A dynamical analysis of the nuclear reactions 3He(d, p)4He and 3H(d, n)4He

I!-! 2.B: 2.C:2.G

Nuclear Physics A169 (1971) 257-274; Not to be reproduced

A DYNAMICAL

by photoprint

ANALYSIS

(@ North-HolIand Publishing Co., Amsterdam

or microfilm without written permission from the publisher

OF THE NUCLEAR

REACTIONS

3He(d, p)4He AND 3H(d, n)*He (I). Two-body states B. DE FACIO Department of Physics, University of Missouri-Columbia, Columbia, MO., 65201 t and R. K. UMERJEE Department of Physics, Texas A and M University, College Station, Texas, 77801 t Received 15 February 1971 Abstract: We present some general expressions for the two-body states of the two stable channels of the five-nucleon systems. The concept of broken symmetries is applied to the polarization, the asymmetry and the vector analyzing powers of the incident deuteron and mass A = 3 particle to extract information on the reaction mechanism by inspection.

1. Introduction

The two nuclear reactions 3He(d, p)4He and 3H(d, n)4He have been studied by many authors ++and are the subject of a great deal of active work ’ “). The purpose of this paper is to provide general formulae so that analyses such as that in ref. ‘) can be carried out including all spin-breaking and orbital-angular-momentum-breaking transitions. The low-energy structure of these reactions suggests the importance of such transitions. The three-body break-up processes are relegated to a future paper ‘I) so that strictly speaking the expressions presented are only valid up to the deuteron break-up threshold. However, as on-the-energy-shell two-body transition amplitudes in the three-body Hilbert space they will form part of the “input” to the more general situation. Contrary to a popular misconception, the “approximate selection rules” derived in sect. 3 from broken symmetries are strictly dynamical because they all depend upon the reaction mechanism. Actually the term “non-dynamical” should in fact be reserved for properties which follow from time-reversal 9 invariance, parity II conservation, or total angular momentum J conservation. Csonka, Moravcsik and Scadron [refs. “-l’)] have treated the FL75 structure of general two-body processes carefully and correctly and we strongly recommend those papers 12-14) to nuclear physicists. 7 Supported in part by the University of Missouri-Columbia Grant 71-1971 at Texas A & M. tt See refs. l-lo) and earlier papers cited there. 257

Research

Council and Air Force

258 1.1. COORDINATE

B. DE FACIO AND SYSTEM

AND

R. K. UMERJEE

CONVENTIONS

Since this is a reaction we follow ref. ‘I) and choose for the orthonormal momentum basis the vectors (2, ti, A) which are defined in terms of q, the initial threemomentum in the c.m. frame, and q’, the final three-momentum in the c.m. frame as follows:

where &Jis the c.m. scattering angle 0 = 4 * 4’. We have interchanged A and fi from ref. ‘I) so that fi can denote the Basle convention normal to the scattering plane. Under the parity transformation 17 we have IT : (1, A, 7%) = (-2, ii, -6~) and II : (S, S’) = (S,S’) wh ere S, S’ denote the initial and final channel spins. Under the time-reversal transformation 7 our momentum and spin basis functions transform according to y: (P, ii, Ijl) = (1, -Ei, -6~) and f: (S, S’) = (-S, -S’). For spin operators we use S, S’ as the channel spin, spinor wave functions as a coupled representation and use crN,nucleon final spin-* spinor for either a proton or a neutron depending on which reaction is being considered, trn, initial 3He or 3H spinor, and Sd as the S = 1 deuteron spin function. Cartesian deuteron tensors are taken from Goldfarb ’ “) and spherical deuteron tensors are taken from Lakin 16). Angular momentum phases are chosen according to the Condon-Shortley 17) convention including spherical harmonics Yy,-” (0,4) and the Clebsch-Gordan coefficients (slvv - v’/ JM) except that following Huby 18) we insert the factor i’ into the angular momentum wave functions to assure time-reversal invariance. Seyler ’ “) has written a careful and thorough paper on the polarization from scattering polarized spin-f particles from unpolarized spin-l particles so we use much of his well chosen notation (combined with some notation and ideas from ref. “)). 2. The transition matrix The entire development presented here is non-relativistic and k denotes the c.m. wave number and C(0) denote the Coulomb scattering amplitude with Ith orbital momentum phase shifts ol, in the incident channel, and o;,, in the final channel (if any). We will use the von Neumann 1“) density matrix p to formulate the scattering observables. The final density matrix p is related to the initial density matrix Pin by

3He(d,

the relation set of channel

p = fip,,fi+

p). 3H(d, n) REACTIONS

where Ii? is the transition

spin functions.

