Effects of Nuclear Correlation on Direct Nuclear Reactions

74

Progress of Theoretical Physics, Vol. 32, No. 1, July 1964

Effects of Nuclear Correlation on Direct Nuclear Reactions

Il-Tong CHEON

DepartJntt:nt of Nuclear Science Kyoto Uniz 1ersity, Kyoto

(Received November 20, 1963)

Effects of nuclear correlations on the differential cross sections for direct reactions of the forms (p, p') and (a, a') are discussed on the basis of the Butler theory. It is shown that the nuclear correlations make no shift of the angular distribution along the scattering angle but make the large-angle scattering increase.

§1. Introduction·

It is interesting to investigate effects of correlations between nucleons in the nucleus on the differential cross section for nuder reactions.

It is well known from both experiment and theory that the nucleon-nucleon interactions are strong and short-ranged. Consequently if the same forces act when nucleons are immersed in a many-body medium, one will very naturally expect to observe under appropriate experimental conditions very appreciable correlations in the nuclear wave function. The correlations are quite essential for the description of many high energy nuclear reactions. 1

)

Recent developments of the theory of the nuclear many-body problem make it possible to have some qualitative and even quantitative insight into the nature of the nucleon-nucleus scattering. Employing the Watson's multiple scattering theory, Dabrowski and Sawicki2

) calculated the well depths of the nuclear optical potential for the case of strongly correlated nuclear matter with the ground state represented by a Jastrow type wave function. T'heir numerical results obtained for the case of a separable nucleon-nucleon interaction show that the effects of the correlations amount to corrections to the Fermi model case up to the order of 10 per cent in the medium and high energy range.

In the present paper, it is particularly discussed how these correlations in the surface region of the nucleus affect the differential cross sections for reactions of the forms (p, p') and (a, a'). As the discussions are limited to the surface region, calculations are based on the Butler theory with the cutoff Born approximation.

It was discussed by Butler3) that predominant contributions came from a direct

process which did not involve the compound nucleus as an intermediate step in the large majority of nuclear reactions which proceeded to a low-lying level of the final nucleus. In fact, he showed that (p, p') and (p, n) reactions leading to low-lying final levels should be equally dominated by direct processes. As is

Dow

nloaded from https://academ

ic.oup.com/ptp/article/32/1/74/1834723 by guest on 21 January 2022

kjfects of Nuclear Correlation on Direct Nuclear Reactions 75

well known experimentally in the cass of (p, d) reactions, direct processes seem to be very important in the formation of such levels of final nuclei.

Effects of the use of the correlated wave function for the nucleus will be discussed on the reactions of the forms (p, p') and (a, a') by the cluster expansion method, under the assumption that these reactions proceed to low-lying final levels through direct processes.

§2. Culculation of the matrix element I(kp, k~)

According to the Butler's theory, the Born approximation matrix element for the direct (p, p') reaction has the following form:

I(kp, k/) = ~de'drndrpCVt(e', Tn)CVo(e', r1l)V';{J."fr't,t(kp', Tp)"frp(kp, Tp), (1)

where CV 0 is the wave function for the ground state of the initial nucleus and CVt is that for the state t of the final nucleus and where e represents the coordi~ nates of all nucleons in the nucleus except the neutron interacting directly with the incident proton, r P and Tn are the position vectors of the proton and a target neutron, respectively, and kP and kP' represent the wave vectors of the incident proton and the outgoing proton, respectively, so that the angle between wave vectors, kv and kv', is the scattering angle. The differential cross section for the direct (p, p') reaction is given in terms of the matrix element I as follows:

(2)

where J.Vlp denotes the mass of the proton and the symbol :Eav represents a summation over final spins and an average over initial spins. In order to avoid unessential complications, spin functions are not introduced explicitly but spms are considered only in the discussions of selection rules.

At first, we shall define the mixed density as follows:

(3)

For CVe and CV0 , we assume the Jastrow type correlated wave function4)

(4)

with 1

¢n (l) = Y Q -YJsn ( dz) 'Xmn ( 'Z"z) (/)an ( Tz), (5)

where ¢P, ( l) is a single particle wave function, YJs and 'Xm are the spin and isospin function, s and m are the corresponding magnetic quantum numbers, Q is the volume of the system, D is the normalization constant and the sum is extended

Dow



76 Il-1~ Cheon

over all the permutations P= (}1 :.'."~i.":.}{) with the proper signs sr, and where

the state independent correlation function J,.(riJ) has the property J,.(riJ)---?0 for rt;-?oo and in the case of a repulsive core of the radius ra in the nucleon-nucleon interaction it satisfies the boundary condition ). (riJ) = -1 for riJ<rc.

