Glen E. Gordon and Charles D. Coryell Mossochusetts Institute
of Technology Cambridge, Mossochusetts 02139
Models for Nuclear Structure of Spherical Nuclei
In the preface to his book on "Operator Techniques in Atomic Spectroscopy," Brian Judd (1) has noted that, in spite of incomplete knowledge of nuclear forces, nuclear physicists have developed powerful techniques for treatment of nuclear structure that in many ways supersede those used in the treatment of atoms and molecules. For this reason, as well as for general background and interest, it is important that chemists have some contact with current developments in nuclear structure theory. Of course, theoretical nuclear physics has borrowed many models and techniques from other fields, but it has also developed methods that have wide application in other areas. Perhaps the most important development of quantum mechanical techniques by nuclear physicists involves the exploitation of the general methods formulated by Giulio Racah (2). In a series of four papers (S), Racah developed powerful techniques, including the algebra of tensor operators and the use of 6-j symbols (now known as "Racah coefficients"), for handling complex atomic or nuclear structure calculations. Although they are too complex for explicit discussion in this paper, Racah's methods are used heavily in all nuclear structure calculations including those discussed below, especially the "effective-interaction" method. The Racah methods have not been as widely adopted in atomic structure calculations, although as Judd notes (I), the techniques and necessary coefficientsare available and the methods are quiteusefulespecially for complicated problems such as the structure of rare earth and actinide atoms. The fundamental internucleon interactions are still not known in the detail that the forces between electrons and between nuclei and electrons are known. Nevertheless, great progress in the understanding of nuclear structure has been made since the development of the nuclear-shell model. In order to limit the scope of the discussion we have chosen to treat only the nuclei that we call "spherical," i.e., those that seem to he describable with the starting assumption of a spherical potential well. We are thus avoiding the regions, principally in the rare earths and actinides, in which a better starting assumption is that of a prolate or oblate spheroidal well. I n a sense, the shell model also works in spheroidal nuclei, hut one has touse single-particleorbitals of a spheroidal well (4) rather than the more familiar orbitals of a spherical well. Much of the impetus for the development of precise nuclear models comes from the high quality of data on This work was supported in part by the U.S. Atomic Energy Commission under Contract AT(3@1)905.
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nuclear structure that are now obtainable. The most important new technique for the nuclear chemist is that of ?-ray spectroscopy with lithium-drifted germanium [Ge(Li)] detectors [see ref. (5)]. The resolution of Ge(Li) detectors is much better than that of NaI(T1) scintillation detectors, which were previously used for most decay scheme studies, allowing the experimenter to obtain much more detailed and accurate data on 7 emission than was formerly possible. Also, the development of tandem Van de Graaff accelerators and sector focused cyclotrons has made possible the careful study of nuclear levels formed in inelastic scattering or nuclear reactions [e.g., ref. @)]. As a result of these and other recent experimental developments, both the quality and quantity of information on nuclear structure have vastly increased over the past four or five years, maKmg possible very detailed comparisons of predictions of nuclear models and experimental observations. Many of the data presented below were obtained with these techniques. I n the following sections we discuss the basic nuclearshell model which assumes no configuration mixing, then the "effectiveinteraction" method which includes some mixing, and finally the extreme model of configuration mixing, the "superconductor model." A more extensive, but quite understandable discussion of nuclear models is contained in Cook's book (7). A somewhat more advanced treatment is included in the nuclear chemistry textbook by Friedlander et al. (8), and a rather sophisticated treatment and introduction to the appropriate quantum mechanical calculations is given by Preston (9). The simple shell model has previously been discussed by Flowers in THIS JOURNAL (10). The monograph by Mayer and Jensen (11) is an easy-tofollow, but rigorous, description of the shell model and its applications. The effective-interaction method is described without detailed mathematics in the review by Talmi (12) and with complete rigor in the textbook by de-Shalit and Talmi (13). The latter is an excellent textbook on the quantum mechanics of the shell model and gives particularly good coverage of Racah's methods. Considerable insight into the relationship of the superconductor model to other models may be obtained from Mottelson's article (14, upon which we have drawn heavily in this paper. A complete, but rather difficult treatment is given by Lane (15). Detailed comparisons of predictions of the superconductor model with experimental data are contained in the paper by Kisslinger and Sorensen (16). The collective model, which we have not covered, is well described at an elementary level by Hyde et al. (17) and in more detail by Nilsson and Nathan (18).
in which M is the nucleon mass. The quantities n and I are the radial and orbital angular momentum quantum numbers, respectively. Just as in atomic spectroscopy, we use the designations s, p, d, f, g . . ., for orbitals having I = 0,1,2,3,4. . . ., and the parity, r,of an orbital is even or odd (+1 or - 1) if 1 is even or odd. The quantity n 1 corresponds to the principal quantum number used in atomic systems, and No is referred to as the "0% cillator" quantum number. At the left of Figure 1 are shown the levels predicted from eqn. (2). Note the high degree of degeneracy of the levels. The degeneracy is largely removed in the center of Figure 1, where we show the levels of a more realistic approximation of the nuclear potential, the "rounded" square well. I n passing from the harmonic oscillator to the rounded square well, the energies of orbitals of high 1 are lowered relative to those of low I because the wave functions of the former are more strongly concentrated a t large r. Since neutrons and protons are diierent particles, they occupy separate has 20 neutrons and 20 sets of orbitals; thus 4010C%o protons filling orbitals up through the ld-2s subshell. Whenever large gaps occur in the spacings of levels, one expects to find special stability in systems filled almost to the gap and special instabilities in species containing particles in the orbitals above the gap. For example, the energy gap between electron orbitals 2p and 3s causes neon to be inert and sodium to have a low ionization potential. Similarly for nuclear structure, we expect to observe elosed-shell phenomena (analogous to those of the noble gases which have closed shells of electrons) in nuclei in which the neutrons or protons are filled up to one of the gaps in the singleparticle orbitals. We do, in fact, observe closed-shell properties in nuclei having 2, 8, or 20 neutrons or protons as predicted by the gaps in the two sets of levels a t the left of Figure 1. But the higher so-called "magic numbers" are found to be 28, 50, 82, and 126 instead of the values (40, 70, and 112) expected on the basis of the two potentials. The closed shell a t 82 neutrons is shown, for example, by the sudden drop in neutron separation energy1in Figure 2 as one goes above N = 82, indicating a gap in neutron orbitals. This difficulty held up the development of useful nuclear models until 1949. Then, the important breakthrough by Haxel, Jensen, and Suess (19) and by Mayer (20) was the assumption of strong spin-orbit coupling: i.e., the "parallel" coupling of 1 with intrinsic nucleon spins to j, j = 1 lies muchlower in energy than the antiparallel coupling j = I - I/?. AS the spin-orbit force is given by a radial function times 1.6, the splitting between couplings should increase with 1 approximately as 21 1. Above the 20 closed shell, the splitting is so large that the positions of the closed shells are modified as shown in Figure 1, giving a natural explanation of the higher magic numbers. Spin-orbit splitting is well known in atomic structure, but the splitting between j = 1 '/, and j = I - '/*is
+
Figure 1. Energies of single-particle orbitals in hormonic-o~cillotor and "rounded square-well" potentialr, the l m e r with and without spin-orbit coupling. Numbers in I 1 indicate orbital capacities and those in [ ] give cumulative capacity up to the given point.
