Two-nucleon stripping processes

I Nuclear Physics 77 (1966) 182--192; (~) North-Holland Publishin# Co., Amsterdam 2.G

I Not to be reproduced by photoprint or microfilm without written permission from the publisher

TWO-NUCLEON STRIPPING PROCESSES

A. Y. ABUL-MAGD and M. EL-NADI

Atomic Eneryy Establishment, Cairo, Eyypt, UAR

Received 13 April 1965

A b s t r a c t : The theory of two-nucleon stripping reactions is developed in the distorted wave Born approximation. The two-nucleon forces responsible for the reaction are taken to be spin and isospin dependent and of finite range. Expressions for the angular distribution and polarization of the emitted particles are given. The angular correlation and circular polarization of ~,-rays following the reaction are considered.

1. Introduction

Since the two-nucleon stripping mechanism 1) was suggested for the reactions in which two nucleons are captured from the incident projectile, a considerable amount of theoretical work has been published in this subject. However the plane wave Born approximation was used in most of these works 2). Recently Henley and Yu 3) developed a general formulation of the two-nucleon stripping reactions taking into account the finite range of the stripping interaction and the distortion of the incident and outgoing waves. Introducing some approximations while calculating the reaction matrix element, the authors obtained results which satisfactorily agree with the ex- perimental data. Hence one may expect that the two-nucleon stripping theory will be improved by replacing the plane waves representing the incident and outgoing particles by the corresponding distorted waves.

In this paper, the distorted wave theory of two-nucleon stripping reactions is further developed. Spin and isospin dependent terms are added to the two-nucleon forces responsible for the reaction and the effect of these additional forces on the selection rules obtained for spin- and isospin independent interaction 4) is investigated. The final expression obtained for the reaction amplitude is correlated with the expressions obtained in the theory of cluster stripping processes 5, 6) in order to apply the methods of calculation used in the analysis of these reactions. Expressions for the angular distribution and polarization of emitted particles and for the angular correlations and circular polarization of y-rays following the reaction (if any) are derived.

The results of this paper are presented for triton and 3He induced reactions. Never- theless, they can be similarly derived for other examples of two-nucleon stripping processes.

182

TWO-NUCLEON STRIPPING PROCESSES 183

2. Formulation

The transition amplitude for the two-nucleon stripping reaction T(c, a)R may be expressed in the distorted wave Born approximation 7) in the following way:

(-) V (+) r l/ta/gR,/~e/~T = <f~ (l'ag)~/Salta,ta~al//SRlZR, tR~RI If~ (eT)~$c~e, lto¢o~/$T/ZT, tT'fT >' (2.1)

wheref~ -) a n d f ~ +) are the wave functions describing the relative motion in the final and initial channels with incoming and outgoing boundary conditions, rtj is the dis- placement of the centre of mass of the particle i from that of the particle j , ~ksj,~, tm is the internal wave function of the particle j which has the spin sj, the spin-projection/~j, the isospin tj and the isospin proj ection z j, and Vis the stripping interaction which will be taken as the sum of the interaction between the two captured nucleons and the nucleons of the outgoing particle a. Now let us expand the wave function of the residual

¢$'~u'~,,'~ of various states of nucleus ~b$,ua, t ~ in terms of the wave functions ~T the target nucleus and all possible shell model functions of the two captured nucleons ~b~:,j,u:, given in the jj-representation

~Ps~a, ta~a = E {nl n21112 J l J2 t11 t12 J~, TZeT s~ I.t~ t~} t ! t v

x A,r(n 1 l 1 J l , n 2 12 J 5 ; JTSTSRtTtR)(STJtT, Jplss/'tlO

x ( t ~ r , ZZltRXs)(jl ~ , JZ #ZIJ/~)ZT=(h, tg)

× ~nl|ljlltt(rl)~n212J21~2(r)Oes'TTl~tT, t'TtT, (2.2)

where the summation indices are enclosed in curled brackets and the coefficients A~r are determined in the nuclear structure theory. The next step is to transform the two captured nucleons wave function into those of the L S representation

= E {m~ rn z ~FS, .C [a } ( [S ] [~a ] [ j , ] [ j 2 ] ) ½ 12

x (Sa, -£adt'lJ#)(/1 rnl, 12 mz[~M)~.lhm~(r~)