matrix

Then Q, the differential

is given by (2 = Tr(p). The ensemble

average

259

(I)

expanded

in a complete

cross section per unit solid angle,

(0)

of a general

operator

0 over spin

space is given by

Q(O = Tr(pfl),

(3)

which is useful for formulating polarization and polarization transfer observables. In the channel spin representation the transition matrix element &‘SSv,,rvdenotes the complex valued probability amplitude in the radial asymptotic region for an outgoing spherical wave with final channel spin s’ and final channel spin projection quantum number v’ which follows from the scattering of an incident plane normalized to unit flux in initial channel spin state sv. The Lane and Thomas “) collision matrix UC, the complete collision matrix, is related to the initial and final Coulomb scattering matrices exp(iw,) and exp(io;,) and the nuclear collision matrix U by the matrix equation UC = exp (iw;,) U exp

(io,).

(4)

Parity conservation requires that the difference in initial and final orbital angular momenta I- I’ must be a positive even integer and the combination of the complex conjugation properties of the spherical harmonics together with the symmetry of the Clebsch-Gordan coefficients under the transformation (vm,M) -+ (-v, --?z~, -M) gives the following symmetry for the transition matrix,

a,_,~,,_, = (-l)S+“-S’-v’~~,y,,SY. The transition matrix for the reactions d+ 3~ --f N-t 4He where 39 is a generic particle which can be either the 3H or 3He particles in the A = 3 iso-doublet and N is the usual nucleon iso-doublet chosen so that (3H and n) and (3He and p) occur together. The relationship of the transition matrix elements to the collision matrix elements

is

A ,,Y,,SY(~2&$) = 71+k-‘(-~(0)8S,S6,,,+

i C (2Zf l)* 0 (sZvO~Jv)(s’Z’v’v- V’(Jv) .I, 1,I’ (6)

where the transition matrix depends on the c.m. energy and in general on two spherical angles in the cm. frame and the collision matrix depends only on the c.m. energy. A general d + 3~ spinor is a 1 x 6 matrix over spin space and a general N + 4He spinor is a 1 x 2 matrix over spin space so that the transition matrix i@ must be a 2 x 6 matrix over spin space and the lower-lying spectroscopic states as generated by a strong spinorbit interaction are listed in table 1 for reference.

B. DE FACIO AND

260

R. K. UMERJEE

TABLE 1 Spectroscopic

N + 4He

channel

s=+

positive parity negative parity

d + 3*

channel

s=+

positive parity negative parity positive parity

s=3

states, assuming a spin-orbit interaction

2S,,2D3,2D,,2G;,2G, ,... 2P,,2Pt,2F;,2F;,2Hp,2HV...

2S,, 2D,, 2D,, 2G;, ‘G,, . . . 2P,,2P,,2F,,2F,,2H,,2HY ,... 4S,, 4D,, 4D,, 4D,, 4D;, 4G,,4G,,4G,,4Gy ,... 4P,, 4P,, 4P,, 4F,, 4F,, 4F;, 4F,, 4H $9 4H $.>4H +9 4H p...

negative parity

We label the transition matrix fi in the channel spin representation sv %T! SIV) a

= 33 9r@)e” 3 - 3 ( 44(Qe2i4

according to

34

t-4

3-t

3t

s2(@ g,(Ojei9

g,(O)e-‘+ - s2(e)

-g4(6)e-2i4 gI(d)e-i9

s,(Q gh e-‘+ 3 (7) -g6 eio ss(e) >

f-f

where we have factored out the azimuthal angle &dependence and g1 -96 are the independent scattering amplitudes and depend upon c.m. energy and the scattering angle 0. From eq. (7) we can see that g1 -g4 represent As = - 1 or quartet to doublet transitions and g5, gs represent As = 0 or doublet to doublet transitions. Using eq. (6) repeatedly we get for the values of g1 -g6 g1 = (2k)-‘rgIei”‘P: +ei~‘~-2

e i”‘*+*(E-i)((21+2j(2E+3))-1

uf’,$, 13

3He(d, p), 3H(d, n) REACTIONS

(I)

261

3(1+1) (2+3)(22+2)(21+3)

g5 = (24~)~’ 1 ei(or+m”)Ppf(l+ l)@fl I=0

tfUfl,f,>,

where we have used the fact that we are studying a reaction to set the Kronecker delta in the channel indices 6,, equal to zero and where in the channel spin indices s’s we have taken 2s’2s to avoid a large number of fractions in the subscripts which label the collision matrix elements. Using table 1 (spectroscopic states) we can use J” conservation to list the lower-lying contributions to the Vi, 2sP2selements. We list these contributions in table 2. In appendix 1 we list the colliiion matrix elements in terms of phase shifts, R-matrix expansions and coupled channel expansions for the most general J” conserving time-reversal invariant interactions. Using the notation si = 9 * P,

s; = 9’ * 2,

s, = s- R,

s; = S’ - R,

sm = s* iii,

s; = S’ * ??t,

r9>

where eq. (9) gives the S, S’ components along the &,rit axis, we can list the most general possible transition matrix in terms of quantities in eq. (9) as A.4= A,S,:+A,S,S;+A,S,S~$A,S,S~+A,S,S~+A,S,”S;,

