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CXLVIII. Neutron-widths and thedensity of nuclear levelsJ.M.C. Scott aa Cavendish LaboratoryPublished online: 22 Apr 2009.

To cite this article: J.M.C. Scott (1954) CXLVIII. Neutron-widths and the density of nuclearlevels, Philosophical Magazine Series 7, 45:371, 1322-1331

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[ 1322 ]

CXLVIII. Ne~#ro~- Widths and the Density of Nuclear Levels

By J. M. C. SCOTT

Cavendish Laboratory *

[Received July 7, 1954]

~UMMARY

The theoretical relation between the level width and t.he mean spacing D of the levels is investigated, using a model with strong configuration- mixing, not strong enough however to destroy the size resonance effect. Rough numerical values of ~'D are obtained using a previous analysis of the size resonance effect in the thermal-neutron cross sections. The lack of sharpness of the surface of the nucleus has been taken into account, and is responsible for a factor of two or three in the calculated level widths.

§ 1. INTRODUCTION

IN the present state of nuclear theory, the quantitative interpretation of resonance reactions can be reduced to the problem of explaining or predicting the partial widths of the energy levels of the compound nucleus. Although in favourable eases, with light nuclei, it may be possible to make a theoretical estimate of the width of an individual level for nucleon emission, it seems that usually we Call only hope to predict tile average width in a statistical sense.

In view of the extensive data on level widths which it is hoped will soon emerge from experiments with fast choppers of the Brookhaven type, it seems desirable to consider what should be expected theoretically, taking into account the size-resonance in nuclei, in order that we may make the best use of the new observations in testing theories of nuclear structure. The model developed here, which may perhaps be called the medium interaction model, involves no new adjustable parameters, so that a coral)arisen with observed level-widths ought to furnish quite a severe test,.

§ 2. WIDTHS AND MEAN SPACING OF LEVELS

There is eertai,fly ~ rough proportionality between the mean level- spacing, D, and the mean ,'educed level-width (when barrier-penetra- bility has been Mlowed for). The method of deriving this relation from thermodynamic reasoning (the evaporation theory of Weisskopf (1937), N. Bohr and Frenkel) does not seem capable of much further refinement. A more promising line of attack is suggested by the refraction picture (Feshbach, Peaslee and Weisskopf 1947, Blatt and Weisskopf 1952,

* Communicated by the Author.

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On Nc:utron-Widths and the Density of Nuclear Levels 1323

K a l m r ~md Peierls 193~). h i Weisskopf's later t r ea tmen t it is supposed t lu~ one e~m write ~ wave funct ion for an incident neutron, just inside the sm'fi~ee of the m~cleus, in the form

¢(r)=(C/~') cos [ A > + ~ ( E > ] . . . . . . . (1)

and its logari thmic der ivat ive at the surface is then expressed in the form

,f(E)=P,[(r~b)'/r~/J],.=t~=--KR t an [I(R~-~(E)]=--KR tan z(E). (2)

The angle z(E) wldch is thus int.r,~duced is monotonic and in equal t,) a multiple of Tr at e~ch resonance.

I f we are prep~m~d to ~ssmne tha t the average vMue of dz/'dE at. the l'es, m~mces in equal to its grneral average, ~,D. then the usual fornmla fi,r th~ a, verage neut ron-width of the levels follows, namely

F= 21:D, TrK . . . . . . . . (3)

This assumt)tion is sometimes described by saying tha t the behaviour of z(E) for nn act, rod nucleus resembles the smooth behaviour tha t u ould be exl)eeted for a simple ,'eft'acting sphere, bu t with a much longer optieM path- length inside.

In order to get a more precise relat ion one needs a more detailed hypothesis about nuclear s t ructure. I t is not difficult to show tha t with some models (3) can be quite misleading. For example, consider for a momen t the most. ex t reme weak-interact ion models, which are used in the theory of a tomic spectra. In sueh a model the wave funet, ion of a s ta te of the compound nucleus in supposed to be quite well described by a zero-order approxim~t ion const ructed out of single-particle wave functions, or else out of ~ one-particle wave flmetion and a wave Nne t ion for the ta rge t or core. l~{ost of the levels will only be able to emit a neu t ron by vir tue of configuration mixing, and will have widths much smaller t han (3). These are the ' compound levels ' of Bohr and Mottel- son (1953) or the " Auger levels ' of Scott (1954 a). I f such a model were correct, the behaviour of Weisskopf 's phase angle z(E) would deviate grea t ly f rom the smo(~th behaviour originally suggested, in a way depend- ing on the optieM pa th length to the centre and back.