× (2.3)

where [S] =- 2S+ 1 and {. . .} is the 9j-symbol. I f the radial parts of the wave functions ~b are taken to be those of an infinite har-

monic oscillator, then the two-particle wave function separates into centre-of-mass and relative motion wave functions 8)

{ml rn2}(/1 rnl, Is m2lLP~/¢)~.m,,(rl)ff,mm2(r2)

= E {nNILmM}<n~l~', n212 : .o~Inl, NL: ~ >

x (LM, Iml~)q~.tm(r)~m~u(R), (2.4)

where <nlli, n212 : .o~lnl, NL : .~> are the Talmi coefficients. Evidently, this separa- tion can be done generally, for any potential well, but the expansion will contain an

184 A. Y. ABUL-MAGD AND M. EL-NADi

infinite number of terms. Nevertheless one would expect that for a reasonable finite well, the expansion coefficients are large only for those terms that remain in the case of an infinite oscillator potential. The use of the infinite oscillator potential in the calculations of direct nuclear reactions is not generally recommended, as the main contribution to the stripping integral comes from the external region 6) where the oscillator wave functions decay rather sharply. To avoid this, the following approximation may be suggested. In the expansion (2.4) the wave function ~b,~m(r) may be taken as the oscillator wave function, while the function ~bNzM(R ), which contributes largely to the cross section, may be taken to correspond to a more realistic potential, say a Saxon-Woods potential. Now using the relation

{../[}(LM, Iml~,/[)(Sa, SO'Jr'I J/0 = (_)~+s+J

l (Scr, Imljlzi)(jpj, LMIJI~), x ~ {JttY}([~] I-j])½ j j

one may write the wave function of the residual nucleus as

~kSR"R,'R~R = E {nl n2 11 12 Jl J2 nNlL~SdTjs~t;mMapl~jp;er}

{ ! ll jj~){ L I } x ( - ) '~+s+1 [~ ] ( [S ] [ j l ]EJ2 ] [ j ] ) ½ Iz j j

~CP

x A.~(nl ll j l , n212j 2 ; JTs'rsRtTta)<nl ll , n212 : ~Z~lnl, NL: ~£a> × Imlj ) x

x XT (tl, (2.5) The Talmi coefficients <nil 1, n212 : .~lnl, NL: .C~°> vanish if the following relations are not valid s):

2nl +l~ +2n2+12 = 2n+I+2N+L, (2.6)

ll + 12 + I+L = even. (2.7)

Moreover, the antisymmetry of the wave function ~,sR,,,t,,, with respect to the ex- change of the space, spin and isospin coordinates of the two captured nucleons re- quires that s)

S + I + T = odd. (2.8)

Now substituting (2.5) into (2.1), one finds

I~...,.o.r = E {nl n2 l112 Jl J2 nNlL.o~SJTjmMalt#~}(-)g +s+ ~

x [.oqrJ([S][jl][j2q[j]) ~ 12 A(n111Jl, nz 12J2" JTsrsrttrtrt) .~ J j

x <n 111, nzI 2 : .oq'lnI , NL: -~)(STltr, J#lsal~R)(Sa, Imlj#j) + (J/£t, LMIJ#) (2.9)

× (tTZT, T~lt~za)~,


where

932 = (f~-)(l~aR)~/sa/ta, tara ~)nIm(r)~)NLM(R)zsa(S1, S2)

× X T , ( t l , t2)lVlf~+)(rcT)¢~a~o, to~o), (2.10)

where all terms in the sum (2.5) vanished except those containing the target nucleus in its initial state, which is a general feature in the theory of stripping reactions.

3. The (aHe, n) and (t, p) Reactions

The wave functions of 3He or the triton may be written as

¢~ovc, tc,o = ,A/" exp { - -T (2¢ 2 +{r2)}X½uc(S 1 s2(O)so)X½,c(tl t2(1)to) , (3.1)

1

r 0

2

Target

Fig. 1. A scbematic diagram for the two-nucleon stripping process. The partic|es ] and 2 are those supposed to be stripped by the target nucleus, while the particle 0 refers to the outgoing particle.

where the coordinates ~ and r are shown in fig. 1, and the index 0 stands for the emitted proton or neutron. Now the stripping integral (2.10) may be rewritten as:

~PJ~=..J/ '~f~-)(~-I- A R) dpntm(r)C~NLM(R)[~7, A + 2

xf~+)(R+½¢) exp {-7(2~ 2+½r2)}~ , (3.2)

where

= (XSo(St, S2)ZT,(tl, t2)Z½go(So)X½,o to)lVlz½uo(sls2(0)So)X½,o(tt t2(1)to)). (3.3)

Now let the stripping interaction be written as:

V = 111o+ V2o, (3.4)

where Vio is the nucleon-nucleon interaction between the captured nucleon i and the emitted nucleon O, and will be assumed to depend on the spin and isospin of these

186 A . Y . A B U L - M A G D A N D M. E L - N A D I

nucleons in the following way:

V,o = Vo exp { - r ?o /a2} [ l+~ la , " ao+~2*," ~o+~3(~," ao)(~," ~o)3. (3.5)

Now substituting (3.5) and (3.4) into (3.3), and using the Wigner-Eckart theorem 9), one finds

17 = (Sa, ½Pol½#¢)(Zz, ½%[½%)Vo[exp {-(~+½r)2/a 2} x 1)--23 a o

-- ~[T]'[}T(T + 1 ) - 23C%6sl } + exp{ - ( ~ - ½r)2/a 2} (3.6)

x { oaT, + a,-, - l ) -

- I ) -

As the two captured nucleons in the reactions considered here are of the same charge, then the projection of their isospin z = _ 1, and hence their isospin T = 1. Now substituting this into eq. (3.2), the matrix element will consist of two terms, one corresponding to 111 o and the other to V 2 o. Replacing the variable r in the latter term by - r , one obtains

= (&r, ½/~ol½/~o)(l~, ½ToI½%)B(S, 1)VoJffdt'~ (3.7) where

~f)21 f drdRd~f~ -)* (~+ A R) * * =

A + 2

1 (¢+½r)2_?(2~2+½r2)}f<r+)(R+½¢) ' (3.8) X exp t--

B(S, 1) = [ 2 - 4 g216so-xfi[2~1--4 ~316sl. (3.9)

Note that if the nucleon-nucleon forces were spin and isobaric-spin independent, i.e. g~ = g2 = ct3 = 0, then the captured pair will be in a singlet spin state, and in this case the selection rules obtained in ref. 4) apply. These rules apply also if~l = ~a = 0 only, that is, when the nucleon-nucleon potential has the form Vo exp{-(r2o/a3)} [1 + g2zix o ] as there will be no forces which can flip the spin of the captured nucleons. It was found 4) that the break down of these selection rules will lead to relatively small two-nucleon stripping cross section. Hence comparing these cross sections with the cross sections of the same reaction when the pair is captured in singlet spin states, may give information about the strength of the spin-dependent forces in the two- nucleon interaction.

Now taking the wave function q~,lm(r) as an oscillator wave function 8)

F 2(n!) l ' r ' r 2 ~nlm(r) = l b 3 F ( ~ . . [ _ ~ ) . ] (~)exp ( - ½ (b) 2} l d : ' ( ~ ) Yl,('),

where LZn + ' is the associated Laguerre function, and carrying out the integration over


r in the expression (3.8), one finds

f dr(a*m(r)exp {~" r/a2} exp {- ( 5 +~,) r2 }

; ( . ) q 2-'-~d-"-'-~:~-' Lr(a ,iJ ' ,J/exp '4->-a7'

x lJ. +½ 4a4b-Yd ' d2 * ^

where dl = 1/4a2+ 1/2b2+½y and dE = 1/4a2+ 1/2b2+½7. Substituting eq. (3.10) into eq. (3.8) one finds

~I.R, = ( ~ ) ' F. 2(nO 1½2-'-'dT"-'-'d~a-a'b -'fdRd~ Lr(n + l + ~)J

• ^ (+) 1 X Ytm(¢)f~ (R"I-a¢) , (3 .11)

where d3 = 27+(1/a3)-(1/4a4d~). The technique of the numerical calculation of the integral (3.11) is given by Austern et al. 5). As the calculation of such six-dimen- sional integrals is very complicated, it seems convenient to apply the WKB method of calculating the stripping integrals 6, ~ o). Introducing the finite-range correction fac- tor 6) D~(Q2) defined by the relation

