15
IL NUOVO CIMENTO VOL. X, N. 3 1° Novembre 1958; On the Optical Model for Nuclear Reactions. ~L. VERLET ~n4 J. GAVORET Laboratoire de Physique, Ecole Normale Supdrieure - Paris (ricevuto il 30 Luglio 1958) ll6sum6. -- La matrice de diffusion au sein de la mati~re nucl~aire indd- finie a dt~ calcul~e exactement dans le module off seules les eorrdlations statistiques sont prises en considdration et grace ~ l'emploi d'une inter- action nucl~on-nueldon s~parable. On en ddduit un potentiel complexe pour la diffusion des nucl5ons par les noyaux. On discute la conver- gence des sdries de perturbation et la validitd du modSle classique de Lane et Wandell. Enfin. en faisant l'approximation de Thomas-Fermi, on montre qu'~ ba.~se dnergie l'absorption a principalement lieu ~ la surface du noyau. 1. - Introduction. In the last few years, great progresses h,~ve been made in our understandin~ of the gross properties of nuclei, mainly due to the work of BRUECKNER, BETHE and others (1-6). They have shown that when the so-called cluster terms may be neglected, one can derive the properties of the infinite nuclear mutter in its ground st~te from an integr~fl equation involving explicitly only two par- tie]es. The effect of the other particles is taken into account both by the average field they produce ,~nd by their presence which, due to the Puuli principle, forbids a number of intermediate states. Though historically the problem of describing the interactions of ~ particle propagating in the nuclear (1) K. A. BRUECKNER, C. LEVINSON and tl. MAttMOUD: Phys. Rev., 95, 217 (1954). (2) K. A. BRUECKNER and C. A. LEVINSON: Phys. Rev., 97, 1344 (1955). (3) K. A. BRUECKNEI% and J. L. GAMMEL: Phys. Rev., 109, 1023 (1958). (4) L. C. Go_~IEs, J. D. WAIJECKA and V. F. WEISSKOPF: ,'l~t~. Phys., 3, 241 (1958). (5) H. A. BETHE: Phys. [~ev., 103, 1353 (1956). (6) C. DE DO~INICIS and P. (!. MARTIN: Phys. Rev., 105, 1417 (1957).

On the optical model for nuclear reactions

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IL NUOVO CIMENTO VOL. X, N. 3 1 ° Novembre 1958;

On the Optical Model for Nuclear Reactions.

~L. VERLET ~n4 J . GAVORET

Laboratoire de Physique, Ecole Normale Supdrieure - Paris

(ricevuto il 30 Luglio 1958)

l l 6 s u m 6 . - - La matr ice de diffusion au sein de la mati~re nucl~aire indd- finie a dt~ calcul~e exactement dans le module off seules les eorrdlations stat ist iques sont prises en considdration et grace ~ l 'emploi d 'une inter- action nucl~on-nueldon s~parable. On en ddduit un potentiel complexe pour la diffusion des nucl5ons par les noyaux. On discute la conver- gence des sdries de per turbat ion et la validitd du modSle classique de Lane et Wandell . Enfin. en faisant l ' approximat ion de Thomas-Fermi, on montre qu'~ ba.~se dnergie l 'absorption a principalement lieu ~ la surface du noyau.

1 . - I n t r o d u c t i o n .

I n t h e l a s t few years , g r e a t p rogresses h,~ve been m a d e in our u n d e r s t a n d i n ~

of t he gross p r o p e r t i e s of nucle i , m a i n l y due to t he w o r k of BRUECKNER, BETHE

a n d o the r s (1-6). T h e y h a v e shown t h a t when the so-ca l led c lu s t e r t e r m s m a y

be neg lec ted , one can de r ive t h e p r o p e r t i e s of t he inf in i te n u c l e a r m u t t e r in

i t s g r o u n d s t~ te f rom an integr~fl e q u a t i o n i n v o l v i n g e x p l i c i t l y on ly two pa r -

t ie]es. The effect of t he o t h e r pa r t i c l e s is t a k e n in to a c c o u n t b o t h b y t h e

a v e r a g e f ield t h e y p r o d u c e ,~nd b y the i r p resence which , due to t he Puu l i

p r inc ip le , fo rb ids a n u m b e r of i n t e r m e d i a t e s t a tes . T h o u g h h i s t o r i c a l l y t h e

p r o b l e m of de sc r ib ing t h e i n t e r a c t i o n s of ~ p a r t i c l e p r o p a g a t i n g in t he nuc l e a r

(1) K. A. BRUECKNER, C. LEVINSON and tl . MAttMOUD: Phys. Rev., 95, 217 (1954). (2) K. A. BRUECKNER and C. A. LEVINSON: Phys. Rev., 97, 1344 (1955). (3) K. A. BRUECKNEI% and J. L. GAMMEL: Phys. Rev., 109, 1023 (1958). (4) L. C. Go_~IEs, J. D. WAIJECKA and V. F. WEISSKOPF: ,'l~t~. Phys., 3, 241 (1958). (5) H. A. BETHE: Phys. [~ev., 103, 1353 (1956). (6) C. DE DO~INICIS and P. (!. MARTIN: Phys. Rev., 105, 1417 (1957).