(10)

where Al -A, are complex scalars. We could apply trace theorems to A x fit but since &#is not diagonal, the generalization of Wolfenstein’s ““) procedure devised by Seyler in ref. lg) is not readily applicable. Let A4denote the transition matrix as transformed by the Clebsch-Gordan transformation which we write in matrix form as

B. DE FACIO AND

262

R. K. UMERJEE

TABLE 2 Collision

G,

matrix elements

1’1

2s’2s

2S’2S

Final

state

Initial

CN+4He) u I+* 1+21,13 u I+3 1f21.13

13 13

1+2 lf2

1 1

2D* 2F*

13 13

with elements

where

4% “5 .

. I 21.13 u’:+ 1 u’? 21,13

state

(d+3X)

l-2

1

2%

4D,

l-2

1

2p,

4F,

given by

the final

M Y’0.Y1Y2

=

state

spin

c(~ov’oI~‘v;)~~~,~~,s”,+yz(~lvl v2lsv1+v,), structure

is greatly

simplified

because

(12) S’(4He)

= 0.

3He(d, p), 3H(d, n) REACTIONS

263

(I)

Repeatedly applying eq. (12) to eq. (7) we get v1 v2 V’O 40

lvl

v’O,vtv2

=

-30

-41

lzl

(

f1

ei9

-f6ezi+

f2

f5ei4

5-I

-)o

g-o

f3 f4eio

f,e-'+ -f3

f5 esi9 -f2

-3-l _f6eb2’# fle-'4

1 ’

(13)

where fr-f are the six independent uncoupled scattering amplitudes and are related to the six coupled representation scattering amplitudes according to

f6 = 949

fl = 919 f2

=

(v3(s2

+

JWP

f3 =

(WJ372

-gJ,

f4

=

W(J%,

-9&

fs

=

(V(s3

-J%d9

(14)

where the fi-f6 amplitudes depend upon c.m. energy and scattering angle as do the quantities gl-g6 in eq. (8). The uncoupled scattering matrix M is especially useful for discussing polarization and polarization transfer observables. 3. Polarization,

analysing

power and broken symmetries

Since Ohlsen, Gammel and Keaton lo) are presenting the expressions for the 3& (d 3N)4He observables in a notation consistent with the present work we focus our attention upon polarization and the 3~ and d analysing powers and the generality of eq. (13) as compared to refs. 5-7) explains both their conclusions and the corrections to their expressions. The unpolarized differential cross section Q can be calculated from eq. (14) as Q” = 4 Tr (&J&J+), so that in terms offi-&

(15)

one has that Q” = 3 2

lfi(k2,

@I’.

i=l

(16)

It is easier to use eq. (13) together with vector and tensor operators from Goldfarb ’ ‘) and Lakin 16) in a direct product (to be introduced) than to generate trace theorems and apply them, so we follow this method. The polarization from an initially unpolarized beam P” is given by Q”P” = $ Tr (MM’s,),

(17)

where the bN operator operates only on the final nucleon spin coordinates and parity conservation requires that only cr2 along fi can be non-zero so that eq. (17) can be

264

B. DE FACIO AND

R. K. UMERJEE

expressed in terms offi---& as

P” =

Tf;ii+,Pm(.&f6 +f2 J; +Jj f4>Iy

(18)

which is the quantity measured by Brown and Haeberli ““) for the 3He(d, p)4He reaction. The vector analysing power of the 3He target was measured by Baker et al. 24) and since then it has been measured more accurately by Watts and Leland [ref. ’ “)I. Since the spin-4 a-operator now operates on the 3.% spin coordinates, one must form a direct product 6 x 6 spin operator u 0 lC3) where lt3) is a 3 x 3 unit operator representing the unpolarized deuteron. Inspection of the labels in eq. (14) shows that

i an=

c 63

l(3)

=

is the correct order and o,A was analysingpower of the deuteron, a 2 x 2 unit operator representing parity conserving vector operator

_-

0

s,

=

--

00 00

00 00

00

01 1 0

00, 0 0 nY

00

00

00

00

01 10 J

0

jz

0

j I

used because of parity conservation. For the vector one must form an operator lC2)18 S, where lt2) is the unpolarized 3# target nucleus and S2A is the in Cartesian coordinates. From eq. (13) we can set 1 0

-1 0 1

0 -1 0

0 0 .o

0 0 0

0 0 0

‘0

an = 1(2)

