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Quadruple correlation functions and their application in double polarization experiments and in few-body problems
2•.B I
Nuclear Phystcs A268 (I 976) 369- 380, (~) North-Holland Pubhshmg Co., Amsterdam Nqt to be reproduced by photoprmt or microfilmwithout written permtsslonfrom the pubhsher
QUADRUPLE CORRELATION FUNCTIONS AND THEIR APPLICATION IN DOUBLE POLARIZATION EXPERIMENTS A N D IN F E W - B O D Y P R O B L E M S
ALBRECHT LINDNER
1 lnstttutJur Theorettsche Phystk der Unwersttat Hambur O Received 8 March 1976 Abstract: We define real functions which form a complete, orthogonal and rotatmnal m v a r m n t basis for all problems depending on four directions Thetr usefulness is demonstrated for processes of the type ~ + b ---, c + d , ~ + b ~ 6 + d , ~t+b --* cl + c 2 + d The specml case of all four directions m one plane has m a n y apphcatlons too, e g m two- and three-body problems
1. Definition Generahzlng Biedenharn's triple correlation functions l, 2)
Pnln2n3(~'~l, ~"~2, ~r'~3) ~
in1 +n2 +n3[[c(nD(~'~l ) x C(~2)(~~2)](n3) x C(n3)(~r~a)](o°) ,
(1)
which form a complete, orthogonal and rotaUonal mvanant basis for all problems depending on three &rectlons, we define m the case of four directions Pnan2(n)n3n,(121, 02, 03, ~4) [ nl "l'n2+?13+n4~lh2n]h4[E[C(nl)(~f~l) = f ' +"~ +"3+"'h,h2h3ha[[C("')(121)
X C(W12)(~¢~2)](//IIx c(n3)(Oa)](n4) x c(n4)(a4)](o O)
x C("~)(122)](") x [C("3l(123) x C(n')(124)](")](o °).
(2)
Both definitions (1) and (2) are given m terms of renormahzed spherical harmomcs a) C~)~(12) =
4rr
(n--m)) Y~)m(12) = ( T- )m X/ (n+m) ) P~'~(cos 0) exp ( -t- im~o).
(3)
The factors guarantee the functions to be real: P.1.2(.).3.,(121,122, t23, 124) = P.,.~(.)n~.,(121,122,123, * 124)"
(4)
In order to avoid factors tl = 2x/r~-+ 1 in the normahzauon [cf. eq. (16) below] we have introduced such factors m the definition (2). The coupling to a scalar guarantees the correlation functions to be mvariant under 369
37o
A LINDNER
rotations. This can also be verified from P.,.~<.).3.,(121, ~2' ~"~3' 04)
-~ lnl+n2-n3-n4hlh2hl'13?14
E ( nl ml~12mm3n14 ~1~1
m2 m/km3 m,, x r"(,,,)*t¢) ~t.-,(n2).to~rtn3)tQ ~r'tn,Oit) "~
(5)
by exploiting the properties of the rotation matrices which appear as transformation coefficients when rotating the spherical harmonics 2. Properties of the quadruple correlation functions The quadruple correlation functions have the foUowlng symmetries
Pnln2(n)n3n4(~-~t, ~'~2, ~'~3, 04.) = Pn3n4(n)nln2(~-~3, ~'~4, ~'~1, ~c~2) = (--)nt+n2+nPn2n,(n)n3n4(~'~2, ~t'~l, ~'-~3,~r'~4)
=~(_).2+.3+.+.,hh,{nl .,
n4
(6)
(7)
n2 n} n3
n' P"'"3("')"~""(~21'f23' f22' f24) (8)
= ( -- )nlPn,n2(n)n3n4( -- ~'~1, ~"~2, ~3, ~c~4)"
(9)
They reduce to triple correlauon functions ff two directions coincide, e.g.
.....
/1
P.l.2t.).3.4(fIx, I22, ~2, (2) = z"+"3+"4nl/12n/13n. 0 n30 04) Pnln2n(Ql'~d2'Q)' (10) or one of the radices n, 0 = l, 2, 3 or 4) is zero, e,g. P.,.2(.).30(fI,, 122, t23, Q4) =
hlh2h3Pnln2.3(Ol,Q2, O3)t~nn3
(11)
Integration over one d~rectlon, e.g t24, results m
f
dQ4P,,,.2(.)nm,,(f21, ~e'~2'~¢~3' ~e'~4) =
4nhlh2haP.,.~.~(~21,~-~2' ~-~3)6nn36n40•
(12)
3. The quadruple correlation functions P.~.2(.).~.,(02, ~23,/24) Until now we have not fixed the coordinate system. Smce the quadruple correlaUon functmns are spherical mvariants, ~t is advantageous to choose the coordmate axes in such a way that only the physically relevant variables occur, i.e. the angles between the directions t2,. Choosing the z-direction along t2x, i e. 0x = qh = 0, and the x-direction (of our fight-handed coorchnate system) such that ~o2 = 0 we obtain
371
QUADRUPLE CORRELATION FUNCTIONS P.~.2o, m,,.(02, ~"~3,~'~4) = inx +n2-n3-n4hl/~2/~h31~4 × I~1111' ~
m+m'
-m-m'/\m
m'
-m-m'
""m+m'lW2'
Collectmg tterms with opposite sign we find for nl +n2 + n3 +n4 even P,,~,,~o,m,,,(02, 03, 04) = l"~ +n2-n3-n4hlfl2hh3h4
( ~ / 'It, xJ~o(2-6"°di"°),
(hi "2 0 m+m'
n
~(n3
n4
-m-m']k,m
m'
n
~
-m-m
× / ( n 2 - ra - m')!(n 3 - m)!(n,t.- m') I
~/(n 2 + m + m')!(n 3 + m)](n 4 + m')! x P~'.2+='I(cos02)P~'.~(cos 03)P~.,'(cos 04) cos (mq~3+ m'q~#)
+ m,m'>O ,,-,..,2(ol × /(n2
m--ra'
.
m'--raJkm
--m'
.)
ra'--m
-- Ira-- re'l)!(n3 -- m)t(n 4 - m')!
\/(n 2 + Ira-- m'D l(n3 + m)!(n 4 + m')] x e/,~-='l(cos 02)P~(co s 03)p~,~(cos 0,) cos (m~o3 -m'cP4)),
(14)
and for n x + n 2 + n 3 + n , odd the same expression but i s m ( ) instead of cos ( ) . Therefore, all quadruple correlation functions of coplanar directions (¢P3, ¢P4 equal to 0 or It) vanish ifn 1 + n 2 + n s + n ~ is odd. This is exploited in sect. 7 Another set of angular varmbles is discussed m the appendix. The quadruple correlation functions form a complete and orthogonal set of (real) functions in the variables 02, t23 and t2." ' t 2 ' 3, [21) P,,,,,~,,m,,,(02, g23,''f2,*)P,,,,,~o,~,,3,,,(O 2, nlnInn3n4
6(o~- o~)_
= 32X2 S~n02
,
,
6(~a -- f23)~(24-- f24)'
(15)
fdO2df23d124 sin 0 2Pntn2(n)n3n4(02, f23, I24)P.i.i(.,).~.a(02, f23, f24) = 32n 26.,.a6.~.~ ¢~..,6.~.~6.,.a.