Nuclear-Shell Model
The nuclear-shell (or "independent-particle") model was the first important step towards quantitative understanding of nuclear levels. The interactions of a specific nucleon with the other nucleons present are approximated as the interaction of the nucleon with a static potential well. For example, if we assume a three-dimensional harmonic-oscillator potential,
+
+
+
the energy eigenvalues obtained are (neglecting zeropoint energy) : e =
&[2(n - I )
The "oscillator frequency,"
+ 11 =
lV&
o,is given by
(2)
Neutron separation energy is the energy required to remove a neutron from the nuclens. Often this quantity is imprecisely called the "neutron binding energy."
Volume 44, Number 7 7, November 1967
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637
o' dd
I "T
N
Figure 2. Neutron separatian energies, S. for species with N - Z = 25 as a function of neutron number, showing shell effect a t N = 82, and S, difference. for even N ond odd N.
generally much smaller than the energy difference between orbitals of different 1. For example, in sodium the 3P1,, - ~SI,,energy difference is about 2.1 ev, whereas the 3Pzh - 3 P y , splitting is only 0.002 ev (corresponding to the 6.04 A differencebetween the lines of the well known sodium doublet at 5890.12 and 5896.16 A). By contrast, in the nuclear case, the splitting is so large that orbitals with j = 1 and 1 - '/* occur in diierent shells for the highest 1 orbitals. For atomic calculations, LS (or "Russell-Saunders") coupling is generally a good assumption, with spin-orbit effects giving very small perturbations. But for all except the lightest nuclei, jj coupling is the better approximation: i.e., for each nucleon, 1 and s arecoupled to j, and the j's of the various nucleons are coupled to total nuclear angular m o m e n t ~ mI. , ~ The maximum population of 1. The j values are indilevels in jj coupling is 2j cated by the subscript in the orbital designation, e.g., d.,, has 1 = 2 and j = 5/2. As the nucleons themselves set up the potential, energies of the orbitals depend on the numbers of each kind of nucleon present. This perhaps seems obvious, but students tend to th'mk of orbitals in all nuclei being ordered as depicted in Figure 1 (just as students of atomic structure too often think of electronic orbitals as being ordered in all atoms as they are in hydrogen or in the order of filling). But the potential well is different for cach nuc.lcui, and eo~tlurnbrrpulsiim among pmtons cause thcir orbituls to bcr ditTcrcnt iron) ncutnm d)itnls. The orbitals shown in Figure 1 are approximately those expected for neutrons in a well having R 10-lZ cm and VO 40 Mev. When we examine a particular region, e.g., 50 to 82 neutron cases discussed below, we often find that the ordering of levels within a major shell is differentfrom that of Figure 1. As the shell model a t this stage represents only a crude
+
first approximation, the very important remaining job is to study the correct;ons to this model, the so-called "residual forces" which are the internucleon forces in addition to those included in the assumption of a potential well. Shell-model calculations since 1949 essentially involve diierent methods of dealing with the residual forces. The effective-interaction and superconductor methods described below represent two extreme starting points for the treatment of these forces. It should be emphasized that the residual forces undoubtedly differ appreciably from the free nucleon interactions ( i . the interactions between two free nucleons that one studies in nucleon-nucleon scattering experiments or in the structure of the deuteron), because a part of that interaction has already been included in the shell-model potential. In order to make detailed comparisons of the shell model with experimental results, we must consider those residual forces. For example, in the calcium isotopes we may consider as a first approximation that the doubly-magic "core" of 4002C&o (20 protons plus 20 neutrons) provides the well for the additional neutrons that occupy the if7,$ orbitals. But what are the interactions among the j,,, neutrons themselves? What are the relative energies of the four possible couplings ( J = 0, 2, 4, and 6)a of the two f,,, neutrons in 4220C&421? The strongest residual force leads to pairwise coupling of nucleon j's to J = 0. Because of large, attractive interaction in that coupling, even-A(mass number)e v e n 4 (atomic number) nuclei have I = 0 in their ground states and special stability relative to odd-A or odd-odd nuclei in which there are one or two nucleons,
+
-
-
'
Throughout the paper we use I for total angular momentum of the nucleus, and J far a group of nucleons. In many eases I and J will have the same value. For brevity, and consistent with nuolertr usage, I will often be called nuclear "spin." a The couplings to odd J are forbidden by the Pauli exclusion principle.