506 L. V E R L E T a n d J . G A V O R E T

medium through a complex potent ia l was examined by WATSON and his co- workers before BRUECKNEn'S work (m)- - see also Ref. (9) and (1°)--explicit

calculations related to this problem have not been pushed very far (11-15). There

appear in fact , when one goes f rom the ground s ta te of nuclear m a t t e r to the

scat ter ing problem, a number of new difficulties which make this p rob lem

ra the r formidable.

1) The decoupling operat ion of the various particles, which enables to

establish a two-body integr~fi equat ion cannot be accomplished in one step. One has first to deeouple the incident part icle f rom the ta rge t nucleus. The integral equat ion for the scat ter ing ampl i tude of the incoming part icle inside the ta rge t still involves the real nuclear states. A second operat ion mus t be

done to disentangle the particles of the ta rge t nucleus. We shall make the

assumpt ion tha t the nucleus m a y be correct ly described by a Fermi gas, so

t ha t we shall escape this problem as well as the difficulties (') connected with

the fact t ha t the real nuclear ground s ta te has, in the general case, m o m e n t u m

components onto the incoming wave function.

2) I f the optical potent ia l is calculated according to WATSON'S work (7-10),

one has to solve a scat ter ing integral equat ion which involves a Green 's funct ion

in which the complex potent ia l to be calculated appears. This par t icular form of the Green's function makes the self-consistency problem quite ~wkward to solve when the imaginary pa r t of the potent ia l is not negligible. In the

present work we shall suppose t ha t "ill particles move in a real well whose m o m e n t u m dependence will be t reated, ra ther empirical ly in the effective mass

approximat ion.

3) When the two colliding particles are in the Fermi sen the scat ter ing

ma t r ix depends very weakly on the absolute m o m e n t u m of the part icles in- volved. I t can then be shown (2,~.~) t ha t it is legi t imate to replace by a sphere the complicated bispherical domain (Fig. 2 ) on which the in tegra t ion over

the in te rmedia te reh~tive m o m e n t u m nms t be made. This approx ima t ion

renders feasible the numerical calculation of the t -matr ix, even when a com-

pl icated nucleon-nucleon interact ion is used (2.3). The s i tuat ion in the seat-

ter ing prob lem is quite different: the t -nmtrix depends s t rongly on the absolute

m o m e n t u m so tha t its ewfluation is in general quite complieated even in the

case where the ta rge t nucleus is t aken as a Fermi gas and where the self-

consis tency problem is left out or t r ea ted in the effective-mass approx imat ion .

(7) K. ~I. WATSON: Phys. Rec., 89, 575 (1953). (s) IN. C. FRANCIS and K. M. WATSON: t)]~ys. Rev., 97, 1336 (1955). (9) G. TAKEDA and K. M. WATSON: Phys. Rev., 94, 1087 (1954).

(10) K. M. WATSON: Phys. Rev., 105, 1388 (1957).

ON T I l E OPTICAL MODEL FOR N U C L E A R I¢EACTIONS 5 0 7

I n fact., ealculat,ions done up to now have resor ted in some w a y to p e r t u r b a t i o n

t h e o r y (1Ha.~5) or to the ex tens ion to low energy of the classical model of

GOLDBERGER (~6) an ex tens ion which has still to be justified (~7,2~).

This paper will be deve loped a round a ea leuNt ion of the t -mat r ix (Sect.. 2). W e used separable nue leon-nueleon in te rac t ions (,2,2) which have the adv~ult,age

of pe rmi t t in o~ an exact, ca lcula t ion ('~) wi thin the f r amework of approx inmt ions

ske tched above, bu t the inconven ien t of their unphys ica l na tu re which renders

r a the r futi le an a t t e m p t to give a precise fit to the exper imenta l da ta . W c

can hope, however , to be able to draw some general conclusions abou t the opt,ieal

model . It: will appea r (Sect. 3) tha, t at low and in t e rmed ia t e e, ner~o~ies, the

complex po ten t i a l thus der ived is fair ly in4ependent, of the precise shape of

the separable po ten t i a l t h a t is used. The ra te of eonvero~ence of the per tur -

ba t ion series depends v e r y s t rong ly on the (( singularit ,y )) of the nuc leon-nuc leon

in te rac t ion . I t will be shown t h a t this convergence is m u c h be t t e r for the

real p a r t of the opt ica l po ten t i a l t h a n for the ima, g imwy part, where it is, in

fact , qui te bad. The nmin effect of the in t roduc t ion of an effective mass of

0.8 M is to decrease v e r y mu(,h the im~ginary p a r t of the po ten t ia l especial ly

at low energy (the r educ t ion fac tor is a round 3 at zero energy).