(16)
Thus, every real, square mtegrable funcUon depending on four &rections in space can be expanded in terms of quadruple correlation funcUons:
f(I21, ~"~2,A"~3,~"~,) =
Z
f.,n2t.)ns.4P.~.2t.).3n.(I21, t22, Q3, f24)
n 1~12nl13n4
Some examples are detailed m the following secnons.
(17)
372
A L1NDNER
4. Spin-correlation: a + b --* e + d In experiments of th~s type we have four directions (m the c.m system). The relatwe motions in the lmtml channel (a versus b) and m the final channel (c versus d), defining ~, and ~f, and m addition the polarization directions ~ and ~b" [These directions are clearly defined 2,4) ff the polanzaUons are produced m external fields However, the concept of polarization dlrecttons can also be used ff the polanzatton is produced m a reaction as discussed m another paper 5) ] Thus we can write da dQf
E t4tninf(n)nanb ~ ttn')'(nb)p I.Q t2f, t2a, Qb), 0 tO n l f l f ( n ) n a n b ~, 17 nlnfnnanb
(18)
the factors t(o"') and t(o"~) gwlng the polarization strengths according to the Madison convention 6) They are connected with the density tensors of the relevant parUcles by 2) p~)(s, s ) = t~)/~ = t(o")C~)(~2)/~
(19)
[It should be noted that the Madison convenUon prescribes different coordinate systems to define the polarizations of particles a and b wRh opposite quant~zat~on dlrectzons. Ohlsen v) (and others) prefer to use only one coordinate system, e g. the projectde hehoty frame Such a conversion of quanUzatmn axes results m a factor ( - ) " m the definmon of t(0") and converts the polarization d~recUon f2, however, th~s does not change the expansion coeffioents m eq. (18) as follows from eq. (9).] In order to connect the expansmn coeffioents w~th the transmon matrix T we rely on our earher results *). Defining the transmon operator as Newton s), namely through S = 1 - 2 r o T , we find m the channel spin representation ~ni + nf - - h a - - r i b ~ninf(n)nanb
n2 h --
- -
k2 S~Sb
$i[iJ/f$ f
If
0 0 ] ( l ' f
Sf
s|liJ'l~
x T,~,~:]]'
o,l
':l t
0
na nb
f ,:!, ,s,
s:;l
nI n
t
?If..] t
t
(20)
× (Eflf(ScSd)Sfl TzlE,l,(SaSb)S,)(Efl'f(ScSd)Sfl Tj,IE,I,(SaSb)S,) ,
and m the j-representation __ int+nf +ha+rib
x
n2 __
1
)n+n,hh,2~n,
Y'. (-),,+S+,o+*d+J,+,e+.'~'f/',ff,~]], (~ ltjtJ'l~j~
nf
n}
I;1"~£"
J' If s~) ~j~ Jf
nf} Sd
QUADRUPLE CORRELATION FUNCTIONS
't
{l s j{s,
'^" ^"(~ ~l 0/11
,I, Sa L
L Sb J
tl~
ri'
ria
rl' t
¢
rl b
?If
t
× (Ef(lfsc)JfsdlTjlE,(l,sa)J,Sb)(Ef(lfsc)JfSdlTs,
373
!
lEt(l,sa)J,Sb)
(21)
.
[The expression for a.,..t.,).f.b would be more simple m this case as can be seen by comparison with eq (8).] Panty conservation reqmres n, +nf to be even.
5. Polarization transfer: a + b ~ c + d In order to descnbe the polanzatxon of emerging parucles (c m this example) in a rotaUonal mvarmnt manner we have defined 2) the "component of the polanzatmn along some arbitrary d~rectxon 12" by
t~")(12) ---- ~ t~)O:)*(~2) = t~")*(f2).
(22)
v
The descnptlon of polanzaUons in terms of these quantities has the further advantage of avoiding redundant parameters 5,9. ~o), m contrast to the conventional approach. We can thus describe the expenment by =
b.,..t.).~.o o P.~..~.~.~.o(t2,, 12., f2f, t2¢),
(23)
niPlannf
and find for the expansion coefficients 4)
b.,..~.).f.o
= l"'-"~'+"f-"c kf2 ~¢h 4k 2 ~a~-~
X~f~fSfStfJJ'
(__)s.+sb+s,+~o+s~+~f+l~+l,+.
2
$iltJ|fllf sil|J'l~s~
l'f n f ~ s f
s'f n c
'f
00:(Sc.
Sc
~If ng
Sd
' .i O](s.
s.
sb t
st
s,,'i} n,
n. t
t
x (Eflf(ScSd)Sfl TjIEJ,(S.Sb)S,)(Efl'f(ScSd)~fl Tj,IEJ,(SaSb)S,) m the channel spin representation and
(24)
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A LINDNER
b.,..(.).,.o = f'+'+"~+"~ k2 Sen 4k2 s.s--~ ~ (_)l|+~t+~+~+lt+j~+~.+s+. liJiJlfJf liJ|J'l~J~
× ~-f~.~fj.~3dy, (~
If' l o f ) {'
0
x
l,l ,JJ
(~
Se
nf
?ic
f
Jf
Sd
O') { l: sa JJl; ~ J' n t j, Sb
0
n~ × (Ef(lfsc)SfSa[
s~ Jji'}/~J J'
1[ If
Ha
TzlE,(l,sa)J,Sb)(Ef(lfs~)sfsal rj.IE,(1,s.)J,sb) !
v
!
(25)
l
m the y-representatmn. 6. Break-n0 reactions initiated by polarized particles: a + b --, c s + % + d
It has been shown m another paper 11) that the angular correlatmn of break-up reactions mmated by unpolanzed particles can be descnbed m terms of tnple correlation functmns (m the c.m. system): d3a -
-
d-Q1df22dE2
E Bnonln2~lonln2enonanu(['~o ' [']1' ~Q2). nonln2
(26)
For polanzed particles ~ we obtain d3a E
d~ld~2dE2
B.o..(.),,,2t (na) o P.o..(.).~.2(~2o, ~,, [21, f22),
(27)
-- .on.nnln2
wah n
B.o~.(.~,.2- i"°+"+"+'24k-~ ~.~--~
(_)lS+l'l+l~+L'+sa+sb+sl+sf+d
E
lomiJl fl2Lscsf lbs|d'lil~L"
^
^ ^' ~ ^'
0
x rfffereLL,]j,
O/is.
s,
Sb
lo
S~
no
n~
l'1 n, !2 1'2 n2 !, 12 o
o/\o
o
o
h
n2
,£
J,n} L sf
× ( ( E 111, E 2 1 2 ) L ( ( S l S2)ScSd)Sf[ T~lEolo(SaSb)Sl) !