QtY~*(~)D,(Q2) = d~ exp {iQ. ~}~exp {-da~2}/J~ +~ 4a4b~d2 Y~m(~),

one gets

_Q2 Dl(Q2) = (27r)~2-t-'d3'-~-~d~it exp {- ~da} l~+~r (16a4b2dld2dad4) ,

where d4 = da - 1/434b2dld2, and performing the same procedure given in ref. 6) one obtains

~rJ~l = ~J~o9~2 Y~*(~), (3.12) where

~ 0 ---- (2~Z) 3 F 2(n!) 1 ½ Lb3r(~-~[+~jj 2-2l-a(dlda)-"-l-~(d2d4)~a-21ilb -l, (3.13)

~==q'fdRf~-'*(~+2R) exp { Q~(~)}g2'l' -Q~(" ) \ 16a4b=d, d2 d3 d,~:

× ~*,.~(R)fI+~(R). (3.14)

] 8 8 A. Y, ABUL=MAGD AND M. EL=NADI

Here Q(R) is the local momentum transfer defined as

= - - - e+ Uo R +U(R) -Uc(R) (3.15) 3h 2

where U 0 and Uc are the optical potentials describing the relative motion in the final and initial channels and U(R) is the potential which determines the function ~bNLM(R ). The vector q = ] K - k is the momentum transfer in the plane wave limit.

4. The (aHe, p) and (t, n) React ions

For these reactions one may write the wave function of the projectile in the form

~Oso~c, to~ ° = ~ exp { -~(2~ 2 +~r2)}Z~r~c(Sl so(O)s2)z½~(t 1 to(1)t2), (4.1)

where Z½uo(SlSo(O)s2) and Z½,c(qto(1)t2) are the spin and isospin wave functions of the incident particle.

Now the matrix element (2.10) will take the form (3.2) but with V defined as

= (~(s~r(sl, S2)XT,(tl, t2)Z½vo(So)X½,o(to)[V[x,}g,(sl so(O)s2)x½ro(t 1 to(1)t2) ). (4.2)

Assuming that the stripping interaction is still defined by eqs. (3.4) and (3.5) and using the Wigner-Eckart theorem the stripping integral breaks into two terms, one corresponding to the potential Va o and the other to V2 o- Changing the variable r by - r in this second term, one may finally write the analogue of eqs. (3.7) and (3.8) but the coefficient B(S, T) becomes

B(S, T) - [S]~[T]~ I-S- T(T + 1)]{[1 + ( - ) ' ]

-~q{3 - ( - ) ' [ 3 - 2 S ( S + 1)1 +ct 2 [1 + ( - ) t [1 + 2T(T+ 1]]

- ~ 3 { 3 - ( - y [ 3 - 2S(S+ 1)] [1 + 2T(T+ 1)]]} (4.3)

and the matrix element ~ will be written as

=(Str , ½/Zol½/~o)(T0, ½Zol½zo)B(S, T)VoJV'~)2x, (4.4)

where 93/~ is given by eq. (3.12). The values of the coefficient B(S, T) is given in table 1 for different values of S and T.

TABLE 1

The values of the coefficient B(S, T)

Reaction T S = 0 S = 1

(SHe, p) 0 ½~/3(c%-- 3~1) ](1--2cq--u3) and (t, n) 1 --½(1--2~2--9cta) ½x/3(~tl+4c%--3cq)

(aHe, n) 1 2--4~, --~/3 (2cts--4~a) and (t,p) l = even l = odd


5. Comparison with Cluster Stripping Theory

In this section the expressions obtained in sect. 3 will be compared with the corresponding expressions obtained for cluster stripping processes. Substituting eqs. (3.12) and (3.7) or (4.4) into eq. (2.9), one finds

I~uR,~o~r = ~ {nNlLJjSmMlt#ja}O(nNlLJjS) x (Sa, Imljl~j)(S~r, ½/~01½/~c)(ST/tT, JyISRItR) (5.1)

× (Jl~j, LMIJ#)Y~*(~)~2(I, LM), where

O(nNILJjS) = ~, {nl n2 ll 12 Jl J2 S~'T)(-)~+s+J[L]

{! 11 Jj~}{L l }Vo.AX~)2 (5.2) ×([S][j l][J2][j]) ~ 12 j j

x A(n111 Jl, nz 12 J2 ; JTSTSR,trtR)(nl 11, r1212 : ~q~lnll NL : .oq ~) × (tTZ T , TzltRzR)B(S, T).

