Treiman-Yang criterion for direct nuclear reactions

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2.F

[

Nuclear Physics 61 (1965) 353--367; (~) North-Holland Publishing Co., Amsterdam

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Not to

be reproduced by photoprint or microfilm without written permission from the publisher

T REIMAN-YANG CRITERION FOR DIRECT NUCLEAR REACTIONS I. S. S H A P I R O a n d V. M. K O L Y B A S O V

Institute for Theoretical and Experimental Physics, Moscow and G. R. A U G S T

Tbilisi State University, Tbilisi, USSR Received 28 F e b r u a r y 1964 T h e A W x ~ B + y + z reactions are considered in t h e pole a p p r o x i m a t i o n . Cases o f applicability o f the T r e i m a n - Y a n g criterion for virtual non-zero spin particles are indicated. A general f o r m u l a is obtained for t h e differential cross section o f the reaction with arbitrary spins o f the particles involved.

Abstract:

1. Introduction The amplitude o f a reaction of the type A + x --, B + y + z

(1)

depends in the general case on five invariant variables. Therefore the mechanism of reaction (1) is a much more complicated problem than that of the binary reaction (A(x, y)B) whose amplitude depends on two variables. However, there is a case when the amplitude of reaction (I) depends on less than five invariants. This case is considered below. The following quantities can conveniently be chosen as five invariant variables in the non-relativistic case: I =

-- (PB--PA) 2 +2(mB--rnA)(E~--EA),

t' = - (py _p=)2 + 2 ( m y - m x ) ( E y - E x ) , s' = -- (pz+py) 2 + 2 ( m z + m y ) ( E z + E y ) ,

(2)

s = - (pA+p~)2+2(mA+mx)(EA+Ex), SB~ = -- (PB + pz)2 + a(mB + mz)(EB + Ez),

where p denotes the m o m e n t u m and E the kinetic energy of the respective particle. The Galilean invariant quantities given by relations (2) are the analogues of the corresponding relativistic variables. 353 January 1965

i.s. SHAPIROet al.

354

The pole mechanism is the simplest mechanism of reaction (1). In this case the amplitude M of reaction (1) is represented by the pole graph of fig 1. If the intermediate particle spin Ji is zero, the amplitude corresponding to the graph of fig. 1 is known 2) to be expressed by the amplitude F of the process A ~ B + i and the amplitude F of the reaction i + x ---, z + y as follows t: i

M = const

F(t)F(s ~ t; t) ,

(3)

t - 2rn i ei where ei is the coupling energy of the particles i and B in the nucleus A ei = mi + m B - mA.

(4)

,q

t"

--

i ~

x J,

Fig. 1. Five-point pole amplitude. F r o m eq. (3) it is clear that the expression for the amplitude can be factorized, i.e. divided into two factors one of which depends only on the invariants relating to the left-hand vertex (on t in the present case) and the other depends only on the invariants s', t' and t which can be formed out of the momenta of the particles contained in the right-hand vertex. In other words, the entire amplitude depends in this case not on five but on three variables t, t' and s'. As indicated by Treiman and Yang 1) there is a simple method of checking the pole character of a reaction. The quantities t, t' and s' (or their relativistic analogues) do not change if in the anti-laboratory system (the system in which Px = 0) the momenta of the particles z and y are rotated about the direction of the sum of these momenta; this direction coincides in the antilaboratory system with the direction of the intermediate particle momentum. Therefore the differential cross section dtr (see eq. (15) below) must remain constant in this rotation. This Treiman-Yang criterion is a necessary condition for the pole mechanism of a reaction. In the relativistic case with a non-zero spin of the intermediate particle the square of the amplitude modulus cannot be factorized and the Treiman-Yang criterion does not apply in the general case. In the theory of direct nuclear processes (see refs. 2, 3)) the graph of fig. 1 decribes a process in which an incident particle x interacts with the cluster i (a nucleon, deuteron, s-particle, etc.) which is emitted virtually by the nucleus A. The pole amt The system of units in which h = c = 1 is used in the paper.

DIRECT NUCLEAR REACTIONS

355

plitude for reactions of type (1) ("five-tail") was considered in ref. *) for zero-spin particles. In the present paper it is shown that factorization is also possible in several non-relativistic cases when the intermediate particle spin is non-zero. These eases are enumerated in sect. 5. Our analysis of the Treiman-Yang criterion requires an expression for the square of the matrix element modulus of reaction (1) averaged over the spin states of the initial and final particles. This expression is derived in sect. 4. In the case of the pole amplitude of the binary reaction ("four-tail") the spins and orbital momenta of the particles involved in the reaction have in fact been considered for separate spin values in many theoretical investigations of stripping and pick-up reactions. The pole amplitude of reaction (1) is considered in ref. 5) for arbitrary spins, but under a rather special assumption (whose physical meaning and sphere of applicability are not clear) about the matrix elements of the process i + x - - , z + y (see the footnote to sect. 4). In the present paper the five-point pole amplitude (five-tail) is investigated in the general form. 2. Amplitude of A --> B + i Decay To write down the amplitude corresponding to the graph of fig. 1 for arbitrary spins of all particles, one must know how the amplitudes of the A ~ B + i and i + x ~ z + y reactions are expressed. Let us introduce a unit vector n~B directed along the relative velocity of the particles i and B (which coincides with the direction of the relative momentum of the particles i and B in the rest system of the particle A). The vector nia is invariant with respect to the Galilean transformation of the reference system. Note that niB can be expressed by the velocities of the particles A and B .~B

