I
2.D
I
Nuclear Physics A!33 (1969) 33~49; (~) North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprintor microfilmwithout written permissionfrom the publisher
AN ITERATIVE M E T H O D FOR THE CALCULATION OF NUCLEAR LEVEL D E N S I T I E S FREDERICK C. WILLIAMS, JR. United States Nat'al Academy, Annapolis, Maryland +
Received 24 March 1969
Abstract: An improved method is presentcd for the calculation of nuclear Icvcl densities from single particle energy levels. The calculation is performed using a statistical approach but avoiding the usual mathematical approximations. Recursion relations arc given for the calculation of coefficients of a finite ordcr partition function for the case of non-interacting fermions and for the case in which the effect of residual interactions is approximated by a diagonal pairing energy for doubly degenerate states. Computcr calculations of the level density arc demonstrated for scvcral simple models exhibiting structure in the single-particle lcvcl scheme and for thc detbrmcd nucleus single-particle Nilsson levels for the nuclei 3°Si, 31p, 32p and "~2Sas a function of deformation paramctcr ~/and pairing cnergy. I. Non-zero values of both parameters are required to obtain reasonable agrcement with observed level densities at or below 8 McV.
1. Introduction In nuclcar reaction theory it is very often necessary to have an accurate estimate of the average density of energy levels of highly excited nuclei as a function of excitation energy and other cGnstants of the motion. There exist formulas for the level density which are based on certain mathematical a p p r o x i m a t i o n s valid only for systems cont a i n i n g large n u m b e r s of particles, but the advcnt of high speed computcrs makcs it possible to use methods of calculation which do not depend on closed formulas. C o n s e q u e n t l y a systcmatic method has bccn developed for the calculation of level densities dircctly from single-particle energy levels. The usual approach to the level density calculatiop, takes the i n d e p c n d e n t particle model as its starting point. It is not at all obvious that ',his model is valid in its simple form for highly excited nuclei but a real test of the a s s u m p t i o n is obscured by the a p p r o x i m a t i o n s uscd to obtain the closed formulas. The present calculations indicate, in fact, that a completely independent particle model overestimatcs tiae level density. "l'hc i n d e p e n d e n t particle a s s u m p t i o n allows one to write a nuclear partition function in a simple form using the single-particle energies. Such a formulation was developed by Bethe 1,2) and Van Lier and Uhlcnbeck 3). Calculation of the level density fl'om the pal'tition function n o r m a l l y proceeds via the D a r w i n - F o w l e r method 4) which involves tx~.o major a s s u m p t i o n s : (i) that the single-particle level scheme can be a p p r o x i m a t e d as a c o n t i n u o u s distribution, and (ii) that the saddle point a p p r o x i m a tion is applicable. + A portion of this work was performed at the University of North Carolina with support from the U.S. Atomic Energy Commission. 33
34
F . C . WILLIAMS, Jr.