A discussion of the wave function involving state dependent correlations of the form

was presented by de-Shalit and Weisskopf.5) With the wave function ( 4), one can expand the mixed density

p(rn) =N~dald'l:'ld(2)···d(N)C{)t*(rnalr-b 2, ... N)C{)o(rnalr-b 2, ... N)

to the Ursell-Mayer cluster2l. 4l-Bl as follows:

where

- ~ <p;, (rn)<pa(r2)<pj, (r2)<p13(rn)},

_ N(N-1)(N-2) ~ ~ P2(rn, rn)- --------3~-'b b b dr2drshC12s) 4 · 3 ! Q o;a/ 1313' rr'

X {(jJ;, (rn) (j)a Cr2) <pif, Cr2) (/113 (rn) (jJ0 (rs) (/J'Y (rs)

- ~ <p;,(rn)<pa(r2)<p:f,(r2)(/JB(rs)(jJj,(rs)(JJy(rn) }.

The h-function is defined by

where

(7)

(8)

(9)

(10)

It is assumed for the final state wave function C{)t that in the uncorrelated part p0 (rn, rn), a single nucleon may be excited to have a higher momentum than the Fermi top and in the correlated parts p1 and p2 , two nucleons may be excited to have higher momenta.

Thus one can write the matrix element as follows:

Dow



Effects of Nuclear Correlation on Direct Nuclear Reactions 77

It is convenient to make the expansion of the matrix element corresponding to that of the mixed density, i.e. Eq. (6).

For convenience, let us make a multipole expansion of the direct interaction V~~' which is assumed to be spin-independent, in terms of the "angular momentum -tensor" operators Vz (r1), rn) as follows:

(13)

where Pz (cos 0) is the Legendre polynomial and (} denotes the angle between the position vectors rv and Tn. Thus the matrix element has the following form

I(kv, kv') = ~ 2f~drndr1)p(rn) Vz(rv, rn)Pz(coso)-r~,t(kv', rv)-rvCkv, rp).

(14)

The mean-free-path of the incident and outgoing particles in the nuclear matter is considered to be very short for the majority of the usual incident energies employed experimentally. This fact means that the wave function -rvCkv, rv) must fall to zero very rapidly as rv decreases inside a core characterized by some radius r 0 • When the coordinate of the proton rv is smaller than (or equal to) r 0 , its wave function pertains to a compound state. The total contribution to the integral (14) with respect to rn, therefore, has to come from the shell whose radius is in the vicinity of r 0 and which has a width dependent on the extent of the surface tail of the nucleon density and on the actual mean-free-path of the proton for capture into a compound state. Let us replace these functions -rvCkv, rv) and -rt,r(kv', rv), by plane waves for rn>ro and set a lower limit to the rn integration at rn = r 0 • Thus we obtain the following result

I(kv, kv') = }y ~ ~r.1~n~drvp(rn) Vz(rv, rn)Pz(cosO) exp [iQrv], (15)

where

Q=kv-kv'. (16)

(a) The matrix element IoU~v, kv')

Now we calculate the matrix element Io Ckv, kv') corresponding to the uncorrelated mixed density p0 (rn, rn)

Io(kv, kp') =~~~t.;/r~n~drv~~~~r(rn)~a(rn) Vz(r1>, rn)P,(cosO) exp [iQrvl.

(17)

Dow



78 1!-T. Cheon

If we chocse the direction of Q as the polar ax1s and if we make the spherical harmonic expansion:

(18)

where (rn, On, (!Jn) are the spherical polar coordinates of the vector rn, then we know that the sum over l is severely limited, often to one term only.