Thc fol lowing Sect ion (Sect. 4) will bc devo t ed to the s t u d y of the clas-

sical approx ima t ion . W e shall give a formula t i (m of the classical model in

which the imag ina ry p a r t of the po ten t ia l is more d i r ec t ly re la ted t h a n usual ly

to the expe r imen ta l ly measu red eross-seetions. We shall show that. the ani-

so t ropic p a r t of the nuc leon-nuc leon poten t ia l ( .ontributes little, even at high

energy. The results for the isotropic p a r t are in qua l i t a t ive agreement, wi th

those ob ta ined recen t ly by VA~ D~:R VEGT a n d JO~KE~'~ (,2~).

The va l id i ty of the classieal model will then be tes ted by compar ing the abso rp t ion po ten t i a l obt,ained f rom the elassi(,al model in which the free

nuc leon-nuc leon ( 'ross-sections are calculat,ed with a separable potent ia l , wi th

the more exact, resul t ob t a ined f rom the t-mat,fix ca lcula t ion (Sect.. 3) using

the same nueleon-nueleon intera(.t,ion. I t will be seen t h a t the agreement.

(11) M. (!INI and S. FUBINI: Nuoco CimeJdo, 2, 443 (1955). (12) K. A. BaU~CK.~ER. R. J. EDEN ~nd N. (!. FRANCIS: Phys. l:er.. 100, 891 (1955). (~3) K. A. BRUECKNEIt: Phy.~. l?ec.. 103, 172 (1956). (la) W. B. RIESl.:NFELD ~nd K. M. WAT.~O~-: Phys. Rev., 102, 1157 (1957). (15) l~. VERLET: ~'~lo~'o Cime~to, 7, 821 (1958) and Thesis to appear in _l~J~. Phys. (16) ~[. I,. GOLDBER(;ER: Phys. Rcv.. 74, 1269 (1948). (17) A. 3[. LANE and C. F. WANDEL: Pl~ys. l?er.. 98, 693 (1955). {~S) ~. CLEMENT]:;L and (~. VILLI: N~eo~,o ('i~e~do, 10, 176 (1955). (19) S. HAYAKAWA..~[. KAWAI and K. K r s t : c m : Progr. Theor. Phys., 13, 415 (1955). (,z0) G. C. MOnRISON. lI. MUIUHJ~AO :rod P. A. B. MURDOCH: Phil. Mag., 46, 795 (1955). (21) A. VAN DnR VE~T ttnd C. C. J~)xKna: Physica, 33. 359 (1957). (,22) y . YA:~[A(¢U('HI: Phys. llev., 95, 1628 (1954).

5 0 8 L. V E R L E T and J . GAVORET

between the two methods is not ve ry good: below 100 MeV the classical ap- proximat ion underest imates the imaginary potent ial by about 50~o.

In the last section, we shall t ry to get an insight in the behaviour of the

imaginary par t of the potent ia l in the surface region which will be t r ea ted in the Thomas-Fermi approximation. I t will appear tha t the absorpt ion is~ at low energy, much bigger in the surface region than in the center of the nucleus and tha t this result is not due to our use of separable potent ia ls al though it m ay be completely altered in a more precise t r ea tmen t of the problem. As it stands, this surface absorption has the advantage of com- pensating the too small volume absorption obtained when an appreciable effective mass effect is taken into account; this surface absorption is made plausible by some recent phenomenological analysis (2~-25).

2. - Calculat ion of the optical potent ia l from the t-matrix.

In t h e eMeulation of the optical potentiM from the t-matrix, the approx-

imations s ta ted in Sect. 1 have been made. The target nucleus is represented by a Fermi gas with N = Z and no

Coulomb forces. In this section and the two following ones the density will correspond to a value for ro of 1.27 fermis and the absolute energy - - A E of the top of the Fermi sea is - - 10 MeV. The effect of w~rying the densi ty will be studied in Sect. 5.

We take for the separable potent ials the simplest possible form which in momen tum space is wri t ten (22):

(1) (p I v Ip') = - - ~ g(p) g(p').

Such a potent ia l acts only in S-states. The normalizat ion of (1) is such tha t the forward pa r t of the scattering"

ampli tude for free nucleons takes the form:

(2) ; dqg2(q) ie]

The t-matrix for the two nucleon sys tem is equal to 4~](p).

(2a) F. E. BJORKLUND, S. FERNBACH and N. SHEridAN Pl~ys. Rev., 101, 1832 (1956). (2a) H. AMSTER: Phys. Rev., 104, 1606 (1956). (25) F. E. BJORKLUND and S. FE~NBACI~: Phys. Rev., 109, 1295 (1958).

ON THE OPTICAL MODEL FO R N U C L E A R REACTIONS 509

We used two kinds of separable potentials:

:1) The Yamaguchi potential (22) for which:

1 g(P) - - p2 + f12 • (Y. potential) .