In such a weak- interact ion model, the pe r tu rba t ion t e rm hi the Hami l ton ian must be so small t ha t it shifts the levels of a configuration by an amoun t smM1 compared with the distance of the next~ configuration with which it can mix. Tiffs eri terion is praet ieMly never satisfied in nuclei. The level spacing reveMed in neu t ron transmission exper iments is often of the order of 1 kev or less, whereas the per turb ing t e rm is cer ta inly far larger t han would be needed to displace the levels f rom their zero-order t)ositions by 1 key. A st.atisticM theo ry of per turba t ions is developed below, for the ease in which the level densi ty is great bu~ the per turbing t e rm not large enough to des t roy ~he influence of the ' optical path' 2KR on the behaviour of z(E), producing deviations from (3) corresponding to systemat ic f luctuations in dz/dE.

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1324 J . M . C . Scott o n

§ 3. WAVE-FUNCTIONS OF COMPOUND STATES IN A MEDIUM-I/gTERACTION MODEL

In order to avoid complication, the following assumptions and simpli- fications will be made. A neutron with energy E is incident on the target nucleus. The compound system can only break up in one way, by re-emission of a neutron with l = 0 ; thus radiative capture is neglected, and inelastic scattering is only possible at higher energies. All angular momenta are taken to be zero.

All co-ordinates describing the internal motions of the target nucleus are denoted by the single symbol r~, and the relative position of the incident neutron by (r, 0, ¢). Antisymmetry requirements are ignored for simplicity. Following current usage, that part of configuration space where tile system can be regarded as completely separated, and where the wave function can be decomposed

~r/(r~, r)-~-Xo(ri). (4~) I/2r--lw(r) . . . . (4) is called a channel.

Tile Hamiltonian H=Hm-4-Htv)-{-H'=HC°~d-H' . . . . (5)

is split up hlto that of the target nucleus (H(i)), that of a neutron in a suitably chosen central field V(r), and a perturbation term.

The equation HqSs(r~, r ) ~ - E ~ . . . . . . . . (6)

taken with the bound~ry condition

( a / a r ) r ¢ ~ = 0 a t ~ ' = R . . . . . . . (7)

possesses a complete ortho-normaI set. of eigenfunctions q)s and resonance energies E~ for the int.ernM region ~'~R (the compound nucleus). There is another comtflete ortho-normal set. of eigenfunetions qs(0) associated with the unperturbed zero-order Hamiltonian, and satisfying

H(°l~Pm~°) ~- (H(i)~-H(~))~P,,,(°~=E~(°)q)~ (°). (8)

These ~wo sets are connected with each other by a unitary matrix {%~}, the elements of which (letermine the level widths of the compound nucleus (as indicated in § 5 (ii)). We therefore wish to estimate the order of magnitude of c~.

Now one can easily deduce from the definitions (6) (S) that

that is to say,

j qSs*H qSm(°) d? . . . . . (9) C, m s ~

This resembles the formula for first-order wave functions in perturbation theory, but is exact. I f the perturbation is very weak and we can assign a small upper bound U to the factor on the right, (9) may give the order of magnitude of %s- But the plausible conclusion that

I c~ [ ~-, U/IE:--E.,<o) I . . . . . (10)

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Neutron-Width~ and the Density of ..¥v( lear Levels 1325

cannot be right when t, his upper t ,ound U is loJrge compared wil h the level spacing D. For t he complete~ess relatiol~

Zl%,, 1 2 = 1 , . . . . . . . . ( I t )

would then give m

We should instead expect the i~]lowing: wh~weas for U ~ D each perturbed state can be, related to a unique unl)cvturbed state (C,~s:'-:8,~s), for U>~D any pertm'b(,d state q~, (,resists la.rg,~ly of a mixture of ,mper- turbed states fi'om the range (E.~--L, E~+D) in roughly equal propo,'l,ions, so tha t ] c,..~]2 is ,~t most of the or.ler of liD, z:k more definite re:salt of this kind will now be obtained (§ 41.