01 10

;z

0 0 0

0 0 0

o-1 1 0

0’ 0 0 0

0 1

fi,

(20)

-1 o/

and as previously mentioned all of the 6 0 Sd cases are presented in ref. ’ “). The vector analysing power of the deuteron and along with its remarkable similarity to the vector analysing power of 3He was first reported by Plattner and Keller “) and will be discussed. Using a generalization of Wolfenstein’s nucleon-nucleon discussion by Seyler I’), one gets for the differential cross section with polarized particles Q = Q”(l++

I’“) = 6D(8)(1+Pi”*

P”),

(21)

which defines D(e) and where Pi” is the constant incident polarization and P” is given by eq. (17). The same equation holds for vector polarized deuterons except that it is now the vector analysing power of the deuteron and in explicit calculation the other direct product is formed as in eq. (20). One observes that our choices of unit vectors

3He(d, p), 3H(d, n) REACTIONS (I)

265

in eqs. (1) and (2) has eliminated the need for the half angle rotations of appendix 1 of ref. lg) and that the there, although exactly correct, would be applicable to our coupled transition matrix of eq. (7). The equations of interest here and especiaIly the &&en symmetry discussion is simplified by using the represen~tion of eq. (12). The vector analysing power of the 3& target and deuteron will be denoted as A,,(“@) and A,(d) respectively and in terms of eqs. (13), (19) and (20) can be written as

(22) (23) Using eqs. (13), (19) and (20) one can express Ay(3Z) and A,(d) in terms off,-&, as

Next we apply the broken symmetry concept, in its simplest sense, to eqs. (18), (24) and (25) to extract information on the mechanism by inspection of the polarization of the reaction nucleon and the vector analysing powers of the 3# and the deuteron. This is useful since the differential cross sections as Hale et al. in ref. lo) have shown are crucial for the numerical fit to all of the data, but are still relatively insensitive to the polarization observables. The effective potential operator Ye, which mediates the reaction produces a large effect in some spectroscopic states and a small effect in others. Its non-dynamical structure, as already mentioned, is established by the invariance principles FnJ but we stress that at low energies, say O-10 MeV, c.m. frame, that the interaction is small in certain states. We call these effects broken symmetries because they arise from the detailed nature of the interaction or dynamics as does isospin. From table 1 the initial channel contains s = 3 and s = f states whereas the final channel contains only s = 4. Thus with ds = s,-si we must investigate the consequences of AS = - 1 and AS = 0. The P = -A relation obtained in refs. 6t 7*’ “) is not an exact relation but is an example of a broken symmetry. The invariant amplitude approach by Tanifuji and Yazaki 26) is similar to the present work except that they start with simple interactions and work up to more general interactions whereas we construct the most genera1 possible transition operator and specialized the general operator to study special cases. In ref. 27) it was found that the most general spin-orbit interactions provide the relation A,,( “2’) - P” = -2A,(d) which is in complete disagr~ment

(26)

with Plattner and Keller in ref. ‘“). Hardy, Baker,

266

B. DE FACIO AND

R. K. UMERJEE

McSherry and Spiger “‘) have also observed the striking similarity between the elastic vector analysing power of the deuteron and the left-right asymmetry for reaction scattering from a polarized 3& target. This similarity arises from strong tensor interactions in the reaction and the elastic d + 3Z channel interaction but one should note that both tensor interactions are those which break both orbital angular momentum and total spin which is discussed further in appendix 2. This is because a spin-orbit operator L* S can only generate azimuthal angle &dependence of at most ekib (or 1) and th e amplitude fs is required. Since the &dependence of the M element containing fs is e ‘Q it is the tensor interaction as discussed in appendix 2. Comparing eqs. (24) and (25) one sees that it is the terms Im(f,f,) which must be small to explain the Plattner-Keller and Hardy et al. experiments and thatf, alone does not explain this result because of additional terms in the elastic d + 3Z channel. This also illustrates the usefulness of studying the general case first instead of working one’s way up to it, as practitioners of direct reactions have always done. Tanifuji in ref. “) first showed the inadequacy of that approach. The P = -A relation can be obtained from eqs. (18) and (24) either if AS = - 1 or if either pair of& and one of (fiYf2), or fi and any one of (fi,f~) vanish. From eqs. (8) and (14) and table 2 we can read off the implications on the reaction mechanism by studying the kinds of collision matrix element U;,, 2s,2s which vanish and which elements remain and which we relegate to appendix 2. Concerning these relations the paper by Keaton, Armstrong, Dodder, Lawerence, McKibben and Ohlsen in ref. lo) obtained this same conclusion on the AS = - 1 “selection rule” or “broken symmetry” but in their discussion they state that “AS = - 1 dominates over most of the angular range, but that AS = 0 competes strongly at zero degrees”. From eq. (8) one sees that the amplitudes in the transition matrix are at fixed energy are a superposition of associated Legendre functions which are smooth functions of angle so we do not understand the statement quoted above. Also they call these relations “non-dynamical”, which is incorrect usage as mentioned in the introduction. One can form any other matrix element for a polarization transfer observable following the procedure used in eqs. (19) and (20) an d using the transition matrix in eq. (13). For practical computations eq. (14) relates thef,‘s to the gi’s which are given for arbitrary YnJ conserving couplings for the first time. This when combined with appendix 1 will enable experimentalists to study their own data numerically so that work like Seiler and Baumgartner ‘) can be extended to arbitrary energies for two body processes. The three-, four-, and five-body equations are being studied 11) for broken symmetries including polarization and polarization transfer and will be submitted soon. Both authors wish to thank Dr. J. L. Gammel for suggesting to them that the reactions studied here are extremely interesting. One of us (B.D.F.) wishes to thank Dr. M. Tanifuji for emphasizing the analytical aspect to this problem over the numerical part and Dr. Gerald G. Ohlsen for sending Los Alamos Report LA-4465-M& which con-