~i - - VB

-

- Ivl-v~l

I~A - - VB

-

.

(5)

fVA--VRI

For an element of the T matrix corresponding to the A --* B + i reaction, we have Ft*luB ~ < J i =

['li , J B , 0iB,

niBIr[jA, ]IA>

2

/li

liB, JiB ,mtB, FiB

(Ji,

,JB,

,ttBlJiB, ~ / i B > ( 0 i B , ~OiBIIiB, m i B >

(6)

X (JIB, /liB, liB, mm[JA, /IA)(JiB, liB,JA, /tA[TIjA, /~A>" The following notations are used in eq. (6): 0~B and ~om are the angles giving the direction of the vector niB, j, and Pa are the spin of the particle a and its projection on the z axis, l m and m~Bare the orbital momentum of the relative motion of the particle i and B and its projection on thez axis (in therest system of the particle A), and JiB is the decay channel spin JiB = Ji "}-JB,Ji + J B - 1 , . . . IJi--JB[,

(6a)

,/-LIB = ~£: "I-it/B •

(6b)

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i.s. SHAPIROet aL

Denoting then = F ( j m , liB),

(6C)

we obtain r ./~A i . . _- -

E JiBs

r ( j , B , liB) E lib

cJlBgIB cJA"A V [** "~ jt/~iJBgtB jlB/,ql31IBmIB J'|tBralB~,'"iB)"

~IlB, tniB

(7)

Here ¢7J3,3. ~ J 1~1J2~2 are the Clebsch-Gordan coefficients, and F ( J i s , lib ) are the form factors corresponding to certain values of the orbital momentum and total spin of the particles i and B. These form factors are functions of the invariant variable t. Their analytical properties were investigated in ref. 3). Eq. (7) for r~/.t"tu~ A was obtained by Timashev, Blokhintsev, and Dolinsky6). 3. Amplitude of the i+x---~ zWy Reaction The expression for the a m p l i t u d e f of the reaction i+x ~ z+y

(8)

through invariant quantities was derived by Bilenky et al. 7). Let us write it in the form (f(nix n ~h~zt~y

E

f(Jix,Jzy, L, ~.)

J l x , J z y , L, X

CJix/~lx CJzY#zY ~jl/tlJxgtx --jz#zJygy

E

(9)

/llx,/tzy, mix, mzy, M /'~jzy/Izy

LM

~ *

~ *

x .~j,~,l~,~, c , _ ~ . i ~ , + ~.z. r,-~.ix("i~)r,+~.z.("z,), Here Y~m -* = ( - 1 ) t - ' ~ -m.

~ . = :Yz., r =

½L if L is even ½ ( L + I ) i l L is odd.

(9a) (9b)

Eq. (9b) holds if the internal parities of the initial (i, x) and final (y, z) particles do not change. If they do change t½(L+I) r = [ ½L

if L is even i f L is odd.

(9c)

The quantity 2 runs through all integral (if r is an integer) or semi-integral (if r is a semi-integer) values for which the trio r - 2, r + 2, L makes a triangle. The unit vectors nix and nzy are directed along the relative velocities of the particles i and x and z and y, respectively. Since the amplitude of reaction (8) is of interest to us in connection with reaction (1) let us express nix and n~ by the velocities of the particles involved in reaction (1) n,x = mA(VA-- Vx) + ms(%-- v.) ImA(VA--Vx) + mB(V,-- Vx)l' n~y -

10z m By

{Vz-- Vy[

(10)

(10a)

DIRECT NUCLEAR REACTIONS

357

The spins of the particles and their projections are denoted in eq. (9) by the same symbols as in eq. (6) (Ja, Pa, where a = i, x, y, z). The quantitiesjix, ~ix andjzy, #zy are the spins of the entry and exit channels and their projections and r - 2 , mix and r + 2, rnzy denote the orbital momenta and their projections of the initial and final particles (in the c.m.s, of the particles z and y). The quantitiesf(jix,jzy, L, 2) are the invariant amplitudes which are functions of the variables s' and t'. (It will be recalled that s' equals, accurately to a factor, the energy of the particles z and y in their c.m.s. and t' is linearly connected with the square of momentum transfer in the same system). For elastic scattering it follows from time-reversal invariance that

/2jix+ l~ L,. f(ji~,jzy,L, 2)=(-1)J'*-J'Y~2j-~y+ljytjzy,Ji~,L,-2 ).

