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Polarization from scattering polarized spin-12 on unpolarized spin-1 particles
I
2.B
] I
Nuclear Physics A124 (1969) 253--272; ( ~ North-Holland Publishin9 Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
P O L A R I Z A T I O N FROM SCATTERING POLARIZED SPIN-½ ON U N P O L A R I Z E D SPIN-1 P A R T I C L E S R. G. SEYLER
The Ohio State University, Columbus, Ohio Received 17 September 1968 Abstract: The most general parity and time reversal invariant form for the spin@-spin-1 scattering amplitude matrix is obtained. The polarization observables (cross section, spin-½ polarization, spin-1 polarization, and spin correlation) which arise upon scattering polarized spin-½ by unpolarized spin-1 particles are expressed in terms of the coefficients of this general form. A phase shift parameterization of these coefficients is also given.
I. Introduction The purpose of this article is to study the structure of the spin-½-spin-1 scattering matrix and to relate to this matrix various polarization quantities which can be measured when a polarized incident spin-½ beam is used. These equations should provide a useful starting point for a complete phase-shift analysis of the nucleon-deuteron scattering problem. The treatment is non-relativistic throughout. The subject of this article has been previously treated, with conflicting results, by other authors: Budianskii 1), Shih et al. 2) and Goldberg 3). The present treatment is an effort to correct and to generalize the works of these authors (BSG). The generalization is the inclusion of the possibility of spin mixing, that is, transitions between doublet and quartet channel-spin states. Of the errors in the earlier works (BSG) those non-numerical in nature are pointed out later. We use, as did each of the authors (BSG), the conventional formalism of the yon Neumann 4) density matrix p. The convenience of this formalism for problems involving a statistical ensemble of different pure spin states is well known as a result of the work of several authors 5) on the very similar problem of nucleon-nucleon scattering. The final and initial density matrices are related by the transition matrix M according to p = MpinM*. Trace p is then the differential cross section (per unit solid angle) herein denoted by Q. The ensemble average ((9) of any operator C is given by
Q(C> = Tr (pC).
(1)
This equation is repeatedly used for the calculation of operator averages in sect. 3. 253
254
R . G . $EYLER
1.1. C O - O R D I N A T E
SYSTEMS
It is convenient to define the following ten unit vectors: /~, along the c.m. momentum of the incident nucleon, /~', along the c.m. momentum of the scattered nucleon, /~[, along the lab momentum of the scattered nucleon,
~'
= (~,'+~)/l~,'+~l,
k = (k'-k)/Ik'-kl, ^
= t~x.~'/l~x =
~:'1 =
/~,
~
A
PxK, =
d,
g = t i x ~ L,
:~ =
~x~,.
(2)
The vectors I£, A and l' are mutually perpendicular, as are the vectors g, 6 and/~[ and the vectors #, ~ and ~. The above definition of the vector K is that of Wolfenstein [ref. 5)] and is opposite to that of Budianskii ~), who chooses K in the direction of the momentum transfer to the deuteron in elastic scattering. We also define 0 = arcos (~./~'), O = arcos (~,'/~[),
(3)
the c.m. and lab. scattering angles, respectively.
2. Transition matrix 2.1. C H A N N E L
SPIN
REPRESENTATION
In the channel spin representation the "transition" or "scattering amplitude" matrix element Ms,v,sv signifies the amplitude (at infinity) of the outgoing spherical wave of channel spin s' with z-axis spin component v' resulting from a unit flux incident plane wave of channel spin s with z-component v. The connection between the transition matrix M and the collision or scattering matrix U is given by Lane and Thomas [ref. 6)] as
M~,v,s,(O) = rc~k-1[_ C(0)6~,6w, + i ~ (21+ 1)~(slvOljv)(s'l'v'v- v'ljv) ill'
x (exp {/(co, + oh,)})(6~,611,--
~vv-v'(, 0, 0)],
TrJ ,~,,'su.1"
(4)
where Ytm is the usual normalized spherical harmonic of Condon and Shortley v) and where the Coulomb scattering amplitude C(O) and the other familiar quantities k and 0h are defined as in ref. 6). The collision matrix U in eq. (4) is the "nuclear" part of the collision matrix and is related to the complete collision matrix U c by the matrix equation 8) U c = exp (i~o)U exp (i~o), where co is the diagonal matrix whose elements are the phase shifts ~o~.
(5)
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
255
Invoking parity conservation which restricts the sum l + l' to being even, it is easy to show that ( - , "V'-s+v'-v M ~._s,v,~.
M~,_v.~_ v -_
(6)
Thus parity conservation, i.e. eq. (6) reduces the number of independent (complex) elements of M from 36 to 18 and permits us to write the matrix M as 31 22
SV
3 2
1 2
~__~
11 22
½__
d j -g a
e k l -f
f l -k e
½
SrV I
2
2
2
2
a 9 j -d m
n
o
p
q
r
L 2
--p
o
--n
m
--r
q
&Z 22 I 2
b h -i c
c i h -b
(7)
It is curious that Budianskii 1), Shih et al. 2) and Goldberg all commit the error of treating c a n d j (in the notation of eq. (7)) as being identical. In the works of Shih et al. and Goldberg, the "equality" of c a n d j is "deduced" by using the alleged relation,
(slv-- l m l J M -
1) = (-)s+t-S(slvmlJM),
which is, however, not generally valid. It should also be pointed out that the authors (BSG) all omit (set equal to zero) the eight spin-mixing M-matrix elements e,f, k, l, m, n, o, a n d p ofeq. (7). The assumed time-reversal invariance of the interaction responsible for the scattering imposes additional restrictions on the elements of the M-matrix is as discussed in subsect. 2.2. 2.2. M O S T
GENERAL
FORM
OF THE
M-MATRIX
The most general form of the M-matrix is determined by requiring that it be invariant under space reflection and rotation and time reversal. The determination procedure is very similar to that used by Wolfenstein and Ashkin 5) in the case of two spin-½ particles. For the spin-½-spin-1 problem the 6 × 6 M-matrix may be expanded in terms of thirty-six linearly independent matrices, which operate in the combined spin space of the two particles. The coefficients of the spin matrices are arbitrary complex functions of the energy and scattering angle. The 36 independent matrices may be obtained by combining the four independent spin-½ matrices (three Pauli matrices and unit matrix) with each of the nine independent spin-1 matrices (five second rank, three first rank and one scalar). Let ~r be twice the spin-½ operator and S the spin-1 operator. We also define the traceless symmetric second rank spin-1 operator
s , = ½(s, sj + s j s , ) - ~ , j .
(8)
256
R.G. SEYLER
By combining 1, a K, a, and ae with each of 1, SK, S., Sp, SK., Set, S,e, SKK and See (note that S., is not independent of the later two operators), we obtain thirty-six independent operators. One observes that of the five operators a, S, K, ~ and t' only and P change sign under space reflection and only K does not change sign under time reversal. This observation may be used to show that of the 36 operators, six violate time reversal invariance, nine violate parity conservation and nine others violate both parity and time reversal invariance. Demanding that the coefficients of these 24 terms equal zero we are left with only twelve terms M = A + Ba. S. + Ca. + D S . + Ear Sv + FaK SK + GSpp + Ha. See -4-ISKK"4-da. Srr + Kai, S.p + La r S t . .
(9)
Budianskii 1) offers a similar twelve coefficient expression for the most general Mmatrix, but two of his coefficients are redundant. His coefficients F and G turned out to be equal and his coefficients H and K should have turned out to be equal. On the other hand, he omitted two coefficients by unjustifiably taking, in the notation of eq. (9),G = IandH= J. The 6 × 6 matrix forms of the operators in eq. (9) may be obtained in the same representations as eq. (7) by the procedure discussed in appendix 1. When these matrices and eq. (7) are substituted into eq. (9) one obtains eighteen equations which express th" M-matrix elements (a through r) in terms of the M-matrix coefficients (A through L). These eighteen equations are given in appendix 3. To express the M-matrix coefficients in terms of the M-matrix elements it is convenient to consider certain traces. For example, from eq. (9), Tr M = Tr A = 6A, where we use the fact that all of the operators (other than the identity operator) on the right side of eq. (9) have zero trace (the trace properties of the spin operators are reviewed in appendix 2). Similarly, we could write T r ( M a , Sn) = B Tr(a,S,a.S,) = 4B; to calculate G, H, I or J is only slightly more complicated, for example, Tr(MSpp) = G Tr S p+ITr(S KS e) = y4 G - ~ I2 . Goldberg and Shih et al. failed to take into account terms such as (in this exa'mple) Tr(SKIcSee). To solve for G, one computes Tr(MSrK) = ~4 I - y G 2 and solves the last two equations simultaneously with the result G = Tr(MSpp) + ½Tr(MSKK). Using trace equations such as these in connection with the matrix expressions for M, eq. (7), and for the spin operators (appendix 1), one may obtain the twelve equations expressing the M-matrix coefficients in terms of the M-matrix elements. These twelve equations are given in appendix 4. The above described procedure could also be employed to express the coefficients of any of the twenty-four operators we discarded in terms of the M-matrix elements. Doing so, we find that the coefficients of the parity non-conserving operators vanish identically. This is because eq. (7) already has parity conservation eq. (6) built into it. On the other hand, the coefficients of the six time-reversal non-invariant operators involve non-trivial equations, which when the coefficients are set equal to zero constitute the time-reversal invariance restrictions on the M-matrix elements. They may
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
257
be written as 0 = 3~(f+ p) + n -- k, 0 = 3~(e+m)+l+o, 0 = [3~(k+n)+f-p]
sin O + 4 ( e + m ) cos 0,
0 = [3~(o-l)+e-m]
sin 0 - 4 ( ] + p ) cos 0,
0 = [3~(h- a) + j + c] sin 0 + 2(g + b) cos 0, 0 = [ 3 ~ ( d + i ) + g - b ] sin O - 2 ( j - c )
cos 0.
(10)
These equations reduce the number of independent M-matrix elements (at any one energy and angle) to twelve in agreement with the number of M-matrix coefficients. Having just seen how the M-matrix coefficients are related to the M-matrix elements (see appendix 4 for the detailed equations), we can now use eq. (4) to obtain the collision matrix and angle dependence of the M-matrix coefficients of eq. (9). We find A
__1
C
--
E+F
2 ~do + ~q o-
B = ½ ( q o - - d o ) - ~ ( b1 2 + + 2q+),
C(O),
1 8 i~d~ +~b~ + ~5q l ,
= ~ ( q o - d o ) + ~ ( b + +2q~), +
+
G+I = -b2 +q2,
D
=
~ d 1 - ~ b2 i
+ ~ 5q l ,
E - F = ½b~ + q~, G-I
= - b ~ +q~,
H+J
= ~(dx + 2 b ~ - q ~ ) + - ? o q L
H-J
= - b 0z + ~1q 3--,
K+L
3 +, = - 2 ( d l + 2 b l - ql) + ~q3
K-L
= b°+q3,
(Ii)
where C(O) is the familiar Coulomb amplitude and where the d, q and b amplitudes (which involve doublet, quartet and both doublet and quartet states, res'pectively) are defined as follows: i do = - -2k - ~ Pt°ez~''[(l+ 1) U,~;~ +I+÷ lUl~t~-(21+ 1)], dl = ~1 qo =
~l
pl
~l e
21~t[-l-ll+½
Tfl--~-I
L""1½14~--"."1½1½1~
i
4k ~ p°e2~"[(l +2) U,÷,~+(/+'+~1)U,~I~+'+~I U l ~ + ( l -
1 ~p:e2UO,~-3(/+2) rrl+~± I - 3 Ut+~ - / + 4 U~l~ ql = 1 ~ / 1+1 '~l~l~:7- ~--- l~z~: 1+1
1 ) U I ~ - 2 ( 2 / + 1)], 3 ( l - 1 ) Ul~t~l 1 '
258
g.G.