3He(d, p), 3H(d, n) REACTIONS

tains

thirty-five

(I)

in ref. lo) and all of those referred

of the papers

(i) Tanifuji and Yazaki and (ii) DeFacio and thank the Research Council at the University fellowship during which the present work was for Air Force Grant 71- 1979 at Texas A&M Appendix

267

to here except

Umerjee. Finally, (B.D.F.) wishes to of Missouri-Columbia for a summer carried out and (R.K.U.) is grateful for summer support. 1

To parameterize the collision matrix U we treat phase shifts in detail and mention coupled channels and the R-matrix. In order that the Coulomb phase shifts enter the transition matrix elements in eq. (8) most simply, we use Stapp’s nuclear bar param2’s 30) has earned its popeterization ‘*). The R-matrix of Wigner and collaborators ularity by the combination of the rigor of its justification and its utility in computations. Coupled channels generalize the usual optical model and anyhow, we present our equations in a form for which one can use real parameters in the Cayley-transform (called the K-matrix), a real symmetric matrix, without doing the coupled potential calculation - which then becomes just two-body rearrangement collision theory. From table 1, one sees that at least three two-body states are coupled together for the spin-parities J” = Ithe three-body threshold there are three 2 92a+ so that below real eigenphases F,, 8,, s3 and three real mixing parameters cl, s2, s3 for each of these J” values. For all other J” values four two-body states couple together so that there are four eigenphases and six mixing parameters for each J” value. In the nuclear bar parameterization the collision matrix is written for each J” state as U = exp (ia)E(c,

. . . Ed) exp (ia),

(A.l.lj

where E(sl . . .+) is the mixing matrix with N = 3 or N = 6 depending upon the J” value and exp(i& is a diagonal matrix with exp[iZi(J”)], i = 1, 2, 3 (4) as entries. The Nf 4He channel is taken as the lowest channel c = 1 and the d + 3X channel is taken as c = 2 and no other stable two-body channel exists. In order to shorten this appendix both the J” and the c.m. energy dependence is to be understood. One cannot use Seyler’s expressions from ref. ’ “) because the N + 4He channel is open for all d+ 3Z? energies. This is the justification for the present appendix. A symmetrical order must be assigned and this order must be specified since we have more than two channels coupled. For J” either +- or 3’ the mixing matrix is E(E~ cz E3)

=

E~(E~>E~(E~>E~(~E~)E~(EZ)E~(E~),

where E,(2&,)

=

EZ(c2) =

cos (2~) i sin (2~~) 0 (

i sin (2~~) COS(2&,) 0

(A.1.2j

268

B. DE FACIO AND

R. K. UMERJEE

0

0 1

i sin Ed 0 .

i sin s3

0

cos Ej1

cos Es

E3(e3) = (

(A.1.3)

Putting eqs. (A.1.3) into (A.1.2) and forming the indicated product gives cl2 E(sI y ~2~~3) =

ezl

c::

cl3 (A.l.4)

e22 e23 s e32

e33

i

where using the notation Cifor COS(Q)when i = 2 or 3 and ei when i = 1 for cos(2ei) and si for sin(ei) when i = 2 or 3 and 3i when i = 1 for sin(2.si) we can give the entries to eq. (A.1.4.) as et1 = (c2 ci + E, sf s: - cz s:) - i(23, s2 s3 c3), e21 = (-c2s2s3-C,c2s2s3)+i31c2c3, es1 = (3, s2 s: - 3, s2 c:) + i(c, c3 s3 - 2, si c3 s3 + cz c3 s3), e12 = (-c2s2s3-&c2s2s3)+i31c2c3, e22 = (h c2 c3 - sl), e23 = (2, c2 c3 s3 -3, c2 s,)+ ic, s2 c3,

e, 3 = (sr s2 sz + c2 c3 s3 - 3, s2 c:) + i(i?, c3 s3 - E, sf s:), e23 = ( -3, c2 s3) + i(t, c2 s2 s3 + c2 s2 c3), e33 = (ci c: -2, si - 1, sg c3 s3) + i(23, s2 s3 c3).

