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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 .