Assumption (i) is clearly invalid for light nuclei as the separation between singleparticle levels often approaches 1 MeV. The use of this approximation can be avoided as was shown by Bloch 5). In addition Margenau 6) pointed out that the saddle point approximation is not very good in the nuclear case, and recently it has come to light that ambiguities arise in its application 7.8). Although the formulas developed from these approximations account reasonably well for general trends of level densities with mass number 9-17) and can be improved on an ad hoc basis by inclusion of corrections for shell and pairing effects they are in general model independent 18) and hence are not very reliable. Although some authors have calculated level densities by direct enumeration and classification of states from one model or another 19-22) this technique is cumbersome and does not lend itself well to wide application. In the next section a simple method of calculation suitable for computer t, se is developed for the independent particle model and extended to the case where interactions are represented by a diagonal pairing energy. Computer programs have been written implementing this method and some of the results are presented in sect. 3. 2. Method of calculation _2.1. N O N - I N T E R A C T I N G
FERMIONS
The partition function for a system of non-interacting fermions occupying energy levels belonging to a set {el} can be written as is)
Z(x, y) = l'-I (1 +xy ~') = i=1
p(N, E)xNy e,
(2.1)
N,E=0
where the coefficients in the summation are the number of ways of obtaining total energy E with N particles in the system. These coefficients are equivalent to the level density of an N-particle system. Instead of considering eq. (2. I) as a relation between an infinite product and an infinite sum, we can detine a finite order partition function Zr(x, y) and a finite order density of states pr(N, E) such that K
K
ZK(X, y) = I-I ( l + x y " ) = • i= I
in which p(0, 0) = I. In the limit
N=O
EK
Y'
px(N, E)xNy ~,
(2.2)
E=O
Z x approaches Z. Noting that
ZK = (1 +xy'~)Zr_l,
(2.3)
it is evident that the recursion relation
pK(N, E) = pK-,(N, E)+ p x - i ( N - l, E--eK)
(2.4)
holds between the coefficients of ZK and Z K_ 1. Use can also be made cf the fact that p(N, E) must be zero for all values of E less than Eo, the ground state energy, given by N
Eo = Z i=l
(2.5)
N U C L E A R I.EVEI. D E N S I T I E S
35
by introducing the excitation energy Q = E - E o to rewrite eq. (2.4) as
pr(N, Q) = p~_ ,(N, O)+p,,:_ , ( N - 1 , O +e,.--eK).
(2.6)
In order to compute the level density of a system containing two kinds of particles, it is only necessary to note that if the proton and neutron systems are independent the grand partition function is a product of neutron and proton factors. In this case the level density of the composite system can be obtained from the t\)rmula for the c o e ~ cients of the Cauchy product of two infinite series: U
p(Z, N, U) = 7. ON(N, Y)pz( Z, U-Q),
(2.7)
O=0
where N and Z are neutron and proton numbers, U is the sy.~tem excitation energy, and a,,~ and Pz are the neutron and proton level densities. 2.2. INTERACTING FERMIONS For our purposes it is convenient to assume that most of the effect of short range interactions at moderate excitation energies can be lumped into a parameter Ai called the diagonal pairing energy, which represents the amount by which the energy of a pair of like particles in the same energy state but with opposite angular momentum projection is depressed compared to the sum of the single-particle energies. The excitation spectrum arising from this assumption bears a crude resemblance to that of the seniority scheme for spherical nuclei. Lang and LeCoutcur 23) and Herring 24) have pointed out that such a pairing energy can be included in the partition function by writing Z in the form ,.1,,
Z = I-I (1 + 2xy~' + ,x23''2*'-a')), ,
(2.8)
i=l
in which each factor represents the possible combinations to be obtained from two degeneratc levels of energy ~. For finite partition functions of the form indicated by eq. (2.8) a recursion relation analogous to eq. (2.4) is
pK(N,E) = PK-,(N,E)+2PK-,(N--1,E--eK)+PK-,(N--2, E--2~:K+AK).
(2.9)
The ground state energy in this case is reduced by the sum of the pairing energies of pairs of nucleons at or below the Fcrmi level, and hence the relation corresponding to eq. (2.6) is more complicated:
PK(N, Q) = PK-,(N, Q) + 2p,,_ , ( N -
1, O + ~.c..' + , , ~ - ~,< - { E ; N ] - [½(:',' - l)]}af,..,., +, ,0
+ p ~ _ ~(N - 2 , Q + et,r0~ + ~)1+ q~,~q- 2e~ +A K- At,.xi ).
(2.10)
The notation [ . . . ] in this equation indicates that the index to be used is the largest integer less than or equal to the quantity enclosed. For two kinds of particles eq. (2.7) still applies.
36
F.C. WILLIAMS,Jr.