(11)

For the reactions involving 0, ½ and 1 spin particles the four-point amplitude is usually written not with the aid of Clebsch-Gordan coefficients as is done in eq. (9) but as an expansion ofinvariants composed of the spin matrices and vectors ni~ and nzy (see e.g. refs. s, 9)). The formulae given in appendix 1 connect the coefficients of the amplitude expansion in these invariants witk the amplitudes f ( j ~ , j~y, L, 2) of the general formula (9) for the elastic scattering of 0 and ½ spin particles on ½ spin particles. Eq. (9) expresses the amplitude of the reaction in the channel spin representation. In some cases it is more convenient to have a representation expressing the amplitude of the reaction through the invariant amplitudes f ' depending on j~z, Jxy, L and 2, whereji~ andjxy are the geometrical sums of the spins of the particles i and z and the particles x and y, respectively. Appendix 2 gives formulae connecting the invariant amplitudesf(j~x,jzy, L, 2) a n d f ' ( j i z , Jxy, L, 2) and establishes relations for the elastic scattering of 0 and ½ spin particles on 1 spin particles between the quantitiesf'(j~,, Jxy, L, 2) and coefficients of the expansion of the amplitude in invariants composed of the spin matrices and vectors n~ and nzy. 4. Expression for Differential Cross Section

The amplitude M of the reaction A + x -+ B + y + z corresponding to the graph of fig. 1 can now be written as follows: M.Bu..y _ P,A P x

2ml ~. t - t o u,

FU.,.f~u,.

--/~A ."/.qP-x

~

_ 2mi

F(j~B, Im)f(j~, j~y, L, 2)

t - - t o JiB, liB, j i x , Jzy, L, ,~

× Z

2

~Jmmn pJAuA

~Jil~lJBl~B ~JlBtltBliBmlB

mix~ mzy, M ( ~ j t x//tx f~Jzy//zy f~jzy//zy X --jl~/tJxYx g'/Jz~/zjy.Uy x'~Jtx~lxLM

LM C r - A mix r+,~ mzy

X YliBraia(niB)Yr_zmix(nix)Yr+;tmzy(tlzy).

(12)

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SHAPIRO et al.

I.s.

Here t to = 2mlei,

ei =

mB+mi--mA.

(13~

The meaning of the other notations is clear from the previous sections. As indicated in sect. 2, the form factor F(jiB, liB) is a function of the invariant t. This dependence, mainly due to the finite size of the interaction range of the nuclei B and i (i.e. the finiteness of the "channel radius" A --* i + B) can be approximated for small values ( - t ) by a Wronskian similar to the one in the Butler formula for the deuteron stripping amplitude 3). This form-factor varies with t no more rapidly in the general case than the pole denominator in eq. (12) and hence one cannot neglect this dependence in the general case by assuming t to be equal to its pole value to•

The invariant amplitudes f(Jix,Jzy, L, 2) depend, as was indicated above, on the invariants s' and t' and the masses of the particles involved in the i + x -+ z + y reaction. The particles x, y and z are free in our case. Instead of the mass of the virtual particle i we must substitute the quantity

mit

2

2~-

= (8i - Pi )

= ( m2 -t- (t-

to)) ~,

(14)

where 8i is the total energy of the particle i, Pi is the momentum of the particle i and to is given by eq. (13). Assuming that I t - tol << ml2 (which we always have in the non-relativistic case) and that the amplitude f(jix,jzy, L, 2) is regular with respect to the variable m'i near m'i = mi and varies sufficiently slowly in this region, we can inf(jiz,jzy, L, 2) replace the variable t by its pole value to. Thus in this approximation the invariant amplitudes of reaction (8) only depend, just as in the case of free particles, on two invariants s' and t'. Since the kinetic energy of the particle i is negative, however, the variables s' and t' in reaction (1) lie in the general case outside the physical region for reaction (8). In the following we consider the reaction A + x --* B + y + z with non-polarized initial particles A and x for the case when the polarization of the particles B, y and z produced is not measured. The differential cross section is given by act = -(2~)4 l-~ I

d3pa dapz dapy 6(p)6(E), (21t)a (2re)3 (27t)3

(15)

where I is the invariant density of the relative beam of incident particles, and [M[2 is the square of the matrix element modulus averaged over spin states of the initial particles and summed over those of the final particles iml2 =

1

£

-'-.A.,Mu"u="" 2.