When spins are included, l has, in general, the following restriction:

(19)

where Ji and Jf are the total spins of the initial and final nuclei, respectively. Making the Legendre polynomial expansion of the exponential and using the wellknown spherical harmonic addition theorem:

= exp [iQr 11 ] = ~ C2n + 1) injn(Qr11)Pn(cos fJp)

n~o

=£1/ 4n(2n+l) t~jn(Qrp)Yno(fJp, <pp), (20) n=tJ

(21)

we obtain

X fa*, (rn !1' rn/) Vt (r 1), rn) :8 V 4n(2n + 1) injn (Qr 11) n=O

where jn is the spherical Ressel's function. The angle integrations may readily be performed to yield the result

(23)

c!f o (l1l1' m1 n; Q)

= ~ r;=drvrp'J jn(Q rp) r= dr,, rn2fa (rnll n11)jJ; Crnl/ m1) Vn (rp, rn), (24) ailXi Ju J r 0

where the C's are the Clehsch-Gordan coefficients. 7) From the property of Ccoefficient, there is R restriction among n, l 1 and l 1':

Dow



Effects of Nuclear Correlation on Direct }•luclear Reactions 79

n+ZL+lt'/even. '(25)

It will be seen later that this matrix element Io (k 11 , kP') IS that which was calculated by Bultler.3 )

(b) The 1natrix element I1 (kv, kv')

Here let us calculate the first order matrix element I1 (kp, kp'). By employing the procedure used in (a), we can readily obtain the result

l1 (kv, kv') = Jv 2j ~r.1~n ~drp P1(cos 0) Vt(r.v, rn)P1 (rn) exp [iQrv]

= (~~21) ~ ~ ~~ 1~ 1~1,Fm212~21 ~~drndrvdr216h(r2n) 21~ 1 V 4n-(2n+1)

xinjn(Qrp)Ylm(On, cpn)Y~(Op, cpp)Yno(Op.cpp) Vl(rp, rn)[fJr(rnl~mDfa(rnllml)

xjt; Cr2z; mDfs Cr2l2 m2) Y~'m/ (On, cpn) Yl1m1 (On, ({Jn) Yt'm/ (02, (/J2) Yt2m2 C02, cp2)

- ! fJ, (rnl~ m;)j"a(r2l1 mt)ft, Cr2l; m~)Ja(rnl2m2) Y~'m/ (On, <pn) Yl1m1 (02, (/)2)

(26)

As the state independent correlation function ..t (r), we shall use the Gaussian type

(27)

Let us expand this function to the following form8 )

-

00

( 1) exp [- ar~n] = exp [-a (r22 + rn2)] V 2n-~ p. + T (2ar2 rn) - 112

X IP.+l/2 (2ar2 rn)P p. (cos 02n)' (28)

where 02n is the angle between vectors r 2 and rn, and

I ( ) - 1 [z~ (-Y(n+r)! +(-)n+1 -z~ (n+r)! J n+ 1/2 z - e L..J { e L..J •

V2n-z r=O r!(n-r)!(2zY r=O r!(n-r)!(2zY

(29)

By using the relation (21), we can write the h (r2n) function as follows:

00 fl.

hCr2n) =~ b [exp{ -2a(r22+rn2)} (4ar2rn)-112 IP.+l/2(4ar2rn) p.=O v= -p.

Thus Eq. (26) becomes

I1Ckp,kp')= (N )2_~~~ b b 2S~b~b4(4n-) 3 inv~{(2n+l_L 2J2 lm l1m112rn2/1fm11 l2fm2l n=O afaf' /3/3' W"O 11 2/ + 1

Dow



80 ll-T Cheon

X ( 4ar2 rn) - 112 IM+l/2 ( 4ar2 rn) -2 exp [-a (r22 + rn2)] (2ar2 rn) -l/2 IM+l/2 (2ar2 rn)}

Xj1; Crnl1' ml')ja(rnlt1nJJft; Cr2/2' m2')JsCr2l2m2)H1

- ! {exp [ -2a(r22+rn2)] (4ar2rn)-112 IM+l;2 (4ar2rn) -2exp [ -a(r22+rn2)]

X (2ar2 rn) - 112 I11-+112 (2ar2rn)} jJ; (rnl/ Jn/)j~ Cr2l1 m1)

Xji;(r2l2' m2')fs(rnl2m2)J-I2J,

where

(31)

H1 = ~d..QndQ2 dQP Y;v (02, (/)2) Y~'m/ (02, (/)2) Yz2m2 (02, (/)2) Y~1m1 1 (On, (/)n)

X Y fi.V (On, <pn) Ytrn (On, <pn) Yr 1m1 (On, <pn) Y;l;tt ( 0 1" <pp) Yno ( (} p, <pp),

H2 = ~dJJndJ22 d.QPY;v (02, (/)2) Y~'m2' (02, (/)2) Yz 1m1 (02,(/)2) Y~'m/ (On, (/)n)

XYz2m2 (0n, <pn)YMv(On, <pn)Yzrn(On, <pn)Y't,n(Op, (/Jp)Yno(Op, <pp).