3. Numerical results 3.1. THE U N I F O R M SPACING MODEL
The uniform spacing model can be used as a first approximation to the singleparticle nuclear states for the purpose of studying level densities. Margenau 6) pointed out many years ago that the levels of a Fermi gas, that is a collection of particles
/-
103
0°
../ /•
1021
• •
P(Q) ~
/.
• Exact 0
I01
,~
/. '°°J
,'o
--~'o
Q (orbitrary unifs)
20
Fig. 1. Comparison ofthe approximate level density formula with exact values in the simple uniform spacing case.
confined to an infinite spherical square well, are too degenerate and too widely spaced to represent well the properties of the nucleus in level density calculations. The uniform spacing model serves well for nuclei because in most cases nuclear states are split e n o u g h so that the level distribution is relatively smooth.
NUCLEAR LEVEL DENSITIES
37
It is interesting to note that the simplest form of the uniform spacing model, the case with one kind of particle and no degeneracy, is equivalent to a combinatorial problem in number theory, the partitioning of integers, first proposed and solved by Euler zs) in 1753. An approximate solution for Euler's problem was later derived by IO
R - Shell E f f e c t Level Density Ratio A:
3"0
B:
8-,I
c:
8-.2
• ;
Present C o l c u l o t i o n
: Eq.(3.3)
.i 0
I I
I 2
n(Shell
Occupation
I 3 Number)
I 4
5
Fig. 2. An example of the Rosenzweig shell effect for shells separated by ten units o f energy with five levels per shell. The splitting 6 varies from 0 to 0.2. The solid circles are exact calculations and the lines are computed from eq. (3.3).
Hardy and Ramanujan 26). Their first approximation is equivalent to Bethe's level density formula
p(V) = exp [(JnZgo U) *] (48)½U
(3.1)
38
I:. c. WILLIAMS, Jr.
in which U is the excitation energy and go the single-particle level density. A comparison of the approximate formula (3.1) and the exact results of Euler is shown in fig. 1. It is evident from the graph that the formula is good to about 10 ~ , but the introduction of structure into the single-particle level scheme destroys this agreement. I0
3
2,
--O
I
•
.
Exact Calculations
Approximation
Rosenzweig's
p (unfilled) r =
I
0
p(filled)
I
1
I
I
I
I
5
tO
15
20
25
30
Q ( a r b i t r a r y units)
Fig. 3. The odd-even level density ratio produced by the Roscnzweig effect for doubly degenerate levels.
Certain of the features which appear in natural single-particle schemes can be built into a uniform spacing model for use with the formulas of subsects. 2.1 and 2.2 so that various effects can be studied separately, although in real nuclei they occur simultaneously and to some extent overlap. The first of these is the Rosenzweig effect 27),
NUCLEAR LEVEL DENSITIES
39
which occurs because of a degenerate level at the Fermi energy. If the Fermi level is g-fold degenerate and contains n particles in the ground state, a number of rearrangem e n t s f o f the particles is possible w h e r e f = g!/(g-n)!n!. Rosenzweig showed that for a uniform spacing model the effect of rearrangements persists with increasing excitation energy. The effect is expressed as a shift in the effective excitation energy, given by dQ = g 12
1 (n_½g)2" 2g
(3.2)
The result is an increase above the normal level density and has a maximum in a halffilled shell. Near the shell closure the level density is the same as for the non-degenerate (
A
B
C
D
E
20 19 18 17 16 15 14 13 12 II I0 9
--of
cf I
"f
Ef
"t ,f
8 7 6 5
4 3 2
I C Each level is doubly degenerate. An even number of particles occupy the system in each case.
Fig. 4. L e v e l s c h e m e s used 1o s t u d y the g a p effect.
case. Rosenzweig has also shown 28) that the enhancement of the level density in the middle of shells persists even if the shell levels are not exactly degenerate but are separated by a small fraction 6. As 6 increases the effect gradually disappears. Rosenzweig's expression for the effective excitation energy in this case is 28) I AQ = ~-~g-- ~-gg(n--½g)2+~5[(n--}g)2--¼g2]+ 2~2g(g+l)(g--l),
(3.3)
where g and n have the same meaning as before and 6 is the ratio of the separation of individual level~ to the inter-shell spacing.