P
t
t
V
x ((Ell1, Ee/e)/~(sls2)sosd)sfl TAEolo(s,sb)s,)
(28)
QUADRUPLE CORRELATION FUNCTIONS
375
In the channel spin representatmn and n h B.o..(., .... = ' - ' ° - " - " ' - ~ 2 4 k o 2 ~a~b 2
__)Io+Jb+Sb+J+Sl
xZooJJs..,(,
+S2+32+1'+
sd+J+n
Sa
1° I n°
0/(l~ × ll'JJ'IJ1
+Jl
E
lo2odllJl1222I IbjbJ'lij'll'2j~,l'
?1a
I~
s lj
I
sa
J2
t
0
0//(12
}
J2
n2
12
s2
n2
× (((Elll sz)jl(E21eS2)Jz)Isal Tjl(Eolosa)josb) t
v
t
t
t
!
!
× (((Elllsl)Jl(E212s2)J2)I Sd[Tj'I(EoIoSa)JOSb)
~t
(29)
m the j-representation. For a two-step process a + b ---, c + d ~ c t + c 2 + d the angular correlatmn o f the final products ~s intimately connected with the polanzatmn o f the intermediate particles c This has been &scussed for unpolanzed particles m the lnmal state ~2) and will be generalized now In the case o f a two-step process the transmon m a m x factorizes into a transmon matrix of the first step and an amphtude of the decay process:
((Eflf((El(slsE)s)ScSa)SflTzlEtl,(SaSb)Sl)
=
~(El(sls2)s)sclals¢)(Eflf(ScSa)sflTjlE,l,(sasOs,). (30)
In this representation we find by comparison wRh eq. (24): nnlna(n)nfnc : ~"--1
~'f S¢~ntna(n)nfnc sll" ~ (-)~° +~
0 0,1(1'
l
x ((El(sls2)s)sJAIso)((El'(sls2)s')sclAls°)*.
(31)
The sum is the same as for unpolanzed particles and has been &scussed already 12).
7. Colflanar case: Application in two- and three-body problems Finally we consider the special case o f all four directions m one plane, which is
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A LINDNER
encountered m many few-body problems, as has been extenswely discussed by Wong and Clement l a). They consider the transformation Pl = s l p ~ + t l P b ,
P2 = s2Pa+t2Pb,
(32)
and look for the bracket (P~P2[Pd~b) m angular momentum representaUon, Le. (p tP2(l112)lm [p~pb~l~lb)l'm').q'hls quantity is proportional to a quadruple correlation function, the angles being determined by the four radial parameters p~, P2, P, and PbIn fact, because of _2 COS
PbP*
COSO,
_2_2
2 2
0 b ~ P b P a __ P, - - ~/l P a - - t~ P b
=
P' p~ p,p~
_
(| = 1, 2),
(33)
2Slt~p~Pb s~p~ + t,Pb COS0b = p2 + S~2p ,2 - - t,2Pb2 p, 2s~p~p~ '
{: tp, =
fit,>0
(34)
(35)
If t, < O,
we have (
I
c, t -- co2~x t [~ (pxp2(lz 12)lmlp~b(IJb)l'm'~ = - --f--) :ll+124-la'J'lb ~ ol,2 ,
x ,~r,5,~,,~[s~t~p~-
S 2 t 2 P l2
q - ( s l t 2 - s2tlXSlS2pa2 - t l t a p b )2 ]
x O(1 -x2)pz,tb~l~hl2(Ob, t21, t22),
(36)
If the states are normalized 13) according to (plp2(lll2)lmlp'~p'2(i'J'~)l' m'> = 6(pl - p'O6(p2- P'2)( (l~12)lml(l'~l'2)l' m').
(37)
lit may be noted that we miss the factor ISlt2 --S2t I [3 m the expression of Wong and Clement ~a). Moreover, they consider mainly a transformaUon (32) of coordinates instead of momenta and claim to get the same result. This is correct if one adopts the same phase convention (t~llrn) = Y~(t}) for coordinates and momenta. However, we would get another phase factor if we had introduced a factor it in the coordinate representation m order to make all operators real when time reversal and rotational mvariant 14).] The quantity x is given by the r.h.s, of eq. (33): The factor O(1 - x 2) guarantees - 1 ~_ cos 0 b _<_ 1 for all values of p,, p, and Pw Parity is conserved since, as has been stated already following eq. (14), !1 + 12+ !a + Ib must be even.
(38)
Besides the examples for the transformation (32) given by Wong and Clement ~a) we mention the angular momentum reduction of the three-body problem ~s) to be briefly discussed now since it is related to the last section. The Jacobi momentum variables of the three-body problem (masse~ m~, m2, ma) are given m the c.m. system by, cf. e.g. Schmid and Ziegelmann 16),
QUADRUPLE CORRELATION FUNCTIONS m3
Pl - - -
m2+ m3
m2
k2
m2 + m 3
377
ks, (39)
m2+m a q l -----m 1 +m2+m
3
kl
m~ m t +m2+m
3
(k2 + k a ) -
In these coordinates particles 2 and 3 appear explicitly as a subsystem. In order to solve the three-body problem one has to change the subsystems 16) [cychc permutatmn of the radices m eq. (39)] Thus one wants to know (qtP~I2P2) or ( E l e l ( L 111)LMIE2e2(L212)L'M') using the same representatmn as m the last sectmn. [This representation mvolves no phase space factors m the calculation of cross sectmns s). Spins may be included afterwards, leading to a recoupling coefficient.] Using E, e, E, - ~ q , , e, = P,--P" (40) we have, with
15) (41)
mlm2 P3 = (m I + maXm2 + m3) ,
the equattons
=-
~1 1+
1-p3)~el, (42)
e2 = - ~/( l - p s ) - ~e2 E~ - e p ~ 3~t2 e l ,
which are of the same form as eq (32) Thus eq. (36) reads (EleI(LIII)LM[E2e2(L212)F-'M')=
(__)L iL'+l'+L2+12 L
~LL,~MM,
(ElelE2e2)~~
x 6(E 1 + e I - E 2 -e2)O(1 -x2)pL,t,O.)L2t2(01, 02, 0, 02 , 7t),
(43)
with X ~ (P3E1 --t-(1 - - P 3 ) e l - - E2)/2%/(p 3 -
p~)E 1el,
COS 01 ---- X,
cos 02 = ( - psEl + (1 - P3)el - E 2 ) / 2 1 ~ , cos 0 z = ( - (1 - P3)El + psel - e2)/2~/~I - ps)El e 2.