We fit all the four low energy scattering constants by using two different

potentials for the singlet and triplet states.

2) A more (,singular ~) potential which fits exactly the effective range

formula (,.~.6). For this potential one ti~s:

1 g(p) - ~/p~. ~ . (E.R. potential) .

We have again chosen two different interactions in the singlet and in the

triplet sta.tes.

The average total cross-section (a,,n+a,,,)/2 obtained with those two po-

tentials is shown in Fig. ]. The experimental data (27) are also given for corn-

parison. For ao~ we took

2z (d%Ud/2)9~o supposing

charge independence and 100C

the isotropy of the pro-

ton-proton cross-sections

~t all energies. We see

tha t both potentials fit 100

fairly well the total cross-

section up to 80 MeV.

Then they lead to cross-

sections which are sub- ~0

stantial ly smaller than

the experimental ones,

especially so in the case

of the Y-potential. To

obtain the t-matrix in the

nuclear medium, one has

still:

1) To take into

(~np *0 n ~mb

f/ab~ ~o0 2o0 3oo 400

Fig. 1 . - The average cross section (anp+an~)/2 as a function of the energy in the laboratory. Curve l: experimental curve (2~) with ~ = 2~(d%p/dtO)~oo. Curves 2 and 3 : cross sections obtained with the E.R.

and Y potentials respectively.

(2~) A. MARTi~ and M. GOURDIN: N'UOVO Cimento, 6, 757 (1957). (~'~) L. BERET~A, C. VILLI and F. FERRARI: Suppl. Nuovo Cimento, 12, 499 {1954).

510 L. VERLET and z. GAVORET

account the ant isymmetr izat ion between the incident nucleon and the target

nucleons. In the case of an S-state separable potential the direct and the ex-

change parts are equal, and we get from this a factor 2.

2) To sum over the spins and isotopic spins of the target nucleons and

to average over the spin of the incident nucleon. This leads to a factor ~ both

for the triplet and singlet spin states.

3) To sum over the A / 4 different states of the Fermi-gas and mult iply

by a factor 1/f2, where ~2 is the volume of the nucleus, which comes from the

normalization of the incident particle wave function. This gives:

- - ~=~ - - k = d u p d p , where u = .

The triplet state contribution to the optical potential is then found to be:

(3) - - 48JrXt ; p d p u d u g ~ ( p )

cl)~ - - .3Ik~ J l + 2 t ( M * / M ) f d q g ~ ( q ) / ( p ~ - - q~ + i s ) "

To this term must be added the singlet contribution which has a similar

form. In this formula M* is the effective mass. The integration domains ~re

as follows (Fig. 2). q must lie outside two

Fig. 2 . - Integration domain for Eq. (3).

ytieal form depen4s on the form of the potential tha t is used.

has generally :

Fermi spheres whose centers are 2u apart. Tile

fact tha t k must lie inside the left Fermi sphere

imposes on p and u the following limits: u . / - - 2 ¢) ~ O - - p 2 varies from k ~ - - p to 5/(ky/ , )÷(k~j , , ) and

p varies from ( k s - - k F ) / 2 to ( k~ , i - k r ) / 2 . For

the potentials tha t we have used, the inte-

gration on q is immediate and leads for (3)

to a denominator of the form 1 - - ( J ~ ÷ i J 2 ) .

The expression for the real par t of the deno-

minator is too long to be given here; its anal-

For J,~ one

J2 = n2~g~'(P)(P 2 ÷ u ~ - - k~)/u ,

when ( p 2 ~ - u 2 - - k ~ ) is positive. Whenever this lust expression is negative,

which may happen when k~ < ~/?,kp, J2 is equal to zero. As in (3) the in-

tegrations on u and p cannot be done analytically, the results we give below

for the optical potential have been computed numerically.

O N T H E O P T I C A L M O D E L F O R N U C L E A R R E A C T I O N S 5 1 1

3 . - R e s u l t s .

Some of the results have been sun lmar ized in Ta.ble I. We have corn-

p u l e d for 4 inc ident par t ic le ener~.ies ( 10 MeV (k v - - kr) , 27 Me w 80 MeV,

230 MeV) and general ly, for b o t h the Y po ten t i a l and the E R potent ia l , tile

real p a r t of the opt ical po ten t i a l given by the two first Born approx ima t ions

and its exac t va lue ; the real p a r t when the Paul i pr inciple is not. t aken in to

a c c o u n t ; the imag ina ry p a r t given in the second Born ' t pp rox ima t ion and

exac t ly ; the imao' inary p a r t when the Paul i principle is neglected and when

the classieal model is used.

TABLE [.