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Figure 3. Moss parabolas for A = 130 ond points for hypothetical oddA species of mars 130 interpolated from moues of species with A = 129 and 131. For convenience we plm mou exes.. M A, where M is the mass of the nevtrol otom and A is mass number I=N f 2). Conversion beween atomic mars unih and Mev is 1 amu = 931.5 Me*.
-
orbital. The 6 function in the integral has the property of giving zero contribution except where /r, - r21 = 0. This allows us to reduce eqn. ( 5 ) to an integral over the single variable r:
Figure 4. Schematic representotion of the couplings of iZthat give maximum ond minimum overlap. At the right is shown the relotive lowering of the energy of lhe isconflgurotian for couplings to various J values as produced b y lhe 6 force. The top line shows the position of all the couplings when the force is "turned off," i.e., g = 0.
respectively, that have no partner of the same type with which to pair. This is shown in Figure 3, where the ground state masses of the isobaric series A = 130 are plotted versus Z. There are two parabolas for the even-A case, one for even-even nuclei, the other for oddodd. We will call the displacement between the two parabolas 2A, where A is the "pairing energy." Since in odd-A nuclei there is but one unpaired nucleon, the hypothetical odd-A parabola that we have interpolated falls about midway between the other curves, i.e., about A above the even-even curve. This effect also manifesk itself in the two curves in Figure 2, one for even N and a lower one for odd N, as it requires about A more energy to remove a neutron that is paired than an odd neutron. The strong pairing tendency probably arises from the very short range of nuclear forces. The limit of short-range forces is the so-called '%force" whose residual potential is given by
From eqn. (6) we see that the coritribution to the integral at any point r is simply proportional to the product of the probability densities of the two wave functions a t that point. This is just the result that we would intuitively expect for a force of zero range. The wave functions of orbitals of large j are strongly concentrated around the plane perpendicular to j as sketched in Figure 4. If particle 1 occupies magnetic substate m, = j, we achieve maximum overlap and interaction by placing particle 2 in suhstate in2 = -j, giving J = 0. (Recall that 7n1 = 7% = j, giving J = 2j is forbidden by the Pauli principle.) With this crude picture, the smallest overlap and interaction occurs for j's approximately perpendicular, giving J j4!Z4 The important feature of the &force is that the J = 0 level is far below the other allowed couplings: in the limit of largej, the interaction energy is four times as great in the J = 0 coupling as for the closest other coupling, J = 2.
-
Figure 5. Levels of single-closed-hell fi/. nuclei. From left to right, h e 6-force prediction for f71aa according to eqn. 17) normalized to experimental 6+ level; observed levelr of 4 % ~ ; levels of frlS4 conflgurdion predicted from 4'C0 via effective interactions; observed levels of "Cr and %a; levelr of fl/,"predicted from "Ca, and observed levelr in "Ca ond 9.
where g is a strength constant and 6(lr, - r21) is a Dirac delta function i f the distance between particles 1 and 2. The 6 function is defined as zero unless its argument is zero, in which case the integral over the argument is unity. The interaction energy, BJ, for the configuration ja (i.e., two particles in the j orbitals) coupled to J is
Fj = liVlg6(!r, - r ~ l ) l j V l= -g
where
+j
SJ+jyrl)fijwa(
1s - s I )dodq
(5)
is the wave function for a particle in the j
Quantitatively, the interaction energy for the 6-force in the configurationj2 is given by:
'We should caution the reader that the sketches of the lefthand side of Figure 4 are much too ela4sicdl, implying as they do that the interaction energy is a symmetrical function about 90" of the angle between5 and& The quantum mechanical result, eqn. (7), shows that the interaction in the antiparallel couplings is much stronger than in the corresponding parallel couplings. Far example, the interaction in the J = 0 coupling is stronger than in the (non-allowed) J = 2j coupling. Volume 44, Number 1 1 , November 1967
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639
where R is the radial overlap integral and (JjO'/*/j'/')is a Clebsch-Gordan coefficient. I n Figure 5 we compare the prediction of eqn. (7) for the J/,2 configuration of 'Va with the known levels. We have normalized g to give the observed energy difference between levels with J = 0 and 6. Although the observed J = 0 level is easily the lowest, the other couplings are not as far removed from it as predicted by the 8 force. Other kinds of forces are clearly involved in theinteraction, although there is a strong tendency towards coupling to J = 0. Because of strong pairing to J = 0, we expect all even-even nuclides to have I = 0 ground states and there are no known violations of the rule. In odd-A nuclei, according to the simplest version of the shell model, all but the one odd nucleon should he coupled to J = 0, giving a nuclear spin I equal to the j of the orbital occupied by the odd nucleon. Low-lying excited states could be formed by moving the uncoupled particle to orbitals just above that occupied in the ground state, or by decouplmg and raising a particle from a lower orbital. The spins and parities of low-lying levels of odd-A nuclei can he understood on the basis of the shell model in most mass regions not involving spheroidally deformed nuclei. One of the great strengths of the shell model is its ability to predict the occurrence of nuclear isomen, that is, long-lived nuclear excited states. Just before the shell closures at 50, 82, and 126, one of the subshells being filled is that of the j = 1 member that has been lowered out of the next higher shell group by spinorbit coupling, i.e., ?/,, h~./,,and i3./,,, respectively. These very high-j orb~talslie near low-j orbitals of the opposite parity. Thus, odd-A nuclei having neutron or proton number just below magic numbers are expected to have low-lying levels involving orbitals of very different j's. As half lives for y-ray transitions increase strongly with increasing spin diierence and decreasing energy diierence between levels, the shell model correctly predicts extensive isomerism just before shell closures. An example of these isomers is provided by the odd-A tellurium isotopes with N = 67-81, in which neutrons are filling the orbitals in order of increasing energy, s ~ / ,hLl,,, , and d./,. The low-lying levels and longlived isomers are shown in Figure 6. At the left, the s'/, and htL/,orbitals are being filled and levels are formed by raising a neutron up to a d8/,orbital. With
+
Figure 6. Low-iying levels of the odd-A tellurium hetopor showing hn/, isomers. Arrows ot the right (or left) of levels indicate that they decoy, o t least in port, by 8- (or 8' and electron capture).