(32)

(33)

The integration over angles in Eqs. (32) and (33) can be immediately performed. Thus we ·have

TJ -a a (-)m2'~[ (2l+1)(2/1+1)(2.l2+1)(2/2'+1) ]112

C(ll I • -. , )

.l-:11- ln mo ~ ( 4rr)3 (2/1' + 1) . 2 2 p., m2 m2 1/

XC(l2l2'p.; OOO)C(ll1ls; 1n m1m+ml)C(lllls; OOO)C(p.lslt'; J.l m+m1m/)

xC(p.lslt'; 000), (34)

H2 =lhnamo(- )m2 ' ~ [-(2l + 1) _(?!__lc± 1) _(?.£2 + l) (2!'2_:±_!1]112

C( ,lf3

• v m J.l +n'l) ~ (4rr) 3 (2!/ + 1) 0

' '

X C(p.lls; 000) C(lsl2lt'; v + m m2 m./) CCZsl2l1'; OOO)C(l1l2' p.; m 1 -nz2' v)

(35)

We may now find for the first order matrix element

X [ J:i211 + l)c~:;: i? C2!~' + 1

) T12 (- )m2'[ccz2l2' p.; m2 -m2'v)C(l2l2' p.; 000)

xC(nl1ls; Om1m1)C(nl1ls; OOO)C(p.lsl1'; v m1m1')C(p.lslt'; 000)

xt!J11(l11n1lt'1nt' l2m2l2'1n2'np.; Qa)- ~ C(p.nls; ).IOJ.i)C(p.nls; 000)

XC(l3l2l/; vm2m/)C(lsl2l/;OOO)C(l1l2'p.; m1 -m2'v)C(l1l2'p.; 1n1 -m2'v)

(36)

Dow



Effects of Nuclear· Correlation on Direct Nuclear Reactions 81

where

c!fu = C"'dr/oodrn\00

dr2rp2rn2r22jn(Qrp) {exp [ -2a(r22+rn2)] (4ar2rn)-112 JJ Jro Jro

X IJ..t+l/2 ( 4ar2 rn) __;_ 2 exp [-a (r22 + rn2) J (2ar2 rn) -l/2 IJ..t+l/2 (2ar2 rn)}

X Vn(rp, rn)f/!; (rnlt' mt')J~(rnl1 mJj~:;*, Cr2l2' m2')j13(r2l2m2),

c!f12= \""'drp\00

drn\ccdr~~rp2 rn2 r22jn(Qrp) {exp [ -2a(r22+rn2)] (4ar2 rn)-112 jo jro jril

X IJ..t+l/2 ( 4ar2 rn) -2 exp [-a Cr22 + rn2)] (2ar2 rn) - 112 IJ..t+l/2 (2ar; rn)}

X Vn(rp, rn)f/!; (rnlt' m/)f,x (r2l11n1)jl; Cr2l2' m2')/sCrnl2m2).

(37)

(38)

The summations over lb l/, l 2 and l2' in Eq. (;36) are quite restricted. For g1ven states a and a', for example, we have

Jlt+Ja+ ~-~mi;=z1<Ji+Ja+ ~ ,

I Jt+Ja+-2~-~ . <l/<Ji+Ja+ 21

• mm

(39)

The allowed values in each case are either even or odd depending on parity considerations. Similar restrictions are placed on [2 and [2'.

§3. Zero-range interaction

It is well known that the zero-range interaction is a good approximation to the neutron-proton interaction in the calculations of pick-up or stripping differential cross sections for the usual energies employed experimentally. ·When this interaction is used in the (p, p') reaction, it is able to write v~r)r in the form

(40)

Since the main contributions to Eq. (11) come from the tail region of the nucleon density distribution, it is reasonable to assume that the effective direct interaction v-;;:1~ does not involve any average influence of other nucleons but is the actual interaction Vpn between the neutron and proton. On this basis we may consider the strength Vo of Eq. ( 40) to be independent of rn and assume that for the region rn?::,ro, the functions /a(rn, l, m) and fs(rn, l, m) have the approximate forms

/s(rn, l, m)~Bs(l, m)hl(iknrn),

f~:;*, Cr2, l, Jn) ~B0 (l, m)h.* (ik2' r2),

(41)