O)(t )

co~p i i

( /)~2)

[ C ~ I XVp

i + 3o MeV

. . . . i _ r

E~,, 10 MeV

- /%"

I

!~i5 __ :32.2 35.0 ]

13.7 ! 17,1 13.7

66.7 i 69.8

21.~ I 21.7 . . . . . i

0 0

0 o

ER ; Y

1.5k F

E R

26.4

15.1

51.5

Y

24.1

4.8

23. ] 48.7

sO MeV 230 MeV

i 2k±, 31@

ER

19.6

8.9

25.7

31.9 34.2 27.6 :~0.4 13.6 17.2

q),(. 0 ' 0 i 97 ! 1105 i S.5 1 S.S

- - : . . . . i ]

Y ER

10.5 11.1

- -0 .6 2.8

8.7 13.7

21.45 i 21.2 i 16.6 17.4 8.1 10.6 i _ _ _

8.5 4.8 10.0 6.5 4.0 4.2

20.6 ! 23.9 12.1 16.1 3.1 6.0

3.4 6.4

2.8 5.35 . . . . . . i

Tile cons idera t ion of this table leads us to the first conclusion t h a t at low

and in te rmed ia te energies the opt ical potent ia l is, to a good approx imat ion ,

independent, of the shape of the separable potent ia l . This independence was

a l ready shown in the sa tu ra t ion p rob lem b y DE DOMINICI,q and MARTIN (6).

At higher energy, the differences are especial ly a p p a r e n t ill tile results for the

imag ina ry p a r t of the potent ia l . B u t it appears also t h a t the wHues of the

imag ina ry poten t ia l ea loula ted wi th the Paul i pr inciple are p ropor t iona l to

those ca lcula ted w i t hou t it when one goes f rom one poten t ia l to the o the r ;

5 1 2 L. V E R L E T a n d J . G A V O R E T

so t ha t the differences in the imaginary pa r t of the potent ia l a t high energy

are direct ly re la ted to the differences in the free nucleon-nucleon cross-sections

as can be expected on the basis of the classical model.

The convergence of the series which gives the real pa r t of the potent ia l is seen to be ra ther good in the case of the Yamaguch i potential . For instance,

for k~. = kF, the ra te of convergence as measured b y the rat io of the second

Born approx imat ion to the first is of the order of 30~o and the sum of the

two first t e rms is only 10% lower than the exact result. I t m a y be no ted t ha t the ra te of convergence of the pe r tu rba t ion series with a nucleon-nucleon local gaussian potent ia l with which a fit for the real pa r t of the optical potent ia l m a y be obta ined (15) is also of the order of 300/0 . On the other hand, the

convergence~ for the imaginary p a r t of the potent ial , is quite bad. Fo r

k~ ~ 1 . 5 k r the exact result is more t han twice the result given by the first

non vanishing t e rm in the pe r tu rba t ion series, i.e. the second Born approx-

imation. We m a y expect t ha t in the case of a regular local in teract ion the

pe r tu rba t ion series has not a s t ructure very different f rom the one obta ined

with ~ <, regular~ separable potential , so t ha t the value for the imag ina ry

poten t ia l t h a t we got in (~5) with the second Born approx ima t ion m a y be substant in l ly below the exact value.

The E.R. potent ia l being more (~ singular )~, the convergence of the corres-

ponding per tu rba t ion series is worse~ especially for the imaginary pa r t of the

po ten t i a l ; for k~, = 1.5k~, the contr ibut ion of the second Born approx imat ion is only 20°/0 of the ex,~ct value.

The effect of the veloci ty dependence of the average potent ia ls is i l lustrated

i , , E., ~MeV) , .

O 50 100 150 200

Fig. 3. - The real part of the optical potential cal- culated with the E.R. separable potential with M*~ M

(curve 1) and M * : 0.8]/ (curve 2).

b y Fig. 3 and 4. We have

p lo t ted the curves for M*/M=landM*/M=0.8 . When an effective mass different f rom M is used the energy scale for the

incident neu t ron m u s t

be modified so as to keep

cons t an t the p r o d u c t

M*(V-- E~). The energy

of the incident neu t ron

is then:

E~ - - M* 2 ~ A E .

Self consistency for neu- trons near the top of the

Fe rmi gas is obta ined for

O N T H E O P T I C A L M O D E L F O R N U C L E A R R E A C T I O N S 513

M*/M= 0.7. We took the

value 0.8 M for the effective -30

mass as a compromise since

a lower v 'due would tend to

overestimate the velocity -2c

dependence for high mo-

menta. The role of the ef-

fective mass is espeeiaUy

striking a t zero energy for -~0

the imaginary potential ; as

has been already noted (1~)

its second Born approxima-

tion is then divided by a o

factor (M/M*) a ~ 2. The ef-

fect on the exact value is

still bigger: the reduction

factor is for the E.R. po.-

tential of the order of 3.

I t would be a little smaller

MeV

Eo(NeVl

Fig. 4. - The imaginary part of the optical potential calculated with theE.R, potential. Curve 1 and 2: result of the t-matrix calculation with M*--M and M*=0.SM respectively. Curve 3: result of the

classical model calculation.

for the Y potential for which the convergence is better.

4 . - T h e c l a s s i c a l m o d e l .