640 / Journal o f Chemical Education
increasing N, the s ~ / ,and , h~.),,orbitals get full and 3/2+ becomes the ground state. I n the extreme case of N = 81, the ground state is represented as one hole in the d8/, orhitals, and the 11/% isomer is made by dropping the hole to the h,,,,, orbitals which lie below. In the ground states of odd-odd nuclei, the shell model predicts that all nucleons except the odd neutron and odd proton are coupled in pairs to J = 0. Brennan and Bernstein (21) have devised a set of rules for predicting the lowest-energy couplmg of odd protons and neutrons in nuclei for which 20 _< A _< 120, and a recent recompilation ($9) of data extends the range up to A = 126 and includes some nuclei near the 2wa2Pbus double closed shell. There are three coupling rules, the first two, R1 and R2, applying when both the neutrons and protons are in "particle" or in "hole" states, a particle state meaning that the subshell in question is less than half full. The third rule, R3, applies to mixed particle hole configurations. The predictions of the lowest couplings for the three cases are given in the table. The
-
Brennan-Bernstein Rules for Spins of Odd-Odd Nuclei. ~ Rule
.4ppIy when
R1
Both particle or hole states L + , ,-= - 1,'
R2
9"
,- ,. ... = * &x
/ ,Mired ,,== 11- hole*+ 112 P
R3
partiale stater
~
,
Successb &No.~correct ~ dprediotioos No. lev& applicable
I = lJp - Jnl
,/2
Both particle or hole states
~
spm(s)"
I = JP or
50/5S
+ Jn
92/102
IJP - J1 .
I=Jp+Jn-1
22/36
The distinction between j and J is as follows: jis the angular momentum of the orbital occupied by the uncoupled particle and J is the tots1 angular momentum of the odd proton or neutron group. Normaliyj = J , but there are anomalous cases in whioh apeouliar oouplmg, e.g.. J = I 1 a observed. For example, in some odd-A silver isotopes, the proton eonfigursfion gl/s-' ha8 pround .tale J = 7 / 2 . "Aeierenoe (MI.
-
ambiguous cases of half-filled subshells are considered either as holes or particle states, whichever will place the case in R1 or R2 rather than R3. For the special case of J , or J , of the prediction is J , J,. The coupling rules are in accord with the theoretical calculations of Schwartz (2.3) and de-Shalit (24), who assumed a residual neutron-proton interaction of the form:
+
which is similar to the 8 force of eqn. (4),except that here we have a term ~ 1 . involving u ~ Pauli spin operators which have eigenvalues - 3 and + l for singlet and triplet spin states, respectively, and or is a parameter giving the relative strength of the spin-dependent part of the interaction. Consider the simple case of the coupling of one odd neutron with one odd proton. The term ( 1 - or) gives the lowest energy for the anti-parallel coupling I = IJ, - J,I, just as for the 8 force of eqn. (4). The second term, oral.u2, produces the lower energy for triplet spin states (i.e., parallel spin alignment), as is observed for the ground state of the deuteron. Cases falling under R1 satisfy both requirements with the coupling I = IJ, - J,I ; thus, this coupling is clearly predicted to be the lowest. Because of its clear-cut p r e diction, R1 is termed a "strong rule." When R2 is applicable there is conflict between the two terms and the rule predicts two low-lying levels, the parallel and anti-
+
parallel alignments I = J , J , and IJ, - J.1. A more detailed calculation would be needed to predict which of these is the ground state. For the particlehole configurations, any J between jJ, - J,( and J , J , is allowed, although there is a tendency for the level with J = J , J , - 1to be lowest in certain ranges of a. If one takes the latter as the prediction for R3, the rule is violated almost as often as obeyed, as shown in the table. The value of a ohtained in the analysis by Brennan and Bernstein is about 0.13. An interesting implication of R2 should be noted. If J , and J , are both greater than R2 predicts two lowlying levels of small energy differences having a spin difference of at least 3 units. These are conditions that produce isomerism, as discussed above for odd-A nuclei. Thus, one expects to find isomers in odd-odd nuclei whenever R2 is applicable and J,, J , > I n the most recent compilation, isomers had been found in 34 out of the 49 nuclei where expected (22). The cases for u~hich isomers are not known are not necessarily violations of the predictions; rather, the proper experiments may not have been done. Thus, R2 is useful for prediction of nuclei in which one may be able to discover isomers.
+
+
Effective Interactions
Above we have seen how the 8-range residual interaction may be used to treat strong pairing effects and, in combination with a spin-spin dependent part, the neutron-proton interactions in odd-odd nuclei. Recently considerable effort towards detailed prediction of nuclear structure with the use of additional or other residual interactions has been made [e.g., ref. (25)l. However, most of this is confined to nuclei having very few nucleons beyond closed shells. Here, we will discuss another technique, the "effective-interaction" method, that is capable of being used to make shellmodel predictions over rather large regions of nuclei and which does not require any assumptions about the shapes of potentials and strengths for various types of interactions (12). Instead of attempting to predict values of the interaction matrix elements for various couplings as in eqn. (4), according to the effective-interaction method one takes them from experimental data. For example, de-Shalit and Talmi (13) have used the tensor methodsof Racah to derive simple equations by which one can use the ohserved energies of t,hef7/,Zlevels in "Ca to predict levels in other single-closed-shell (s.c.s.) nuclei that have several f7/, particles (i.e., nuclei having a closed shell of one type of particle and several of the other type in f7/, orbitals). I n Figure 5 we compare the predicted and observed levels of other 6.c.s. nuclei in the calcium r e g i ~ n . ~The levels that are thought to arise primarily from f,,, configurations are shown as heavy lines and are connected from one nuclide to the next. I n some of the more complicated shell-model configurations, it is possible to have more than one level of a given J, e.g., the configuration f~/,4can have two levels each of J = 2 and J = 4 plus four other allowed couplings. Thus, to construct and Mecullen, et al., ( $ 6 ) have made a very detailed comparison of experimental data on d l nuclides having N and Z between 20 and 28 with effective-interaction predictions bssed on the assumption of only f7/* ueutron and proton configurations.