(44) (45) (46) (47)
[The expression (43) falls for Ps = 0 and I, but this case is trivial, cf. eq. (42).] In the coplanar case another frame than that used in sect. 3 seems to be more appropriate 1 s): Defining the z-direction perpendicular to this plane we can avoid
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A LINDNER
associated Legendre polynomials of the angular variables since C~(½~, q~) = f + m
V/(n-m)v(n+m)v
2"(~(n-- m))!(~n + m)) vexp zm~o,
(48)
l f n - - m Is even (otherwise the spherical harmonic vanishes) Choosing the x-direction along t ~ we thus have (since the quadruple correlation function is real and n l + n2 + n3 + n4 Is even in the coplanar case) hlh2hhah4
P.,.~(.).~.,(q~2, ~°3, q~4) - 2.,+.~+.3+.,
cos (m2q~2 - m 3 ~ 3-m4tp4 )
~
m2m3m4 mlm × (111 t12 11~(113 114 11)~=1%/(11,--r11,)l(n,-I-m,)! mI
m2
m/km 3
m,,
m
,
(49)
(~11,-m,))T(~(nl+m,))l '
with n , - m, even The quadruple correlation function In eq (36) appears in the form Pt.tbt~)llZ2(0b, slgntt 01, sight2 02),
(50)
and the one in eq (43) in the form PLIII(L)L212(01, 02, -- 02)
(51)
8. Conclusion In sect 1 we have defined a complete set of real functions which form an orthogonal and rotational mvanant basis for all problems depending on four directions m space. The symmetries of these functions and their connection with Bledenharn's triple correlation functions have been studied m sect 2 These quadruple correlation functions are very useful as has been shown m sects 4 to 7 for some examples For example, one does not have to worry about which coordinate system to use in double polarization experiments smce the only relevant parametersf.,.2~.).3., do not depend on that choice (This is of course only true if the frames can be rotated into each other; we still stick to the c m. system.) In fact we describe polarizations using the "natural quantlzatlon axes" as termed by Ohlsen 7). Moreover we avoid redundant parameters m the expansion 10) As an example let us consider the spin-correlation experiment of two spin-½ particles where only s-waves contribute in the initial and final channel (e g nucleonnucleon scattering at energies below 10 MeV) From eq. (20) we immediately infer that the spm-correlaUon Is then solely determined by the two coefficients ~2
aoo~o)oo - 4k 2 (31Tx12+ ITo12), 7~2
ao0to)ax - 4x/~k2 (17"112-[To[2),
(52)
QUADRUPLE CORRELATION FUNCTIONS
379
and the correspondmg angular functions reduce to P0ow)oo = 1,
Poo~o~l1 = xf3Pl( c°s 0.b) = ,f3 COS0,b,
(53)
where 0ab is the angle between the polarization directions 12~ and 12b" In the conventional approach we would have an expansion in terms of associated Legendre polynomials, i e. I, Pl(cos 0a)Pl(cos 0b) and P~(cos 0,)Pl(cos 0b) COS (tp~--tpb) However, eq (14) reads in this example Pooto~l 1(12,, 12f, 12a, 12b) = x/'3(Pl( cos O.)PI(cos Oh)+ PI(COS Oa)Pl(cos Oh) COS(¢p~-- ~Pb))
(54) Thus we already know (from rotational invariance) a linear relation between the conventional expansion coefficients in this case.
Appendix THE QUADRUPLE CORRELATION FUNCTIONS
~fl1.2{.}.3.4(~2, 01 fl~)
Besides the angular variables discussed in sect. 3 one sometimes uses another set, e.g. for the description of polarization transfer according to the Madison conventmn 6). One defines two coordinate systems with the same y-direction (perpendicular to 121 and 123) but different z-directions, one along 121, the other (primed) one along 123, cf fig. 1. Giving 122 in the first frame and 124 in the second one (as 12~), we derive from eq. (5) Pn,n=(n)n3n4(Q2, O, •t•t 4) = zn' + rl2 -"a--n41~lfi2fifiafi 4 X E m.'
n2
m
n
n3
--m]\O
n4
H
m'
--m
#'~(n2)*t'tt~ ~Nn
[~Q'~f"(n4)[("~t ~
Z'
y= y'--z.z' Fig I The definltmn of 02, 0 and 0~ The variables ~02 and q~] are defined correspondingly
(55)
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On the other hand one may use eq. (14) hawng expressed the old set of variables (Oz, 03, tO3, 04, tP4) m terms of the ne~v ones (02, tO2, O, 0~, ~p~.) 02 = 02;
03 = 0,
(103 = --~02, (56)
cos04 = cos 0~, cos O - s i n 0:, sm Ocos q~,
q~4 = (~04--q~2
References 1) L C Bledenharn, m Nuclear spectroscopy B, ed F Ajzenberg-Selove (Academic Press, New York,
! 96o) 2) 3) 4) 5) 6) 7) 8) 9) 10) Ii) 12) 13) 14) 15) 16)
A Lmdner, Nucl Phys A2M (1974)477 D M Brink and G R Satchler, Angular momentum (Oxford Umv Press, 1968) A Lmdner, Nucl Phys A!55 (1970) 145 A Lmdner, Nucl Phys A261 (1976) 253 H H Barschall and W Haeberh, ed, Polarization phenomena m nuclear reactions, Proc Third lnt Syrup, Madison 1970 (Umv Wtsconsm Press, 1971) G G Ohlsen, Rep Prog Phys 35 (1972) 717 R G Newton, Scattering theory of waves and particles (McGraw-Hall, New York, 1966) A Lmdner and H Ebmghaus, Nucl Phys A230 (1974) 487 M Dohrendorf and A Lmdner, to be pubhshed A Lmdner, Nucl Phys AI99 (1973) I l0 A Lmdner, m Proc 4th lnt Symp on polarization phenomena m nuclear reactions, Zurich 1975, ed W Gruebler and V Komg (Birkhauser, Basel, 1976) C W Wong and D M Clement, Nucl Phys AI83 0972) 210 A Bohr and B R Mottelson, Nuclear structure, v o l l (Benjamin, New York, 1969) R Bahan and E Br6zm, Nuovo Cim 61B (1969) 403 E W Schmld and H Zlegelmann. The quantum mechamcal three-body problem (Vleweg, Braunschwelg, 1974)
Nuclear Phystcs A268 (I 976) 369- 380, (~) North-Holland Pubhshmg Co., Amsterdam Nqt to be reproduced by photoprmt or microfilmwithout written permtsslonfrom the pubhsher
QUADRUPLE CORRELATION FUNCTIONS AND THEIR APPLICATION IN DOUBLE POLARIZATION EXPERIMENTS A N D IN F E W - B O D Y P R O B L E M S
ALBRECHT LINDNER
1 lnstttutJur Theorettsche Phystk der Unwersttat Hambur O Received 8 March 1976 Abstract: We define real functions which form a complete, orthogonal and rotatmnal m v a r m n t basis for all problems depending on four directions Thetr usefulness is demonstrated for processes of the type ~ + b ---, c + d , ~ + b ~ 6 + d , ~t+b --* cl + c 2 + d The specml case of all four directions m one plane has m a n y apphcatlons too, e g m two- and three-body problems
1. Definition Generahzlng Biedenharn's triple correlation functions l, 2)
Pnln2n3(~'~l, ~"~2, ~r'~3) ~
in1 +n2 +n3[[c(nD(~'~l ) x C(~2)(~~2)](n3) x C(n3)(~r~a)](o°) ,
(1)
which form a complete, orthogonal and rotaUonal mvanant basis for all problems depending on three &rectlons, we define m the case of four directions Pnan2(n)n3n,(121, 02, 03, ~4) [ nl "l'n2+?13+n4~lh2n]h4[E[C(nl)(~f~l) = f ' +"~ +"3+"'h,h2h3ha[[C("')(121)
X C(W12)(~¢~2)](//IIx c(n3)(Oa)](n4) x c(n4)(a4)](o O)
x C("~)(122)](") x [C("3l(123) x C(n')(124)](")](o °).
