess involving four field operators at the same space- time point. Such a theory cannot be renormalized. So it Nuclei with incomplete shells are usually not spheri- was postulated that a new charged particle W was in- cal. Therefore their orientation in space is a significant volved which interacted both with leptons and with observable. We may consider the rotation of the nucleus , by interactions such as as a whole. The moment of inertia ␪ is usually quite ¯ ¯ ¯ large; therefore, the rotational energy levels which are ␾eW␾␯ , ␺pW␺n . proportional to 1/␪ are closely spaced. The lowest exci- This W particle was discovered at CERN and has a mass tations of a nucleus are rotations. of 80 GeV. These interactions, involving three rather and Ben Mottleson have worked exten- than four operators, are renormalizable. The high mass sively on rotational states and their combination with of W ensures that in ␤-decay all the operators ␺n , ␺p , intrinsic excitation of individual nucleons. There are also ␾␯ , ␾e have to be taken essentially at the same point, vibrations of the nucleus, e.g., the famous vibration of all within about 10Ϫ16 cm, and the Fermi theory results. neutrons against all protons, the giant dipole state at an A neutral counterpart to W, the Z particle, was also excitation energy of 10–20 MeV, depending on the mass found at CERN; it can decay into a pair of electrons, a number A. pair of , or a pair of baryons. Its mass has been Many nuclei, in their ground state, are prolate sphe- determined with great accuracy, roids. Their rotations then are about an axis perpendicu- lar to their symmetry axis, and an important character- m͑Z͒ϭ91 GeV. (33) istic is their quadrupole moment. Many other nuclei The difference in masses of Z and W is of great theoret- have more complicated shapes such as a pear; they have ical importance. The mass of Z has a certain width from an octopole moment, and their rotational states are which the number of species of neutrinos can be deter- complicated. mined, namely three: ␯e , ␯␮ , and ␯␶ .

IX. WEAK INTERACTIONS X. NUCLEOSYNTHESIS Fermi, in 1934, formulated the the first theory of the on the basis of Pauli’s neutrino hypoth- It is an old idea that matter consisted ‘‘originally’’ of esis. An operator of the form protons and electrons, and that complex nuclei were gradually formed from these (see Salpeter, 1999). (Mod- ¯ ¯ ␾e␾␯␺p␺n (31) ern theories of the big bang put ‘‘more elementary’’ par-