identify the separate wave functions, one needs an additional quantum number. Because of the strong tcw dency for painvise coupling to J = 0, one generally uses the "seniority scheme,"in which the additional quant,rnn number, seniority (symbol v), is equal to the number of nucleons not paired to J = 0. I n '%Cs,the J = 0 ground state has u = 0, and the J = 2 , 4 , 6 levels have v = 2. The s.c.s. nuclei having f,/,4 configurations, i.e., 522nCr28 and 44zoCazr, can also have v = 4 levels with J = 2,4, 5, and 8. The effective interaction treatment predicts that the u = 2 levels lie at the same heights relative to the J = 0 ground state in all of the even, s.c.s., f ~ /nuclei, , and the positions of the v = 4 levels predicted from 42Caare shown in Figure 5. I n the fp/,3 configurations, the J = '/%ground state has v = 1 and the excit,ed members, u = 3. We see that there is rather good agreement between the positions of the predicted and experimental levels in the s.c.s. f ~ /nuclei, , However, a more detailed examination shows our analysis to he too sirnpliied. There a l x several levels, e.g., at 1.84 and 2.42 Mev in '%a, that cannot be explained as belonging to the f~,," configurations. The calculations of Federman and Talmi (27) suggest that these two levels result from collective excitations of the ' T a nucleus. Furthermore, the observed 3/2 level a t 0.59 Mev in 43Cais much lower than the predicted one (1.32 Mev) and there is a second a/2 level at 2.05 Rlev. Recent calculations (27, 28) show that the wave function of the 0.59-R4ev level is largely made up of an fThZpa/, configuration and that the level a t 2.05 Alev is mostly the expectedf~/,3level. Whenever levels of the same spin and parity lie rather close together thcrc is likely to be some configuration mixing, and the t,wo levels involved are "pushed" apart from their unpcrturbed positions. One should not be disturbed by the observation of considerable configuration mixing. I n view of t,hc great con~plexityof the nuclear many-body problem, we should be quite surprised if the eigenfunctions obtained from our simple model were those of the true nuclear Hamiltonian. The amazing result is that the shell model works a t all for this complex problem. Frobahly the most important reason for its relative success is that. the Pauli principle excludes many interactions that would otherwise complicate the structure. Configuration mixing is quite a general result for atomic, molecular, and nuclear calculations as, except for a very simple system such as the hydrogen atom, on(! always works with Hamiltonians that are only approximations of the true Hamiltonian. Thus, the true wave functions are always linear combinations of the eigcw functions of the approximate Hamiltonian. Onc ean deal wit8htheuse of very complicated Hamiltonians t,liat. closely approach the true ones and therefore obt:~in nearly true eigenfunctions or deal with the use of simplc, approximate Hamiltonians and eigenfunctions and allow for considerable mixing. The latter course is followed in most nuclear calculations to date. The effective-interaction method is capable of dealing with configuration mixing: the mathematical techniques for inclusion of more than one configuration arc well known but, until recently, the amount of data available scarcely exceeded the amount needed to establish the values of the interaction matrix elements. H o w ever, considerable progress has recently been made i l l Volume 44, Number 1 I , November 1967
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641
handling configuration mixing in the calcium isotopes (27, 28) and the region of mass 90 (29-81). I n the calcium isotopes, considerable improvement in fitting levels was ohtained by use of configurations j5/,"-'p8/, and j>/,"-2pa/,2 in addition to f~/,". I n references (29) and (SO); 88asSrjo is treated as a double-closed shell nucleus, although the 38 protons represent only a closed suhshell--p,/, and j7/, (see Fig. 1). The fint few added neutrons are presumed to occupy the 2&/, orbitals, hut added protons may occupy either the 2p1/,or lg./, orbitals. A great deal of information on nuclides in this region has recently been ohtained so that there are sufficiently more data than parameters to make comparisons of prediction with data meaningful. I n Figure 7 we compare the results of the calculations with observed levels in two typical nuclei. The generally good
Figure 7. Comparison betreen experimental and predicted (by effective interadion melhod) levels of OSMoond OaZr.