(2)
Both definitions (1) and (2) are given m terms of renormahzed spherical harmomcs a) C~)~(12) =
4rr
(n--m)) Y~)m(12) = ( T- )m X/ (n+m) ) P~'~(cos 0) exp ( -t- im~o).
(3)
The factors guarantee the functions to be real: P.1.2(.).3.,(121,122, t23, 124) = P.,.~(.)n~.,(121,122,123, * 124)"
(4)
In order to avoid factors tl = 2x/r~-+ 1 in the normahzauon [cf. eq. (16) below] we have introduced such factors m the definition (2). The coupling to a scalar guarantees the correlation functions to be mvariant under 369
37o
A LINDNER
rotations. This can also be verified from P.,.~<.).3.,(121, ~2' ~"~3' 04)
-~ lnl+n2-n3-n4hlh2hl'13?14
E ( nl ml~12mm3n14 ~1~1
m2 m/km3 m,, x r"(,,,)*t¢) ~t.-,(n2).to~rtn3)tQ ~r'tn,Oit) "~
(5)
by exploiting the properties of the rotation matrices which appear as transformation coefficients when rotating the spherical harmonics 2. Properties of the quadruple correlation functions The quadruple correlation functions have the foUowlng symmetries
Pnln2(n)n3n4(~-~t, ~'~2, ~'~3, 04.) = Pn3n4(n)nln2(~-~3, ~'~4, ~'~1, ~c~2) = (--)nt+n2+nPn2n,(n)n3n4(~'~2, ~t'~l, ~'-~3,~r'~4)
=~(_).2+.3+.+.,hh,{nl .,
n4
(6)
(7)
n2 n} n3
n' P"'"3("')"~""(~21'f23' f22' f24) (8)
= ( -- )nlPn,n2(n)n3n4( -- ~'~1, ~"~2, ~3, ~c~4)"
(9)
They reduce to triple correlauon functions ff two directions coincide, e.g.
.....
/1
P.l.2t.).3.4(fIx, I22, ~2, (2) = z"+"3+"4nl/12n/13n. 0 n30 04) Pnln2n(Ql'~d2'Q)' (10) or one of the radices n, 0 = l, 2, 3 or 4) is zero, e,g. P.,.2(.).30(fI,, 122, t23, Q4) =
hlh2h3Pnln2.3(Ol,Q2, O3)t~nn3
(11)
Integration over one d~rectlon, e.g t24, results m
f
dQ4P,,,.2(.)nm,,(f21, ~e'~2'~¢~3' ~e'~4) =
4nhlh2haP.,.~.~(~21,~-~2' ~-~3)6nn36n40•
(12)
3. The quadruple correlation functions P.~.2(.).~.,(02, ~23,/24) Until now we have not fixed the coordinate system. Smce the quadruple correlaUon functmns are spherical mvariants, ~t is advantageous to choose the coordmate axes in such a way that only the physically relevant variables occur, i.e. the angles between the directions t2,. Choosing the z-direction along t2x, i e. 0x = qh = 0, and the x-direction (of our fight-handed coorchnate system) such that ~o2 = 0 we obtain
371
QUADRUPLE CORRELATION FUNCTIONS P.~.2o, m,,.(02, ~"~3,~'~4) = inx +n2-n3-n4hl/~2/~h31~4 × I~1111' ~
m+m'
-m-m'/\m
m'
-m-m'
""m+m'lW2'
Collectmg tterms with opposite sign we find for nl +n2 + n3 +n4 even P,,~,,~o,m,,,(02, 03, 04) = l"~ +n2-n3-n4hlfl2hh3h4
( ~ / 'It, xJ~o(2-6"°di"°),
(hi "2 0 m+m'
n
~(n3
n4
-m-m']k,m
m'
n
~
-m-m
× / ( n 2 - ra - m')!(n 3 - m)!(n,t.- m') I
~/(n 2 + m + m')!(n 3 + m)](n 4 + m')! x P~'.2+='I(cos02)P~'.~(cos 03)P~.,'(cos 04) cos (mq~3+ m'q~#)
+ m,m'>O ,,-,..,2(ol × /(n2
m--ra'
.
m'--raJkm
--m'
.)
ra'--m
-- Ira-- re'l)!(n3 -- m)t(n 4 - m')!
\/(n 2 + Ira-- m'D l(n3 + m)!(n 4 + m')] x e/,~-='l(cos 02)P~(co s 03)p~,~(cos 0,) cos (m~o3 -m'cP4)),
(14)
and for n x + n 2 + n 3 + n , odd the same expression but i s m ( ) instead of cos ( ) . Therefore, all quadruple correlation functions of coplanar directions (¢P3, ¢P4 equal to 0 or It) vanish ifn 1 + n 2 + n s + n ~ is odd. This is exploited in sect. 7 Another set of angular varmbles is discussed m the appendix. The quadruple correlation functions form a complete and orthogonal set of (real) functions in the variables 02, t23 and t2." ' t 2 ' 3, [21) P,,,,,~,,m,,,(02, g23,''f2,*)P,,,,,~o,~,,3,,,(O 2, nlnInn3n4
6(o~- o~)_
= 32X2 S~n02
,
,
6(~a -- f23)~(24-- f24)'
(15)
fdO2df23d124 sin 0 2Pntn2(n)n3n4(02, f23, I24)P.i.i(.,).~.a(02, f23, f24) = 32n 26.,.a6.~.~ ¢~..,6.~.~6.,.a.