SEYLER
i
4k ~ (3P~-l(l+ 1)P°)e~'°''
q~ =
[ /+2 rr,+, x (/+1)(2/+3) "'~'~
/+3 r,,+ ½ I-2 ,_, /-1 ,_,1 /(2/+3) "~'~ (/+1)(2/-1) U,~t~+ /(2/~1-~ U,~,~
+2x/J(Pe+(l+l)(l+2)P°)e'(°"+~'*~'
1 F / l + 3 \ ~ ,+~ 2/+3
Eta)
/' l '~rr,+~
i q2 = ~-~ ~ (P~+l+l(l+ l)P°,) e2'o'
I-2 t_~: l - 1 t-~] 1+2 rrl+e /+3 rrl+ ½ I"~l~l~ - "Jl~l~ Ut~t~ + l(21-1~) Ut~z~ (1+ 1)(2/- 1) l+1)(21+3) 1(2t+3) 1 2 + ~2 (P,+,-3(/+ 1)(/+ 2)P°+,)e '(~'+ .... ) - 2/+3 X
Ii
X
F(/+3~½rrt+, ± [ l '~*,r,+* ,,, i + 2 . ~ l } T
-
-
7
q~- = 1 1~ ~ (5Pz3- 3(1-1)(1 + 2)P~)e2i°'' 1 [/+~rr,+÷ 3rr,+~± 3(2/+3) (21+3) "l~l~-- -f '~t~t~ ~- (l+ 1)(2/-- 1)
21+3 t-~] l(21-1) Uliti
+ lO~/3(P3t+(l+ 2)(l+ 3)P~)ei(~'+o'÷~) l 2l+3
x [((/+ 1)(/+ 3)) -~ UI++~,~-(l(l + 2)) -~ vI~.~j, q;
X
=
-
_
12k ~ (Pt3+' +l(l-1)P]+l)e2"°'
_ _
3 E/@l rrt+ ~ 3 rrt+~± 3(2/+ 3) Ut~t~ 2/+3 ,,,~,:~-- ~- ,-,q,lT (1+1)(2/--1) -
+~/3(3p3+l-51(l+ 3)P~+l)e'('°'+'~'÷~) 2 2l+3 x [((/+ 1)(/+ 3))-½uI++~t~ - ( l ( l + 2)) -~r v,+~,~], l+~
21+3 I(21-1)
t-~] Uz~-t~
7
259
POLARIZATION FROM SCATTERING POLARIZED SPIN-~"
1 ~p~+lei(~,+o~+2) F/21+3\~ t+½
/2/+3\~
l+~
b• = - ~ ~ (3P 2 - l(1 + 1)P°)e 2i~'E(/(2/+ 3))- ~ ulI,~ - ((l +
1)(2/-
i
q
1))- ~ UI;,~]
3 ½ -~ t+~ -~ t+~ + ( p 2 +(l + 1)(1 + 2)P°)e '(°n+"n +2) ( 2 / ~ +3) [ ( / + 2) Ut+2,14z--(1q-1) Ut+2xtck], i
b2 = 2k ~ (P2I + l(l + 1)Pt°+ 1)d'°"[(l(21 + 3))- ~ UI~-,~- ((l + 1)(2/- 1)) -~ uli,~] 1 (PLI - 3 ( / + 1)(I+2)P ° , ) e '(~n+.... ~(2/+3) -~
× [(t+2)-~wl++~,~-(l+l~
~,+
(12)
~,~j.~
The associated Legendre polynomials P~ appearing in eq. (12) are related to the usual spherical harmonics by the equation, _ r n ~ x~ Y:"(0, qS) = (_)~(,.+Iml) V2 l + 1 ( / - [ m l ) ! ] pl~l(cos 0)e,~,.
L4£
(13)
(l+lml)!A
Eqs. (I1) and (12) are consistent with the similar equations of Csonka et al. given on p. 1331 of ref. 9).
3. Polarization observables 3.1. DIFFERENTIAL CROSS SECTION Before treating the case of primary interest in which the incident spin -1 beam is polarized, we briefly consider the case of an unpolarized beam. The incident (in the c.m. system) spin-1 beam is in this paper taken to be unpolarized. The density matrix P~'nfor the case of unpolarized incident spin -1 and spin-1 beams is equal to one-sixth times the six dimensional identity matrix. The final density matrix p~' resulting from the scattering of these initially unpolarized beams is thus equal to one-sixth of the product M M t. The unpolarized differential cross section QU is then 1 equal to ~Tr M M t. By using eq. (9) and the trace properties of the spin operators (appendix 2) we can express this cross section in terms of the M-matrix coefficients,
Q" = [AIe +[cIa +$([DI2 +[BIE +IEI2 +[F[e)+~(IGI2 +[HI2 +IIIE +IjI2 ) + ~([K]2 + i l i a ) _ 2 Re (GI* + H J*).
(14)
The spin-½ polarization P" produced by scattering initially unpolarized beams is, from eq. (1), Qupu = ~ Tr (MMta), (15)
260
R.G.
SEYLER
which yields Q"P" = ~2Re[AC*+Z3BD * + 2v G H • + ~2I J • - ~ 1G J * - ~-IH*].
(16)
We now give attention to the case in which the incident spin-½ beam has arbitrary polarization pi.. In this case, the initial density matrix has the form Pin = 1( 1 + p i , . a).
(17)
The final density matrix is then given by pf = ~ ( M M t + P in. MtrM*),
(18)
and the differential cross section Q a = Q u + p i , . ~ Tr (MaM*).
(19)
Although is it not obvious, the traces in eqs. (19)and (15) are equal. This equality is dependent on the fact that the M-matrix is time reversal invariant, as has been shown by Wolfenstein and Ashkin 5). Thus one can use eq. (15) to re-express eq. (19) in the familiar form, a = a"(1 +p~n. pu). (20) 3.2. POLARIZATION OF THE SCATTERED SPIN-½ PARTICLE The spin-½ polarization P produced by scattering a spin-½ beam of polarization p~n is, by use of the eqs. (18) and (1), seen to be QP = ~ Tr ( M M t a ) + p i . 1 Tr (MtrMta),
(21)
which with the aid of eq. (15) may be rewritten as Q p = Q u ( p . + ~ . pin),
(22)
where following Oehme 5), we define the dyadic ~, ~jk = f " ~ " ~ = Tr (trjMtrkM*)/Yr ( M M t ) .
(23)
An alternative form of eq. (22) is given by Wolfenstein (1954) 5) as Q p = Q ~ ( t ~ ( p n + ~ p i n ) + ? ~ ( d p i n + . ~ p i n ) + k~,L ( d ,e .i. + ~ ,P~i. ),
(24)
where the unit vectors are defined by eq. (2). Upon comparing eqs. (22) and (24), the following identifications may be made: 2.. = ~,
Z.z = 0,
~-~k'Ln = O,
Zk,Lz = , ~ ' ,
Zs. = O,
%~. = d ,
~k'Lx =- ~ " Zsx = ~ .
(25)
When expressed in terms of the M-matrix coefficients the five Wolfenstein parameters
POLARIZATION FROMSCATTERING POLARIZEDSPIN'~-
261
become the following: = 1-271/Q" , QU~
= 74 cos 0 - 7 3 sin 0 + 7 2 cos ( 0 - 0 ) ,
Q u d ' = 74 cos 0 - y 3 sin 0 - 7 2 c o s ( O - O ) , QUa, = ?4sin 0 + ] 2 3 COS 0 - - 7 2 s i n ( 0 - O ) , Qud
= - 7 ~ sin O - y 3 cos 0 - 7 2 sin ( 0 - 0),
(26)
where in order to simplify writing the equations we have defined 71 =
~[FI2 + 2IEI2 +~[LI2 +~IKI 2,
72 = ~IFI2-21EI2 +~ILI2-1]K[ 2, 73 =
2 I r a ( C A * + ~2B D * +vHG 2 * + v2J l * - F1J G * - F1H I * ),
7, =
[A[ 2 - [ C ]
2 +~IDI 2 - ~ I e l
2 +2[[G[ 2 -Int 2 + [II z - I d l z - Re ( G * I -
H*J)]. (27)
F r o m eq. (26) it follows immediately that d + :~' = ( d ' -
~ ) tan ( 0 - O),
(28)
which demonstrates the lack of independence of these four quantities. 3.3. POLARIZATION OF THE SCATTERED SPIN-1 PARTICLE
In general, eight quantities are required to completely specify the polarization properties of a spin-1 beam. Following Lakin 10) these quantities may be taken to be the expectation values of the following set of irreducible tensor operators: 1 / 6Sz, T~ 0 = zV
T1 _+1 = ~ ½ 4 3 ( S x + i S-r ) ,
7"20 = 42(~S~-~),
T2+_~ = ½,/3(S~+~S~7,
T2--1 =
'~ 1 4 3 [ ( S x
-{- i S y ) S z + S z ( S x -{- i S y ) ] .
(29)
It is seen that TK-Q = (--)QT~Q. These operators are therefore equivalent to a set of eight hermitian operators and thus their expectation values are expressible in terms of eight real quantities. Since it is convenient to use the spin operator components of eq. (9) we rewrite the operators of eq. (29) in the KnP system of coordinates.
Tlo = ½x/6(Se cos ½ 0 - SK sin ½0), 7"1±j = -T-½x/3(Se sin ½0 + SK COS ½0 + iS,), 72o = 3~/2[(See + SKK) + (See-- SKr) cos 0-- 2Set sin 0], T2+ 1 =
+½43[(Spp--SKK) sin O+2Ser cos O+2i(Se. cos ½ 0 - S t . sin ½0)],
T2 +2 = ¼x/3[3(Sep + SKK)- ( S e e - Srr) cos 0 + 2Set sin 0 +4i(Se. sin ½0+St. cos ½0)].
(30)
262
R.G. $EYLER
Thus, from eq. (30) it is seen that the expectation value (TKe) is simply related to the expectation values of the various Sj and Ski operators. For these latter expectation values we find, 3 Q ( S . ) = Re [4(AD* + CB*) + (EK* + F L * ) - ](DI* +DG* + BH* + B J*)] + p i . Re [4(AB* + CD*) + 2EF* + {KL* - ~(DH* + D J* + BG* + BI*)]~ 3Q(SeK) = Im [DI* + GD* + HB* + BJ* + ½EK* + ½LF*]
+ pin Im
[HD*+D J* + GB* + BI* + EF* + 1LK*],
9Q(Spe + sK~) = lEE* + FF* - ~(GG* + HH* + II* + J J*) + ¼(KK* + L L * ) - 2(BB* + DD*)] + 2 Re [CH* + C J* + AG* + AI* + ~(HJ* + GI*)] + Pi."2 Re [AH* + GC* + A J* + IC* - 2DB* - ~(IJ* + GH* + 2G J* + 2IH*)], 3 Q ( S p p - SrK) = lEE* - F F * - ½(GG* + HH* - I I * - J J * ) - ¼ ( K K * - LL*)] + 2 Re [ C H * - C J * + A G * - A I * ] + Pi."2 Re [AH* + GC* - A J* - IC* + ½(lJ* - GH*)], 3 Q ( S e ) = f~e"Re [4AE* +DK* + 4GE* - 2IE* + 2BF* -- JL*] + P~" Im [4CE* + BK* + ~-ne* - ]JE* + 2DF* - IL*], 3Q(SK) = n~" Im [4FC* + LB* + {FJ* - }FH* + 2ED* -- KG*] +P~"Re [4FA* + LD* + ~FI * * - ]FG* + 2EB* - KH*], 3 Q ( S p . ) = P I e " R e [ A K * + D E * - H F * + { B L * + i l K ' * -~GKI l
*
1
+P~" Im [CK* + B E * - G F * +~DE + i l K
*
*]
-~HK*],
3Q(SK,,) = P~,"Im[FB*+LC* - E I * + 2~K D * + , Lx H * - ~ L~ J *] + pin Re [FD* + L A * - E J* + ½KB* + X3LG*- 1LI*].
(31)
It is seen that the last four of the expectation values in eq. (31) vanish for an unpolarized incident spin-½ beam. We have described the spin-1 polarization by the tensors ( T r q ) in the xyz coordinate system. This same polarization would be described in a different coordinate ' \ where as in eq. (29) T(o = \ 2"~S: , etc. These two system x'y'z' by tensors (T;~Q,/ sets of tensors are related by the equation ( T/~e') = E ( TrQ)D~Q'(c~, fl, '/), e where (c~, fl, y) are the Euler angles of the rotation which takes the unprimed into the K primed coordinate system, and the quantities Doe, are elements of the rotation matrix [ref. ,2)].