(16)

(2jA+ 1)(2j,+ 1) . . . . . . 0..,.... • As the amplitude of fig. 1 was considered in ref. 5) it was assumed that the quantitiesf~i~7 _.,.,,f "z"y* can be neglected as compared with the diagonal terms fl/~'#Y z As indicated above the physical mean--/tlPx " ing a n d sphere o f applicability o f this a s s u m p t i o n are unclear.

359

DIRECT NUCLEAR REACTIONS

Using the graph method of summing the Clebsch-Gordan coefficients 1o) the following expression can be finally obtained from eq. (12): IMI 2 -

4mi2 1 • ( t - to) z (4z) ~ J,B, n~, J,x, j,,~, ,,,B, J,,., Jzy, L, ~.

×

F*(j~B, l~)f(Jix,

r(ji,, tin)

L', 2'

Jzy, L,

2)f*(j~x, Jzy,

]-~, '~')

(17)

x ( - 1)r+"+t'BA(/iB, liB,.' • • Jzy) Z C,,~c°o,',~,o a, b, c

X [--,a 0 "~r'-~'o,-~o

b0 C,,+a,o~+~oR(a,

b, c)F~bc.

The summation is performed over all a, b and c that are allowed by the ClebschGordan coefficients. In eq. (17) use is also made o f the notations 2j~y+ 1 A(liB, l~n . . . . J=Y) -- 2j,,+l {(2jm+l)(2j~B+l)(2Ji~+l) x (2j[~ + 1 ) ( 2 L ' + 1 ) ( 2 L + 1)(2liB + 1)(2/i'B + 1)(2r'--2~.' + 1)

(18)

x ( 2 r - 2 2 + 1 ) ( 2 r ' + 2 2 ' + 1 ) ( 2 r + 2 2 + 1)}~,

R(a,b,c)=(_l)JA+t,.+j~-j.-L'-j~,,ll,.

Jm JA}

tj[B [JiB

× (Ji

Ji.t

JBllJi'

JiB

C )[Jix Ji

Jix

(19)

l~B

Jxl / Jix

L

jzy

a

r'-2'

c)[L'

r-~

Jix

C /(b

r'+2'

r+2

.¢

.

On the right-hand side of eq. (19) we have the product of four @symbols and one 9j-symbol. Note that there is no summation over c. The quantities F~b~ are rotation-invariant combinations of the vectors nix, n~y and niB

Fa,c =

S

(a

ma ,rob,me \ m a

b

mb

c)Yama(nix)Ybmb(nzy)Ycmc(niB)

mc /

'

(20)

where ( a ,,~ b too) c is the Wigner coefficient. It can readily be seen that ¢q. (17) only contains such F~bc for which the sum a+b+ c is an even number. The explicit form of the quantities F~bc form some simpler cases is given in appendix 3.

5. Factorization of the Expression for

]MI 2

The expression for [M[ z can be factorized only when it does not depend on s and sBz, i.e. only when it does not contain angles between naB and n~x and between n m and nzy. According to eq. (17) this is the case only when c = 0. Let us see under what conditions c is always zero. First of all note that because of the Clebsch-Gordan coefficient C~°oviuo in eq. (17) c may only have an even number since lib and l'13have the

I.S. SHAPIROet

360

al.

same parity. Furthermore, the coefficient R(a, b, c) (19) contains 6j-symbols which do not vanish only when the trios {liB, liB, e}, {JiB,JIB, e}, {Ji,Ji, c}, {Jix,J[x, C} and {L,L', c} makes triangles lo). Therefore i f j i = ½, for example, c is certainly zero since it is an even integer and the trio ½, ½, e makes a triangle. Several other cases for which e can only be zero are similarly ascertained. These cases are enumerated below. (1) The intermediate particle spin ji is 0 or ½. The caseji = 0 is trivial. It can readily be seen why factorization is always possible if the intermediate particle spin is ½. Omitting the factor inessential in this case and the subscripts #x,/~y,/~z we can write the amplitude of reaction (1) in this form M~] = E -F'u" - / t A Jfr : *

(21)

v

where v is the spin projection of the particle i assuming the values ½ and -½. Then we have M~B 2 =

~.f~f,( ~ r~,Br~'~,) ~ / t A --/~A ]"

(22)

The quantity in brackets is the matrix of second order with respct to the subscripts v and v' and can be represented using the Pauli matrices as

~., _..r'""r¢""*_.. =

aa,,,,, + b . o-,~,,,.