(16)
Thus, every real, square mtegrable funcUon depending on four &rections in space can be expanded in terms of quadruple correlation funcUons:
f(I21, ~"~2,A"~3,~"~,) =
Z
f.,n2t.)ns.4P.~.2t.).3n.(I21, t22, Q3, f24)
n 1~12nl13n4
Some examples are detailed m the following secnons.
(17)
372
A L1NDNER
4. Spin-correlation: a + b --* e + d In experiments of th~s type we have four directions (m the c.m system). The relatwe motions in the lmtml channel (a versus b) and m the final channel (c versus d), defining ~, and ~f, and m addition the polarization directions ~ and ~b" [These directions are clearly defined 2,4) ff the polanzaUons are produced m external fields However, the concept of polarization dlrecttons can also be used ff the polanzatton is produced m a reaction as discussed m another paper 5) ] Thus we can write da dQf
E t4tninf(n)nanb ~ ttn')'(nb)p I.Q t2f, t2a, Qb), 0 tO n l f l f ( n ) n a n b ~, 17 nlnfnnanb
(18)
the factors t(o"') and t(o"~) gwlng the polarization strengths according to the Madison convention 6) They are connected with the density tensors of the relevant parUcles by 2) p~)(s, s ) = t~)/~ = t(o")C~)(~2)/~
(19)
[It should be noted that the Madison convenUon prescribes different coordinate systems to define the polarizations of particles a and b wRh opposite quant~zat~on dlrectzons. Ohlsen v) (and others) prefer to use only one coordinate system, e g. the projectde hehoty frame Such a conversion of quanUzatmn axes results m a factor ( - ) " m the definmon of t(0") and converts the polarization d~recUon f2, however, th~s does not change the expansion coeffioents m eq. (18) as follows from eq. (9).] In order to connect the expansmn coeffioents w~th the transmon matrix T we rely on our earher results *). Defining the transmon operator as Newton s), namely through S = 1 - 2 r o T , we find m the channel spin representation ~ni + nf - - h a - - r i b ~ninf(n)nanb
n2 h --
- -
k2 S~Sb
$i[iJ/f$ f
If
0 0 ] ( l ' f
Sf
s|liJ'l~
x T,~,~:]]'
o,l
':l t
0
na nb
f ,:!, ,s,
s:;l
nI n
t
?If..] t
t
(20)
× (Eflf(ScSd)Sfl TzlE,l,(SaSb)S,)(Efl'f(ScSd)Sfl Tj,IE,I,(SaSb)S,) ,
and m the j-representation __ int+nf +ha+rib
x
n2 __
1
)n+n,hh,2~n,
Y'. (-),,+S+,o+*d+J,+,e+.'~'f/',ff,~]], (~ ltjtJ'l~j~
nf
n}
I;1"~£"
J' If s~) ~j~ Jf
nf} Sd
QUADRUPLE CORRELATION FUNCTIONS
't
{l s j{s,
'^" ^"(~ ~l 0/11
,I, Sa L
L Sb J
tl~
ri'
ria
rl' t
¢
rl b
?If
t
× (Ef(lfsc)JfsdlTjlE,(l,sa)J,Sb)(Ef(lfsc)JfSdlTs,
373
!
lEt(l,sa)J,Sb)
(21)
.
[The expression for a.,..t.,).f.b would be more simple m this case as can be seen by comparison with eq (8).] Panty conservation reqmres n, +nf to be even.
5. Polarization transfer: a + b ~ c + d In order to descnbe the polanzatxon of emerging parucles (c m this example) in a rotaUonal mvarmnt manner we have defined 2) the "component of the polanzatmn along some arbitrary d~rectxon 12" by
t~")(12) ---- ~ t~)O:)*(~2) = t~")*(f2).
(22)
v
The descnptlon of polanzaUons in terms of these quantities has the further advantage of avoiding redundant parameters 5,9. ~o), m contrast to the conventional approach. We can thus describe the expenment by =
b.,..t.).~.o o P.~..~.~.~.o(t2,, 12., f2f, t2¢),
(23)
niPlannf
and find for the expansion coefficients 4)
b.,..~.).f.o
= l"'-"~'+"f-"c kf2 ~¢h 4k 2 ~a~-~
X~f~fSfStfJJ'
(__)s.+sb+s,+~o+s~+~f+l~+l,+.
2
$iltJ|fllf sil|J'l~s~
l'f n f ~ s f
s'f n c
'f
00:(Sc.
Sc
~If ng
Sd
' .i O](s.
s.
sb t
st
s,,'i} n,
n. t
t
x (Eflf(ScSd)Sfl TjIEJ,(S.Sb)S,)(Efl'f(ScSd)~fl Tj,IEJ,(SaSb)S,) m the channel spin representation and
(24)
374
A LINDNER
b.,..(.).,.o = f'+'+"~+"~ k2 Sen 4k2 s.s--~ ~ (_)l|+~t+~+~+lt+j~+~.+s+. liJiJlfJf liJ|J'l~J~
× ~-f~.~fj.~3dy, (~
If' l o f ) {'
0
x
l,l ,JJ
(~
Se
nf
?ic
f
Jf
Sd
O') { l: sa JJl; ~ J' n t j, Sb
0
n~ × (Ef(lfsc)SfSa[
s~ Jji'}/~J J'
1[ If
Ha
TzlE,(l,sa)J,Sb)(Ef(lfs~)sfsal rj.IE,(1,s.)J,sb) !
v
!
(25)
l
m the y-representatmn. 6. Break-n0 reactions initiated by polarized particles: a + b --, c s + % + d
It has been shown m another paper 11) that the angular correlatmn of break-up reactions mmated by unpolanzed particles can be descnbed m terms of tnple correlation functmns (m the c.m. system): d3a -
-
d-Q1df22dE2
E Bnonln2~lonln2enonanu(['~o ' [']1' ~Q2). nonln2
(26)
For polanzed particles ~ we obtain d3a E
d~ld~2dE2
B.o..(.),,,2t (na) o P.o..(.).~.2(~2o, ~,, [21, f22),
(27)
-- .on.nnln2
wah n
B.o~.(.~,.2- i"°+"+"+'24k-~ ~.~--~
(_)lS+l'l+l~+L'+sa+sb+sl+sf+d
E
lomiJl fl2Lscsf lbs|d'lil~L"
^
^ ^' ~ ^'
0
x rfffereLL,]j,
O/is.
s,
Sb
lo
S~
no
n~
l'1 n, !2 1'2 n2 !, 12 o
o/\o
o
o
h
n2
,£
J,n} L sf
× ( ( E 111, E 2 1 2 ) L ( ( S l S2)ScSd)Sf[ T~lEolo(SaSb)Sl) !