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
263
3.4. SPIN CORRELATION Additional information concerning the M-matrix may be obtained by measuring in coincidence certain components of the polarization of the scattered spin-1 and spin-½ beams• The expectation value of the product S • l a . "~ m is " referred to as the spin correlation function C(lm), C(lm) = (S, rr,.). (32) By scattering a beam and measuring its asymmetry one can determine only those components of the (vector) polarization which are perpendicular to the propagation direction of the beam in the laboratory *. Thus for the spin-½ particle only the polarization components along the directions fl and g (both perpendicular to k~) can be conveniently measured. And for the spin-1 particle, since it emerges 0-n the lab. system) along the - K direction, only the polarization components along the directions A and P need be considered• On the basis of these facts we may limit our consideration to the four spin correlation functions, C(nn), C(Ps), C(Pn) and C(ns), that is, we write ( S a ) = C(nn)fl~ + C(Ps)~'g + C(Pn)l'fl + C(ns)~L (33) The parity requirement that the first two correlation functions be scalars and the last two pseudoscalars may be met by noting that p i . . / t ( = p~n) is a scalar whereas pin. i~(= pin) and p i . . ~ x / ~ ( = p i . ) are pseudoscalars. These facts justify our writing
QC(nn) = Q u(C.~+C..Py), y in
QC(Pn) = ,q~EC~ ~ t P, P~
in
~ y m i n x), QC(Ps) = Q u ( C v,+cp~ry QC(ns) = O ~(C.sP~ z in x in +C.sP~),
x i n ), + Cv.P~
(34)
where the eight correlation coefficients C.., etc. are functions of the scattering angle and energy. I f the incident spin -1- beam is unpolarized the only measurable coefficients are C.. and Ces. In order to obtain the M-matrix dependence of the last six coefficients of eq. (34), it is convenient to express them in terms of some other coefficients, whose M-matrix dependence is more easily given, as ( O - ½0)Cee,
C p s = COS ( O - - ½ 0 ) C p K - - s i n 1
1 n CeY~ = cos (O - ~O)Cer- sin (O - ._O)Cpp, z
Cpn =
n
1 P " COS ~ O f p n - - s m
1
K
~OCpn ,
x 1 P 1 K Cp, -- sin ~OCp. + cos ~OCp., P " zOC.K)~ ~ " ( 0 - ½0) ¢OC.Ksm sin
C~=cos(O-k0)(cos'
P • ~OC.p), l K ~OC.psm
X (COS 1
C.5 = cos (O-½0)(sin 7OC,, ~ P K + COS 7OC.K) 1 K • -- sln ( 0 -- ½0) •
1
P
X ( s i n ~OCnp"}- c o s
1 K ~OC.p).
(35)
• The polarization component along the laboratory propagation direction can be measured, if on first rotates (with a magnetic field, for example) the spin vector.
264
R . G . SEYLER
The M-matrix coefficient dependence of the various spin correlation coefficients is obtained by evaluating the expectation value in eq. (32) and using the equation
QC(lm) = Qu[Ctm+ Z f"'lm't j Din'] j A' J
(36)
with the appropriate choices for the variables l, rn a n d j . Carrying out this procedure one finds
3Q"C.. = Re ( c q - ~ 2 ) ,
3QUCpp
P 3Q u C.K = im(cq-c~z),
3 Q "C"pK=Im(o~6--0:5),
u
K
--=
Re ( ~ 5 - % ) ,
3Q C.p = I m ( ~ l + e 2 ) ,
3flucK~Pn = I m (0~5 ~- ~6),
3Q"CpK = I m (~4--%),
3Q"C~,, =
3 Q "C"pp = R e (~3--0~4),
3 Q uc",,p = Re (~7-c~8+~9-cq0),
u
P
31qucK ,,K
3Q Cp,, = Re (c~3 + e4),
=
R e (g7--l--o~8--l--~9-l-~xlO),
Re (~7-0~8-(x9--1-0~10),
(37)
where the c~i are abbreviations of the following expressions:
cq = - 2 E F * - ½KL*, ~2 =
- - 4 ( A B * +DC*)+ ~a(GB* + IB* +DH* +D J*),
ct3 = 4CE* + - } H E * - ~ J E * + BK*,
c~, = 2 D F * - I L * ,
ct5 = 4AE* + { G E * - ~ I E * + D K * ,
c~6 = 2BF*--JL*,
~7 = 4 D A * - ~ ( D G * +DI*), ~9 = E K * ,
~8 = 4 B C * - ~ ( B H * + BJ*), (38)
~1o = F L * .
4. P h a s e
shifts
A necessary prerequisite to a phase-shift analysis of scattering data is, of course, a set of equations expressing the observables in terms of the phase shifts. In the previous section the observables are expressed in terms of the M-matrix ceeffcients. And in eqs. (11) and (12) the M-matrix coefficients are expressed in terms of the collision matrix elements. In the section we complete the circuit by expressing the collision matrix elements in terms of phase shifts. We achieve this goal by a simple generalizatiov of the Blatt and Biedenharn 11) method of parameterizing the two-nucleon collision matrix. The two-nucleon collision matrix U in addition to being symmetric and unitary is diagonal in j, s and parity and thus its only off-diagonal elements are of the form U/sl s where l' -- l_+ 2. Therefore, the largest submatrix of U corresponding to particular values of j, s and parity is 2 x 2. Blatt and Biedenharn parameterize this sub-matrix
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
U j by writing
U j = (u J) ~ exp
(2it~J)(uJ),
265
(39)
which is clearly unitary and symmetric for u j real and orthogonal and 6J real and diagonal. They 1~) introduce the real angle J and write u j in the obviously orthogonal form U j = V COS 8 j " sin eJ 1 I _ - s i n e J cose j " (40) The angle eJ is called the mixing parameter (it is a measure of the mixing or coupling between partial waves of different orbital angular momentum), and the two elements (angles) of the phase shift matrix 6 j are called the eigenphase shifts (rather than phase shifts of particular partial waves, since they have the latter interpretation only in the absence of mixing, i.e. when ej = 0). For spin-½-spin-I scattering, the matrix U need not be diagonal in channel spin. Thus for a particular value o f j it is necessary to consider two 3 x 3 matrices, • U J~
=
J / Us:_+½.~jq=kk Uj+_~j+_½~
(41)
U jj+½~72j+_½4 J ~ Uj+_½~2j+_4x{..]
~ L U j+_½~j.T-~2
which have parity ~z = (-)J-+*. Extending eq. (39) to 3 × 3 matrices we write U j~ = (uJ~) ~ exp (2it~J~)(uJ~),
(42)
where the phase shift matrix is now given by ~J~ =
~
o Ool
~ij ± ~ 0
•
(43)
6J±~
In eq. (43) the eigenphase shifts are labeled with the l and s subscripts of the partial wave whose phase shift they would be in the limit of no partial wave mixing. One obvious method of parameterizing the real orthogonal matrix u j" is to apply three successive Blatt and Biedenharn "rotations", (44)
Ujn = I~Jr~wJnxJn,
vJ'~ =
L 0 0
cos eJ~ -sinJ~
• i cos
01 sin0 ' l ,
wj~ =
L-sin
o1
sin eJ~ , coseJ'~J
~J~
0
(45)
(46)
cos ~J~l
• V c o s q ~. sin~/~. 07 xJ~ = [ - s i n r f l ~ cosrfl ~ ~ J . 0 L o
(47)
266
R. G. SEYLER
A c o m p a r i s o n of eqs. (45), (46) and (47) with eq. (41) reveals that for given values of i and re, e j~ is a measure of spin mixing without orbital (angular m o m e n t u m ) mixing between the partial waves, whereas (J~ is a measure of orbital (without spin) mixing and ,7j~ is a measure of the mixing (or coupling) between partial waves which differ in both their spin and orbital q u a n t u m numbers. We label f r o m 1 to 3 the rows and columns of the matrix u J~ and use eqs. (44)-(47) to obtain u~'] = cos ~J~ cos ~J", J= = /'/12
J= U13
s i n t/~n c o s (J~,
= sin ( ~ ,
j/t u21
- c o s ej~ sin r f ~ - s i n eJ~ sin (J= cos ~f=,
j1t Uz2 = cos ej~ cos q J ~ - s i n aJ~ sin (J~ sin q~=, u 23n 3 =
J~ /'/31 j~x
tl32
s i n 8 jn c o s ( j n ,
= sin ej" sin ~ / ~ - c o r e j" sin ~J" cos rff~, - s i n ej~ cos q ' ~ - cos e J= sin ( ~ sin ql~,
u~3 = cos ej~ cos r/j~.
(48)
U p o n substituting eqs. (43) and (48) into eq. (42) one obtains the desired result, namely the phase p a r a m e t e r dependence of the collision matrix elements of eq. (41), U~-v~j:~
H J~ l l U J~ lI
• i J~ J~ exp(2ifiJ_+~)+,,J~,,J~ t431 " 3 1 exp (2i6~_+÷~), exp (216j_z_k~)q-u2t/'/21
U Jj ~ k k j + _ ~ ½ = UI1HI2 J~ J~ exp (2tfj~ff~) " J J~ u22 J~ exp (2i3~± ~)+/,/31 J~ uJ3~2 exp (2i6~+~), + Uzt VJ~j+_~
~ exp ( 2 t"f j j~ _ ~ ) + u 2 , J~ u zj=3 exp ( 2 i ~ + ~ ) + u 3 ~J~ uJ3~3 exp (2i6J+~_), = u lJ~ l u~3
UJ+~&i+~ = u ~ u~2 ~ exp (2t6j-v_~k) "j i~ u22 j~ exp (2ifij +-I-~)''~--32-3z "''j~ exp (2i6~i±~), + u22 U ~j+_~j++.~:g~ = ,,J~ ~ 1 2 ~"1~3 exp (2i6~ ~ ) Uj4-½~jq-~ 2 = H13U13
jn
j~r
+ U2z u23 exp
. -j jn jn (210j+_½½) "}- U32 U33
exp (2i~+_~_),
(21(~jgk~k)'-~-U23 U23 e x p (210j+~½)'q-U33U33 exp
(49) The other three elements of the matrix U ~= need not be written since U j= is symmetric. Although for j = ½ the collision submatrices are only 2 × 2, these matrices m a y also be obtained f r o m the above equations provided one adopts the natural procedure of setting equal to zero any collision matrix element U~,~ for which (i) the quantities l s a n d j (or l ' s ' and j ) do not f o r m a triangle or (ii) the orbital q u a n t u m n u m b e r 1 or (l'), as c o m p u t e d f r o m the j - d e p e n d e n t equations, is negative. It is interesting to ask how m a n y phase p a r a m e t e r s enter if for the scattering there is a m a x i m u m eifective value of orbital angular m o m e n t u m l ..... . F o r I.... = 0 only
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
267
two phase shifts enter; for lmx = 1, nine phase parameters are present; for lm,~ = 2, twenty phase parameters are involved; and each additional increase (by one) of/max introduces twelve additional parameters. One can confidently predict that the various polarization experiments will play an important role in the determination of the parameters. For the analysis of elastic scattering at energies where other reactions are possible the above phase parameterization must be amended to permit the loss of flux to these other reaction channels. Thus at these energies the elastic scattering submatrices U j= [eq. (41)] although still symmetric are no longer unitary. The above equations may still be used provided one permits the various phase parameters to become complex subject to the constraints that the eigenvalues of the (hermitian) matrix UJ=(UJ=) * do not exceed unity. These eigenvalues may be obtained by diagonalizing this matrix by a unitary transformation. This step is non-trivial for complex mixing parameters since the orthogonal matrix u of eq. (41) is then not unitary. The above constraints lead to the requirement that the imaginary parts of the phase shifts be non-negative and that a single inequality involving phase shifts as well as mixing parameters be satisfied. Specifically, for a 3 x 3 matrix U, such as eq. (41), where for brevity we designate the diagonal elements of UU* as a, b and c and the non-diagonal elements above the principal diagonal by d, e and f (the other three elements follow from hermiticity) the inequality takes the form
( 3 x - 1 ) ( X + l ) ___
(50)
where X = ~1 Tr UU* = l ( a + b + c ) and Y = } ( a b + b c + c a - d d * - e e * - f f * ) . The condition that X < I follows and therefore need not be separately imposed. For the case of a 2 x 2 matrix such as eq. (39) one requires that ~l2 = e x p ( - 4 Im 6_+) < 1 and that lexp ( 2 i 6 + ) - e x p (2i6_)12[sinh (Ira 2e)] 2 < ( i - r / 2 ) ( 1 _ q2_). (51) This equation has been given by Arvieux ~3) although a factor ½ is missing from the right hand side of his eq. (14). The additional inequalities given in his eqs. (12) and (13) are unnecessary, since they are automatically satisfied when eq. (51) is satisfied. As a final point, we remind the reader that the nucleon-deuteron scattering phase shift analyses published to date ~3) have been based on the assumption that channel spin is conserved. This assumption has not been made in the present work. If this assumption is invoked the number of independent M-matrix coefficients is reduced from twelve to eight; in particular, one can show that channel spin conservation implies that E-G 2J-4H+3K
= F-I
= B,
= 2H-4J+3L
= 12C-6D.
(52)
The channel-spin conservation hypothesis leads (by use of the above equations, for example) to relations between certair~ polarization and spin correlation quantities.