We shall make now a fur ther approximation which leads to the classical

model: supposing tha t the wave length of the incident particle is small com-

pared with the interpartiele distance, the absorption potential is proportional

to the sum of the cross-sections with the restriction tha t the possible final

states should be permit ted by the Pauli principle. One sees easily, using the

same method and notations as in Sect. 2, tha t the imaginary potential is then:

(4) cG . . . . ~2Mk- t~j dp du dq ua (p, p .q) ~(p - - q) .

a(p, p.q) is the average of the differential cross-sections for neutron proton

and proton proton collisions. The integration domains are the same than in

Sect. 2, and there is also the restriction p2+u2--k~> O. We approximate

the differential cross-section by the expression:

(5) ,~(p, p "q) = a0(p) + ~l(p)Pl(eos 0) + adp)Pdeos 0).

The quantities ao(p), (~I(P), (12(P) are directly related to experiment.

33 - II N u o v o Cimento.

5 1 4 L. VERLET a n d j . GAVORET

The imaginary pa r t of the poten t ia l corresponding to the isotropic t e rm

in the cross-section is easily calculated. For the in tegrat ion on q, one has :

f d 2~p (6) q(~(p--q)= ~ , (u ~÷p~-k~).

After an immedia te in tegrat ion on u, one has an expression for the imag ina ry

p a r t of the potent ia l where the only in tegrat ion to be done involves quan- tit ies direct ly related to exper iment (*).

The results are shown in Fig. 5. They are qual i ta t ively similar to those obta ined b y VAN DER VEGT and JONKER (21). The differences arise ma in ly

f rom the fact t ha t we did not make the assumpt ion an, ~ = }a~ .

20

10

L

.tm (#,~ MeV

, Elab. 4~o 9Io 1~o 19o

J

J

Fig. 5. - Imaginary potential obtained with the classical model using the experimental cross sections of Fig. 1 when only the isotropic part is included (curve 1) and when

the anisotropy is t~ken into account (curve 2).

The contr ibut ion to the imaginary potent ia l due to the /)1 t e rm in (5)

cancels out. The P2 contr ibut ion has been calculated using the exper imenta l

values (.~7) of a2(P); it is found to be fair ly small. The imaginary potent ia l

with the P2 correction included is shown in Fig. 5. The smallness of the ~ni-

so t ropy correction leads us to th ink t ha t the addit ion of anisotropic t e rms

to the separable potent ials would not essentially change the results of the

t -matr ix calculation for the imaginary potential .

(*) Similar expressions will be derived for r~ and K-mesons in a forthcoming paper.

ON T I l E O P T I C A L M O D E L F O R N U C L E A R R E A C T I O N S 5 1 5

We shall now introduce in the classical expression (4) the free nucleon-

nucleon cross-sections calculated with the separable potentials of Sect. 2

(curves 2 and 3, Fi~. 1) and compare it with tile imaginary par t of the po-

tential obtained from the t-matrix. We thus single out the effect of the clas-

sical approximation with the restriction that we have still to show that the

results obtained do not depend essentially on the fact tha t the potentials

which were used are separable. The results are tabulated for both potentials

in Table I and they are shown, for the E g potential in Fig. 4. I t is seen tha t

below 100 MeV the <(exact ,) and the classical calculations differ by a factor

of the order of 2. The failure of the classical model to lead to more than semi-

quant i ta t ive results is of course not unexpected. From these results one is

also tempted to conclude that above 100 MeV, the classical model still under-

estimates the absorption and tha t it is altogether better to negle('t the Pauti

principle than to take it into account through the Goldberg'er approximation.

We shall see, however, t h ~ this last point, rests on the special form of the

potentials tha t we have used. To see this, we have calculated the ratio between

the second Born approximation for the t-matrix with and without the exclusion

principle, both for local and separable potentials. In Table I I we have given

TABLE II. Ratios of various appro.rimations o/ the t-matrix calculated u, ith the t 'auli principle taken, in.to account to the same qactntities when the exclusion, principle is left out.

I Yuk~wa i ER potentiM I ¥ potential

I I i EA '

: (MeV)

1

27 i_ i

80

2

1.5

G ~l lSSi~l l

Dir. Exch.

3 4

0.32 0.37

0.47 0.65

0.47 0.71 230

Dir. ] Exch. '2'~Born I

5 I 6 - - 7

0.31 0 . 3 4 0 . 4 2

0.45 0.58 0.61

0.53 0.58 0.82

Exact

8

0.79

0.94

0.94

L, , . I t Class. !2 Bornl Exact' Clas~.

. . . . 1 1 ~ ! 12- - - 9 ll)

0.36 0 . 3 4 1 0 . 7 5 ! 0.35

] 0.51 0.60 0.89 0.63

0.83 0.75 0 . 9 1 o.82 - - - - ] - -

these ratios for the direct and exchange terms of the Gaussian and Yukawa

potent,ials (15) which fit the low energy-data (columns 3-6 of Table I I ) and

for the E R and Y separable potentials (columns 7 an6 10 of tile same table).