§ 4. A THEORY OF S'nJ¢o~ PFI~TURBATIO:NS (' MED1U-~I ] NTEI~ACTION ')

The term strong perturbations is perhaps paradoxical, for an effect tha t can be t reated as a perturber.ion must be small compare([ with something: what is assumed her( is tha t the levels are very c[osely spaced (e.g. D-----100ev), and t]m% whereas the per turbat ion is much too strong merely to displace the levels by an araount ~ D and instead mixes up m a n y adjacent levels, it is nevertheless weak enough for the contaminat ion by distant levels to be slight. TI~_us, if the per turbat ion is gradually increased from zero t o its fldl value, a per turbed level q5 (energy E~) retains hardly anyth ing of the individual characteristics of q). (0) except of course its seri~l num ~er s which is preserved in consequence of the non-crossing rule of v. Neumann and %Vigner.

The result we need is not an ex~.~t algebraic theorem about the mfi tary mat.rix ~c~s } connecting the eig(,nstates of two t Iermit ian operators which (lifter slightly, but r~ther :nv, flves probt~bility theory in a way which makes a physical argumenl, more approprigte. As a convenient s tart ing point we m a y tt~ke the autolysis given by Weisskopf and Wigner for the problem of line-broadening by radigtion damping. Jus t as with the radiation rich[ in a large enclosure, H (°) has m a n y closely spaced eigenvalues, gnd only certain ~Lverage propert.ics of the eigenfnnctions are required.

We ~malyse the syst, em H(°)+H ' first in terms of the eigenfunctions ¢(0) of H (°), and then in terms of those of H(°)@I{ '.

Starting from the equation

ih &~,~at= (H(°)~-H')¢ . . . . . . . (12)

and seeking a solution of the form

¢=~a~(t)qs,~(°/exp (--iE,,~())t/h) . . . . . (13)

one obtains the equations

/~k=(1/ih)H'koao exp [i(Ek(°)--Eo(°))t/h], . . . . (14 a)

do-~(1/ih)ZHom'a ~ exp [i(Eo(°>--E,~(°))ti"h] . . . . . (14 b)

The initial conditions are %(0)=1, with all the other a~(0)=0. The usual text-book procedure is to put a0( t )=l in (i4 a) and to ignore (14 b),

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1326 J . M . C . Scott o n

which is good enough for small t. Weisskopf and Wigner (1930), however, showed how to extend the solution to larger wdues of t. Their procedure is to t ry put t ing

% = e x p (--~gt,k) . . . . . . . . (15)

Then by integrating (] 4 a), 1 - -exp (iE a.--iEo--½g)tffi

azfl)=H,,o' E~_Eo+½i f f . . . . . (16)

Tiffs s~ttisfies (14 b) too, provided that J, iy is equal to

.2] Ho, / 12 1 - -exp ( i E o - - i E ~ + {,q)(:h: ,,, E , ,_Eoq_.~i( t =;~,-IHo.,' ]°,'D.

which is t~ssumed t,() be in(lel)en(lent, ,d" m. or a.t any rnte constant when aver~ged over troy smMl rauge of enevoies. We t.herefi)re define g by

g--2r~Avl l to , , [ l" .D . . . . . . . (17)

The distributi(m of the pr,~bability of oc('ui)at.ion of the states when t >> ]i '(./follows from ( l 6)

I ,,,,,.)I" 1H.,0' I - - o ( t oc) (Em--Eo) ~ 4 g

and one can check that ( g/2vr

2 I%(o~) I~= J (E__t;0V---+¼g~ d E = 1, (lS)

as it should be. In the original application to the emission of light, g gives the natural

breadth of the emit t ing state. The expression under the integral sign in (1 s) gives the probabi l i ty distr ibution of the energy of the system.

Now let ¢ be analysed instead into eigenflmetions of H, i.e. the t ru ly s ta t ionary states of the system. Since the initial condition of the syst, e m is

(/3o (°)-= 2 Co.~q5 . . . . . . . . . . . (19) x

our solution (13) must. be

¢=Zco.,,qs,. exp (--iEd,:k) . . . . . . (20)

The probabi l i ty distr ibution of the energy of the sys tem can thus be expresse(l a,lternatively in terms of t.he local average of ] cos ]2/D. Hence

Av ]c,,.~ p = gD 1 2 - - ~ ( E , _ _ E , ) 2 + { g 2 . . . . . . (°I)

This relation (21) supersedes (10), and in fact (21) yields (10) as a limiting case ; (21) is of com'se now consistent with (11). w]fich (10) was not.