(23)

.UA, ~ B

Hence we have

Z M~aA2=f+(a+b'a)f, I r A , l,t B

where

}

<24,

Due to parity conservation b must be a pseudovector built up of the momenta of the particles entering the left-hand vertex. It is obvious, however, that there is no such pseudovector (the left-hand vertex is characterized by the only vector nla )Therefore b = 0 and hence factorization is possible. With a higher intermediate particle spin, expansion (23) would contain higherorder tensors which are not zeros in the general case. Note that in the case Ji = 0 and Ji = ½ factorization is possible for pole graphs with a larger number of particles at the right-hand vertex than in the graph of fig. 1, and the pole character of the corresponding reactions can be checked by the TreimanYang criterion. If the main contribution to the left-hand vertex corresponding to the A ~ i + B reaction comes from any form factor F(jla, liB) (or a group of form factors with the same lib or JiB) factorization is also possible in the two cases: (2) when lib = 0, i.e. the orbital momentum of the relative motion of the particles i and B in the nucleus A is zero (in this case the direction nia does not enter at all into amplitude (12)),

D I R E C T N U C L E A R REACTIONS

361

(3) when the summary spin of the particles i and B (JIB) is 0 or ½. If one term with the invariant function f(Ji~,Jzy, L, 2) predominates in amplitude (9) factorization is possible in the following cases (in addition to those enumerated above): (4) when L = 0 and hence reaction (8) only occurs from the s state to the s state since r - 2 = r + 2 = 0 corresponds to the zero value of L (if the internal parity of the particles remains unchanged); (5) whenji ~ = 0 orji ~ = ½. This case would correspond tojiz = 0, ½with a different choice of the system of the invariant amplitudes (see appendix 2) when they are given by the quantities Jiz, J'~y, L, 2. Let us also note a special case when the amplitudes with r + 2 = 0 predominate, i.e. reaction (8) occurs in such a way that the particles z and y are produced mainly in the s state. There is no factorization in this case since I-MI 2 depends on s but the Treiman-Yang criterion is still applicable since the expression does not depend on SBz •

Though the Treiman-Yang criterion only applies to some particular cases described above, these cases embrace a wide range of reactions, including reactions in which the intermediate particle is a nucleon, e.g. the (p, 2p) and (p, pn) reactions as well as reactions on such nuclei A in which the cluster i and the nucleus B are in the s state. In the cases of factorization eqs. (17)-(20) become much simpler, and IMI 2 can be expressed by the sum of Legendre polynomials of the scattering angle cosine in the c.m.s, of the i + x -o z + y reaction _ IMI2

4mi2 1 ( t - to) 2 (4n) 3 (j,~.~t,s IF(jIB, /i~)]2)

×

Z

• " f(Ji ,Jzy, L, 2)f *(j" x,jzy, L,

jtx, jzy, L,~,,2"

2jzy+l (2ji + 1)(2j~+ 1)

x [ ( 2 r - 2 2 + 1 ) ( 2 r - 2 2 ' + 1 ) ( 2 r + 2 2 + 1 ) ( 2 r + 2 2 ' + 1)] ~ x ( _ l ) Z ' - 2 + L E too ~'~r-2" 0 r - 2 a

fr '

x |r+2

r+2'

c,ao 0 ~-~r+2'

(25)

0 r+2 0

P,(cos0),

while IMI z is not zero only if the triads {/m,JiB,JA}{Jin,Ji,Js}, {Jl, Jlx,Jx}, {Jix, Z, Jzy} satisfy the triangle rule. It should be recalled that 0 is the angle between the vectors nix and nzy given by the relations (10) and (10a). The cosine of this angle is a function of s', t' and t cos 0 =

z(s', t', t).

(26)

As indicated above the invariant amplitudesf(jix, Jzy, L, 2) also depend on the variables s', t' and t. We are interested (see sect. 4) in the amplitude of reaction (8) on

i. $. SHAPIROet al.

362

the " m a s s surface" o f the particles i (i.e. when t = to). Assuming that t = t o in eqs. (25) and (26) we can express [ n l 2 through the differentialcross section do"/dI2 (s,' t') of reaction (8) in the c.m.s, of the particles * z and y IM[ 2 -

4nm2I mzyp(t_to)2 (~,~,Z,,. IF(JiB ,/ia)[2 ) d-odo"(s', t'),

(27)

while

do- (s', C) -

do-

(s', cos 0'),

(28)

d~ where cos

0' =

z(s', r, to).

(28a)

In eq. (27) I is the density of the relative flux of the particles i and x (29) I=

]/~iix ( 2 ( m r + m y )

Q')'

where Q' = m i + m x - m z - my.

(29a)

Here p denotes the m o m e n t u m o f the particles y and z in their c.m.s.

x/ my mzs'

p - - my + mz

(30)

The corresponding reduced masses are denoted mzy and mix. The results expressed by eqs. (17) and (25) are obtained for the pole amplitude of reaction (1). F r o m these one can readily obtain the corresponding f o r m u l a e for the pole amplitude of reaction (8) and those cases when the expression for [M] 2 corresponding to it can be factorized. We n o w give the expression for ]M] 2 o f reaction (1) in the case when the i + x ~ z + y process is elastic scattering o f particles with spins ½ IMI2 -

m2 1 ( t - t o ) 2 (an) 3 (j,~.y~.~IF(JiB,/m)l 2)

sin201f212+ 31f412+ 9 sin20lfs]2 + 3(3 cos20 + 1)1f612+ 9 ( 3 + cos20)] f712 + 6~/~ cos o ( f 6 f * x {Ifx] 2 + 9

(31)

+ f*fT)}.