268
R.G.
SEYLER
These relations are the subject of a very recent publication by Arvieux and Raynal who find that the relations take on a particularly simple form when the helicity formalism is employed. The interested reader is referred to their article 14) for the details. The relation between the M-matrix in the helicity representation M ' ( O ) of ref. [(14)] and our M ( O ) [eq. (7)] is easily given, Ms,v,sv = Z dcm, '' 6,mM~,m,sm, '
(53)
m'm
where d(0) is the reduced rotation matrix 12) of order s'. The collision matrix of ref. 1 4 ) differs from our eq. (49) by the phase factor (i)2s,-2s+v-t. The author takes this opportunity to express his gratitude to Dr. Louis Brown for his discussion and careful reading of the manuscript. He is also grateful to Drs. Walter Trfichslin and Tom CIegg for their work in the checking of equations.
Appendix I EXPLICIT MATRIX FORMS OF THE SPIN OPERATORS To find the explicit form of the spin operators in the channel spin representation one expresses the channel spin wave functions in terms of spin-½ and spin-1 wave functions according to the familiar equation 1 m v-m Zsv = Z (21mv-mlsv)z÷Z 1 .
(AI.1)
m
One then computes the matrix elements of the spin-1 operators and of (twice) the spin-½ operators between the various channel spin states [eq. (AI.1)]. One finds, in the notation of eq. (7)
io ooo Sz =
i O"z
---~
~
0
0
-~,/2
0 0
-½ 0
0 0
-~,/2 o
o -~,/2
0 -1 o
o
o
0 o 0 ~,/2 o
o
0 - - ,3 0 o ~/2
o
0 o -1 o o
0 o 0 -~ o
2 ,
-~ J
°1
~,/2 0| o
J
POLARIZATIONFROMSCATTERINGPOLARIZEDSPIN-½
Sx =
I!
Ix/+
[o
~x
Sy
=i
~,,=i
0 32 0
0 I,,/'2
a 0
0 x/'3
0 -Ix/2
%/3
0
0
°
- 61x/2 0 0 - ,~/{
0
-7'3
0
0
43
o
0
2
2 0
0 ,/3
o -,/~ o
o o -~/2
,/~ ~/2 o
o o #~
269
162 --41 / '
0
0 ] -,/'~ 0 ] 0 -3x/2 / 3x/2
0
]
o o -3
,/~ ] ' o
o
-,~/3
o
o
-,/1
o ]
~/3 0
0 32-
2 0
0 -~/3
0 -ix/2
-+x/Z] 0 ,
o
o
,/3
o
o
-,/t
,,/1 0
0 Ig2
+,/2 0
0 x/1
0 ~
-~ 0
o
-,/3
o
o
~/3 0
0 ~
2 0
0 -~/½
o
o
,/3
o
o
o
o
- x / az
-3
o
0
-3x/2
0
,/~
o ]
0 3x/2] 3,/2 0
,/~
"
oJ
The components of the spin operator S (or a) along the K, • and 1' axes are found by using the vector rotation equation, S, S,,
=
rosoOO SoOl[] Lsin½0
1 0
Sy . cos½0 .] S=
The matrix representations of the operators S u are obtained by performing the matrix multiplication indicated in eq. (8). Appendix 2 TRACES OF TIIE SPIN OPERATORS
An important feature of the trace properties of operators is that they are independent of the representation chosen to express the operators. It is well known that the trace of any component of a spin operator Si (or al) is zero. Using this fact and the
270
R.G. SEYLER
familiar equation
(A2.1)
(a" d,)(~, b) = ~" [~+ ia" ~ix b,
where ~i and ~ are arbitrary unit vectors, it is easy to see that Tr aia j = G for i # j and that Tr a~2 = Tr 1( 6 ) = 6. F o r the spin-1 operators, the similar result that Tr SISj = 0 for i ¢ j applies, as m a y be easily verified using the explicit representation of appendix 1. F o r i = j we use the fact that Tr S 2 = ~ T r S • S = 2 T r 1(6~ = 4. Just as eq. (A2.1) m a y be used to limit the a dependence of an expression to terms linear in a so m a y the equation, ( S . a ) ( S . b)(S" ~) --- ¼i[(S. ~ x ~ ) ( S . ~ ) + ( S . +(s.
a)(s./;×
~)-(s.
~× a ) ( s . b ) - ( s ,
~)(S. ~× ~ ) + ( S .
~x~)(S.
a)
t;)(s. ~x a)]
+½(tl. l;)(s. ~ ) + ½ ( b . ~)(S. t/)+½i(tl x/~. ~),
(A2.2)
be used to limit the S dependence to terms second order in S. F r o m eq. (A2.2) it follows that Tr SjSkS~ = }i() x/¢- i ) T r 1~6~ = 2iejk~. The repeated application of eq. (A2.2) permits one to evaluate the trace of expressions containing four or m o r e components of S. We summarize the trace properties of the six dimensional spin operators: Tr a; = 0, Tr tTja k = 6(~jk , Tr aJ(S)
= 0,
Tr S~ = 0, Tr S i S k = 46jk, Tr S;k = O, Tr S~SkS t = 2iejkZ, T r SjSkl = O, 2 Tr S jk S ~ = (~jl (~km"~-t~jm 6kl -- ~(~ jk t~Im
= Tr S~kS, Sm, Tr Sj Skl Stun ~-- ½i[Sjk m 6in d- e jR n (~lm -~ ~jlm (~kn "q- 8jln (~ltm~"
(A2.3)
Appendix 3 M-MATRIX ELEMENTS IN TERMS OF M-MATRIX COEFFICIENTS Substituting the M - m a t r i x eq. (7) and the matrix form of the spin operators (appendix 1) into eq. (9) and c o m p a r i n g left and right sides element by element we obtain
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
271
the following eighteen equations: a+h = 2A+~(B+E+F), amh
=
- ] B + ½ ( E + F)+~(G + I)+½ cos O E 2 ( E - F ) + ( G - I ) ] ,
b + g = 3 -} sin 0 [ 2 ( E - F ) + G - I ] , b-g
= - i 3 - ~ [ 2 ( C + D ) + ½ ( K + L ) + i 6 ( H + J ) - ½ cos O ( L - K + J - H ) ] ,
c+j
= 3- ~ [ - 2B + E + F +½(G + I ) + ½ cos O(2(F- E)+ I - G ) ] , i3 -½ sin O [ J - H + L - K ] ,
d+i
= - i [ +~(C +D)+~;(K + L)+ a-~(H + J)+ ½ cos O ( J - H + L - K ) ] ,
d-i
= ½i[2(C + D) -
(K + L) -
~-(H + J)],
e+m
6 -½ sin O [ E - F + I - G ] ,
e--m=
i6- ~[2D - 4C + ½(K + L) - ½(H + d) + ½ cos O(K - L + 2(J - H))],
f+p
= ½i6- + sin 0 E 2 ( J - H) + (K - L)],
f--p
= 6-~[- - 2 B + E + F -
G - I - cos O ( E - F + I -
G)],
k + n = ~ , j 2 [ 2 B - E - F +G + I - 3 cos O(E--F + I--G)], k-n=
¼ix/2 sin 0 1 2 ( J - H) + K - L], ½v/2 sin O [ E - F + I - G ] ,
l+o
=
I-o
-- aix/2 [ D - 2C + ¼(K + L ) - ~(H + J ) - ¼ cos O ( K - L + 2 ( J - H))], q= A-~(B+E+F), ½i[C- 2D + K + L - ~(U + J)].
Appendix 4 M-MATRIX COEFFICIENTS IN TERMS OF M-MATRIX ELEMENTS Using eq. (9) and the trace properties of the spin operators (appendix 2) one sees that Tr M = 6A, Tr MSep = ~-G-~I, Tr M % S . = 4B,
Tr MSt
Tr M % = 6C,
Tr M a n See = - } H - z3J,
Tr M S . = 4D,
Tr Man SK~
Tr Map Sp = 4E,
Tr M a e S e . = K,
Tr M a r S r = 4F,
Tr MaKSKn = L.
=
-- ~H +
4,i,
272
R . G . SEYLER
U p o n s u b s t i t u t i n g i n t o the a b o v e e q u a t i o n s the M - m a t r i x e l e m e n t s eq. (7) a n d the m a t r i x f o r m s o f the spin o p e r a t o r s ( a p p e n d i x I) the f o l l o w i n g e q u a t i o n s result:
A = ½[a+h+q], B = - ( 7 2 ) - ~[ - (k + n) + 2~/2(q - h) - x/3(p - f ) + x/6(c + j ) ] ,
C = ~i[2i - r - ~ j 2 ( o - l) + x/3(b - g) + ~ / 6 ( e - m)], D = ( 7 2 ) - ½ i [ - ( l - o) + 2~/'2(i + r ) - \ / 3 ( e - m) + ~/6(b - O)], E + F = ( 2 4 ) - * [ - (p - f ) + x/~2(c +j) - ½\/3(k + n) + x/6(a + ½h - 4 q ) ] , E - F = ( 2 4 ) - + [ s i n 0{(e + m) + 2~/2(b + 9) - x/3(l + o)} + cos O{(p--f) - x/~2(c + j ) - - x/3(k + n) + \/6(a - h)}], G + I = ½[a - h + x/2(k + n) + x/3(c + j ) + x / 6 ( p - f ) ] ,
G-I
= 6 - ~ sin O [ - ( e + m ) + x / 2 ( b + g ) + x / 3 ( l + o ) ] + ½ cos O[a - h + x/2(k + n) - ½x/3(c +j) - ½~/6(p - f ) ] ,
H + J = ( 1 2 ) - ~i[b - 9 + ~//2(e - m) + ½x/3(8r - 7i + 9d) + ½x/6(l - o)], H - J = 3 - ~i[sin O { ( c - j ) + l x / 2 ( f + p) + ½ x / 6 ( k - n)} +
cos
O{b - g +
- m) -
+ d) - , / g ( l
- o)}],
K + L = 6 - ½i[-- ( e - m) + x/2(b - 9 ) - ½x/3( / - o) + ½ x / 6 ( 3 d - i - 4r)], K - L = 6 - ~ i [ s i n 0{ - ( f + P) + 2 x / 2 ( c - J ) + c o s 0{ - ( e -
x/3( k - n)}
m)+x/2(b -9)+x/3(1-
o ) - x / 6 ( d + i)}].
References 1) G. M. Budianskii, ZhETF (USSR) 33 (1957) 889; JETP (Soy. Phys.) 6 (1958) 684 2) H. D. Shih, L. F. Lo, S. Shu and N. N. Haung, Acta Phys. Sinica 16 (1960) 324 3) H. S. Goldberg, Lawrence Radiation Laboratory Report UCRL-11526 (1964); thesis, University of California (1964) 4) J. yon Neumann, Mathematische Grundlagen der Quantenmechanik (Dover, New York, 1943) p. 174; H. Weyl, Theory of groups and quantum mechanics (Dover, New York, 1931) p. 78 5) L. Wolfenstein, Phys. Rev. 75 (1949) 1664; Phys. Rev. 76 (1949) 541; Phys. Rev. 96 (1954) 1654; Phys. Rev. 98 (1955) 766, 1870; Ann. Rev. Nucl. Sci. 6 (1956) 43; L. Wolfenstein and J. Ashkin, Phys. Rev. 85 (1952) 947; R. H. Dalitz, Proc. Phys. Soc. A65 (1952) 175; D. R. Swanson, Phys. Rev. 89 (1953) 740; R. Oehme, Phys. Rev. 98 (1955) 147, 216 6) A. M. Lane and G. R. Thomas, Revs. Mod. Phys. 30 (1958) 257 7) E. U. Condon and G. H. Shortley, Theory of atomic spectra (University Press, Cambridge, 1935) 8) A. M. Lane and R. G. Thomas, ibid. p. 290 9) P. L. Csonka, M. J. Moravcsik and M. D. Scadron, Phys. Rev. 143 (1966) 1324 10) W. Lakin, Phys. Rev. 98 (1955) 139 11) J. M. Blatt and L. C. Biedenharn, Revs. Mod. Phys. 24 (1952) 258; Phys. Rev. 86 (1952) 399 12) M. E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957) p. 52 13) R. S. Christian and J. L. Gammel, Phys. Rev. 91 (1953) 100; W. T. H. van Oers and K. W. Brockmann, Nucl. Phys. A92 (1967) 561; J. Arvieux, Nucl. Phys. A102 (1967) 513 14) J. Arvieux and J. Raynal, Nucl. Phys. A100 (1967) 472
2.B
] I
Nuclear Physics A124 (1969) 253--272; ( ~ North-Holland Publishin9 Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
P O L A R I Z A T I O N FROM SCATTERING POLARIZED SPIN-½ ON U N P O L A R I Z E D SPIN-1 P A R T I C L E S R. G. SEYLER
The Ohio State University, Columbus, Ohio Received 17 September 1968 Abstract: The most general parity and time reversal invariant form for the spin@-spin-1 scattering amplitude matrix is obtained. The polarization observables (cross section, spin-½ polarization, spin-1 polarization, and spin correlation) which arise upon scattering polarized spin-½ by unpolarized spin-1 particles are expressed in terms of the coefficients of this general form. A phase shift parameterization of these coefficients is also given.