40
F . C . WILLIAMS, Jr.
A comparison of this formula and a simple case is shown in fig. 2. ResultsWfor three A level schemes are shown. One is highly degenerate, one intermediate, andIone uniformly spaced. The computed level density ratio at a given excitation energy is plotted as a function of shell occupation number n. The calculations of the present work are 10 4
10 3
_
l
p(Q)
i0 2
Letters designate level schemes shown in Fig. 4
I01
I0
2=0
i 30
Q (arbitrary units)
Fig. 5. The computed level density corresponding to the level schemes of fig. 4. shown as dots and values derived from the Rosenzweig approximation, formula (3.3), as a solid line. The agreement is satisfactory. Experimental evidence indicates that such effects are actually observed as N or Z varies within a major shell 29,30). The Rosenzweig effect appears to be at least partly responsible for odd-even changes in the level density for neighboring nuclei. Calculations have been performed on a
,~UCLEAR LEVEL DE.~S~T,ES
41
level scheme where each state is doubly degenerate and separated from neighboring states by two units of energy. When the Fermi level is occupied by two particles in the ground state, the level density is almost identical with the results for no degeneracy. When the Fermi level is half-filled, however, the level density is enhanced. The cairo
QQ 2q e 0 0
0O
r
Degeneracy Effect with Pairing Energy /% " 2 units
~(unfqled) r. j p(filled)
.I
I
I
I0
I
15
I
20
[
25
30
O (arbitrary units)
Fig. 6. The odd-even level density ratio with pairing energy included in the calculation. culated ratio of the two is shown in fig. 3 along with a smooth curve representing values of the ratio derived from eq. (3.2). The approximation is good at large Q. This odd-even effect is enhanced by the introduction of the pairing energy, as is shown below. Between major shells there is usually a large energy gap in the single-particle level scheme, at least for spherical nuclei. The existence of this gap helps produce a large
42
F.C. WILLIAMS, Jr.
reduction in the level density at shell closure. A series of level schemes with uniform spacing and a gap of various sizes and positions is shown in fig. 4. The level density calculated for each scheme is shown in fig. 5 as a function of excitation energy. The reduction of the level density increases rapidly with size of gap. It is also clear from IO3!
Without 102 _
p (Q)
I0 u
density in odd particle number system.
Level
I I0 0 0
I
I
I
2
I
3
I
I
I
l
4 5 6 7 Q(arbitrary units)
B
I
9
II0
I
I
II
12
113
Fig. 7. The level density for the odd particle system with and without pairing energy.
fig. 5 that the existence of one unfilled level just above the Fermi level causes the level density to increase again. Thus, at shell closure, the gap effect and the disappearance of the Rosenzweig shell effect cooperate to reduce the level density drastically compared to mid-shell nuclei.
NUCLEARLEVELDENSITIES
43
The inclusion of a pairing energy in the partition function as outlined in subsect. 2.2 also reduces the calculated level density. Numerical calculations were performed for a system of doubly degenerate levels with uniform spacing and a pairing energy of 2 units. Odd particle number systems exhibit a reduction in level density, but the re-
-~~/~
N(~)
10 2
~
~
_
TOTAL STATES UP TO 8.0 MeV
iOI
1 --6
I -4
I -2
I 0
I 2
I 4
I 6
-r/(Deformotion)
Fig. 8. Calculatedtotal numberof levels below 8 MeV for ~°Sias a functionof deformationand pairingenergy. duction for even particle number systems is much greater. Thus the odd-evcn shift produced by the Rosenzweig effect is enhanced and persists to higher energy. The ratio of the " o d d " level density to the "even" level density is plotted in fig. 6. Compare this with fig. 3. The " o d d " level density with and without pairing energy is shown as
4,-1
F.C. ".VILLIAMS, Jr.