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 S14 Hans A. Bethe: Nuclear physics ticles, like quarks, even earlier, but this is of no concern Many supernovae explosions have taken place in the here.) At a certain epoch, some neutrons would be galaxy, and so galactic matter contains a fair fraction Z formed by of elements beyond C, called ‘‘metals’’ by astrophysi- cists, viz., ZӍ2%. This is true in the solar system, HϩeϪ Nϩ␯. (34) → formed about 4.5 billion years ago. New stars should These neutrons would immediately be captured by pro- have a somewhat higher Z, old stars are known to have tons, smaller Z. Stars of Mу3M᭪ are formed from galactic matter NϩH Dϩ␥, (35) that already contains appreciable amounts of heavy nu- → 56 and the deuterons would further capture protons, giving clei up to Fe. Inside the stars, the carbon cycle of 14 3He and 4He. This sequence of reactions, remarkably, nuclear reactions takes place, in which N is the most leads to a rather definite fraction of matter in 4He nu- abundant nucleus. If the temperature then rises to about clei, namely 100 million degrees, neutrons will be produced by the reactions 4HeϷ23%, (36) 14Nϩ4He 17Fϩn, nearly all the rest remaining H. Traces of D, 3He, and → 17 4 20 7 Oϩ He Neϩn. (38) Li remain. → Again remarkably, there exist very old stars (in globu- The neutrons will be preferentially captured by the lar clusters) in which the fraction of 4He can be mea- heavy nuclei already present and will gradually build up sured, and it turns out to be just 23%. This fraction de- heavier nuclei by the s-process described in the famous pends primarily on the number of neutrino species article by E.M. and G. R. Burbidge, Fowler, and Hoyle which, as mentioned at the end of Sec. IX is three. in Reviews of Modern Physics (1957). In stars like the sun and smaller, nuclear reactions Some nuclei, especially the natural radioactive ones, take place in which H is converted into He at a tempera- U and Th, cannot be built up in this way, but require the ture of the order of 10–20 million degrees, and the re- r-process, in which many neutrons are added to a leased energy is sent out as radiation. If, at later stages nucleus in seconds so there is no time for ␤ decay. The in the evolution, some of the material of such a star is conditions for the r-process have been well studied; they 9 lost into the galaxy, the fraction of 4He in the galaxy include a temperature of more than 10 K. This condi- increases, but very slowly. tion is well fulfilled in the interior of a supernova a few In a star of three times the mass of the sun or more, seconds after the main explosion, but there are addi- tional conditions so that it is still uncertain whether this other nuclear processes occur. Early in its life (on the is the location of the r-process. main sequence), the star produces energy by converting H into He in its core. But after a long time, say a billion years, it has used up the H in its core. Then the core XI. SPECIAL RELATIVITY contracts and gets to much higher temperatures, of the order of 100 million degrees or more. Then ␣ particles For the scattering of nucleons above about 300 MeV, can combine, and for the equation of state of nuclear matter of high density, special relativity should be taken into account. 3 4He 12Cϩ␥. (37) A useful approximation is mean field theory which has → Two 4He cannot merge, since 8Be is slightly heavier than been especially developed by J. D. Walecka. two 4He, but at high temperature and density, 8Be can Imagine a large nucleus. At each point, we can define ¯ exist for a short time, long enough to capture another the conserved current i␺␥␮␺ where ␺ is the 4He. Equation (37) was discovered in 1952 by E. E. Sal- baryon field, consisting of protons and neutrons. We peter; it is the crucial step. also have a scalar baryon density ¯␺␺. They couple, re- 12 4 Once C has formed, further He can be captured spectively, to a vector field V␮ and a scalar field ␾ with and heavier nuclei built up. This happens especially in coupling constants gw and gs . The vector field is identi- the inner part of stars of 10 or more times the mass of fied with the ␻ meson, giving a repulsion, and the scalar the sun. The buildup leads to 16O, 20Ne, 24Mg, 28Si, and field with the ␴ meson, giving an attraction. Coupling on to 56Ni. The latter is the last nucleus in which the ␣ constants can be adjusted so as to give a minimum en- particle is strongly bound (see Sec. VII). But it is un- ergy of Ϫ16 MeV per nucleon and equilibrium density stable against ␤ decay; by two emissions of positrons it of 0.16 fmϪ3. transforms into 56Fe. This makes 56Fe one of the most The theory can be generalized to neutron matter and abundant isotopes beyond 16O. After forming all these thus to the matter of neutron stars. It can give the elements, the interior of the star becomes unstable and charge distribution of doubly magic nuclei, like 208Pb, collapses by gravitation. The energy set free by gravita- 40Ca, and 16O, and these agree very well with the distri- tion then expels all the outer parts of the star (all except butions observed in electron scattering. the innermost 1.5M᭪) in a supernova explosion and thus The most spectacular application is to the scattering makes the elements formed by nucleosynthesis available of 500 MeV protons by 40Ca, using the Dirac relativistic to the galaxy at large. impulse approximation for the proton. Not only are

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 Hans A. Bethe: Nuclear physics S15 cross section minima at the correct scattering angles, but Green, E. S., 1954, Phys. Rev. 95, 1006. polarization of the scattered protons is almost complete, Pudliner, B. S., V. R. Pandharipande, J. Carlson, S. C. Pieper, in agreement with experiment, and the differential cross and R. B. Wiringa, 1997, Phys. Rev. E 56, 1720. section at the second, third, and fourth maximum also Rutherford, E., J. Chadwick, and C. D. Ellis, 1930, Radiations agree with experiment. from Radioactive Substances (Cambridge, England, Cam- bridge University). REFERENCES Salpeter, E. E., 1999, Rev. Mod. Phys. 71 (this issue). Siemens, P. J., 1970, Nucl. Phys. A 141, 225. Bergervoet, J. R., P. C. van Campen, R. A. M. Klomp, J. L. de Stoks, V. G. J., R. A. M. Klomp, M. C. M. Rentmeester, and J. Kok, V. G. J. Stoks, and J. J. de Swart, 1990, Phys. Rev. C 41, J. de Swart, 1993, Phys. Rev. C 48, 792. 1435. Till, C., 1999, Rev. Mod. Phys. 71 (this issue). Brown, G. E., and T. T. S. Kuo, 1966, Nucl. Phys. 85, 140. Walecka, J. D., 1995, Theoretical Nuclear and Subnuclear Burbidge, E. M., G. R. Burbidge, W. A. Fowler, and F. Hoyle, Physics (Oxford, Oxford University). 1957, Rev. Mod. Phys. 29, 547. Wiringa, R. B., V. G. J. Stoks, and R. Schiavilla, 1995, Phys. Drell, S. D., 1999, Rev. Mod. Phys. 71 (this issue). Rev. E 51, 38.

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