Dow



82 Jl- T. Clzeon

where h1 is the spherical Hankel function of the :first kind, tr determines the binding energy ti}~r2 /21\1n of the neutron, and coefficients A and R are constants independent of rn and r2. For the matrix element, therefore, we may obtain the following form

xc!f'oCldt' 171-tn; Q),

where

(42)

S' o= ~ Aa (!1 nZt) A~, Cl/ mt) r= drn rn2 jn(Qrn) hll (il.:nrn) ht/ (iKn1 rn). ( 43) a,a/ J,o

_/(2lt+l)(2l2 +1)(2l2 '+1) I . XV (2-z

1' ~l) -------- L C(l1 nl:,, nL1 Omt)CCltnl3 ; 000)

xC(l2l'/f-1.; rn2 -m2'v)C(l2l2'f-l.; OOO)C(lsf-1.!1'; mtvm/)CClstJ.lt'; 000)

xS~tCltm1lt'mt' l2m2l2'm2'f-l.n; Qa) --!-C(l2nl:,; m201n2)C(l2nl3; 000)

X C(l1l2' f-1.; m1- m2' v )C(l1l2' f-1.; 000) CC!slt!/; m2 1-m1t') CCls tJ.lt'; 000)

xS~2Cltm1l1'm/ l2m2l2' rn2'f-l.n; Qa) J, where

c!f~1 =A;, Cl1' m 1')Aa Cl1 m1) R'J (!._/ m:/) BBCl2 m2) \"" drndr2 r-,? r 22 jn(Qrn)

Jro

(44)

X [exp{ -2a(r}+rn2J} (4ar~ru)-112 IM112 ( 11ar2 rn) -2exp{ -a(r22 +rn2)}

X (2ar2 rn) -l/2 I/J.-t-1/2 (2m-2 rn) l hit' ( iKn1 rn) hit CiKn Tn) hzz' (ik2' r2) hzz (ik2 r2)' ( 45)

S~2 =A~, Clt' mt') Aa Cl1 m1! B~, Cl2' m2') BBCl2 m.J r= drndr2 rn2 r22 jn(Qrn) .to X fexp{ -2a(r2LJ-rn2

)} (4ar2rn)-112 l,"+1/s(4ar2 rn) -2exp{ -a(r22 +rn2)}

X (2ar2 TnJ -l/2 IIL+l/2 (2ar2 r,J l hit' (iten 1 rn) hzt (ite2 r2) hz/ (ik2' r2) hzz ( ikn rn). ( 46)

In the case of !1 ::-=- 0, Eq. ( 42) represents the matrix element obtained by Butler.3

) The matrix element 11 (kv, k 11') is, therefore, the first order correction term due to effects of correlations.

§4. Numerical results and discussion

In the present paper, we do not give any comparison between theory and experiment but investigate the effects of the correction term 11 (kp, kP') on the

Dow




differential cross section by giving suitable values to the energy of incident protons and binding energies of nucleons.

It is assumed that there will, in general, be one predominating term in the summations over a, a', [j and /3' in Eqs. (43) and (44). Moreover, it is assumed that the summations over l 1 and l2 are limited to one value each and in particular case l1=l2=0. We shall calculate the cross section, when l 1 =l2 =l1'=0 and l2' = 2, under the assumption that the final state of the nucleus is in a low-lying level. In calculating the matrix element 11 , one must sum up over all the values of lt' and l2' which satisfy the selection rule (39) and the parity conservation. But it will only increase the differential cross section obtained under the above restriction on l2', without changing its shape.

Making some approximations ·for integrals, which are shown in Appendix, one may write the matrix elements as follows:

( Q Kn + Kn1

• Q ) X cos r 0+-o- sin ro , (47)

vs,ro~lf-})2 A*(O, O)A(O, O)B*(2, O)B(O, 0)(1+~+ k' 23. 2 )

KnKn {:2 2 ro t2 /?,2 ro 2 l o

X exp [- (~en+Kn'+ k2+ k2')roJ [ ( 1- Q:r02)( P1 --~R1) + ;0

( P2- ! R2)],

where (48)

p 1

= _1_(3 _ 23_ + _ _j~-) (~en+ K1 n) sin Qr0 + Q cos Qr0

a 4aro2 16a2 ro'1 (k2 + k2') [(~en+ Kn') 2 + Q 2] Q

_J-_(l+-3-+-- 3 ) (8ar0 +~en+,~n'2_~inQ?~+Q C1<?~_Q!j]_ a 4ar02 16a2 r 0

4 (8aro+k2+/?-2) [(Saro+~en+Kn ) 2+Q2]