a function of excitation energy in fig. 7. The reduction by pairing is still fairly large here. It seems possible to explain qualitatively most of the observed variation of level densities with mass number by some combination of the simple situations described l0 4 A
(MeV)
0.0
N(~/1103 _ ~
2.0
32
p
TOTAL STATESUP TO8.0 MeV i0z
I 01
I
-6
I
--4
I
f
--2 0 "g(Deformation)
I
2
f
4
I
6
Fig. 9. C a l c u l a t e d t o t a l n u m b e r o f levels b e l o w 8 M e V for a2p as a f u n c t i o n o f d e f o r m a t i o n a n d pairing energy.
above. Direct comparison with experiment, however, should be made using a more sophisticated single-particle model. Such results are presented below.
NUCLEARLEVELDENSITIES
45
3.2. CALCULATIONS BASED ON THE NILSSON MODEL
The a p p l i c a t i o n o f the Nilsson model 31) in a naive way to excited states has obvious limitations. Nevertheless, the results indicate that there is some value in ass u m i n g that the position o f single-particle levels is stable. Energy levels for nuclei with IO4 (MeV)
~
2.0
iO2 --
3z S Total States up to 8 0 MeV
lob
-6
l
-4
l
1
-2
l
0
k
+2
l
+4
I
+6
,q(Deformotion) Fig. 10. Calculated total n u m b e r o f levels below 8 M c V for 32S as a Function o f d e f o r m a t i o n aad p a i r i n g energy.
30 _-< A =< 32 were computed from Nilsson's tabulation for a range of energies ~ithin _ 10 MeV o f the Fermi level for values o f the d e f o r m a t i o n p a r a m e t e r q = - 6 , - 4 , - 2 , 0 , + 2 , + 4 , + 6 , using heb o = 41A -s" MeV with A = 32. The energy scale varies only a b o u t 2 o~ from A = 30 to 32 so that the same levels are satisfactory for all nuclei
46
F . c . WILLIAMS, Jr.
in this region. The actual deformation 6 is related to the shape parameter r/ by the equation 6 = nq, rc being the relative strength of the spin orbit coupling. Little experimental information is available on the absolute magnitude of the deformation for the nuclei in question, so that q was treated as a parameter and assumed to remain constant for all values of the excitation energy. In addition the Coulomb energy was ignored. Constant pairing energy A was assumed for all energy levels. Calculations were performed with. A = 0, 1, 2 and 3 MeV. It is found that for the mass range in question: (i) the number of states predicted for a given nucleus is a strong function of both deformation and pairing energy and is most often reduced by increasing either
102 i
102
N(E)
N(E) iO l
I0 i
3osi
i
_~
r/=+6
31p 7"/=*4
A=2.0 MeV
iO (
0
I
2
T
J
4 6 E (MeV)
I
8
Z~=3.0 MeV
IO
~0 °
0
I
2
I
4
I
6 E (MeV)
f
8
I0
Fig. I I. Cumulative number of lcvels as a function of excitation cncrgy for 3°Si.
Fig. 12. Cumulativc number of levels as a function of excitation encrgy for 3zp.
H e a v y line is calculated curve.
H e a v y line is calculated curvc.