(A.1.5)

The product of eqs. (A. 1.4) and (A. 1.5) as in eq. (A. 1.1) is straightforward but long and can most easily be carried out numerically, so we write the elements of eq. (A. 1.1) for each i, j as uij = ei’dr+G)eij,

(A.1.6)

where i, j can run from 1 to 3 independently and are the channel indices. For four states coupled the mixing matrix E(sl . . . c6) for each J” value is given by E(q . . . ~6) = &i(~2

. . . s,j)E1(2~#26(~2

. . . ~6) ,

(A.1.7)

where B,, is the transpose of E26 and where one has E26(&2

* * * 4

=

E2(&2)E3(&3)E4(&4iE5(Eg)E6(Es),

j-J,&,

* . . ~2)

=

E~(E~)E~(&~)E~(E~)E~(E~)E~(Ez),

(A.1.8)

3He(d, p), 3H(d, n) REACTIONS

269

(1)

with /

E,(W =

cos

!

(24 (2E1)

i sin 0

0

i sin (25) cos (24

0

0 0

0 0

1 0

0

’

11

Using the matrices in eq. (A.l.9) to perform the products indicated by the second line of eq. (A.1.8) one finds for the matrix Ez6 using the notation Of ci, tl, Si, $I as defined for eq. (A.1.5), one gets

(A.l.lO)

270

B. DE FACIO

AND

R. K. UMERJEE

with entries aij given by all

=

w6,

azl

=

-(s2c3s6+c2s4s5c6)-is2s3c4s5s6,

agl

=

-c2s3c4s5c6+i(c2s3s6-s2s4S5c6)9

all

=

i(s3s6+c3c4s5c6),

al2

=

0,

az2

=

C2CqvlS2S3S4,

a32

=

a42

=

zc3s4,

a13

=

w6,

a23

=

s2s3c4s~s6+i(s2c3c6-c2s4s~s6)~

a33

=

(cZc3c6

a43

=

-

aI

=

is,,

a24

=

-s,s,c4c,+1c,s,c,,

a34

=

--2s4c5+lc2s3c4c3,

a44

=

. .

.

-c2s,s,+~S,c,,

fs2s4s5s6)~-ic2s3c4s5s6)~

c3c4s5s6

f

is3c6,

(A.l.ll)

c3c4c5.

-

The mlxmg matrix Ez6 is the transpose of Ez6 as given in eq. (A.l.lO) with elements given by eq. (A.l.11) so with E16 as the four by four complex mixing matrix whose elements are written as yij and which is defined to be El6 =

El(2El)E26(EZ.

. . E6)

z

(A.1.12)

(Yij),

with the Yij elements given by 711

=

y2l

=

(~~c5c6+3~~2~3~4~5s6)-i(~~s2~3~6f~l~~S4S5C6), -

(tlc2s4s5c6

A

+

&s2c3s6)

+

i(3,c,c,j

-i?$2s$4s5&j),

.*

YlZ =

~,~2s3~4+~~l~2~4,

yz2

=

?,c,c,-it,s,s,s,,

713

=

(3~czs4s5s6

723

=

(~~~~~~~~~~~~-8~~~~~)+i(~~~~~~~~-~~~2~~~g~g),

714

=

-~~~~~~c~+i(~~~~-3~~~~3~4~5),

724

=

-(Els2s3c4c5+~1s5)+i~lc2s4c5,

yij =

aij

-s,s,c3c,)+i(~lCgSg+S2S3C4SgS6),

(all other ij).

(A.1.13)

271

3He(d, p), ‘H(d, n) REACTIONS (I) Upon

substituting

eq. (A.1.7)

the transpose

of eq. (A.l.lO)

one gets the elements

The 0th element

of the mixing

and eqs. (A.l.12)

of the full mixing matrix

matrix

and (A.l. 13) into

for four coupled

states.