I. Introduction The purpose of this article is to study the structure of the spin-½-spin-1 scattering matrix and to relate to this matrix various polarization quantities which can be measured when a polarized incident spin-½ beam is used. These equations should provide a useful starting point for a complete phase-shift analysis of the nucleon-deuteron scattering problem. The treatment is non-relativistic throughout. The subject of this article has been previously treated, with conflicting results, by other authors: Budianskii 1), Shih et al. 2) and Goldberg 3). The present treatment is an effort to correct and to generalize the works of these authors (BSG). The generalization is the inclusion of the possibility of spin mixing, that is, transitions between doublet and quartet channel-spin states. Of the errors in the earlier works (BSG) those non-numerical in nature are pointed out later. We use, as did each of the authors (BSG), the conventional formalism of the yon Neumann 4) density matrix p. The convenience of this formalism for problems involving a statistical ensemble of different pure spin states is well known as a result of the work of several authors 5) on the very similar problem of nucleon-nucleon scattering. The final and initial density matrices are related by the transition matrix M according to p = MpinM*. Trace p is then the differential cross section (per unit solid angle) herein denoted by Q. The ensemble average ((9) of any operator C is given by
Q(C> = Tr (pC).
(1)
This equation is repeatedly used for the calculation of operator averages in sect. 3. 253
254
R . G . $EYLER
1.1. C O - O R D I N A T E
SYSTEMS
It is convenient to define the following ten unit vectors: /~, along the c.m. momentum of the incident nucleon, /~', along the c.m. momentum of the scattered nucleon, /~[, along the lab momentum of the scattered nucleon,
~'
= (~,'+~)/l~,'+~l,
k = (k'-k)/Ik'-kl, ^
= t~x.~'/l~x =
~:'1 =
/~,
~
A
PxK, =
d,
g = t i x ~ L,
:~ =
~x~,.
(2)
The vectors I£, A and l' are mutually perpendicular, as are the vectors g, 6 and/~[ and the vectors #, ~ and ~. The above definition of the vector K is that of Wolfenstein [ref. 5)] and is opposite to that of Budianskii ~), who chooses K in the direction of the momentum transfer to the deuteron in elastic scattering. We also define 0 = arcos (~./~'), O = arcos (~,'/~[),
(3)
the c.m. and lab. scattering angles, respectively.
2. Transition matrix 2.1. C H A N N E L
SPIN
REPRESENTATION
In the channel spin representation the "transition" or "scattering amplitude" matrix element Ms,v,sv signifies the amplitude (at infinity) of the outgoing spherical wave of channel spin s' with z-axis spin component v' resulting from a unit flux incident plane wave of channel spin s with z-component v. The connection between the transition matrix M and the collision or scattering matrix U is given by Lane and Thomas [ref. 6)] as
M~,v,s,(O) = rc~k-1[_ C(0)6~,6w, + i ~ (21+ 1)~(slvOljv)(s'l'v'v- v'ljv) ill'
x (exp {/(co, + oh,)})(6~,611,--
~vv-v'(, 0, 0)],
TrJ ,~,,'su.1"
(4)
where Ytm is the usual normalized spherical harmonic of Condon and Shortley v) and where the Coulomb scattering amplitude C(O) and the other familiar quantities k and 0h are defined as in ref. 6). The collision matrix U in eq. (4) is the "nuclear" part of the collision matrix and is related to the complete collision matrix U c by the matrix equation 8) U c = exp (i~o)U exp (i~o), where co is the diagonal matrix whose elements are the phase shifts ~o~.
(5)
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
255
Invoking parity conservation which restricts the sum l + l' to being even, it is easy to show that ( - , "V'-s+v'-v M ~._s,v,~.
M~,_v.~_ v -_
(6)
Thus parity conservation, i.e. eq. (6) reduces the number of independent (complex) elements of M from 36 to 18 and permits us to write the matrix M as 31 22
SV
3 2
1 2
~__~
11 22
½__
d j -g a
e k l -f
f l -k e
½
SrV I
2
2
2
2
a 9 j -d m
n
o
p
q
r
L 2
--p
o
--n
m
--r
q
&Z 22 I 2
b h -i c
c i h -b
(7)
It is curious that Budianskii 1), Shih et al. 2) and Goldberg all commit the error of treating c a n d j (in the notation of eq. (7)) as being identical. In the works of Shih et al. and Goldberg, the "equality" of c a n d j is "deduced" by using the alleged relation,
(slv-- l m l J M -
1) = (-)s+t-S(slvmlJM),
which is, however, not generally valid. It should also be pointed out that the authors (BSG) all omit (set equal to zero) the eight spin-mixing M-matrix elements e,f, k, l, m, n, o, a n d p ofeq. (7). The assumed time-reversal invariance of the interaction responsible for the scattering imposes additional restrictions on the elements of the M-matrix is as discussed in subsect. 2.2. 2.2. M O S T
GENERAL
FORM
OF THE
M-MATRIX
The most general form of the M-matrix is determined by requiring that it be invariant under space reflection and rotation and time reversal. The determination procedure is very similar to that used by Wolfenstein and Ashkin 5) in the case of two spin-½ particles. For the spin-½-spin-1 problem the 6 × 6 M-matrix may be expanded in terms of thirty-six linearly independent matrices, which operate in the combined spin space of the two particles. The coefficients of the spin matrices are arbitrary complex functions of the energy and scattering angle. The 36 independent matrices may be obtained by combining the four independent spin-½ matrices (three Pauli matrices and unit matrix) with each of the nine independent spin-1 matrices (five second rank, three first rank and one scalar). Let ~r be twice the spin-½ operator and S the spin-1 operator. We also define the traceless symmetric second rank spin-1 operator
s , = ½(s, sj + s j s , ) - ~ , j .
(8)
256
R.G. SEYLER
By combining 1, a K, a, and ae with each of 1, SK, S., Sp, SK., Set, S,e, SKK and See (note that S., is not independent of the later two operators), we obtain thirty-six independent operators. One observes that of the five operators a, S, K, ~ and t' only and P change sign under space reflection and only K does not change sign under time reversal. This observation may be used to show that of the 36 operators, six violate time reversal invariance, nine violate parity conservation and nine others violate both parity and time reversal invariance. Demanding that the coefficients of these 24 terms equal zero we are left with only twelve terms M = A + Ba. S. + Ca. + D S . + Ear Sv + FaK SK + GSpp + Ha. See -4-ISKK"4-da. Srr + Kai, S.p + La r S t . .
(9)
Budianskii 1) offers a similar twelve coefficient expression for the most general Mmatrix, but two of his coefficients are redundant. His coefficients F and G turned out to be equal and his coefficients H and K should have turned out to be equal. On the other hand, he omitted two coefficients by unjustifiably taking, in the notation of eq. (9),G = IandH= J. The 6 × 6 matrix forms of the operators in eq. (9) may be obtained in the same representations as eq. (7) by the procedure discussed in appendix 1. When these matrices and eq. (7) are substituted into eq. (9) one obtains eighteen equations which express th" M-matrix elements (a through r) in terms of the M-matrix coefficients (A through L). These eighteen equations are given in appendix 3. To express the M-matrix coefficients in terms of the M-matrix elements it is convenient to consider certain traces. For example, from eq. (9), Tr M = Tr A = 6A, where we use the fact that all of the operators (other than the identity operator) on the right side of eq. (9) have zero trace (the trace properties of the spin operators are reviewed in appendix 2). Similarly, we could write T r ( M a , Sn) = B Tr(a,S,a.S,) = 4B; to calculate G, H, I or J is only slightly more complicated, for example, Tr(MSpp) = G Tr S p+ITr(S KS e) = y4 G - ~ I2 . Goldberg and Shih et al. failed to take into account terms such as (in this exa'mple) Tr(SKIcSee). To solve for G, one computes Tr(MSrK) = ~4 I - y G 2 and solves the last two equations simultaneously with the result G = Tr(MSpp) + ½Tr(MSKK). Using trace equations such as these in connection with the matrix expressions for M, eq. (7), and for the spin operators (appendix 1), one may obtain the twelve equations expressing the M-matrix coefficients in terms of the M-matrix elements. These twelve equations are given in appendix 4. The above described procedure could also be employed to express the coefficients of any of the twenty-four operators we discarded in terms of the M-matrix elements. Doing so, we find that the coefficients of the parity non-conserving operators vanish identically. This is because eq. (7) already has parity conservation eq. (6) built into it. On the other hand, the coefficients of the six time-reversal non-invariant operators involve non-trivial equations, which when the coefficients are set equal to zero constitute the time-reversal invariance restrictions on the M-matrix elements. They may
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
257
be written as 0 = 3~(f+ p) + n -- k, 0 = 3~(e+m)+l+o, 0 = [3~(k+n)+f-p]
sin O + 4 ( e + m ) cos 0,
0 = [3~(o-l)+e-m]
sin 0 - 4 ( ] + p ) cos 0,
0 = [3~(h- a) + j + c] sin 0 + 2(g + b) cos 0, 0 = [ 3 ~ ( d + i ) + g - b ] sin O - 2 ( j - c )
cos 0.
(10)
These equations reduce the number of independent M-matrix elements (at any one energy and angle) to twelve in agreement with the number of M-matrix coefficients. Having just seen how the M-matrix coefficients are related to the M-matrix elements (see appendix 4 for the detailed equations), we can now use eq. (4) to obtain the collision matrix and angle dependence of the M-matrix coefficients of eq. (9). We find A
__1
C
--
E+F
2 ~do + ~q o-
B = ½ ( q o - - d o ) - ~ ( b1 2 + + 2q+),
C(O),
1 8 i~d~ +~b~ + ~5q l ,
= ~ ( q o - d o ) + ~ ( b + +2q~), +
+
G+I = -b2 +q2,
D
=
~ d 1 - ~ b2 i
+ ~ 5q l ,
E - F = ½b~ + q~, G-I
= - b ~ +q~,
H+J
= ~(dx + 2 b ~ - q ~ ) + - ? o q L
H-J
= - b 0z + ~1q 3--,
K+L
3 +, = - 2 ( d l + 2 b l - ql) + ~q3
K-L
= b°+q3,
(Ii)
where C(O) is the familiar Coulomb amplitude and where the d, q and b amplitudes (which involve doublet, quartet and both doublet and quartet states, res'pectively) are defined as follows: i do = - -2k - ~ Pt°ez~''[(l+ 1) U,~;~ +I+÷ lUl~t~-(21+ 1)], dl = ~1 qo =
~l
pl
~l e
21~t[-l-ll+½
Tfl--~-I
L""1½14~--"."1½1½1~
i
4k ~ p°e2~"[(l +2) U,÷,~+(/+'+~1)U,~I~+'+~I U l ~ + ( l -
1 ~p:e2UO,~-3(/+2) rrl+~± I - 3 Ut+~ - / + 4 U~l~ ql = 1 ~ / 1+1 '~l~l~:7- ~--- l~z~: 1+1
1 ) U I ~ - 2 ( 2 / + 1)], 3 ( l - 1 ) Ul~t~l 1 '
258
g.G.