quantity; and (ii) too many states are predicted for the nuclci studied unless moderately large values of both pairing energy and deformation are assumed. In figs. 8 and 9 the calculated total number of states with energy less than or equal to 8 MeV is plotted for 3°Si and 32p as a function of deformation and pairing energy. Clearly the effect of the pairing energy is more pronounced for the even nucleus. The odd-even level density ratio is in the expected range except for zero deformation and pairing energy. The decrease in level density and hence total numbers of states as the magnitude of the deformation increases is due to the gradual diminution of the Rosen-
NUCLEAR LEVEL DENSITIES
47
zweig effect because of the destruction of the degeneracy of the shell states as the potential is deformed. At slightly higher particle numbers, at the closure of the d t_ subshell, this decrease will disappear because the gap effect will predominate over the Rosenzweig effect. In that case the level density is expected to be lower for small deformations, a result in accord with the experimental observation that closed shell nuclei tend to have zero deformation and large level spacing. Even in the case of 32S, the maximum is less distinct. This is illustrated in fig. 10 with another total states plot. A direct comparison of the calculated level density with observed levels can be made. Combinations of the Nilsson levels produce nuclear states with all possible
102
I0 ~
N(E)
N(E)
i01
I01
3ZS
A=3.0
I0
I 2
I 4
E (!lllv)
I 6
MeV
I 8
I0
Fig. 13. C u m u l a t i v e n u m b e r o f levels as a function o f ex citation energy for 32p. Heavy line is calculated curve.
I0
_1 2
4
E (MIV)
A-3.0
MeV
6
8
I0
Fig. 14. C u m u l a t i v e n u m b e r of levels as a function o f e xc i t a t i on energy for 32S. Heavy line is calculated curve.
values of the Z-projection of angular momentum M, so that, unless one calculates the distribution by total angular momentum the observed energy levels must be counted ( 2 J + 1) times, where J is the experimentally determined angular momentum. Unfortunately this value is not always known, and for some nuclei, values are missing for many levels. However, the general properties of the spin distribution are well known and the range of probable spin values can be estimated. The spin cut-off factor a is expected to be about 2 in this mass region, so it is reasonable to suppose that most spin values will lie between 1 and 3. Plots of the cumulative number of levels, N(E), are shown in figs. I 1 through 14. Energy levels and spins are taken from the tables of
4~
F . C . V~ILLIAMS9 Jr.
Endt and Van der Leun 32) and K u n z and Schintelemeister 33) for the nuclei 3°Si, 3tp, 32p, and 32S. The experimental count covers a range of values, where the lower limit assumes J = 1 and the upper limit J = 3 for unknown spins. On the same graphs are shown the calculated N(E) for the set o f parameters q and A which best reproduced the experimental curves. In general the slopes of the calculated and experimental curves are in better agreement than the absolute numbers although in some cases the numerical agreement is satisfactory. 104
PI~
I.O
;.o i0 ~
'1'
L
-6
-4
I
I
31 z p
-2 0 2 "17 ' (DEFORMATION)
I
I
4
6
Fig. 15. A v e r a g e level d e n s i t y at 8 MeV for 32p as a f u n c t i o n o f d e f o r m a t i o n a n d p a i r i n g energy.