P
t
t
V
x ((Ell1, Ee/e)/~(sls2)sosd)sfl TAEolo(s,sb)s,)
(28)
QUADRUPLE CORRELATION FUNCTIONS
375
In the channel spin representatmn and n h B.o..(., .... = ' - ' ° - " - " ' - ~ 2 4 k o 2 ~a~b 2
__)Io+Jb+Sb+J+Sl
xZooJJs..,(,
+S2+32+1'+
sd+J+n
Sa
1° I n°
0/(l~ × ll'JJ'IJ1
+Jl
E
lo2odllJl1222I IbjbJ'lij'll'2j~,l'
?1a
I~
s lj
I
sa
J2
t
0
0//(12
}
J2
n2
12
s2
n2
× (((Elll sz)jl(E21eS2)Jz)Isal Tjl(Eolosa)josb) t
v
t
t
t
!
!
× (((Elllsl)Jl(E212s2)J2)I Sd[Tj'I(EoIoSa)JOSb)
~t
(29)
m the j-representation. For a two-step process a + b ---, c + d ~ c t + c 2 + d the angular correlatmn o f the final products ~s intimately connected with the polanzatmn o f the intermediate particles c This has been &scussed for unpolanzed particles m the lnmal state ~2) and will be generalized now In the case o f a two-step process the transmon m a m x factorizes into a transmon matrix of the first step and an amphtude of the decay process:
((Eflf((El(slsE)s)ScSa)SflTzlEtl,(SaSb)Sl)
=
~(El(sls2)s)sclals¢)(Eflf(ScSa)sflTjlE,l,(sasOs,). (30)
In this representation we find by comparison wRh eq. (24): nnlna(n)nfnc : ~"--1
~'f S¢~ntna(n)nfnc sll" ~ (-)~° +~
0 0,1(1'
l
x ((El(sls2)s)sJAIso)((El'(sls2)s')sclAls°)*.
(31)
The sum is the same as for unpolanzed particles and has been &scussed already 12).
7. Colflanar case: Application in two- and three-body problems Finally we consider the special case o f all four directions m one plane, which is
376
A LINDNER
encountered m many few-body problems, as has been extenswely discussed by Wong and Clement l a). They consider the transformation Pl = s l p ~ + t l P b ,
P2 = s2Pa+t2Pb,
(32)
and look for the bracket (P~P2[Pd~b) m angular momentum representaUon, Le. (p tP2(l112)lm [p~pb~l~lb)l'm').q'hls quantity is proportional to a quadruple correlation function, the angles being determined by the four radial parameters p~, P2, P, and PbIn fact, because of _2 COS
PbP*
COSO,
_2_2
2 2
0 b ~ P b P a __ P, - - ~/l P a - - t~ P b
=
P' p~ p,p~
_
(| = 1, 2),
(33)
2Slt~p~Pb s~p~ + t,Pb COS0b = p2 + S~2p ,2 - - t,2Pb2 p, 2s~p~p~ '
{: tp, =
fit,>0
(34)
(35)
If t, < O,
we have (
I
c, t -- co2~x t [~ (pxp2(lz 12)lmlp~b(IJb)l'm'~ = - --f--) :ll+124-la'J'lb ~ ol,2 ,
x ,~r,5,~,,~[s~t~p~-
S 2 t 2 P l2
q - ( s l t 2 - s2tlXSlS2pa2 - t l t a p b )2 ]
x O(1 -x2)pz,tb~l~hl2(Ob, t21, t22),
(36)
If the states are normalized 13) according to (plp2(lll2)lmlp'~p'2(i'J'~)l' m'> = 6(pl - p'O6(p2- P'2)( (l~12)lml(l'~l'2)l' m').
(37)
lit may be noted that we miss the factor ISlt2 --S2t I [3 m the expression of Wong and Clement ~a). Moreover, they consider mainly a transformaUon (32) of coordinates instead of momenta and claim to get the same result. This is correct if one adopts the same phase convention (t~llrn) = Y~(t}) for coordinates and momenta. However, we would get another phase factor if we had introduced a factor it in the coordinate representation m order to make all operators real when time reversal and rotational mvariant 14).] The quantity x is given by the r.h.s, of eq. (33): The factor O(1 - x 2) guarantees - 1 ~_ cos 0 b _<_ 1 for all values of p,, p, and Pw Parity is conserved since, as has been stated already following eq. (14), !1 + 12+ !a + Ib must be even.
(38)
Besides the examples for the transformation (32) given by Wong and Clement ~a) we mention the angular momentum reduction of the three-body problem ~s) to be briefly discussed now since it is related to the last section. The Jacobi momentum variables of the three-body problem (masse~ m~, m2, ma) are given m the c.m. system by, cf. e.g. Schmid and Ziegelmann 16),
QUADRUPLE CORRELATION FUNCTIONS m3
Pl - - -
m2+ m3
m2
k2
m2 + m 3
377
ks, (39)
m2+m a q l -----m 1 +m2+m
3
kl
m~ m t +m2+m
3
(k2 + k a ) -
In these coordinates particles 2 and 3 appear explicitly as a subsystem. In order to solve the three-body problem one has to change the subsystems 16) [cychc permutatmn of the radices m eq. (39)] Thus one wants to know (qtP~I2P2) or ( E l e l ( L 111)LMIE2e2(L212)L'M') using the same representatmn as m the last sectmn. [This representation mvolves no phase space factors m the calculation of cross sectmns s). Spins may be included afterwards, leading to a recoupling coefficient.] Using E, e, E, - ~ q , , e, = P,--P" (40) we have, with
15) (41)
mlm2 P3 = (m I + maXm2 + m3) ,
the equattons
=-
~1 1+
1-p3)~el, (42)
e2 = - ~/( l - p s ) - ~e2 E~ - e p ~ 3~t2 e l ,
which are of the same form as eq (32) Thus eq. (36) reads (EleI(LIII)LM[E2e2(L212)F-'M')=
(__)L iL'+l'+L2+12 L
~LL,~MM,
(ElelE2e2)~~
x 6(E 1 + e I - E 2 -e2)O(1 -x2)pL,t,O.)L2t2(01, 02, 0, 02 , 7t),
(43)
with X ~ (P3E1 --t-(1 - - P 3 ) e l - - E2)/2%/(p 3 -
p~)E 1el,
COS 01 ---- X,
cos 02 = ( - psEl + (1 - P3)el - E 2 ) / 2 1 ~ , cos 0 z = ( - (1 - P3)El + psel - e2)/2~/~I - ps)El e 2.