,~ 5. APPLICATION OF TIlE I)ERTUIIBATION THEORY TO ELASTIC SCATTERING

(i) The (;e'~eraI Nuclear Dispc~rsion Form~da

For the sake of convenience a derivat ion of the nuclear dispersion formula will be given here with the simpli~'hlg assumptions of § 3 (elastic

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Neutro ' , - |F id ths and the De,~sity of Nuc lear Levels 1327

scattering of neutrons, I = . . etc). Detailed discussions can be found in the literature.

The actual wt~ve function ~P" at, ~ny chosen energy E (.ml be decomposed according to (4) in the external region or channel, and it, satisfies

H~(r~, r)=E~ . . . . . . . . (22)

everywhere. In th,~ interna] region it can be expanded in a series

tI't= S b.Jo.~. , . . . . . . . . . (23)

which *m~y be non-lmifornllv convergen~ on tile boundary r : R , and in fget will b , : the derivative w'(R) (of. eqn. (4)) cannot be obtained by term-by-tevuI ([iffereu~i~tion. el. (7).

In order t,o lwing in the deriva.tive w'(R) , one c~m use (6) and (22) to obtain

When the left-hand side ,,f (24) is simplified by means of (;teen's theorem, the only non-va.nishing part comes from the term

2 M r CP f f - - r qo f f . r T

in the integrand" integrating this by ptu'ts, using (4) and (7). the relation (24) becomes

w'(R) l, 2 R ( r b ~ : E s _ _ E 2 M ~/(4~) j J c P * Xo dri d w = - - ti2 ¢~* w'(R) , (25) ' 2M E~--E

where R . m ~ ( / ) . ° . . . . .

and can be considered as dm ~verage of q5 over the entrance ( r = R ) of t ha t channel whivh corresponds to incidence and elastic scattering.

Insert ing the value of T(r,.. R) fi'om (23) and (25) in (4) now gives the s tandard dispersion formula connecting the wave function with the reduced widths ;e.(' :

w'(R) --~ Es--E' . . . . . . ( 2 7 )

h 2

r ~ = 2~<z I¢~ I ~ . . . . . . . (es)

From this, the Brei t -Wigner formula for slow neutron sc'~ttering can be deduced, and the neuirou width P~ identified as F~=2]cyse which in this case has a simple physical in terpre ta t ion: F~ is proportion,%l to I¢.~12 which is a me,~sure of the probabil i ty of finding the neutron in the clmnnel entrance, and to the ve]ocitv h/c 'M with which it might escape.

(ii) App l i ca t ion to the, M e d i u m I'n.teractio~ Model

I t will great ly simplify the analysis if we adopt tile assumption of a sharp boundary to the nucleus (the necessary correction can be made afterwards), and if ,aTe fi~rther assume tha t the reaction r~dius is no t

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1328 J. 1~I. C. Scott o n

appreciably larger, so tha t r=R represents bo th the surface of the nucleus and the sphere necessary to seal off the internal region f rom the react ion channels.

The Hamil toni~n can be divided into par ts as in § 3 :

H=H(O+H(V)+H ', (5)

when H (o is the exact Hami l ton ian of the target nucleus, and has eigen- functions X1

H (i)xt(r ~)= Ej(')Xi. E0(O can be convenient ly t aken to be zero. H (v) describes a neu t ron in a sui tably chosen potent ia l well

H (v)= -- (h~; 2M)V 2-F V(r),

so tha t H ' can be t rea ted as a per turbat ion. We now have two complete or thonormal sets of functions in the

internal region, as in (6), (7) and S : namely the eigenfunctions of the unper tu rbed problem (H (o -FH (v)) which c~n be wri t ten more explici t ly as

%(r) {2"~a/2sinKnr rV0-Q ) =xj(r,) \R] (29)

where K,~=(nq-½)rr/R ( n = 0 , 1, 2 . . . . ), (30)

and the eigenfunetions of the ~ctual problem (H) which are

a~ = ~ c~,,,~*q~j, (o). j , n

When q)~ is analysed in this w,%y into a combinat ion of the Xiv,~, the only terms which contr ibute anyth ing to its ' m e a n ' value ¢.~ at the channel entrance are the terms in X0V,, ( t a ~ e t left in its ground state). Thus f rom (26)

G = (2/R)1/2 :'co,,,~*. n

The statistical result (21) t, hen gives, for tim ~verage value of the reduced width,

h 2 1 Av y~2=- MR AV Z [co,,s [2= 2MffD Z (31) whiR ,=0 (K~'--K~2)+ M"g2/h,4 ,

1/,) l where K denotes the int.ernal wave number [2M(II-I-E~)] -~h, and W is the well depth.