Other experimental information is available on the nucleus 32p in the form of a level spacing value from neutron resonances. The average resonance spacing is found to be 21.8 keV 34). Since p-wave scattering is possible in the energy range considered, i.e. 0-1 MeV, correction via the usual spin-distribution formula with a = 2 leads to a value of 418 levels/MeV for the total density o f levels of all M-values at about 8 MeV. This density of states is represented in fig. 15 by a heavy horizontal line. The computed average level density at 8 MeV is also plotted as a function o f deformation and pairing energy. It can be seen that agreement can be obtained for several combinations of the parameters, but only one, r/ = + 4 , A = 3 MeV produces coincidence with the observed bound states (fig. 13). 4. Conclusion The method developed in this paper should help to clarify the interpretation of previous level density calculations. In particular it seems possible to concludc that a completely independent particle model with any reasonable single-particle level scheme will seriously overestimate level densities for nuclei near mass 30. Even inclusion of
NUCLEAR LEVEL DENSITIES
49
long range quadrupole-quadrupole forces in an SU3 model appears not to remedy this situation zz), and the present results indicate that some form of coupling similar in effect to a zero-range residual interaction is needed for satisfactory correction of independent particle values. The calculations exhibited here produce in a natural way most of the effects known to be present in observed level densities. Nuclear deformation is shown to have a markcd influence on calculated level densities, and the Rosenzweig effect is shown to have important consequences. It remains to be seen whether the method developed has widespread applicability, but the results obtained so far are encouraging. The author wishes to express his deep appreciation to Prof. Eugen Merzbacher of the University of North Carolina for his initial suggestion of the problem, for his encouragement at various stages, and for numerous enlightening discussions. References l) 2) 3) 4~ 51 6) 7) 8) 9) 10) I11 121 131 141 151 161 171 181 191 20) 21 ) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34)
H. A. Bethe, Phys. Rev. 50 (19361 332 H. A. Bethe, Rev. Mod. Phys. 9 (19371 69 C. van Lier and G. Uhlenbeck, Physica 4 (19371 531 D. Ter Haar, Elements of statistical mechanics (Holt, Rinehart anti Winston, New York, 1961 ) C. Bloch, Phys. Rcv. 93 (19541 1094 H. Margenau, Phys. Rev. 59 (1941) 627 A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (19561 1446 D. W. Lang, Nucl. Phys. 77 (19661 545 J. M. B. l.ang and K. J. LeCouteur, Proc. Phys. Soc. A67 (19541 586 T. D. Newton, Can. J. Phys. 34 (19561 804 A. G. W. Cameron, Can. J. Phys. 36 (19581 1040 K. J. LcCouteur and D. W. Lang, Nucl. Phys. 13 (19591 32 D. W. Lang, Nucl. Phys. 26 (19611 434 A. V. Malyshev, JETP (Soy. Phys.) 18 (19641 221 V. Anufrienko, B. Devkin, G. Kotel'nikova, Yu. Kt, labukhov, G. Lovchikova, O. Sal'nikov, L. "l'imokhin, V. Trt, bnikov, N. l:etisov, J. Nucl. Phys. (Soy. Phys.) 2 (19661 589 S. P. Kapchigashev and Y. P. Popov, J. Nucl. Phys. (So*,'. Phys.) 4 (19661 686 O. Sal'nikov, N. Fctisov, G. Lovchiko,,a, G. Kotel'nikuva, V. Anufrienko, B. Dcvkin, J. Nucl. Phys. (Soy. P h ) s . ) 4 (19661 1155 T. Ericson, Adv. in Phys. 9 (19601 425 C. Critchtield and S. Oleska, Phys. Re','. 82 (19511 243 H. W. Newson, M. M. Duncan, Phys. Rex'. Left. 3 (19591 45 N. Rosenzweig, private communication K. W. C. Stewart, Nucl. Phys. AI00 (19671 74 D. W. Lang and K. J. LcCouteur, Nucl. Phys. 14 (19591 21 J. R. Herring, Ph.D. thesis, University of North Carolina (1959), unpublished L. Euler, Opera Omnia Scr. I, Vol. II, Comm. Arith. (B. G. Teubncr, Leipzig, 1911) 254 G. l-l. Hardy and S. Ramanujan, Proc. London Math. Soc. Series 2, 17 (19181 75 N. Rosenzweig, Phys. Rev. 108 (19571 817 N. Rosenzwcig, Proc. lnternat. Conf. on the study of nuclear structure with neutrons, ed. M. Neve de Mevergnies, P. Van Assche and J. Vervier (North Holland, Amsterdam, 1966) 309 A. Ewart, M. Blann, Nucl. Phys. 72 (19651 577 M. Blann, Nucl. Phys. 80 (19661 223 S. G. Nilsson, Kgl. Matt. Fys. Medd. Dan. Vid. Selsk. 29 (19551 16 P. M. Endt and C. van der Leun, Nucl. Phys. AI05 (1967) 1 W. Kunz and J. Schintlemeister, Nuclear tables (Pergamon Press, Oxford, 1967) D. J. Hughes and R. B. Schwartz, Neutron cross sections, second edition BNL-325 (19581