SEYLER
i
4k ~ (3P~-l(l+ 1)P°)e~'°''
q~ =
[ /+2 rr,+, x (/+1)(2/+3) "'~'~
/+3 r,,+ ½ I-2 ,_, /-1 ,_,1 /(2/+3) "~'~ (/+1)(2/-1) U,~t~+ /(2/~1-~ U,~,~
+2x/J(Pe+(l+l)(l+2)P°)e'(°"+~'*~'
1 F / l + 3 \ ~ ,+~ 2/+3
Eta)
/' l '~rr,+~
i q2 = ~-~ ~ (P~+l+l(l+ l)P°,) e2'o'
I-2 t_~: l - 1 t-~] 1+2 rrl+e /+3 rrl+ ½ I"~l~l~ - "Jl~l~ Ut~t~ + l(21-1~) Ut~z~ (1+ 1)(2/- 1) l+1)(21+3) 1(2t+3) 1 2 + ~2 (P,+,-3(/+ 1)(/+ 2)P°+,)e '(~'+ .... ) - 2/+3 X
Ii
X
F(/+3~½rrt+, ± [ l '~*,r,+* ,,, i + 2 . ~ l } T
-
-
7
q~- = 1 1~ ~ (5Pz3- 3(1-1)(1 + 2)P~)e2i°'' 1 [/+~rr,+÷ 3rr,+~± 3(2/+3) (21+3) "l~l~-- -f '~t~t~ ~- (l+ 1)(2/-- 1)
21+3 t-~] l(21-1) Uliti
+ lO~/3(P3t+(l+ 2)(l+ 3)P~)ei(~'+o'÷~) l 2l+3
x [((/+ 1)(/+ 3)) -~ UI++~,~-(l(l + 2)) -~ vI~.~j, q;
X
=
-
_
12k ~ (Pt3+' +l(l-1)P]+l)e2"°'
_ _
3 E/@l rrt+ ~ 3 rrt+~± 3(2/+ 3) Ut~t~ 2/+3 ,,,~,:~-- ~- ,-,q,lT (1+1)(2/--1) -
+~/3(3p3+l-51(l+ 3)P~+l)e'('°'+'~'÷~) 2 2l+3 x [((/+ 1)(/+ 3))-½uI++~t~ - ( l ( l + 2)) -~r v,+~,~], l+~
21+3 I(21-1)
t-~] Uz~-t~
7
259
POLARIZATION FROM SCATTERING POLARIZED SPIN-~"
1 ~p~+lei(~,+o~+2) F/21+3\~ t+½
/2/+3\~
l+~
b• = - ~ ~ (3P 2 - l(1 + 1)P°)e 2i~'E(/(2/+ 3))- ~ ulI,~ - ((l +
1)(2/-
i
q
1))- ~ UI;,~]
3 ½ -~ t+~ -~ t+~ + ( p 2 +(l + 1)(1 + 2)P°)e '(°n+"n +2) ( 2 / ~ +3) [ ( / + 2) Ut+2,14z--(1q-1) Ut+2xtck], i
b2 = 2k ~ (P2I + l(l + 1)Pt°+ 1)d'°"[(l(21 + 3))- ~ UI~-,~- ((l + 1)(2/- 1)) -~ uli,~] 1 (PLI - 3 ( / + 1)(I+2)P ° , ) e '(~n+.... ~(2/+3) -~
× [(t+2)-~wl++~,~-(l+l~
~,+
(12)
~,~j.~
The associated Legendre polynomials P~ appearing in eq. (12) are related to the usual spherical harmonics by the equation, _ r n ~ x~ Y:"(0, qS) = (_)~(,.+Iml) V2 l + 1 ( / - [ m l ) ! ] pl~l(cos 0)e,~,.
L4£
(13)
(l+lml)!A
Eqs. (I1) and (12) are consistent with the similar equations of Csonka et al. given on p. 1331 of ref. 9).
3. Polarization observables 3.1. DIFFERENTIAL CROSS SECTION Before treating the case of primary interest in which the incident spin -1 beam is polarized, we briefly consider the case of an unpolarized beam. The incident (in the c.m. system) spin-1 beam is in this paper taken to be unpolarized. The density matrix P~'nfor the case of unpolarized incident spin -1 and spin-1 beams is equal to one-sixth times the six dimensional identity matrix. The final density matrix p~' resulting from the scattering of these initially unpolarized beams is thus equal to one-sixth of the product M M t. The unpolarized differential cross section QU is then 1 equal to ~Tr M M t. By using eq. (9) and the trace properties of the spin operators (appendix 2) we can express this cross section in terms of the M-matrix coefficients,
Q" = [AIe +[cIa +$([DI2 +[BIE +IEI2 +[F[e)+~(IGI2 +[HI2 +IIIE +IjI2 ) + ~([K]2 + i l i a ) _ 2 Re (GI* + H J*).
(14)
The spin-½ polarization P" produced by scattering initially unpolarized beams is, from eq. (1), Qupu = ~ Tr (MMta), (15)
260
R.G.
SEYLER
which yields Q"P" = ~2Re[AC*+Z3BD * + 2v G H • + ~2I J • - ~ 1G J * - ~-IH*].
(16)
We now give attention to the case in which the incident spin-½ beam has arbitrary polarization pi.. In this case, the initial density matrix has the form Pin = 1( 1 + p i , . a).
(17)
The final density matrix is then given by pf = ~ ( M M t + P in. MtrM*),
(18)
and the differential cross section Q a = Q u + p i , . ~ Tr (MaM*).
(19)
Although is it not obvious, the traces in eqs. (19)and (15) are equal. This equality is dependent on the fact that the M-matrix is time reversal invariant, as has been shown by Wolfenstein and Ashkin 5). Thus one can use eq. (15) to re-express eq. (19) in the familiar form, a = a"(1 +p~n. pu). (20) 3.2. POLARIZATION OF THE SCATTERED SPIN-½ PARTICLE The spin-½ polarization P produced by scattering a spin-½ beam of polarization p~n is, by use of the eqs. (18) and (1), seen to be QP = ~ Tr ( M M t a ) + p i . 1 Tr (MtrMta),
(21)
which with the aid of eq. (15) may be rewritten as Q p = Q u ( p . + ~ . pin),
(22)
where following Oehme 5), we define the dyadic ~, ~jk = f " ~ " ~ = Tr (trjMtrkM*)/Yr ( M M t ) .
(23)
An alternative form of eq. (22) is given by Wolfenstein (1954) 5) as Q p = Q ~ ( t ~ ( p n + ~ p i n ) + ? ~ ( d p i n + . ~ p i n ) + k~,L ( d ,e .i. + ~ ,P~i. ),
(24)
where the unit vectors are defined by eq. (2). Upon comparing eqs. (22) and (24), the following identifications may be made: 2.. = ~,
Z.z = 0,
~-~k'Ln = O,
Zk,Lz = , ~ ' ,
Zs. = O,
%~. = d ,
~k'Lx =- ~ " Zsx = ~ .
(25)
When expressed in terms of the M-matrix coefficients the five Wolfenstein parameters
POLARIZATION FROMSCATTERING POLARIZEDSPIN'~-
261
become the following: = 1-271/Q" , QU~
= 74 cos 0 - 7 3 sin 0 + 7 2 cos ( 0 - 0 ) ,
Q u d ' = 74 cos 0 - y 3 sin 0 - 7 2 c o s ( O - O ) , QUa, = ?4sin 0 + ] 2 3 COS 0 - - 7 2 s i n ( 0 - O ) , Qud
= - 7 ~ sin O - y 3 cos 0 - 7 2 sin ( 0 - 0),
(26)
where in order to simplify writing the equations we have defined 71 =
~[FI2 + 2IEI2 +~[LI2 +~IKI 2,
72 = ~IFI2-21EI2 +~ILI2-1]K[ 2, 73 =
2 I r a ( C A * + ~2B D * +vHG 2 * + v2J l * - F1J G * - F1H I * ),
7, =
[A[ 2 - [ C ]
2 +~IDI 2 - ~ I e l
2 +2[[G[ 2 -Int 2 + [II z - I d l z - Re ( G * I -
H*J)]. (27)
F r o m eq. (26) it follows immediately that d + :~' = ( d ' -
~ ) tan ( 0 - O),
(28)
which demonstrates the lack of independence of these four quantities. 3.3. POLARIZATION OF THE SCATTERED SPIN-1 PARTICLE
In general, eight quantities are required to completely specify the polarization properties of a spin-1 beam. Following Lakin 10) these quantities may be taken to be the expectation values of the following set of irreducible tensor operators: 1 / 6Sz, T~ 0 = zV
T1 _+1 = ~ ½ 4 3 ( S x + i S-r ) ,
7"20 = 42(~S~-~),
T2+_~ = ½,/3(S~+~S~7,
T2--1 =
'~ 1 4 3 [ ( S x
-{- i S y ) S z + S z ( S x -{- i S y ) ] .
(29)
It is seen that TK-Q = (--)QT~Q. These operators are therefore equivalent to a set of eight hermitian operators and thus their expectation values are expressible in terms of eight real quantities. Since it is convenient to use the spin operator components of eq. (9) we rewrite the operators of eq. (29) in the KnP system of coordinates.
Tlo = ½x/6(Se cos ½ 0 - SK sin ½0), 7"1±j = -T-½x/3(Se sin ½0 + SK COS ½0 + iS,), 72o = 3~/2[(See + SKK) + (See-- SKr) cos 0-- 2Set sin 0], T2+ 1 =
+½43[(Spp--SKK) sin O+2Ser cos O+2i(Se. cos ½ 0 - S t . sin ½0)],
T2 +2 = ¼x/3[3(Sep + SKK)- ( S e e - Srr) cos 0 + 2Set sin 0 +4i(Se. sin ½0+St. cos ½0)].
(30)
262
R.G. $EYLER
Thus, from eq. (30) it is seen that the expectation value (TKe) is simply related to the expectation values of the various Sj and Ski operators. For these latter expectation values we find, 3 Q ( S . ) = Re [4(AD* + CB*) + (EK* + F L * ) - ](DI* +DG* + BH* + B J*)] + p i . Re [4(AB* + CD*) + 2EF* + {KL* - ~(DH* + D J* + BG* + BI*)]~ 3Q(SeK) = Im [DI* + GD* + HB* + BJ* + ½EK* + ½LF*]
+ pin Im
[HD*+D J* + GB* + BI* + EF* + 1LK*],
9Q(Spe + sK~) = lEE* + FF* - ~(GG* + HH* + II* + J J*) + ¼(KK* + L L * ) - 2(BB* + DD*)] + 2 Re [CH* + C J* + AG* + AI* + ~(HJ* + GI*)] + Pi."2 Re [AH* + GC* + A J* + IC* - 2DB* - ~(IJ* + GH* + 2G J* + 2IH*)], 3 Q ( S p p - SrK) = lEE* - F F * - ½(GG* + HH* - I I * - J J * ) - ¼ ( K K * - LL*)] + 2 Re [ C H * - C J * + A G * - A I * ] + Pi."2 Re [AH* + GC* - A J* - IC* + ½(lJ* - GH*)], 3 Q ( S e ) = f~e"Re [4AE* +DK* + 4GE* - 2IE* + 2BF* -- JL*] + P~" Im [4CE* + BK* + ~-ne* - ]JE* + 2DF* - IL*], 3Q(SK) = n~" Im [4FC* + LB* + {FJ* - }FH* + 2ED* -- KG*] +P~"Re [4FA* + LD* + ~FI * * - ]FG* + 2EB* - KH*], 3 Q ( S p . ) = P I e " R e [ A K * + D E * - H F * + { B L * + i l K ' * -~GKI l
*
1
+P~" Im [CK* + B E * - G F * +~DE + i l K
*
*]
-~HK*],
3Q(SK,,) = P~,"Im[FB*+LC* - E I * + 2~K D * + , Lx H * - ~ L~ J *] + pin Re [FD* + L A * - E J* + ½KB* + X3LG*- 1LI*].
(31)
It is seen that the last four of the expectation values in eq. (31) vanish for an unpolarized incident spin-½ beam. We have described the spin-1 polarization by the tensors ( T r q ) in the xyz coordinate system. This same polarization would be described in a different coordinate ' \ where as in eq. (29) T(o = \ 2"~S: , etc. These two system x'y'z' by tensors (T;~Q,/ sets of tensors are related by the equation ( T/~e') = E ( TrQ)D~Q'(c~, fl, '/), e where (c~, fl, y) are the Euler angles of the rotation which takes the unprimed into the K primed coordinate system, and the quantities Doe, are elements of the rotation matrix [ref. ,2)].