The summat ion can be performed by means of the formula

Imtaf_jFi( ~ n ~ = 0 [(n-JK1)2~T2--~2F(~2] 2-~4~2¢2' (32)

which can ei ther be proved f rom Mitt~g-Leffler 's theorem or der ived from the series cot z=-~l/(z--m~). The result is

DR t an (~+i~) _ DR ~ sinh 2~ Av y.~'~= - - I m ~r ~ - i a ~ ~2+a2 cos 2~-Fcosh 2a

sin__2 \ . . . . (33) X 1 s inh2~ 2~ J"

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~eutron-Widths a~d the Density of Nuclear Levels 1329

Here ~ and a are positive numbers defined b y means of

~.-~i~-~(K2-~-iMg/~)~/2R . . . . . . . (34)

In pract ice a 2 ~ 2, so tha t ~ K R , and the first two factors in (33) yield D/TrK ; 2~ is large enough for the last factor to be neglected. The mean level width for neu t ron emission is then found to be

- - - 2D/~ sinh 2a (35) /~2kY~----- ~rK eosh 2 a + c o s 2 K R

which agrees with the Weisskopf formula (3) in the limit ~ --> ~ . I t is now necessary however to allow for the effect of the sim~)lifying

assumption made above, t i lat ~he nucleus had a sharp boundary . The gradual na ture of the t ransi t ion of the field V(r) from the well-depth

- - W to zero can be taken into account as in Scott (1954 b), and has two effects : it replaces the factor 1/K by a larger constant S, and it adds a constant to the phase angle KR, so tha t ~_KR--~z. In the exper imenta l reports, the veloci ty-var ia t ion of F is often allowed for and the value of F(1)-=F~/(E1/E) is quoted, calculated for a s tandard energy of E l=-1 ev. Wi th this convention,

.[,(1) 2klS sinh 2a Av ~ - - -- . . . . (36)

cosh 2~+e o s 2~ '

where ~=KR--:c=.(O'6A1/S--O'S)rr and ]Q=(2ME~)l /2/ / /=2.2 × 109 cm -x.

§ 6. NUMERICAL VALUES AND DISCUSSION

The first factors in (36) correspond to the 2kl/~rK of the sharp-boundary theory (Blat t and Weisskopf 1952, and eqn. (35)), whicil would be 1.4 × l0 '-~ with a well dep th of 20 ~Iev as used ill the l i terature, or 0.9 × 10 -4 with the value 51 •ev which we believe to be more correct (Scott ]954 b). With the diffuse bounda ry however, using S----18.9× I0 -1~ cm as in the la t te r paper, the actual value of the factors in (36) is r a the r larger, namely 2-6X 10 -~

The w~lue of the interact ion-~trength ~ can be es t imated by the me thod given previously (Scott 195~ b), bu t olfly roughly because of inadequate s ta t is t ics : for nuclei with 5 0 < : N ~ 8 2 , tanha----0-16±0.04, and for N~S2, t anh ~ 0 . 7 ~ 0 . 2 5 . This suggests t ha t F(~)/D should have a m ax im u m of 4~:1 near A-=--165, a somewhat larger m a x i m u m at A ~ 5 6 , and a broad min imum of 0.44_-0-1 a t A=-102. This is in fair agreement with prel iminary results f rom Brookhaven (Harvey, Por te r and Hughes 1954).

Whert the widths of enough levels have been measured it m a y be possible to work backwards and deduce ~ fi'om F~. But , in considering exper imenta l values of I~,,/D, the statist ical uncertaint ies in both D and /~,~ when deduced f rom a sample of say 5 or 10 levels mus t be borne in mind, as we]] as the exper imenta l error of an individual T',, which often depends on an assumed value for the radiat ion width, so tha t the task of determining F,,/D accura te ly will be difficult and laborious.

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1330 J . M . C . Scott on. the

In certain cases, e.g. Zn, the predicted mean level widths are ra the r filsensitive to the choice of a value for ~. and a be t te r predict ion is possible. For Zn, (36) gives / ' (1)/D=3-7 X l0 a, with a probable error of :-- 10'~." o arising most ly f rom S. This m a y be compared with (5~2) X l0 -4 f rom four levels in eTZn observed by Dahlberg a,nd Bollfilger (1954).