E is given by eij: 4

eij

=

c

(A.1.14)

ClkiYkj?

k=l

which can easily be calculated in terms E~-E~ using the &mentS c(kiand Ykj from eqs. (A. I. 11) and (A. 1.13). We refrain from listing the explicit dependence on the Ci’s and si’s because it would double the length of this paper while only adding long equations which should be formed on an electronic computer anyhow. Finally, for each J” value the [jth element of eq. (A.l.l) is uij = ei(dl’G)eij. To parameterize

eq. (A. 1.1) so that elastic unitarity

(A.1.15) for the S-matrix

S means that the

relation S’S holds below the deuteron

break-up

threshold

s+s holds at all energies, the S-matrix as

the so-called

matrix

and inelastic

unitarity (A.1.17)

= ss+ 6 1,

K-matrix

s = (1 -iK)_l(l so that the collision the relation

(A.1.16)

= ss+ = 1,

of scattering

theory is defined in terms of

+iQ,

U of eq. (A.1.15) is related to the K-matrix

(A.1.18) according

to

I/ = (1 -iK)-1K = (1 +KZ)-l(K+iKZ).

(A.1.19)

Below the deuteron break-up threshold energy the S-matrix satisfies the elastic unitarity condition of eq. (A.1.16) so that S is unitary and all eigenphases are real. The K-matrix defined in eq. (A.1.18) is called the Cayley transform of a unitary matrix S and if K is a real symmetric matrix, S is exactly unitary. Thus for placing parameters into the collision matrix U in eq. (A.1.19) the K-matrix is an optimum parameterization because it assures one that his approximate S-matrix is exactly unitary. Above the deuteron break-up threshold energy the S-matrix satisfies the inelastic unitarity condition of eq. (A.1.17) and now the eigenphases have become complex and the K-matrix is no longer real. Below threshold for a two-body process the submatrix of the closed channel c’ consists of zero terms because in the asymptotic radial region r - COthe channel wave function is proportional to e-ikc”r although the closed channel must be fully coupled to the open channels as was first pointed out in ref. 2 “). The details of a generalized optical model or coupled-channel model are applied

272

B. DE FACIO AND

to the three two-body channels of and we follow their exact notation. matrix @ in eq. (6) of the present final scattering states ~cksm,) and

R. K. UMERJEE

the four-nucleon system by Hale and Umerjee 31) In ref. 31) their eqs. (l)-( 16) specify the transition work in terms of overlap integrals of initial and (c’k’s’m,l as

JLl~., sm, = (c’k’s’ms,Icksm,),

(A.1.20)

where the initial and final states are specified in ref. 31) for all possibilities of open and closed channels, so that we do not repeat them. A more useful dynamical analysis of the collision operator is in terms of the Rmatrix of Wigner and collaborators provided that the single-level approximation is not made. Buttle 32) has shown that R-matrix methods can give the same numerical results as coupled-channel calculations in both resonant and non-resonant energy regions. Since the two R-matrix and one distorted wave methods developed in ref. 32) are faster and more stable than coupled-channel methods, as well as numerically simpler, all three of Buttle’s methods deserve attention especially (in our opinion) his method II which includes both nearby and distant levels. From ref. 30) one has the reactance matrix Q(E), defined in terms of the collision matrix U as U = (l-iQ)-l(l+iQ),

(A.1.21)

and since the R-matrix depends upon closed channels, the sub-matrix R, over channel space refers only to the open channels and if I? is (h/M, VJ times the Hamiltonian H and 1 is the boundary value I divided by b, the channel radius, then the collision matrix U can be written in terms of the reduced R-matrix R, according to the relation

The resonance formulae and various approximations to eq. (A.1.22) are discussed in detail in ref. 30). Of course, one retains the usual R-matrix problems, (i) of the failure of one level formulae to fit accurate data, and (ii) the R-matrix phase shifts lying below phenomenological phase shifts on the higher-energy side of a resonance. Appendix 2

The dynamical content of table 2 can be expressed in terms of the quantum numbers of the most general scalar potential using an idea which Hale et al. in ref. ’ “) ascribe to Gammel. Unfortunately, this idea holds only for local energy-independent potential operators, whereas the wild fluctuations in potential parameters with incident energy imply a strong intrinsic non-locality for the channel potential. With SA as the three deuteron adjoint spinors and S as the three Pauli spinors the spin-orbit factor is L. Si, the spin-spin factor is S . S, and the tensor factor is given by 3(Si *P)(S *P) Si *S so that in terms of the magnitude of orbital angular momentum 1 the factors multiplying the radial matrix elements are given in table 3.

sHe(d, p), 3H(d, n) REACTIONS

273

(I)

TABLE3 Quantum numbers in local, energy independent

A. Spin-orbit

B. Spin-spin

L - s;