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
263
3.4. SPIN CORRELATION Additional information concerning the M-matrix may be obtained by measuring in coincidence certain components of the polarization of the scattered spin-1 and spin-½ beams• The expectation value of the product S • l a . "~ m is " referred to as the spin correlation function C(lm), C(lm) = (S, rr,.). (32) By scattering a beam and measuring its asymmetry one can determine only those components of the (vector) polarization which are perpendicular to the propagation direction of the beam in the laboratory *. Thus for the spin-½ particle only the polarization components along the directions fl and g (both perpendicular to k~) can be conveniently measured. And for the spin-1 particle, since it emerges 0-n the lab. system) along the - K direction, only the polarization components along the directions A and P need be considered• On the basis of these facts we may limit our consideration to the four spin correlation functions, C(nn), C(Ps), C(Pn) and C(ns), that is, we write ( S a ) = C(nn)fl~ + C(Ps)~'g + C(Pn)l'fl + C(ns)~L (33) The parity requirement that the first two correlation functions be scalars and the last two pseudoscalars may be met by noting that p i . . / t ( = p~n) is a scalar whereas pin. i~(= pin) and p i . . ~ x / ~ ( = p i . ) are pseudoscalars. These facts justify our writing
QC(nn) = Q u(C.~+C..Py), y in
QC(Pn) = ,q~EC~ ~ t P, P~
in
~ y m i n x), QC(Ps) = Q u ( C v,+cp~ry QC(ns) = O ~(C.sP~ z in x in +C.sP~),
x i n ), + Cv.P~
(34)
where the eight correlation coefficients C.., etc. are functions of the scattering angle and energy. I f the incident spin -1- beam is unpolarized the only measurable coefficients are C.. and Ces. In order to obtain the M-matrix dependence of the last six coefficients of eq. (34), it is convenient to express them in terms of some other coefficients, whose M-matrix dependence is more easily given, as ( O - ½0)Cee,
C p s = COS ( O - - ½ 0 ) C p K - - s i n 1
1 n CeY~ = cos (O - ~O)Cer- sin (O - ._O)Cpp, z
Cpn =
n
1 P " COS ~ O f p n - - s m
1
K
~OCpn ,
x 1 P 1 K Cp, -- sin ~OCp. + cos ~OCp., P " zOC.K)~ ~ " ( 0 - ½0) ¢OC.Ksm sin
C~=cos(O-k0)(cos'
P • ~OC.p), l K ~OC.psm
X (COS 1
C.5 = cos (O-½0)(sin 7OC,, ~ P K + COS 7OC.K) 1 K • -- sln ( 0 -- ½0) •
1
P
X ( s i n ~OCnp"}- c o s
1 K ~OC.p).
(35)
• The polarization component along the laboratory propagation direction can be measured, if on first rotates (with a magnetic field, for example) the spin vector.
264
R . G . SEYLER
The M-matrix coefficient dependence of the various spin correlation coefficients is obtained by evaluating the expectation value in eq. (32) and using the equation
QC(lm) = Qu[Ctm+ Z f"'lm't j Din'] j A' J
(36)
with the appropriate choices for the variables l, rn a n d j . Carrying out this procedure one finds
3Q"C.. = Re ( c q - ~ 2 ) ,
3QUCpp
P 3Q u C.K = im(cq-c~z),
3 Q "C"pK=Im(o~6--0:5),
u
K
--=
Re ( ~ 5 - % ) ,
3Q C.p = I m ( ~ l + e 2 ) ,
3flucK~Pn = I m (0~5 ~- ~6),
3Q"CpK = I m (~4--%),
3Q"C~,, =
3 Q "C"pp = R e (~3--0~4),
3 Q uc",,p = Re (~7-c~8+~9-cq0),
u
P
31qucK ,,K
3Q Cp,, = Re (c~3 + e4),
=
R e (g7--l--o~8--l--~9-l-~xlO),
Re (~7-0~8-(x9--1-0~10),
(37)
where the c~i are abbreviations of the following expressions:
cq = - 2 E F * - ½KL*, ~2 =
- - 4 ( A B * +DC*)+ ~a(GB* + IB* +DH* +D J*),
ct3 = 4CE* + - } H E * - ~ J E * + BK*,
c~, = 2 D F * - I L * ,
ct5 = 4AE* + { G E * - ~ I E * + D K * ,
c~6 = 2BF*--JL*,
~7 = 4 D A * - ~ ( D G * +DI*), ~9 = E K * ,
~8 = 4 B C * - ~ ( B H * + BJ*), (38)
~1o = F L * .
4. P h a s e
shifts
A necessary prerequisite to a phase-shift analysis of scattering data is, of course, a set of equations expressing the observables in terms of the phase shifts. In the previous section the observables are expressed in terms of the M-matrix ceeffcients. And in eqs. (11) and (12) the M-matrix coefficients are expressed in terms of the collision matrix elements. In the section we complete the circuit by expressing the collision matrix elements in terms of phase shifts. We achieve this goal by a simple generalizatiov of the Blatt and Biedenharn 11) method of parameterizing the two-nucleon collision matrix. The two-nucleon collision matrix U in addition to being symmetric and unitary is diagonal in j, s and parity and thus its only off-diagonal elements are of the form U/sl s where l' -- l_+ 2. Therefore, the largest submatrix of U corresponding to particular values of j, s and parity is 2 x 2. Blatt and Biedenharn parameterize this sub-matrix
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
U j by writing
U j = (u J) ~ exp
(2it~J)(uJ),
265
(39)
which is clearly unitary and symmetric for u j real and orthogonal and 6J real and diagonal. They 1~) introduce the real angle J and write u j in the obviously orthogonal form U j = V COS 8 j " sin eJ 1 I _ - s i n e J cose j " (40) The angle eJ is called the mixing parameter (it is a measure of the mixing or coupling between partial waves of different orbital angular momentum), and the two elements (angles) of the phase shift matrix 6 j are called the eigenphase shifts (rather than phase shifts of particular partial waves, since they have the latter interpretation only in the absence of mixing, i.e. when ej = 0). For spin-½-spin-I scattering, the matrix U need not be diagonal in channel spin. Thus for a particular value o f j it is necessary to consider two 3 x 3 matrices, • U J~
=
J / Us:_+½.~jq=kk Uj+_~j+_½~
(41)
U jj+½~72j+_½4 J ~ Uj+_½~2j+_4x{..]
~ L U j+_½~j.T-~2
which have parity ~z = (-)J-+*. Extending eq. (39) to 3 × 3 matrices we write U j~ = (uJ~) ~ exp (2it~J~)(uJ~),
(42)
where the phase shift matrix is now given by ~J~ =
~
o Ool
~ij ± ~ 0
•
(43)
6J±~
In eq. (43) the eigenphase shifts are labeled with the l and s subscripts of the partial wave whose phase shift they would be in the limit of no partial wave mixing. One obvious method of parameterizing the real orthogonal matrix u j" is to apply three successive Blatt and Biedenharn "rotations", (44)
Ujn = I~Jr~wJnxJn,
vJ'~ =
L 0 0
cos eJ~ -sinJ~
• i cos
01 sin0 ' l ,
wj~ =
L-sin
o1
sin eJ~ , coseJ'~J
~J~
0
(45)
(46)
cos ~J~l
• V c o s q ~. sin~/~. 07 xJ~ = [ - s i n r f l ~ cosrfl ~ ~ J . 0 L o
(47)
266
R. G. SEYLER
A c o m p a r i s o n of eqs. (45), (46) and (47) with eq. (41) reveals that for given values of i and re, e j~ is a measure of spin mixing without orbital (angular m o m e n t u m ) mixing between the partial waves, whereas (J~ is a measure of orbital (without spin) mixing and ,7j~ is a measure of the mixing (or coupling) between partial waves which differ in both their spin and orbital q u a n t u m numbers. We label f r o m 1 to 3 the rows and columns of the matrix u J~ and use eqs. (44)-(47) to obtain u~'] = cos ~J~ cos ~J", J= = /'/12
J= U13
s i n t/~n c o s (J~,
= sin ( ~ ,
j/t u21
- c o s ej~ sin r f ~ - s i n eJ~ sin (J= cos ~f=,
j1t Uz2 = cos ej~ cos q J ~ - s i n aJ~ sin (J~ sin q~=, u 23n 3 =
J~ /'/31 j~x
tl32
s i n 8 jn c o s ( j n ,
= sin ej" sin ~ / ~ - c o r e j" sin ~J" cos rff~, - s i n ej~ cos q ' ~ - cos e J= sin ( ~ sin ql~,
u~3 = cos ej~ cos r/j~.
(48)
U p o n substituting eqs. (43) and (48) into eq. (42) one obtains the desired result, namely the phase p a r a m e t e r dependence of the collision matrix elements of eq. (41), U~-v~j:~
H J~ l l U J~ lI
• i J~ J~ exp(2ifiJ_+~)+,,J~,,J~ t431 " 3 1 exp (2i6~_+÷~), exp (216j_z_k~)q-u2t/'/21
U Jj ~ k k j + _ ~ ½ = UI1HI2 J~ J~ exp (2tfj~ff~) " J J~ u22 J~ exp (2i3~± ~)+/,/31 J~ uJ3~2 exp (2i6~+~), + Uzt VJ~j+_~
~ exp ( 2 t"f j j~ _ ~ ) + u 2 , J~ u zj=3 exp ( 2 i ~ + ~ ) + u 3 ~J~ uJ3~3 exp (2i6J+~_), = u lJ~ l u~3
UJ+~&i+~ = u ~ u~2 ~ exp (2t6j-v_~k) "j i~ u22 j~ exp (2ifij +-I-~)''~--32-3z "''j~ exp (2i6~i±~), + u22 U ~j+_~j++.~:g~ = ,,J~ ~ 1 2 ~"1~3 exp (2i6~ ~ ) Uj4-½~jq-~ 2 = H13U13
jn
j~r
+ U2z u23 exp
. -j jn jn (210j+_½½) "}- U32 U33
exp (2i~+_~_),
(21(~jgk~k)'-~-U23 U23 e x p (210j+~½)'q-U33U33 exp
(49) The other three elements of the matrix U ~= need not be written since U j= is symmetric. Although for j = ½ the collision submatrices are only 2 × 2, these matrices m a y also be obtained f r o m the above equations provided one adopts the natural procedure of setting equal to zero any collision matrix element U~,~ for which (i) the quantities l s a n d j (or l ' s ' and j ) do not f o r m a triangle or (ii) the orbital q u a n t u m n u m b e r 1 or (l'), as c o m p u t e d f r o m the j - d e p e n d e n t equations, is negative. It is interesting to ask how m a n y phase p a r a m e t e r s enter if for the scattering there is a m a x i m u m eifective value of orbital angular m o m e n t u m l ..... . F o r I.... = 0 only
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
267
two phase shifts enter; for lmx = 1, nine phase parameters are present; for lm,~ = 2, twenty phase parameters are involved; and each additional increase (by one) of/max introduces twelve additional parameters. One can confidently predict that the various polarization experiments will play an important role in the determination of the parameters. For the analysis of elastic scattering at energies where other reactions are possible the above phase parameterization must be amended to permit the loss of flux to these other reaction channels. Thus at these energies the elastic scattering submatrices U j= [eq. (41)] although still symmetric are no longer unitary. The above equations may still be used provided one permits the various phase parameters to become complex subject to the constraints that the eigenvalues of the (hermitian) matrix UJ=(UJ=) * do not exceed unity. These eigenvalues may be obtained by diagonalizing this matrix by a unitary transformation. This step is non-trivial for complex mixing parameters since the orthogonal matrix u of eq. (41) is then not unitary. The above constraints lead to the requirement that the imaginary parts of the phase shifts be non-negative and that a single inequality involving phase shifts as well as mixing parameters be satisfied. Specifically, for a 3 x 3 matrix U, such as eq. (41), where for brevity we designate the diagonal elements of UU* as a, b and c and the non-diagonal elements above the principal diagonal by d, e and f (the other three elements follow from hermiticity) the inequality takes the form
( 3 x - 1 ) ( X + l ) ___
(50)
where X = ~1 Tr UU* = l ( a + b + c ) and Y = } ( a b + b c + c a - d d * - e e * - f f * ) . The condition that X < I follows and therefore need not be separately imposed. For the case of a 2 x 2 matrix such as eq. (39) one requires that ~l2 = e x p ( - 4 Im 6_+) < 1 and that lexp ( 2 i 6 + ) - e x p (2i6_)12[sinh (Ira 2e)] 2 < ( i - r / 2 ) ( 1 _ q2_). (51) This equation has been given by Arvieux ~3) although a factor ½ is missing from the right hand side of his eq. (14). The additional inequalities given in his eqs. (12) and (13) are unnecessary, since they are automatically satisfied when eq. (51) is satisfied. As a final point, we remind the reader that the nucleon-deuteron scattering phase shift analyses published to date ~3) have been based on the assumption that channel spin is conserved. This assumption has not been made in the present work. If this assumption is invoked the number of independent M-matrix coefficients is reduced from twelve to eight; in particular, one can show that channel spin conservation implies that E-G 2J-4H+3K
= F-I
= B,
= 2H-4J+3L
= 12C-6D.
(52)
The channel-spin conservation hypothesis leads (by use of the above equations, for example) to relations between certair~ polarization and spin correlation quantities.