fits ohtained indicate the power of the technique. Note, in particular, that the calculations correctly predict the presence of a long-lived isomer in 831\foof I = 21/2+ (resulting from the proton configuration g./,2 with J = S coupled to a d5/,neutron). The isomer has a long half life because there are no levels below the 21/2+ with spins sufficiently high that a y r a y can be emitted with low angular momentum change (the highest available is 13/2+ requiring A1 = 4). There is a 17/2+ level obtained in the configuration, and if it lay below the 21/2+, A1 would he only 2 and no long-lived isomer would he obtained. As shown, the calculations indicate that the 17/2+ level lies slightly above the 21/2+. I n summary, the effective-interaction method can provide useful and accurate predictions of nuclear levels in regions where sufficient data are available to evaluate all of the matrix elements needed for inclusion of the important configurations. Presently, precise data on levels in all mass regions, but particularly for elements below the rare earths, are being obtained in such volume that detailed effectiveinteraction calculations in many regions should soon become possible. A note of caution, however, should be made. One should not be overwhelmed by good predictions of energy levels. As noted above, moderately good agreement in positions of energy levels in the calcium region is ohtained with the assumption of pure fi/, configurations, although there is 642
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Journal of Chemical Education
clear evidence for mixing of configurations involving one or two neutrons in pa/, orbitals. A certain amount of mixing can he absorbed in the effective-interaction method without manifesting itself prominently via energies of levels. The problem of "concealed" configurations in the effective-interaction treatmentis perhaps best illustrated by the calculations of Cohen et al. (32) on "pseudonium" nuclei. These are hypothetical nuclei of A = 4 0 4 5 in which the last 4-12 neutrons occupy the d,/, and f,/, orbitals which are defined as degenerate. They then calculated the positions of the low-lying levels in the nuclei with the use of a simple analytical residual force. Because of the assumed degeneracy of the d a / , and j7/, orbitals, most of the levels of the hypothetical pseudonium nuclei are of very mixed character. To test the ability of the effective-interaction treatment to show up configuration miximg, the levels were treated via effective interactions assuming only j7/, configurations. The five parameters needed were determined by a least-squares fit to 31 energy levels. The resulting fit to the energy levels had an rms error of only 161 kev even though d.,, mixtures were completely ignored! It is generally known that energies of levels are often not too seriously affected by mixing, hut electromagnetic transition rates and nuclear multipole moments are normally quite sensitive to admixtures of other configurations. However, in the example of the pseudonium nuclei, even the electromagnetic properties were rather well fitted by the effective-interaction calculation that ignored d,/, mixing. I n summary, although the effective-interaction technique is very powerful in many applications, one must exercise great care to ensure that all important configurations are included in the calculation. We have discussed some of the great strengths of the shell model in terms of systematizing existing knowledge of nuclear levels and predicting as yet unmeasured quantities. Furthermore, the orbitals provided by it may be used as the starting point for other models such as that discussed below. However, there are some mass regions, most notably 150 < A < 180 and A > 220, in which the following properties do not seem easily explainable with the shell model: low-lying (-40-100 kev) excited states in even-even nuclei, very large electric quadrupole moments, large E2 transition rates, and observed spins of low-lying levels. These features are well handled by the collective model in which the nucleus is assumed to be spheroidally deformed. Superconductor Model
The superconductor model uses the shell-model orbitals as a starting point and considers as residual interactions a strong pairing force, similar to the 6 force treated above, and a long-range "Pzforce." The name "superconductor model" derives from the fact that the mathematical treatment is in part similar to that developed by Bardeen, Cooper, and Schrieffer (55) for superconductivity in metals. Let us assume that the short-range force is the hypothetical "pairing force;" that is, in the configuration j2, there is no residual interaction except for the coupling J = 0. If we place two neutrons in a given orbital, say gv,, we can form the J = 0 coupling by placing them in the magnetic suhstates ml = -mz = 7/2. If the poten-
tial well is spherical, there is no reason for the neutrons to remain in those substates. In fact, they can maximize their pairing interaction by occupying all of the substates with equal probability, the only requirement being that m, = -mz in order to conserve total angular momentum. By occupying all substates with equal probability, the neutrons give an overall spherically symmetric probability distribution. Now consider placing two neutrons in the shell 50 5 N 5 82. If the orbitals of the 5 subshells available, i.e., g7/,, d8,?, s>,,, hu/,, and d./,, were degenerate, the pair of neutrons would occupy all substates of all the orbitals with equal probability, again maintaining ml = -m2 and J = 0. In real nuclei, however, the single-particle orbitals are not degenerate, but lie at various energies r j above an arbitrarily defined zero energy as shown in Figure 8.
Pairing
calculated occupancies of neutron orbitals in '14Cd. Note the qualitative similarity between the occupancy of nucleons as a function of energy and that of noninteracting fermions (i.e., particles obeying Fernli-Dirac statistics) occupying a set of energy levels at a given temperature. Note that the number of quasiparticles in this discussion has much the same meaning that seniority has in shell-model discussions, namely, the number of particles not coupled to J = 0. The principal difference between the two approaches is that according to the simple shellmodel description, particles fill the orbitals discretely, whereas, in the superconductor model, there is a continuous occupancy function, eqn. (9). If we add a neutron to l14Cd,into which orbital does it go to attain the lowest energy configuration? Again there are counteracting effects: e j of the added particle is minimized by placing it in the lowest orbital, but the particle interferes least with the pairing interactions if i t enters the highest orbital, i.e., the one which is least occupied by the paired particles. Quantitatively, these effects balance so as to give minimum energy when the odd particle is placed in the orbital closest to the Fermi energy, X (thus, the analogy with chemical potential). The energies of the singlequasiparticle levels are given by Ej
Shell model
= d ( L j
- A)Z
f As
(10)
For the lowest single-quasiparticle level, s j - X = 0, and E j = A. This result is obtained because we measure the energies of the levels from the quasiparticle "vacuum," i.e., the ground state of the hypothetical even-even nucleus having the same number of nucleons. Likewise, the ground state of an odd-odd nucleus would have energy Ej. Ejv = 2A. In Figure 9 we have plotted the energies El of t,he single-quasiparticle levels
+
figure 8. Leh: Single-neutron energy levek in the region of A = 114Right: Occupancy V" of the levels in '"Cd based on tho pairing model and on the simple shell model.
(Here we are taking a careful look at nuclei having A -114; therefore, the ordering of levels with the shell is not exactly the same as obtained in the overall diagram of Figure 1.) There are thus two counteracting effects: the single-particle energy is minimized if the neutrons are placed in the lowest orbitals; but the pairing interaction is maximized if all substates of all orbitals are occupied equally. The role of the pairing interaction is thus somewhat analogous to that of entropy in statistical thermodynamics and the resulting occupation probabilities in the nuclear case are derived with the use of an "undetermined multiplier," similar to the method of deriving the Boltzmann distribution. As a result of these calculations, the occupancy V j Zof each substate is predicted to be
in which A is essentially the pairing energy defined above (see Figs. 2 and 3) and X is called the Fermi energy or chemical potential. In Figure 8 we show the
Figure 9. The ringie-quoriparticle energies, Ej, as a function of energy Minimum of the curve occurs of the ringle-neutron levels, r j , in I'Cd. a t r = A, where E = A. At the far right we show the energies of the ~ingl~-quosiparticle levels relative to that of the lowest lr1/,1 member.