I++

E

l-5

1

l-3

1-2

-JJ3 1+1 z

(31(21+ 3))+ (3@+1)(2~-I))+

potentials

si * s

-J3 -X/3

C. Tensor 3(Si * P)(S * P) -s,

O

0

O

0

0

-

*s

( &+

1

($J

--

I

+

( 1 21-1

This eliminates the specification of an independent local “potential” for each J” state and limits the number of parameters sharply. At higher energies where the number of J” states required increase this will facilitate computations. The criticism above concerning non-locality is crucial when one uses the M-matrix to calculate three- (and more) body final states because at a fixed energy an equivalent local potential usua1l.v exists and in the numerical treatment of a two-body reaction one seldom goes beyond this local potential. From table 3 one can see which interactions contribute to each U&, element and it is important to observe that the tensor term breaks both orbital angular momentum I and channel spins s, s’ because the first term does not commute with the channel spin for the + 0 3 spin system which is a significant difference from the 4 @ _5 spin system where the tensor interaction (cl . P)(cz . P) does commute with total spin s = +(a, +a,). Moravcsik ‘“) has discussed the spinorbit potential of the second and third kinds (p. 52 et seq.) and in ref. ’ “) DeFacio and Umerjee were discussing part of the tensor potential of term C in table 3 as a product of aspin-orbit potential of the second kind and an orbital tensor, i.e. I-breakingpart, but if one notes the simultaneous I and s breaking implicit for this tensor their discussion is necessary. In closing it should be mentioned that Gammel was studying Fourier transforms (the amplitudes in ref. I’) equivalent to eq. (10) of this paper) when he produced table 3 which again emphasizes the value of the generality in refs. 12-r4). If one combines our eqs. (8) and (14) then Csonka, Moravcsik and Scadron r2) have calculated ourfi’s and we then agree with their amplitudes.

274

B. DE

FACIO

AND

R. K. UMERJEE

References 1) F. Seiler and E. Baumgartner, Nucl. Phys. A153 (1970) 193 2) B. De Facie, R. K. Umerjee and J. L. Gammel, Phys. Lett. 25 (1967) 449 3) B. De Facie, R. K. Umerjee and J. L. Gammel, Phys. Rev. 151 (1966)819 4) L. J. B. Goldfarb and A. Hug, Helv. Phys. Acta 38 (1965) 541 5) A. Hug, Helv. Phys. Acta 39 (1966) 507 6) M. Tanifuji, Phys. Rev. Lett. 15 (1965) 113 7) I. Duck, Nucl. Phys. 80 (1966) 617 8) B. P. Ad Yasevich and V. G. Antonenko, Sov. J. Nucl. Phys. 10 (1970) 378 9) V. A. Khangulyan, Sov. J. Nucl. Phys. 11 (1970) 676 10) Proc. Third Int. Symp. on polarization in nuclear reactions, ed. H. H. Barschall and W. Haeberli (University of Wisconsin Press, 1971) 11) B. De Facie, to be published 12) P. L. Csonka, M. J. Moravcsik and M. D. Scadron 143 (1966) 1324 13) P. L. Csonka, M. J. Moravcsik and M. D. Scandron. Ann. of Phys. 41 (1967) 1 14) P. L. Csonka, and M. J. Moravcsik, Phys. Rev. 152 (1966) 1310 15) L. J. B. Goldfarb, Nucl. Phys. 7 (1958) 622 16) W. Larkin, Phys. Rev. 98 (1955) 139 17) E. U. Condon and G. W. Shortley, Theory of atomic spectra (University Press, Cambridge, 1965) 18) R. Huby, Proc. Phys. Sot. A67 (1954) 1103 19) R. G. Seyler, Nucl. Phys. Al24 (1969) 253 20) J. von Neumann, Mathematical foundations of quantum mechanics (Princeton University Press, 1955) p. 178 et seq. 21) A. M. Lane and R. Thomas, Rev. Mod. Phys. 30 (1950) 257 22) L. Wolfenstein, Ann. Rev. Nucl. Sci. 6 (1956) 43 23) R. I. Brown and W. Haeberli, Phys. Rev. 130 (1963) 1163 24) S. D. Baker, G. Roy, G. C. Phillips and G. K. Walters, Phys. Rev. Lett. 15 (1965) 115 25) G. R. Plattner and L. G. Keller, Phys. Rev. Lett. 29B (1969) 301 26) D. M. Hardy, S. D. Baker, D. H. McSherry and R. J. Spiger, Nucl. Phys. A160 (1971) 154 27) M. Tanifuji and K. Yasakik, Prog. Theor. Phys. 40 (1968) 1023 28) M. J. Moravcsik, The two nucleon interaction (Clarendon Press, Oxford, 1963) 29) T. Teichmann and E. P. Wigner, Phys. Rev. 87 (1952) 123 30) G. Breit, Handbuch der Physik 41/l (Springer Verlag, 1959) p. 107 et seq. 31) G. M. Hale and R. K. Umerjee, Nucl. Phys. A156 (1970) 570 32) P. J. A. Buttle, Phys. Rev. 160 (1967) 719