We see that at fairly low energies these ratios are much the same whatever the

potential is. On the opposite, at high ener~o'y the reduction due to the Pauli

principle is much big~'er for a local regular potential than for a separable

potential. The reduction factor for the classical model, calculated with se-

parable potentials (columns 9 and 12), is much the same than the one given

by the second Born approximation of the ~, exact ~ t-matrix calculated with

the same potentials. The exact result is much less affected by the Pauli

516 L. VERLET and 5. GAVORET

principle than the second Born approximat ion . This obvious fac t is i l lus t ra ted

b y the figures of columns 8 and 11. The unsensivi ty of the second Born resul t

to the nucleon-nucleon interact ion, added to the fact t ha t the reduct ion fac tor

is a lmost the same in the classical model and in the second order calculation,

seems to point to the safe conclusion t h a t the classical model overes t imates

bad ly the role of the Paul ; principle a t tow and in te rmedia te energies. As ~t high energy the effect of the Paut i principle depends s t rongly on the kind of in terac t ion t ha t is used, one cannot draw any conclusions f rom these a rguments as to the val idi ty of the Goldberger approx imat ion a t high energy.

5. - Behaviour of the imaginary potential in the surface region.

We shall now t ry to get an insight on the behaviour of the optical potent ia l

in the region of the nuclear surface. We shall make the Thomas -Fe rmi ap-

proximat ion, i.e. we define in each point a local Fe rmi gas whose m o m e n t u m k F

depends on the local densi ty ~:

G = ( 3 ~ 2 Q / 2 ) ~.

W e shall assume tha t the m a t t e r distr ibution is identical with the charge distri- but ion given b y Hofs tad te r ' s exper iments (.,s). As we are only interested in

the surface region, we can choose any heavy nucleus. For instance, for gold,

one has:

169 (7) e(r) = 1 + exp [(r - - 6.38)/0.535] '

in which r is expressed in fermis. In each point , we can define a local t -mat r ix in the same way as in Sect. 3, and thus a local complex potential . I n this

section we shall take AE = 7 MeV, which, tak ing (7) into account, gives, for

the auxi l iary well, the value of 45 MeV when M * / M = 1.

We cannot expect , with this method, to obta in any th ing of interest for

the real pa r t of the well, whose radial behav iour on the other hand is ra ther

well explained to day (29). The potent ia l lies abou t 1 fermi outside the m a t t e r

dis t r ibut ion for two reasons:

1) The sa tura t ing character of the nuclear forces makes the relat ion

be tween the potent ia l and tile densi ty non-l inear so t h a t the poten t ia l lies

outside the densi ty b y abou t ½ Fe due to this effect; as we used, for the sake

(2s) B. HAI~N, D. G. RAVENHALL and R. HOFSTADTER: Phys. Rev., 101, 1131 (1956). (29) L. WILETS: Phys. Rev., 101, 1805 (1956).

ON T H E O P T I C A L M O D E L F O R N U C L E A R R E A C T I O N S 517

of simplici ty forces which are not sa tura t ing (al though they give a t normal

densi ty the r ight order of magni tude for the binding energy (6)), we c~nnot

take this effect into account.

2) The finite range of the force, which is manifes ted especially by the

direct pa r t of the first Born approx imat ion (30); this effect, when proper ly

t rea ted yields another ½ Fe for the potent ia l extension. Due to the unrealist ic

character of the separable potent ia l , we cannot hope to reproduce correct ly this effect either.

On the other hand, we can hope t o have some insight on the imaginary potent ia l a t low energy because we proper ly take into account the Pauli prin-

ciple which is very impor t an t in this case, and because the finite range effect should be smaller than for the real potent ia l as only Born approximat ions

higher than the first one are non-vanishing. We p lo t ted in Fig. 6 and 7 the

, ) ( r ) I

05

0 - -

_20 ~-

(1)

2 h 8 rF~rm, l f L

Fig. 6. - R~dial density (curve 1) and r~dial beh~viour of the imaginary potential irt the Thomas-Fermi approximation for E,,=O (curve 2), E~= 10 MeV (curve 3) and

E , ~ 30 MeV (curve 4) with M*/M= 1.

imaginary potent ia l for M*/M = 1 and M*/M--0.8 respect ively for several

values of the incident part icle energy. We see t ha t the absorpt ion takes place

main ly on the surface a t low energy. This is due to the fact tha t in the sur-

face region there is a compet i t ion between two effects: a decrease in the

densi ty which finally, as the value of r increases, makes the ~bsorption go to

zero, and the action of the Paul i principle which becomes less and less im-

(s0) S. D. DRELL: Phys. Rev., 100, 97 (1955).