X exp [- 8aro2 ]

+ _i_(1 + _3_ + _ 3 ) ( 4aro + Kn + Kn') sin Qr0 + Q cos Qro a 2ar0

2 4a2r 04 C4aro+k2+k2') [(4aro+Kn+~en') 2 +Q2]

X exp [ -4aro2 ], · (49)

p2=_1_(3 _ __23 + _ 45 ) (~en+~en') cosQr0 -Q sinQr0

a 4aro2 16a2 ro4 (k2 + k2') [(~en --!-~en') 2 + Q 2] Q 2

_ _1_(1 + _3_+ 3 ) (8aro+~en;t·~en') cosQr0 -~ sinQr0 a 4ar0

2 16a2ro4 (8aro+k2+k2) [(8aro+~en+Kn ) 2+Q2] Q 2

X exp [ - 8ar02 ]

+_!-(1+_3_+ 3 ) (4ar0 .-l-Kn+~en') cosQro-Q_sinQ_r0"----

a 2aro2 4a2 ro 4 ( 4aro k2 + k2') [ ( 4aro + Kn + Kn') 2 + Q 2] Q 2

X exp [- 4ar02] , (50)

Dow



84 Il-T. Cheon

(51)

(52)

In Eqs. ( 49), (50), (51) and (52), the first term is a leading part. Considering only these leading terms, I1 (kv, kv') has the following form

I(k k ')-[4VoA*(O,O)A(O,O) {-( + ')·}J·r 1 p, p - · 1 n exp Kn Kn 7 0 llnKn ro.o~

X [(I----~-) {~~n+_~rn') ~inQJ_j_J_±-:-Q cos9_ro __ l__ (~r2 +~rn') si~Qro±Q~_osQro_} Q 2 ro2 (k2+!?2') L(~rn+Kn') 2 +Q2 \Q 4 (!?n+/?2') [(~r2+~r,/) 2 +Q2JQ

+ ;0 { ~i~-t-}:~~ ~(:.~i~75~!i:J~732 - -l- {k!: ~:~~~-~-(:~-:/>~~iQ~32}] , (53)

where

_v5(N-1) * ( 3 3) r--4

z:z:-, _ 2 ,:;-B (2, O)B(O, 0) 1 + T,-~-+k-, 2 _ 2 1<2 1<2 1 o .o.: K2 1 o ,2 1 o

X _1__(3-4

23 2- + -

16 4~ _

4) exp [- (!?2 + k2 ')ro]

a aro a Jo

Figure 1 shows angular distributions for the direct reaction (p, p') with the incident energy of 30 MeV and with the values of Kn, ~r 2 , !?2 , ~r,/ and !?n obtained by assuming that binding energies of the nucleons responsible for the scattering are 13 MeV and with the value of !?2 ' corresponding to the binding energy in the final state which is assumed to .be 10 MeV. In order to determine the value of a in our first order approximation, we must know the values of factors B* (2, 0) X B(O, 0) .9 ) But a suitable value of a is used, because in the present paper we want to know only the characteristic effects of the correlations between nucleons in the nucleus on the differential eros:> section wihtout any comparison between theory and experiment. We cannot choose the arbitrarily small value for a, because it should be unnatural that such a correlation as is given in this paper exists strongly when the distance between nucleons is larger than the nuclear radius. Thus it has a severe restriction, ar0

2> 1.

The effect of the correlations becomes larger as the factor r increases. From Fig. 2, one can readily know that r has the largest value in the case ·of a=

(23 + 2-.f 31 ) /12r02

•

In Fig. 1, the full curve is the zeroth order angular distribution in the case of r 0 =3 fermi, and the dotted curve represents the angular distribution containing the effects of the first order correlation with r= (20)-1 and r 0 =3 fermi. The oscillatory nature of the angular distribution is sensitive to the value of r 0 , and if the larger value than 3 fermi is used for r 0 , the angular distribution shifts rapidly towards smaller scattering angles.

Dow




As is shown in Fig. 1, the large-angle scattering has a tendency to increase slightly without shifting the positions of peaks due to the correlations between nucleons. When the heights of these two angular distributions are normalized at 20°, the peak of the dotted curve is higher than that of the full curve by 17.6

22

21 I I I I I I

20 \ I I I

-:.:-. I ·a I ;::l

>, 19 I

..... oj

.t; :..0 .....

t1:S '-" -::::: 4.§ u <!)