of "5Cd calculated from eqn. (10) using the parameters suggested by Kisslinger and Sorensen (16), X = 1.86 Mev, A = 1.30 Mev. As shown for 42Cain Figure 5 , neither the pairing interaction nor the 6 force is a good approximation by itself: clearly one needs additional residual interactions to obtain a better approximation. Finiterange nuclear forces may be expressed in terms of the expansion Volume 44, Number 1 1, November 1967
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in which v(rI,r) is a salar function of the lengths of radial vectors, and P.(cos WE) is the nth order Legendre polynomial of the cosine of the angle w , ~between the radial vectors. The first contributing term of this expansion that is dependent on the relative orientation of j~and jn is Ps( cos WD) = (3 eosBw12 - 1)/2. Most calculations with the superconductor model employ as an additional residual force the Pzterm of eqn. (ll), calling it the "I% force." Just as in the case of a 6 force, application of the Pzforce to a jz configuration gives J = 0 as the lowest energy coupling, but it is not so far removed from the other couplings as for the 6 force shown in Figure 4. Application of the Paforce to an even-even nucleus affects most strongly the 2+ two-quasiparticle lev&. A state made up of a linear combination of many such levels is brought down near the ground state. In an even-even nuclide having both valence protons and neutrons, one would obtain two such states, one of excited neutrons and the other of protons, in the absence of a strong P, force between neutrons and protons; thus, IGmlinger and S o r e m , for example, assumed that the strength constant of the n-p P g foroe is the same as that for the p-p and n-n interactions. The properties of the 2+ level are much like those of a classically vibrating nucleus, i.e., a nucleus that is, on the average, pherical but with prolate and oblate vibrations as shown in Figure 10. It is often called the "one-phonon" level, and
Sphere
Inverted Peor
Figure 10.
Schema* picture of qumdrupoia ana octupde vibration. nvsleur thatir, on tho overage,rpherical.
d
its energy is one vibrational phonon, h. At an energy 2 h , we would predict a triplet of "two-phonon" levels having the allowed couplings of two 2+ phonons: 0,2, 4+. These levels result from application of the PIforoe to the four quasiparticle levels. At 3 h , one expects a quintuplet, 0, 2,3,4, 6+ of three-phonon levels. A h , not far above the two-phonon levels, one generally observes a 3- octupole vibrational level. In Figure 11we compare the observed levels of l14Cd with those p r e dieted according to the vibrational picture. Although
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there are some discrepancies, crude agreement with the vibrational picture is obtained. Below we discuss in more detail the limitations of the vibrational picture. What is the effect of the Pzforce on the levels of an odd-A nucleus? As a specific example consider ""3.3, which we may naively depict as a IWd "core" plus an odd neutron. We have seen that we can make onequasiparticle excited states by moving the odd neutron to various singl&particle levels. Also, we should be able to excite the "4Cd core to the one-phonon level at an energy f% (550 kev). As a zeroth approximation, in l16Cdwe should expect to find, at about 550 kev above the ground state, a degenerate set of levels consisting of all the possible couplings of the 2+ phonon with the spin vector of the quasiparticle, in this case only two % and %, (see Fig. 12). Other excited states I states can be constructed by adding a 2+ phonon to the other onequasiparticle states as shown in I3gure 12. Additional levels could be constructed by ooupling one octupole phonon or two quadrupole phonons to the one-quasiparticle states, but these have not been treated in detail. Note that in the zeroth-order we predict five 8/e+ levels below 1.5 Mev in II6Cd. There is considerable codguration mixing as the wave function of each level contains contributions from four other states and the energies of the mixed states are somewhat different from those of the crude approximation. Kisslinger and Soremen's (16) calculations take &mixing intoaccount andtheirpredictedlevelsfor lL5Gdare shown in Figure 12. At the righbhand side of F i 12 are shown the levels of "=Cd as observed in the p d a y of "6Ag ($4)and from the nuclear reactions ll'Cd(d,p) and IWd(d,t) (6,36). Although the model does not predict the level positions exactly, it does e x p k many observ~r tions such as the large numbers of lnw-lying levels of the same spins and parities, here, in particular, the several % and %f levels. Became of selection rule8 on the 6 and y decay processes and on the nuclear reactions, some levels have undoubtedly been mimed in the ex-
-
the surface of.
644
Figure 11. levels of "'Cd predided f r m the rvpsrcanductor model. At the I&, Wc-quoriparlicls bveb f r w application of the 6 force only, senler. vibrational ~ r s d l c t inormlirsd ~ to ernsrimontol first excited
Figure 12. Superconductor model predictians for 'lSCd. At left, zerothorder predidion of levels bared on coupling of the various ringle-quoriparticle levels (see Fig. 91 with a 2 + phonon Center; More refined prediction (ref. (1611 including configuration interaction between levels of same spin and pority. At right, the experimental data from reference 161. (341. and 1351.