Here for the sake of brevity the amplitudes f(Jix, Jzy, L, 2) are denoted by one index (see eq. (A.4) of appendix 1). i- There is another approach u) to the "pole approximation". In ref. u) the invariant amplitudes

fp(s', t', t) (p denotes the total set of the indices Jix, Jzy, L and 2) are'considered as functions of s', t' and cos 0. In the transition to the mass surface the substitutionfp(s', z(s', t', t), t) --->fp(s', z(s', t', t), to) is made, i.e. t is replaced by to in one case and is not in the other. This approach seems inconsistent to the present authors. Using the method of ref. n) we would obtain a formula similar to eq. (27) but with da/d.Q(s', cos 0).

363

DIRECT NUCLEAR REACTIONS

6. Conclusion Eqs. (15) and (17) solve in principle the problem of calculating the differentiaI cross section for reaction (1) in the pole approximation. Of special interest are the cases when the expression for IM] 2 can be factorized, and eq. (17) can be simplified into eq. (25) since here the pole character of the reaction can be checked by the Treiman-Yang criterion described in sect. 1. Practically, the application of this criterion requires a comparison of the differential cross sections for different momenta of the particles z and y corresponding to the rotation of the momenta of these particles in the anti-laboratory system (denoted kz and ky) about the direction n = (kz+ky)/lkz+ky I. Let us give the relations making it possible to determine the momenta pz(~O) and py(q~) into which the momenta p~ and py pass after the above rotation by the angle tp. The momenta in the laboratory and antilaboratory systems are connected by the relation

k,, = p~-(m~/mx)Px

(v -- z, y).

(32)

After the rotation about n by the angle tp the vector kv passes into kv(q0 k~(tp) = k ~ - s i n ¢pk~x n - 2 sin2½~0(k~-n(n, kv)).

(33)

Going over again to the laboratory system we obtain

pv(~o) = kv(q 0 + m~/mxp x.

(34)

According to the above it is the quantity IM] 2 that must be invariant in the transformation (32)-(34). In particular, if the differential cross section daa/dg2zdf2ydEy is measured experimentally the quantity invariant with respect to this transformation is mz - -+ my Pz -- Px COS Ozx+ py COS Ozy IMI z -

I(2~) 3

4m Bmy

mz

d%

Pz2Py

(35)

dr2z dOy dEy '

and it is the values o f this quantity that should be compared. The Treiman-Yang criterion is a necessary but not a sufficient condition for the pole mechanism of a reaction. Note that the criterion allows the s dependence of the differential cross section, which is not the case when true factorization is possible. Therefore the pole character of the reaction can be checked more rigorously by investigating the dependence of the differential cross section on s; this requires experiments for different energies of incident particles. It can be expected that the pole mechanism predominates in several cases in the region of small momentum transfers. Therefore it is in this region that the TreimanYang criterion should above all be applied. Despite the simplicity of the Treiman-Yang criterion its application requires sufficiently elaborate experimental data. As far as we know the reaction (p, 2p)

364

L S. SHAPIRO et al.

on light nuclei has been investigated most thoroughly (see ref 12) and its extensive list of references). However, so far the experiments have been conducted mainly for the case when the momenta p~, py and p~ are co-planar, which rules out the use of the Treiman-Yang criterion• An elaborate experimental study of the non-coplanar case would therefore be highly desirable.

Appendix 1 Consider the scattering of a particle with spin ½ on a particle with spin 0. Let ii = j~ = ½ and j~ = jy = 0. We introduce the notations /'1 = f ( J ~ = ½,Jzy = ½, L = 0, 2 = 0),f2 = f(Ji~ = ½,J~y = ½, L = 1, ). = 0). (A.1) Then we can obtain from eq. (9) the following expression for the amplitude:

1 f = 7 {A + ix/~ sin 0f2 n" a}, 4re

(A.2)

where 0 is the scattering angle and n

--

-

nix

-

(A.3)

)< nzY

[nix X nzy I

If two particles with spin ½ are scattered the amplitudesf(ji~, j~y, L, 2) can be numbered as follows: f~ = f(0, 0, 0, 0),

f2 = f ( 0 , 1, 1,0),

f3 = f(1, 0, 1, 0),

f4 = f(1, 1, 0, 0),

fs = f(1, 1, 1, 0),

f6 = f ( 1 , 1, 2, --1),

f7 = f(1, 1, 2, 0),

fs = f(1, 1, 2, 1).

(A.4)

For elastic scattering it follows from eq. (11) that ~3f2 = --f3,

f6 = f s .