On the other hand, the duster stripping theory 6), where the captured pairis assumed to form a cluster b in the projectile, gives the following expression for the transition amplitude:

II~aUR. Utter = E (ebJbLSb Jb lMbMm, mb m}Oeb(SclSa Jb ISb) x O,b(SRIST Jb LSb)(Sa Ila, Jb mblSo/~o)(Sb/~b, l m l Jb mb) X (Sb#b, LMIJbMb)(Srl~r, JbMblSSl~R)Y~m(q)~eb(l, LM). (5.3)

Now performing the following re-arrangement in eq. (5.1):

(Sa, Imljt~j)(Sa, ½/~ol½P~)Y~*(~)

= (--)J \~S]I (j l t j ' l .m lSa ) (½~o , Sal½~o)Y, - . ( 4 ) (5.4)

and comparing this equation with eq. (5.3), one finds that the two amplitudes have the same structure. Hence the expressions obtained for the cross section and for spin tensors characterising the spin state of the emitted particles and residual nuclei obtained in the cluster stripping theory can be rewritten for the two-nucleon stripping process after some changes in the notations. The cross section for a two-nucleon stripping process is

(~)_ M.TM.Rk[SR] ~ In,N l,Jj S M, b I 6ss,~,j, ~~~" (2.h2) 2 K [st] (n N'l J'j'S'M flJ \4rcES][S ] [b]]

x (--)r+L'+M'+~+S+b+P+J-J'O(nNIL J)S)O*(n'N'I'L'Jj'S) (5.5)

{~ L' Jl[ j' l' b} × j bit I j (EM', L -MIbfl)(l'O, lOlbO)Y~(q)~(l ', EM')9"~(I, LM).

190 A . Y . A B U L - M A G D A N D M. E L - N A D 1

Choosing the quantization axis parallel to K x k, one obtains the following expression for the polarization of the outgoing nucleons:

e~ = ~ ~ (2.h2) 2 r t . N' l 'CJj s M' # J

x ([b,][ j ] [ . ; ] [ l ] [ l ' ] ]~(_)~+,,+~,+,+~+,+j,+, [g-2J I

x O(nNILJjS)O*(n'N'I'L'Jj'S'){L E J}{S S' 1} (5.6) j b l ½ ½ ½

{ bl b2 1 } I S (l'0, lOIb20)(EM', L-Mlblfl)(10, blfllb2fl)

x j l' S' J

x Y~a(q)g)2(l, LM)~*(I', EM').

The polarization of the residual nucleus will be characterized by the spin tensors <TRP(SR)> where R __< 2SR. The expression for this spin tensor is the following:

(da)-IV{~]}M~TM°tk In,N,IL, Jj,SMblb21 <TR'(SR)> = d-~ (2nh2) z K ~" In N IlSJj S'M'fllfl2J 6ss"

x ( [ j ] [ f ] [J] [J ' ] [ l ] [ r ] ]*(_)~+~,+,+~+,+,+,._j ,+,~ \ [s ] [s ' ] I

x O(nNILJjS)O*(n'N'I'EJ'j'S)" {J' J Rt[J' l' S I (5.7) sR sit ST][l j bz

x L j (EM', L-Mlblfll)(l'O, 101620)(blfll,Rplbzflz) I2 j'

x r~p~(~)~(l, LM)g)~*(l', EM'). The angular correlation function 11) IV(O, 4) and the degree of the circular polarization 12) C(O, qb) of ?-rays emitted in the direction (0, ~b) and measured in coincidence with the emergent nucleon, is related to the spin tensors <TRP(SR)> in the following way:

W(O, 4) = 2 {22', R = even, p} \[R]! <TRP(SR)>FR(22'SR°sR)YRo(O' ~0), (5.8)

2 {22', R = odd, p}<Tm'(sR))c~cx, FR(LL'SrtoSR)[R]-½YRp(O, (o) (5.9) C(O, 4) = E {22', R even, p}< TRP(srt)>CxCa, FR(22'SRoSR)[R]-*YRo(O, q~) '

where F R are the ?-emission coefficients and c z the reduced matrix elements for the 2~-pole v-rays.