I t is not possible to compare t, he present theory with the cloudy crystal ball of Feshbach. Porter and \Veisskopf, since (let,~ils of the applicat ion of the ]att(,r to the resona, nce region are not ye t available to us. Some of our f()rmulae, e.g. (:14) ;tnd (35) do indeed suggest, an oI,t, ic~l analogue with an ext inct ion roeflicient, oc ~:R or a emnplex potent ia l W@~i : /bu t it shouhl be no ted thg~t, e and g do not correspond to the neu t ron being lost from the entrance cha.nnel t.o other channels (inelastic scat ter ing or radiat ive c~pl,ure).

It is no tewor thy theft, the u.rgument from t.he graph of :(E), which gave (3),also suggests theft when two ,)r three levels ]ml)pen to be u]msuMly oh)s(, together (£'~+1--15',.~11). their widths sll()uhl be below average, whereas the present analysis does not, lead one t.() expect any such eorrel~tion. This point c(mld probably be l ested experimental ly . It, is also m~tural to suppose tha t the real and inmginary parts of c o ..... in (31) will fl)llow ~ Gaussian distr ibution, and t,heretbre tha t F will follow dP=(1 / I ' ) exp (--/7,,T') dF, 1)ut this does not seem t.o be striel,ly deducible f rom the considerations given here.

I t is ~ pleasure to t hank l)rofessor D. J . Hughes for several valuable discussions.

A I ' P E N ] ) [ X

Esti.m.(Itio~, ~f ~ j)'om the Thermal Ncatteri'~g Data

In this appendix the probabi l i ty dis tr ibut ion of the coherent scat ter ing length will be obt~fined. I t is a consequence of the present theory which was quot.ed wi thout proof in Scott, (1954 b) (referred to as l I ) . According to I[, § 3, this problem is equiwdent to finding the dis tr ibut ion of the values of fi, the phase of the neu t ron wave fnnet ion when extra- polated inwards to the centre.

The relation between the nota,tion of I1 and t.hat of Blat,t, and "Weisskopf (1952) is as follows :

Phase just outside 1)h~se extrapolated to centre Dittbrenec ()f these Phase shift, ill traversing nuclear surf~ce

I I = S . (1954 b) B. & W. (1952

--0 5

KR--~--~ OL -- TT

z(0)-}~,

K R 0

The result of the present paper can be interpreted in terms of the graphs of the Feshbaeh-Weisskopf functions z(E) and ~(E), and their systematic

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Ne~ttro.n- Widths and the Den,sity of Nttclear Levels 1331

fluctuations in slope. These fluctuations of average slope are periodic in ~=z- -~=z- - I (R+-~ . The results (33) ~md (35) gave the average value (y2/S) of dE/dz at those points where z is a multiple of ~ (i.e. the resonances)

dE dE D sinh 2a Av ~ ~ A v dz -- ~ cos 2~+cosh 2a"

Now if the energy of a thermal neutron, E~--0, fails r~mdomly among the resonance levels, the corresponding value of ~, n,~mely ~(0)=/3+~-~, does no~ fall randomly on the scale of ~ : instead, the chance tha t it lies in a r,mge d~ is proportiom~l to dE/d~, and the probabil i ty distr ibution of fi is therefore

sinh 2~ dfi eosh 2~--cos 2/?"

This is the formula which was used in I I to est imate a.

]{EFERENCES

BLATT, J. 1V[., and WEISSKOPF, V. F., 1(}5 °, Th, or~tical N~clear Physic.% 1st ed. (New York : Wiley), esp. pp. 398-4(}6.

Bo•R, A., and MOTTELSON, B., 1953, Dan. mat.fys. Medd., 27, No. 16. I)A~LBER(}, D. A., and BOLLINt;ER, L. IV[., 1954, Bull. Amer. Phys. Soc., 29,

No. 4, 57. ti~ESH:BACH, ~-~., I~EASLEE, D. C., and WE[SSKOPF, ~2. F., 1947, Phys. Rev., 71,

145. HARVEY, J. A., PORTEn, C. E., and HUtCHeS, D. J., 1954, Bull. Amer. Phys. Soe.

29, No. 4, 57.* K~(Prm, P. L., and PEIERLS, P~., 1938, Proc. Roy. Soc. A, 166, 277. SCOTT, J. M. C., 1954 a, Phil. Mag., 45, 44l : 1954 b, Ibid., 45, 751. WEISSKOI?F, V. F., 1937, Phys. Rev., 52, 295. WEISSKOPF, V. F., and WIGNER, E., 1930, Z. Phys., 63, 54.

*Now reprir~ted in Phys. R,v., 95, 645.

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