(A.5)

Then from eq. (9) we obtain the following expression for the amplitude f :

16=f = (A +3A)+ ( - f l +f, + 2~/if6 +2,/3 cos 0f7),,1. ~2

+ix/2f2 sin 0 n .

( a l - a 2 ) + 3 i sin

Ofsn" (trl +az)

-3~/~(f6 +,,/~-o f7)0 + cos o)(N. ax)(N. ~ ) - 3 , / i ( f 6 - ~ / ~ f , ) ( 1 - c o s o)(x. ~,)(K • ~2),

(A.6)

where N -

nix'~-nzY Inix +

and n is given by eq. (A.3).

nzyl '

K ~-- n i x - n z Y [nix-- n z y [ '

(A.7)

365

DIRECT NUCLEAR REACTIONS

With the aid of eq. (A.6) the invariant amplitudesf(ji~,jzy, L, 2) used in this paper are connected with the well-known coefficients which are usually used for the scattering amplitude of particles with spin ½. For the case of identical particles (or nucleons) we have f2 = 0.

Appendix 2 As indicated in sect. 3, when the process i + x ~ z + y is elastic scattering, the following representation different from eq. (9) can conveniently be used for its amplitude: (f(nix, n~Y]iv.u, //~igx =

E

f'(Jiz, Jxy, L, 2)

x

I~LM

Z #ix, #zy, mix, mzy, M

j i z , Jxy, L, ~

i,~ LM

~ *

[

cJ~z CJ~u" ~jl~lJlz#lz~Jx#xJxy#xy

~ Yr"*

(A.8)

The amplitudes f'(jiz,j'~y, L, 2) can be conveniently expressed through the sum of the amplitudes f(Ji~,Jzy, L, 2), used in eq. (9), with the aid of the corresponding 9-j symbols:

f'(Jiz, Jxy, L, ~,) =

(2jxY+ 1)(2jiz + 1) [(2jz -'{-1)(2jy "-{-1)(2L + 1)] ~"

(A.9)

Ji J* Jiz] X ~, (--1)2J'*(2jzy+l)(2jix+l) ~ Jix Jzy L lf(Jix,jzy,L,,~). J'~'J*Y tj~ jy Jxy)

I

From time-reversal invariance 7, 13) it follows that f'(Jiz,Ly, L, 2) = (-- 1)z+J~z+JxYf'(jiz,Ly, L, --2).

(A.10)

We can obtain formulae analogous to eqs. (17)-(20) which would includef'(ji,, Jxy, L, 2) instead off(Jix,j~y, L, 2). For this purpose the right-hand sides of eq. [17) should be multiplied by (_l)L+r' (2jy+l)(2j~+l) 2 F ( 2 L + l ) ( 2 E + l ) 7 '~ (2Jx+ 1)(2jxy+ 1)2 [ _ ~ ~j

(A.11)

after the substitutions Jtx ~Jiz, Jzy -'*Jxy, Jx -~Jz.

(A.12)

Consider the case of scattering of a particle with spin 1 on a particle with spin ½.

366

The

I.s.

SHAPIRO e t al.

amplitudesf'(jiz,j~y, L, 2) are numbered as follows (Jl = J~ = 1,j~ = jy = ½): ./'1 = f ' ( 0 , 0, 0, 0),

f2 = f ' ( 1 , 0, 1, 0),

fa = f ' ( 2 , 0, 2, - 1 ) ,

f4 = f ' ( 2 , 0, 2, 0),

f5 = f ' ( 2 , 0 , 2, 1),

f6 = f ' ( 0 , 1, 1,0),

f7 = f ' ( 1 , 1, 0, 0),

f8 = f '

f9 = f ' ( 1 , 1, 2, - 1),

f l 0 = f ' ( 1 , 1, 2, 0),

f a t = f ' ( 1 , 1, 2, 1),

f~2 = f ' ( 2 , 1, 1, 0),

A s = f ' ( 2 , 1,2, - 1 ) ,

./'14 = f ' ( 2 , 1, 2, 0),

f~s = f ' ( 2 , 1, 2, 1),

f16 = f ' ( 2 , 1, 3, - 1), f17 = f ' ( 2 , 1, 3, 0),

f18 = f ' ( 2 , 1, 3, 1).