It is interesting to note, that if one wishes to treat (3He, n) or (t, p) reactions using the cluster stripping approach from the very beginning, then one has to put j = 0,


l = 0 and S = 0. On the other hand, if one uses the method developed in the present work, which permits the two captured nucleons to move independently, then other j , l and S states would appear. The most important difference between the two for- mulations is in the choice of the stripping interaction. In the two-nucleon stripping theory, the stripping interaction is taken as the sum of the interaction between the emitted nucleon and each of the captured two nucleons. On the other hand, in the cluster-stripping treatment one takes the stripping interaction to be a function of the distance between the centre of masses of the two clusters forming the projectile which will be in our case the captured diproton and the emergent neutron. The residual forces after this latter choice, i.e. the difference between the two stripping interactions, will cause the captured diproton to be excited, which will lead to the inclusion of other values of j , l and S.

For the purpose of illustration, let us consider the reaction 2sSi(t, p)3°Si. The ground state of the residual nucleus 3°Si, may be considered to consist of a core of 28Si and two neutrons in the 2s~ shell coupled to d = 0 and T = 1. Thus one may write:

= E (nNlm}Xoo(Sz, s2)xl,-i(tl, t2) (_ ) ' - -

x ls; Olnl, NI; 0). (5.10)

Now using the values of the Talmi coefficients calculated by Balashov and Eltekov s), one obtains

x {~;dPtoo(r)C~too(R)+½~/~dP2oo(r)(9ooo(R ) T 1

+½46~booo(r)¢200(R)--3 X {m}( - ) ra~o2m(r )~o2- ra (R) ) "

(5.11)

All these terms may contribute to the cross section. On the other hand, in the cluster stripping approach, one writes for the wave function of the 3°Si nucleus:

~ks...,,~.~(3°Si) = O~k~..T, tT.~(2SSi)zoo(Sl, S2)ZI - 1(t l , t2)q~/.=o(r)

x q~r b=o(R)+ terms not contributing to the cross section. (5.12)

The function q~/b=o(r) stands for the relative motion of the dineutron supposed to form a cluster and the proton in the triton. This function has no nodes and its shape is discussed in the work x 3). Hence, the contributing term in eq. (5.12) corresponds to the third term in eq. (5.11). Other terms in the expression (5.11) have no analogues in eq. (5.12) and their contribution to the cross section will represent the difference between the two-nucleon and cluster stripping processes.

192 A. Y. ABUL-MAGD AND M. EL-NADI

References

1) M. E1-Nadi, Proc. Phys. Soc. A70 (1957) 62 2) M. E1-Nadi and M. E1-Khishin, Proc. Phys. Soc. 73 (1959) 705;

N. C. Newns, Proc. Phys. Soc. 76 (1960) 489; M. EI-Nadi and H. Sherif, Nuclear Physics 28 (1961) 331; M. E1-Nadi and H. Sherif, Proc. Phys. Soc. 80 (1962) 1041; N. K. Glendenning, Nuclear Physics 29 (1962) 109

3) E. M. Henley and D. V. L. Yu, Phys. Rev. 133 (1964) B1445 4) A. Y. Abul-Magd and M. Ei-Nadi, Phys. Lett. 13 (1964) 328 5) N. Austern, R. M. Drisko, E. C. Halbert and G. R. Satchler, Phys. Rev. 133 (1964) B3 6) A. Y. Abul-Magd, M. E1-Nadi and G. L. Vysotsky, Nuclear Physics 71 (1965) 606 7) N. Austern, in Selected topics in nuclear theory (IAEA, Vienna, 1963);

W. Tobocman, Theory of direct nuclear reaction (Oxford University Press, London, 1961) 8) M. Moshinsky, Nuclear Physics 13 (1959) 104;

V. V. Balashov and V. A. Eltekov, Nuclear Physics 16 (1960) 439 9) M. E. Rose, Elementary theory of angular momentum (John Wiley, London, 1957)

10) P. J. A. Buttle and L. J. B. Goldfarb, Proc. Phys. Soc. 83 (1964) 701 I1) R. Huby, M. Y. Refai and G. R. Satchler, Nuclear Physics 9 (1958) 107 12) G. R. Satchler, Nuclear Physics 16 (1960) 674 13) M. EI-Nadi and F. Riad, Nuclear Physics 50 (1964) 33

Two-nucleon stripping processes

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