(44) (45) (46) (47)
[The expression (43) falls for Ps = 0 and I, but this case is trivial, cf. eq. (42).] In the coplanar case another frame than that used in sect. 3 seems to be more appropriate 1 s): Defining the z-direction perpendicular to this plane we can avoid
378
A LINDNER
associated Legendre polynomials of the angular variables since C~(½~, q~) = f + m
V/(n-m)v(n+m)v
2"(~(n-- m))!(~n + m)) vexp zm~o,
(48)
l f n - - m Is even (otherwise the spherical harmonic vanishes) Choosing the x-direction along t ~ we thus have (since the quadruple correlation function is real and n l + n2 + n3 + n4 Is even in the coplanar case) hlh2hhah4
P.,.~(.).~.,(q~2, ~°3, q~4) - 2.,+.~+.3+.,
cos (m2q~2 - m 3 ~ 3-m4tp4 )
~
m2m3m4 mlm × (111 t12 11~(113 114 11)~=1%/(11,--r11,)l(n,-I-m,)! mI
m2
m/km 3
m,,
m
,
(49)
(~11,-m,))T(~(nl+m,))l '
with n , - m, even The quadruple correlation function In eq (36) appears in the form Pt.tbt~)llZ2(0b, slgntt 01, sight2 02),
(50)
and the one in eq (43) in the form PLIII(L)L212(01, 02, -- 02)
(51)
8. Conclusion In sect 1 we have defined a complete set of real functions which form an orthogonal and rotational mvanant basis for all problems depending on four directions m space. The symmetries of these functions and their connection with Bledenharn's triple correlation functions have been studied m sect 2 These quadruple correlation functions are very useful as has been shown m sects 4 to 7 for some examples For example, one does not have to worry about which coordinate system to use in double polarization experiments smce the only relevant parametersf.,.2~.).3., do not depend on that choice (This is of course only true if the frames can be rotated into each other; we still stick to the c m. system.) In fact we describe polarizations using the "natural quantlzatlon axes" as termed by Ohlsen 7). Moreover we avoid redundant parameters m the expansion 10) As an example let us consider the spin-correlation experiment of two spin-½ particles where only s-waves contribute in the initial and final channel (e g nucleonnucleon scattering at energies below 10 MeV) From eq. (20) we immediately infer that the spm-correlaUon Is then solely determined by the two coefficients ~2
aoo~o)oo - 4k 2 (31Tx12+ ITo12), 7~2
ao0to)ax - 4x/~k2 (17"112-[To[2),
(52)
QUADRUPLE CORRELATION FUNCTIONS
379
and the correspondmg angular functions reduce to P0ow)oo = 1,
Poo~o~l1 = xf3Pl( c°s 0.b) = ,f3 COS0,b,
(53)
where 0ab is the angle between the polarization directions 12~ and 12b" In the conventional approach we would have an expansion in terms of associated Legendre polynomials, i e. I, Pl(cos 0a)Pl(cos 0b) and P~(cos 0,)Pl(cos 0b) COS (tp~--tpb) However, eq (14) reads in this example Pooto~l 1(12,, 12f, 12a, 12b) = x/'3(Pl( cos O.)PI(cos Oh)+ PI(COS Oa)Pl(cos Oh) COS(¢p~-- ~Pb))
(54) Thus we already know (from rotational invariance) a linear relation between the conventional expansion coefficients in this case.
Appendix THE QUADRUPLE CORRELATION FUNCTIONS
~fl1.2{.}.3.4(~2, 01 fl~)
Besides the angular variables discussed in sect. 3 one sometimes uses another set, e.g. for the description of polarization transfer according to the Madison conventmn 6). One defines two coordinate systems with the same y-direction (perpendicular to 121 and 123) but different z-directions, one along 121, the other (primed) one along 123, cf fig. 1. Giving 122 in the first frame and 124 in the second one (as 12~), we derive from eq. (5) Pn,n=(n)n3n4(Q2, O, •t•t 4) = zn' + rl2 -"a--n41~lfi2fifiafi 4 X E m.'
n2
m
n
n3
--m]\O
n4
H
m'
--m
#'~(n2)*t'tt~ ~Nn
[~Q'~f"(n4)[("~t ~
Z'
y= y'--z.z' Fig I The definltmn of 02, 0 and 0~ The variables ~02 and q~] are defined correspondingly
(55)
380
A LINDNER
On the other hand one may use eq. (14) hawng expressed the old set of variables (Oz, 03, tO3, 04, tP4) m terms of the ne~v ones (02, tO2, O, 0~, ~p~.) 02 = 02;
03 = 0,
(103 = --~02, (56)
cos04 = cos 0~, cos O - s i n 0:, sm Ocos q~,
q~4 = (~04--q~2
References 1) L C Bledenharn, m Nuclear spectroscopy B, ed F Ajzenberg-Selove (Academic Press, New York,
! 96o) 2) 3) 4) 5) 6) 7) 8) 9) 10) Ii) 12) 13) 14) 15) 16)
A Lmdner, Nucl Phys A2M (1974)477 D M Brink and G R Satchler, Angular momentum (Oxford Umv Press, 1968) A Lmdner, Nucl Phys A!55 (1970) 145 A Lmdner, Nucl Phys A261 (1976) 253 H H Barschall and W Haeberh, ed, Polarization phenomena m nuclear reactions, Proc Third lnt Syrup, Madison 1970 (Umv Wtsconsm Press, 1971) G G Ohlsen, Rep Prog Phys 35 (1972) 717 R G Newton, Scattering theory of waves and particles (McGraw-Hall, New York, 1966) A Lmdner and H Ebmghaus, Nucl Phys A230 (1974) 487 M Dohrendorf and A Lmdner, to be pubhshed A Lmdner, Nucl Phys AI99 (1973) I l0 A Lmdner, m Proc 4th lnt Symp on polarization phenomena m nuclear reactions, Zurich 1975, ed W Gruebler and V Komg (Birkhauser, Basel, 1976) C W Wong and D M Clement, Nucl Phys AI83 0972) 210 A Bohr and B R Mottelson, Nuclear structure, v o l l (Benjamin, New York, 1969) R Bahan and E Br6zm, Nuovo Cim 61B (1969) 403 E W Schmld and H Zlegelmann. The quantum mechamcal three-body problem (Vleweg, Braunschwelg, 1974)