268
R.G.
SEYLER
These relations are the subject of a very recent publication by Arvieux and Raynal who find that the relations take on a particularly simple form when the helicity formalism is employed. The interested reader is referred to their article 14) for the details. The relation between the M-matrix in the helicity representation M ' ( O ) of ref. [(14)] and our M ( O ) [eq. (7)] is easily given, Ms,v,sv = Z dcm, '' 6,mM~,m,sm, '
(53)
m'm
where d(0) is the reduced rotation matrix 12) of order s'. The collision matrix of ref. 1 4 ) differs from our eq. (49) by the phase factor (i)2s,-2s+v-t. The author takes this opportunity to express his gratitude to Dr. Louis Brown for his discussion and careful reading of the manuscript. He is also grateful to Drs. Walter Trfichslin and Tom CIegg for their work in the checking of equations.
Appendix I EXPLICIT MATRIX FORMS OF THE SPIN OPERATORS To find the explicit form of the spin operators in the channel spin representation one expresses the channel spin wave functions in terms of spin-½ and spin-1 wave functions according to the familiar equation 1 m v-m Zsv = Z (21mv-mlsv)z÷Z 1 .
(AI.1)
m
One then computes the matrix elements of the spin-1 operators and of (twice) the spin-½ operators between the various channel spin states [eq. (AI.1)]. One finds, in the notation of eq. (7)
io ooo Sz =
i O"z
---~
~
0
0
-~,/2
0 0
-½ 0
0 0
-~,/2 o
o -~,/2
0 -1 o
o
o
0 o 0 ~,/2 o
o
0 - - ,3 0 o ~/2
o
0 o -1 o o
0 o 0 -~ o
2 ,
-~ J
°1
~,/2 0| o
J
POLARIZATIONFROMSCATTERINGPOLARIZEDSPIN-½
Sx =
I!
Ix/+
[o
~x
Sy
=i
~,,=i
0 32 0
0 I,,/'2
a 0
0 x/'3
0 -Ix/2
%/3
0
0
°
- 61x/2 0 0 - ,~/{
0
-7'3
0
0
43
o
0
2
2 0
0 ,/3
o -,/~ o
o o -~/2
,/~ ~/2 o
o o #~
269
162 --41 / '
0
0 ] -,/'~ 0 ] 0 -3x/2 / 3x/2
0
]
o o -3
,/~ ] ' o
o
-,~/3
o
o
-,/1
o ]
~/3 0
0 32-
2 0
0 -~/3
0 -ix/2
-+x/Z] 0 ,
o
o
,/3
o
o
-,/t
,,/1 0
0 Ig2
+,/2 0
0 x/1
0 ~
-~ 0
o
-,/3
o
o
~/3 0
0 ~
2 0
0 -~/½
o
o
,/3
o
o
o
o
- x / az
-3
o
0
-3x/2
0
,/~
o ]
0 3x/2] 3,/2 0
,/~
"
oJ
The components of the spin operator S (or a) along the K, • and 1' axes are found by using the vector rotation equation, S, S,,
=
rosoOO SoOl[] Lsin½0
1 0
Sy . cos½0 .] S=
The matrix representations of the operators S u are obtained by performing the matrix multiplication indicated in eq. (8). Appendix 2 TRACES OF TIIE SPIN OPERATORS
An important feature of the trace properties of operators is that they are independent of the representation chosen to express the operators. It is well known that the trace of any component of a spin operator Si (or al) is zero. Using this fact and the
270
R.G. SEYLER
familiar equation
(A2.1)
(a" d,)(~, b) = ~" [~+ ia" ~ix b,
where ~i and ~ are arbitrary unit vectors, it is easy to see that Tr aia j = G for i # j and that Tr a~2 = Tr 1( 6 ) = 6. F o r the spin-1 operators, the similar result that Tr SISj = 0 for i ¢ j applies, as m a y be easily verified using the explicit representation of appendix 1. F o r i = j we use the fact that Tr S 2 = ~ T r S • S = 2 T r 1(6~ = 4. Just as eq. (A2.1) m a y be used to limit the a dependence of an expression to terms linear in a so m a y the equation, ( S . a ) ( S . b)(S" ~) --- ¼i[(S. ~ x ~ ) ( S . ~ ) + ( S . +(s.
a)(s./;×
~)-(s.
~× a ) ( s . b ) - ( s ,
~)(S. ~× ~ ) + ( S .
~x~)(S.
a)
t;)(s. ~x a)]
+½(tl. l;)(s. ~ ) + ½ ( b . ~)(S. t/)+½i(tl x/~. ~),
(A2.2)
be used to limit the S dependence to terms second order in S. F r o m eq. (A2.2) it follows that Tr SjSkS~ = }i() x/¢- i ) T r 1~6~ = 2iejk~. The repeated application of eq. (A2.2) permits one to evaluate the trace of expressions containing four or m o r e components of S. We summarize the trace properties of the six dimensional spin operators: Tr a; = 0, Tr tTja k = 6(~jk , Tr aJ(S)
= 0,
Tr S~ = 0, Tr S i S k = 46jk, Tr S;k = O, Tr S~SkS t = 2iejkZ, T r SjSkl = O, 2 Tr S jk S ~ = (~jl (~km"~-t~jm 6kl -- ~(~ jk t~Im
= Tr S~kS, Sm, Tr Sj Skl Stun ~-- ½i[Sjk m 6in d- e jR n (~lm -~ ~jlm (~kn "q- 8jln (~ltm~"
(A2.3)
Appendix 3 M-MATRIX ELEMENTS IN TERMS OF M-MATRIX COEFFICIENTS Substituting the M - m a t r i x eq. (7) and the matrix form of the spin operators (appendix 1) into eq. (9) and c o m p a r i n g left and right sides element by element we obtain
POLARIZATION FROM SCATTERING POLARIZED SPIN-½
271
the following eighteen equations: a+h = 2A+~(B+E+F), amh
=
- ] B + ½ ( E + F)+~(G + I)+½ cos O E 2 ( E - F ) + ( G - I ) ] ,
b + g = 3 -} sin 0 [ 2 ( E - F ) + G - I ] , b-g
= - i 3 - ~ [ 2 ( C + D ) + ½ ( K + L ) + i 6 ( H + J ) - ½ cos O ( L - K + J - H ) ] ,
c+j
= 3- ~ [ - 2B + E + F +½(G + I ) + ½ cos O(2(F- E)+ I - G ) ] , i3 -½ sin O [ J - H + L - K ] ,
d+i
= - i [ +~(C +D)+~;(K + L)+ a-~(H + J)+ ½ cos O ( J - H + L - K ) ] ,
d-i
= ½i[2(C + D) -
(K + L) -
~-(H + J)],
e+m
6 -½ sin O [ E - F + I - G ] ,
e--m=
i6- ~[2D - 4C + ½(K + L) - ½(H + d) + ½ cos O(K - L + 2(J - H))],
f+p
= ½i6- + sin 0 E 2 ( J - H) + (K - L)],
f--p
= 6-~[- - 2 B + E + F -
G - I - cos O ( E - F + I -
G)],
k + n = ~ , j 2 [ 2 B - E - F +G + I - 3 cos O(E--F + I--G)], k-n=
¼ix/2 sin 0 1 2 ( J - H) + K - L], ½v/2 sin O [ E - F + I - G ] ,
l+o
=
I-o
-- aix/2 [ D - 2C + ¼(K + L ) - ~(H + J ) - ¼ cos O ( K - L + 2 ( J - H))], q= A-~(B+E+F), ½i[C- 2D + K + L - ~(U + J)].
Appendix 4 M-MATRIX COEFFICIENTS IN TERMS OF M-MATRIX ELEMENTS Using eq. (9) and the trace properties of the spin operators (appendix 2) one sees that Tr M = 6A, Tr MSep = ~-G-~I, Tr M % S . = 4B,
Tr MSt
Tr M % = 6C,
Tr M a n See = - } H - z3J,
Tr M S . = 4D,
Tr Man SK~
Tr Map Sp = 4E,
Tr M a e S e . = K,
Tr M a r S r = 4F,
Tr MaKSKn = L.
=
-- ~H +
4,i,
272
R . G . SEYLER
U p o n s u b s t i t u t i n g i n t o the a b o v e e q u a t i o n s the M - m a t r i x e l e m e n t s eq. (7) a n d the m a t r i x f o r m s o f the spin o p e r a t o r s ( a p p e n d i x I) the f o l l o w i n g e q u a t i o n s result:
A = ½[a+h+q], B = - ( 7 2 ) - ~[ - (k + n) + 2~/2(q - h) - x/3(p - f ) + x/6(c + j ) ] ,
C = ~i[2i - r - ~ j 2 ( o - l) + x/3(b - g) + ~ / 6 ( e - m)], D = ( 7 2 ) - ½ i [ - ( l - o) + 2~/'2(i + r ) - \ / 3 ( e - m) + ~/6(b - O)], E + F = ( 2 4 ) - * [ - (p - f ) + x/~2(c +j) - ½\/3(k + n) + x/6(a + ½h - 4 q ) ] , E - F = ( 2 4 ) - + [ s i n 0{(e + m) + 2~/2(b + 9) - x/3(l + o)} + cos O{(p--f) - x/~2(c + j ) - - x/3(k + n) + \/6(a - h)}], G + I = ½[a - h + x/2(k + n) + x/3(c + j ) + x / 6 ( p - f ) ] ,
G-I
= 6 - ~ sin O [ - ( e + m ) + x / 2 ( b + g ) + x / 3 ( l + o ) ] + ½ cos O[a - h + x/2(k + n) - ½x/3(c +j) - ½~/6(p - f ) ] ,
H + J = ( 1 2 ) - ~i[b - 9 + ~//2(e - m) + ½x/3(8r - 7i + 9d) + ½x/6(l - o)], H - J = 3 - ~i[sin O { ( c - j ) + l x / 2 ( f + p) + ½ x / 6 ( k - n)} +
cos
O{b - g +
- m) -
+ d) - , / g ( l
- o)}],
K + L = 6 - ½i[-- ( e - m) + x/2(b - 9 ) - ½x/3( / - o) + ½ x / 6 ( 3 d - i - 4r)], K - L = 6 - ~ i [ s i n 0{ - ( f + P) + 2 x / 2 ( c - J ) + c o s 0{ - ( e -
x/3( k - n)}
m)+x/2(b -9)+x/3(1-
o ) - x / 6 ( d + i)}].
References 1) G. M. Budianskii, ZhETF (USSR) 33 (1957) 889; JETP (Soy. Phys.) 6 (1958) 684 2) H. D. Shih, L. F. Lo, S. Shu and N. N. Haung, Acta Phys. Sinica 16 (1960) 324 3) H. S. Goldberg, Lawrence Radiation Laboratory Report UCRL-11526 (1964); thesis, University of California (1964) 4) J. yon Neumann, Mathematische Grundlagen der Quantenmechanik (Dover, New York, 1943) p. 174; H. Weyl, Theory of groups and quantum mechanics (Dover, New York, 1931) p. 78 5) L. Wolfenstein, Phys. Rev. 75 (1949) 1664; Phys. Rev. 76 (1949) 541; Phys. Rev. 96 (1954) 1654; Phys. Rev. 98 (1955) 766, 1870; Ann. Rev. Nucl. Sci. 6 (1956) 43; L. Wolfenstein and J. Ashkin, Phys. Rev. 85 (1952) 947; R. H. Dalitz, Proc. Phys. Soc. A65 (1952) 175; D. R. Swanson, Phys. Rev. 89 (1953) 740; R. Oehme, Phys. Rev. 98 (1955) 147, 216 6) A. M. Lane and G. R. Thomas, Revs. Mod. Phys. 30 (1958) 257 7) E. U. Condon and G. H. Shortley, Theory of atomic spectra (University Press, Cambridge, 1935) 8) A. M. Lane and R. G. Thomas, ibid. p. 290 9) P. L. Csonka, M. J. Moravcsik and M. D. Scadron, Phys. Rev. 143 (1966) 1324 10) W. Lakin, Phys. Rev. 98 (1955) 139 11) J. M. Blatt and L. C. Biedenharn, Revs. Mod. Phys. 24 (1952) 258; Phys. Rev. 86 (1952) 399 12) M. E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957) p. 52 13) R. S. Christian and J. L. Gammel, Phys. Rev. 91 (1953) 100; W. T. H. van Oers and K. W. Brockmann, Nucl. Phys. A92 (1967) 561; J. Arvieux, Nucl. Phys. A102 (1967) 513 14) J. Arvieux and J. Raynal, Nucl. Phys. A100 (1967) 472