(1, 1, 1,0),

(A.13)

Then the a m p l i t u d e f c a n be written as f

= ez,* #ZyfB~,Zxei, ~, +

(A.14)

where ez and el are the spin functions of the vector particle (~t, fl = x, y, z are the axes of coordinates), Zx and Xy are the spin functions of the particle with spin { before and after the scattering, and

4nf~ = (.1"1-~/2f3 -~/~ cos Of,)6e~-½ sin Of2(NaK~-N~K~) +½(1 + c o s O)(.,/~f, +x/2f3)NtjN,+~(1 - cos O)(-x/~f, +x/2f3)Kt~K, + / s i n 0 (f6 -- T4"~'fl 1 x 2 -- Y42" 1 ~-f l 6 -- ~ 4 3 COS Ofw)a'n6~, + i(1x/6f7 -- ½x/5f9 ---}x/6 cos Ofl o)eaar trr + ½i(1 + cos 0) (x/5f9 + x/~-fl o) a " Neaar Nr + ½i(1 - cos 0) (x/~f9 - x/~f* o) a " Kea~r Kr

-Yx/gf16-T~x/5fw)(naa~,+n~,tra) + 1 i sin 0.1",3{[a x N]aK~ + [a x N]~Ka + [a x K]eN,+ [a x K]~Np} +½sin 0(1 + c o s 0)(~/~f16 +½x/5f17){tr " N(n~,N~+naN~,) +a" nNpN~} +½ sin 0(1 - c o s 0)(x/~f, 6 -½x/5flv){tr" K(n,,Kp+naK,,)+a. nKpK~,}.

+i

sm 0(lg4~-fl

2

(A.15)

Here we have for elastic scattering f3 = f s ,

f8

--

0,

f9 = f x l ,

]'13 = --f15,

]'14 = 0,

f*6 = f 1 8 . (A.16)

The vectors n, N and K are given by eqs. (A.3) and (A.7), and e~a~ is a completely antisymmetrical tensor of third rank. Note that the first four terms in eq. (A.15) correspond to the elastic scattering amplitude of a spin 1 particle on a spin 0 particle. Appendix 3

Let us givethe explicitformulae for the quantities F, bc in eq. (17) for some a, b and c:

ma, r o b ,

me

~a

ITlb

D~lc

DIRECT NUCLEAR REACTIONS

367

F,,bO = 6( a, b) ( - 1)a(2a + 1)~ P~(cos 0), (4n) k

(A.18)

where 0 is the angle between n t and n 2, 3 Fl12 = (-~

F132 =

F222 =

- -

"5- /1 4 2 [ ( 1" /13)(/12"/13)--1(/11 . /12)],

(A.19)

2(3n)~ [5(n1 • nz)(n2" na) 2-2(/11" n3)(n2 • / 1 3 ) - ( n l • n2)],

(4n)~J14

{-9(/11"/12)(nl

(A.20)

"/13)(/12"/13) + 3[(/11"/12) 2 -Jr-(/11"/13) 2 + (112 ° /13) 2] -- 2},

(A.21)

3 '-r- (4n)-1~ 1-35(nl. na)2(n2, n3)2_20(nx./12)(/11 . /13)(/12"/13) F22, = ~x/1-~ ..[...[..2(!I1 • /12)2--5(/11 • /13)2--5(/12 • n3)2--[-1],

(A.22)

x/3 F13 4 - 2(47r)~ [35(/11 " /13)(/12 " /13) 2- 15(/11' /12)(/12" /13) 2 --15(/11" na)(n2"/13)+3(nx

"/12)].

(A.23)

Explicit formulae for Fabc with other a, b and c can be obtained using eq. (12) of ref. 14). References 1) S. B. Treiman and C. N. Yang, Phys. Rev. Lett. 8 (1962) 140 2) I. S. Shapiro, JETP 41 (1961) 1616; Nuclear Physics 28 (1961) 244; Selected topics in nuclear theory (IAEA, Vienna, 1963) p. 85-155 3) I. S. Shapiro, JETP 43 (1962) 1068; Nuclear Physics 38 (1962) 327 4) I. S. Shapiro and V. M. Kolybasov, Nuclear Physics 49 (1963) 515 5) J. Ruelle-Lardinois, Nuovo Cim. 28 (1963) 724 6) S. F. Timashev, L. D. Blokhintsev and E. I. Dolinsky, private communication. 7) S. M. Bilenky, L. I. Lapidus, L. D. Pusikov and R. M. Ryndin, Nuclear Physics 7 (1958) 646 8) L. Wolfenstein and J. Ashkin, Phys. Rev. 85 (1952) 947; R. H. Dalitz, Proc. Phys. Soc. A65 (1952) 175 9) R. Blankenbecler, M. L. Goldberger and F. R. Halpern, Nuclear Physics 12 (1959) 629 10) A. P. Yutsis, I. B. Levinson and V. V. Vanagas, Mathematical formalism of the theory of angular momentum (Vilnus, 1960) 11) E. Ferrari and F. Selleri, Nuovo Cim. 21 (1961) 1028 12) T. Berggren and G. Jabob, Nuclear Physics 47 (1963) 481 13) A. M. Baldin, V. I. GoldanskyandI. L. Rosental, Kinematics of nuclear reactions (Moscow, 1959) 14) V. I. Ritus, JETP 37 (1959) 217