,<fl

"' "' 0 5 . ..... u

~ (:1 <!) .....

:£:l 4 Ci

3

Scattering angle fJ

Fig. 1. The full curve is the zeroth order angular distribution in the case of ro= 3 fermi. The dotted curve represents the angular distribution containing the first order correction. It is drawn in the case ofT= (20)- 1 and ro=3 fermi.

Dow



86 Il-1~ Cheon

y

5

4

3

2

1

0 0.1: 0.2 0.3 0.4 0.5 0.6 0. 7 0.8 a I

: -91 (=--4); minimum:~~

ro ro

per cent. If the heights of these curves are normalized at smaller angles, this tendency will increase. These facts are very interesting.

Effects of nuclear correlations on the surface region of the nucleus are never large, but it seems that those effects on the inside of the nucleus reflect strongly on the angular distribution.

For the direct (a, a') reaction we can obtain the differential cross section by making a simple modification to the treatment of the direct (p, p') reaction if we do not consider the structure of a-particle.

Acknowledgments

The author expresses his thanks to Prof. M. Kobayasi and Prof. T. Marumori for their advices and kind encouragements and to Dr. T. Sasakawa and Dr. S. Nagata for useful discussions.

Appendix

~ "" 1 1 ~00

1 --exp( -tcr)dr::::::::.- exp(-tcr)dr=--exp( -tcr0). ro r - - ro ro tcro

(A1)

(A2)

(A3)

Dow



E.]fects of }luclcar Correlation on Direct Nuclear Reactions 87

r= \"' cos~y exp [ -2a(x-y) 2 -kx-~.:y] dxdy Jro Jro xy

1 [ 1.:2 ]\00

; ( k )]-1 :::::::::: 4

a exp Sa- (~.:+k)ro ,),.;>-'-ly2 -r0 y-4; cosQy

X exp[-2a( y- ro + 4~.:; Y]dy

:::::::::: r)k~.: [ cosQro --Q2 (~.: 2 +Q2)-1 {cosQr0 +-~-sinQr0} J exp [- (~.:+k)r0]. (A4)

In this approximation, this integral is independent of a.

\"" r= sin~y exp [ -2a(x+y) 2 -kx-~.:y] dxdy Jro Jro xy

= ~: dy ·y-2 sinQy exp[ i: + u~-~.:)y J~: dx·x- 1 exp[ --2a( x+ y+--:~ )2

]

1 [ 1.:2 ]\oo _ . [ k J-1 [ , ( K )2] :::::::4a exp 8~-+ (~.:-k)r0 J,"x 2 smQx x+r0 +4~~- exp -2a x+y+4:~[

:::::::r(} [8ar0 +k] - 1 [ (8ar0 + ~.:) 2 + Q 2

] -1

[ (8ar0 +~.:)sin Qro + Q cosQro]

X exp [ -8aro2- (~.:+k)ro]. (AS)

References

1) K. A. Brueckner, R ]. Eden and N. C. Francis, Phys. Rev. 98 (1955), 445. 2) ]. Dabrowski and J. Sawicki, Nucl. Phys. 13 (1959), 621. 3) S. T. Butler, Phys. Rev. 106 (1957), 272. 4) R. Jastrow, Phys. Rev. 98 (1955), 1479. 5) A. de-Shalit and V. F. Weisskopf, Ann. of Phys. 5 (1958), 282. 6) E. E. Sal peter, Ann. of Phys. 5 (1958), 183.

S. D. Drell and K. Huang, Phys. Rev. 91 (1953), 1527. I. Iwamoto and M. Yamada, Prog. Theor. Phys. 17 (1957), 543. C. D. Hartogh and H. A. Tolhoek, Physica 24 (1958), 721, 875, 896.

7) M. E. Rose, Elementary Theory o.f A_ngular A1omentum. 8) G. N. Watson, A Treatise on the Theory of Bessel .function (Cambridge University Press,

1922), 369. 9) S. T. Butler, Proc. Roy. Soc. (London) A208 (1951), 559; Phys. Rev. 88 (1952), 685.

Dow



Effects of Nuclear Correlation on Direct Nuclear Reactions

Documents

Transcript of Effects of Nuclear Correlation on Direct Nuclear Reactions