perimental study. The agreement obtained for 'lSCd does not represent an isolated case for which the predictions happen to coincide well with observations. There are rather large regions of nuclides in which the agreement is con~parable,for example in the odd-A iodine isotopes (e.g., see ref. (56)). The strength of the superconductor model is that il. successfully predicts the large numbers of low-lying levels of the same spin and parity that are observed. Furthermore, it also predicts many E2 transitions that, are observed to be much faster than expected on the basis of the shell model. I n terms of the superconductor model a 5 / z + 5/2+ transition, for example, can proceed not only via the jump of a quasiparticle from a d~,,to an s~,, orbital, hut also by the 2+ + Of phonon transitions that areallowed viasomeof theconfigurations mixed into the wave functions (37). As the latter are collective transitions, i.e., they involve a large number of the protons, they are often much faster than a single quasiparticle transition. It remains to be seen whether or not the shell model with considerable mixing can explain the fast E2 rates. The shell and superconductor models represent extreme limits of the same model. Most shell-model calculations to date have involved only pure configurations or a mixture of a small number of configurations, e.g., the Auerbach-Talmi treatment of the 90Zrregion (39). Conversely, the superconductor wave functions represent a very complicated mixture of configurations, as suggested by the occupancy diagram of Figure 8. As one mixes in more configurations in shell-model calculations they should approach the superconductor treatment, although the former quickly becomes intractable. I n spite of its usefulness, the superconductor model has a number of weaknesses. First, the notion of vibrational phonons is a very poor approximation-much poorer, for example than the rotational hand structure observed in deformed nuclei. Even in "Cd, which is one of the best known examples of a vibrational nucleus,
-
the two-phonon triplet is far from degenerate and occurs considerably above 2 fiw. Also, two levels occur very close to the presumed triplet. There are very few nuclides for which more than two of the predicted three members are known. The third member often seems to be missing. A few years ago one could suppose that the third member had been missed because of the inadequacy of the experiments performed. However, in view of the many excellent results now being obtained from inelastic scattering experiments, one should probably 'admit that the predicted third member often does not occur in the expected energy region. Possibly the shell model with extensive configuration mixing will be able to explain the "vibrational" levels via effective-interact,ion calculations. Note in Figure 7 that the 2nd and 3rd excited states of g2Zrare in much better agreement with the experimental data than is the vibrational picture. A second major difficulty in the superconductor model, a t least as treated by Kisslinger and Sorensen (16), is the inadequate handling of the n-p force. I n that treatment valence neutrons and protons are assumed to interact only via the P,force. The analysis of spins of odd-odd nuclides discussed above suggests that there is also a short-range attraction as well as a spindependent interaction between neutrons and protons. Much of the very recent theoretical work on nuclear structure involves attempts to calculate the positions and properties of levels in complex nuclei with the use of "realistic potentials" (i.e., internucleon potentials that reproduce the results of two-nucleon scattering experiments) and, often, with the Hartree-Fock method (58, 30). As yet most of these calculations are restricted to nuclei a t or very near closed shells or subshells, for example, the calcium region (do), near 90Zr (58), the tin isotopes (dl), and the lead region (35). Although this t.ype of calculation is still rather primitive, the results so far are encouraging, and we can expect considerable activity in this field over the next few years. Acknowledgment
We are grateful to the many students who, by their probing questions, have forced us to re-examine and clarify our ideas about nuclear structure. We are indebted to Professor John Rasmussen, Professor Gordon Goles, and Dr. James Ball for their constructive criticisms of the manuscript. We thank the former Editor, Professor Willlam Kieffer, for his encouragement, pat,ience, and helpful suggestions. Literature Cited ( 1 ) JUDD,B. R., "Operator. Techniques in Atomic Spectroscopy," MeGraw-Hill Book Co., Inc., New York, 1963. (2) TALMI, I., Nucl. Phya., 83. l(1966). (3) R ~ c i mG., , Phya. Rev., 61,186 (1942); Bid., 62,438(1942); ibid., 63, 367 (1943); ibid., 76, 1352 (1949). (4) NILSSON, S. G., Mat. Fys. Medd. Dan. Vid. Selsk., 29, No. 16 11955). ~-~..,~
(5)
HGLANDER, J. M., Nuel. Inslr. and Methods, 43,65 (1966).
(6) SILVA,R. J., (1964).
AND
GORDON, G. E., Phys Rm., 136, B618
(7) COOK,C. S., "Structure of Atomil Nuclei," D. Van N + strand and Co., Inc., Princeton, N. J., 1964. (8) FRIEDLANDER, G., KENNEDY, J. W., and MILLER,J. M., "Nuclear and Radiochemistry," (2nd ed.) John Wiley and Sons, Inc., New York, 1964.
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(9) PRESTON, M. A., "Physia of the Nucleus," Addison-Wesley, Reading, Mass., 1962. B. H., J. CHEM.Enuc., 37,610 (1960). (10) FLOWERS, (11) MAYER,M. G. AND JENSEN,J. H. D., "Elementary Theory of Nuclear Shell Structure," John Wiley and Sons, Inc., New York, 1955. TALMI,I., Rev. Mod. Phys., 34, 704 (1962). DE-SEALIT. A.. AND TALMI.I.. "Nuclear SheU Theorv." , DE-SHALIT, ~ c s d e m i cpress, Inc., ~ e ~ ko r k 1963; Nuel. Phys., 7, 225 (1958). MOTTELSON, B. R., i n "Proceedings of the International School of Physics, Course XV, Nuclear Spectroscopy," (Editon G. RACAH),Academic Press, Inc., New York, 1962, pp. 44-99. (15) LANE,A. M., "Nuclear Theory," W. A. Benjamin, Inc., New York, 1964. L. S.. AND SORENSEN. R. A.. Rev. Mod. Phus.. (16) . . KIBSLINGER. " , 35,853 (i963). ' (17) HYDE,E. K., PERLMAN, I., AND SEABORG, G. T., "The Nuolesr Properties of the Hemy Elements," PrenticeHall, Inc., Englewood Cliffs, N. J., 1964, vol. 1, ch. 3. (18) NILSSON,S. G., AND NATHAN,O., in "Alph*, Beta., and Gamma-Ray Spectroscopy," (Editor: K. SIEGBMN), North-Holland Publishing Co., Amsterdam, 1965, vol. 1, ch. 10. (19) HAXEL,O., JENSEN, J. H. D., AND SUESB,H. E., Phw. Rev., 75, 1766 (1949). M. G.., Phm. Rev.., 75.1969 11949). (201 . . MAYER. " . . . (21) BRENNAN, M. H., AND BERNSTEIN, A. M., Phys. Rev., 120, 927 (1960). (22) RAGAINI,R. C., M.I.T., unpublished eompilstion, 1967.
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