5 1 8 L. V E R L E T and J . GAVORET

p o r t a n t when one goes towards the surface of the nucleus. The predominance

of this second effect a t low energy explains the surface absorpt ion. When

the energy rises, the Paul i principle plays a smaller role so t h a t the decrease

of the densi ty when one goes towards the outside tends to be the ma in effect.

~(r)

05~

(1)

L 8 rre~=,

Fig. 7. - Same as Fig. 6 except that M*/M--0.8.

About these results, we wish to make the following remarks :

1) The surface absorpt ion does not seem to be due to our use of the separable potent ia l , a l though it m a y be exaggera ted by it. We have made the same t r e a t m e n t of the surface behaviour of the imaginary potent ia l with a local gaussian potent ia l t r ea ted in second Born approximat ion . At zero energy of the incoming part icle it gives an absorpt ion on the surface which,

a t its m a x i m u m , is twice as big as it is in the center. On the other hand, the

ra t io of the second Born approx imat ion to the exact value, as given with the

E R separable potent ia l with M*/M = 1 , goes f rom 0.25 a t the center to 0.1

in the region of m a x i m u m absorpt ion, always a t zero energy. This shows

t h a t we m a y expect a bigger contr ibut ion of higher Born approx imat ions in

the surface region than in the center~ so t ha t an exact calculation of the

t -ma t r ix with a local regular po ten t ia l would ?Ave a surface absorpt ion still

more pronounced than when only the second Born approx imat ion is considered. A t zero energy the classical model gives also a surface absorpt ion whose

m a x i m u m is 3 t imes wha t it is at the center of the nucleus.

2) The exper imenta l si tuation, a l though it is still not comple te ly clear, is not in contradic t ion with a surface absorpt ion (23-25). In reference (2~) ex-

ON THE OPTICAL MODEL FOR N U C L E A n REACTIONS. 5 1 9

p e r i m e n t a l n e u t r o n e l a s t i c c ross -sec t ions are a n a l y z e d in t e r m s of a rea l Saxon-

t y p e p o t e n t i a l a n d of an i m a g i n a r y p o t e n t i a l of t h e fo rm (a~):

• [ ( r ; ) l }~ exp - - R0 2 ,

w h e r e Ro ~ r~A ~. The consta .nts a r e : b = 0.98 F e , r0 ~ 1.25 F e , VII is e q u a l to

7 3IeV, 9.5 MeV a n d 11 MeV when the i n c i d e n t n e u t r o n e n e r g y is e q u a l to

4.1 MeV, 7 MeV a n d 11 MeV r e s p e c t i v e l y . I t m a y be seen t h a t t h e r e su l t s of

th is e x p e r i m e n t a l ana lys i s a re in a g r e e m e n t , as fa r as o rde rs of m a g n i t u d e

a r e conce rned , w i t h t h o s e t h a t we d e d u c e d t h e o r e t i c a l l y .

3) A p a r t f r om t h e T h o m a s - F e r m i a p p r o x i m a t i o n , some o t h e r a p p r o x -

i m a t i o n s which h a v e been m a d e in th i s work m a y h~ve an i m p o r t a n t inf luence

on t h e shape of t h e ~tbsorpt iou p o t e n t i a l :

- - The v e l o c i t y d e p e n d e n c e has been a s s u m e d to be t he s ame t h r o u g h o u t

t he surface. I t is o b v i o u s l y n o t the ease as t he v e l o c i t y d e p e n d e n c e t e n d s to

d i s a p p e a r when one goes t o w a r d s t he ou t s ide of the nucleus . I t w o u l d seem

n a i v e l y t h a t th is effect t e n d s to g ive a b igge r sur face a b s o r p t i o n t h a n the one

which is c a l c u l a t e d w h e n the s a m e v e l o c i t y d e p e n d e n c e as in t he cen te r of

t h e nuc leus is e x t e n d e d th rouo 'hout .

- - W e h~ve n e g l e c t e d ( ' luster t e rms which are p r o b a b l y i m p o r t a n t in t he

su r face r eg ion (~) b u t we d id n o t f ind any ea sy w a y to a p p r e c i a t e t h e i r in-

f luence.

(a,) There is also an l" s force, which is i r re levant for the present discussion. (a~o) We are indebted to Dr. DE I)O~IINIC18 for having pointed out this fact to us.

R I A S S U N T 0 (*)

La matrice di diffusione dentro la rnateria nucleare indefinita 5 s ta ta calcolata esat- tamente nel modello in eui si prendono in considerazione solo le eorrelazioni statistiche- e con l ' impiego di un ' interazione nueleone nucleone separabile. Se ne deduce un poten- ziale complesso per la diffusione dei nucleoni da par te dei nuclei. Si diseute la con- vergenza delle serie di perturbazione e la validi tg del modello classico di Lane e Wandell . Infine, applicando l 'approssimazione di Thomas-Fermi, si dimostra che alle basse energie l 'assorbimento avviene principalmente alla superfieie del nucleo.

(*) Tradt t z i (n~e st. c ~ r a d e l l a Red~z io t~e .