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Triple correlation functions for describing three-body reactions and polarization experiments
1 1.B:Z.B (
Nuclear Physics A230 (1974) 477-486; Not to be reproduced
by photoprint
@
or microfilm
TRIPLE CORRELATION FOR DESCRIBING
North-Holland without
Co.. Amsterdam
permission
from the publisher
FUNCTIONS
THREE-BODY
AND POLARIZATION
Publishing
written
REACTIONS
EXPERIMENTS
ALBRECHT LINDNER 1. Znstitut fiir Theoretische Physik der Unioersitiit Hamburg Received 21 January 1974 (Revised 11 March 1974) Abstract: The triple correlation functions P.o.l,Z (Q,, 9, , Q,) form a rotational invariant basis for all experiments depending on three directions. Their usefulness for the description of threebody reactions and polarization experiments is demonstrated. In the latter case the sum nofn1+n2 may be odd, pertaining to the well-known right-left asymmetries. Numerical methods are given to calculate these functions and the relevant expansion coefficients. The directions are tabulated which are best suited for experiments.
1. Applicabiiities of triple correlation functions Every physical problem depending on three directions in space, e.g. Q,, R, and Q,, can be described in terms of Biedenharn’s ‘) triple correlation functions, ~flOnlnl(Qo 2 Q2,3a,>
since these functions form a complete orthogonal system, as shown in sect. 4. (Here, -__ fi = J2n+l.) Until now, triple correlation functions have been considered in some detail only for even no, n, and n2 [refs. r* 2)], pertaining to the sequential decay of intermediate states with sharp parities. Triple correlation functions can be applied in a much wider range. They can be used in the description of all complete three-body scattering experiments “) (where parity conservation still requires the sum no + nl + n2 to be even) as well as in all those polarization experiments where three directions are relevant (ki, k, and the polarization direction). In the latter case, as shown implicitly in an earlier paper 4), the sum n,+n, +n2 may be odd, e.g. in the case of right-left asymmetries. This is detailed in sect. 5. 477
478
A. LINDNER
2. Symmetries and other triple correlation functions
Biedenharn’s triple correlation functions are real and very symmetrical, as follows from eq. (lb):
If one of the indices n,, n, or n2 is zero, one obtains a Legendre polynomial:
In the literature functions with different norm and phase are called triple correlation functions too. Yet, they are of lower symmetry and are not real, if n,+n, +n, is odd: Biederharn and Rose “) use Anon,n2(Q05Q, 7 Q,) = ~-“0-“‘+“2(479-*fio fi, fiZ,P”,,,“,(%, 0, Q2); 3
Devons and Goldfarb “) use ~“On,n2(%5 Q, >52,) = ~no+n’-“2&l fh fiZP”on,n2(Qo 9 Q, 3 Q,); Rose “) uses
3. The triple correlation functions P,_,nln@l, 02, q2) Until now we did not fix the coordinate system. The triple correlation functions are spherical invariants ‘). Choosing the z-direction along 8, and the x-z plane of our right-handed coordinate system to go through 52,, we retain the angles 8,, 0, and (p2 as (the only physically relevant) variables:
x
P;(COS
e,)qcos
8,)
exp
(hcp,).
(4
These triple correlation functions possess the symmetries
Pnond4 e2,(~~1 = cde2, el, q2) = (-)no+nl +~~~no.ln2(e1 , e2, - ~7~). 7
Pa> (w
TRIPLE CORRELATION
Evidently,
if no +n, +n,
479
FUNCTIONS
is even,
min (nl, 14
and, if n,+n, Konl”*(&
+n, is odd,
343 cpz) =
The real expansion
coefficients
A In+1 =
with the starting
min (III, nt) mz, 4n(~o 9 n,> %)c@
-
&I sin mcpz-
w
follow from the useful recurrence relation “)
A,-,+[no(no+l)-nI(nI+l)-n,(n,+1)+2mZ]A, (n,-m)(n,+m+l)(n,-m)(n,+m+l)
(7)
’
values
A
=
_
1
if no-i-n, +n,
4PXcos
n,(n,+l)-n,(n,+l)-n,(n,+l)~
02
is even, or
A, = 0, A, = J.,(.,+:).,(.,+l)
II:
= _ J(no+n,+nz+l)(-no+n,
“I’
4
+n,)(n,-nn,+n,)(no+n,-nn,+l) n,(n1+
l)u,(u*
+ 1) X
no n, 0
0
n,--I 0
1’ (8’3) 1’
if no+n,+n, is odd. The associated Legendre polynomials PT(cos 0) can be calculated by recurrence relations too, which is a well-known procedure ‘): Especially suited for eq. (6) is the recurrence relation ‘) p::+'(cose) = 2 m cot ep,m(cose) -(n
+ m)(n - m 4- l)P;-‘(~0s
but only for analytical expressions, since the relation 8 z 0. Using computers one should apply ‘)
cxx) =
is numerically
1 [(2n-l)xP,“_,(x)-(n+m-1)P,“_z(x)], n- m
e),
unstable
for sin
480
A. LINDNER
with Pz(cos 0) = (2m - l)! ! sinm0,
P;+ 1(~~~0) = (2m + l)!! cos 0 sinm 8. Often useful is the value at 8 = 90”: i”-”
e(o)
=
i
()
(n+m-l)!! -__ (n-m)!!
if
n-m
= 0,2,4,.
if
n-m=1,3,5
..
,....
(9
In this case the sums in eq. (6) run only over even or only over odd m. This is applied in eq. (18). 4. Expansion into triple correlation functions In their range of definition, the triple correlation functions form a complete orthogonal system’ :
=
8n
‘tel -e;) s(e2-e;) sin 8,
sin
e
2
(The integers no, nl and n2 must of course satisfy the triangular conditions.) Therefore, every real, square integrable function f(e,, 8,, (p2) in the range 0 5 Bi $ R, 0 5 40, 6 27~can be expanded according to (11)
In order to evaluate the coefficients, it is useful to start with a Fourier series in q2:
f(e, , e2, cpz) = C (“$:i mL0
“)COS mcp, + b,(@i ,0,) sin mq,) .
(12a)
PIlO
Since the associated Legendre polynomials of given order m are orthogonal to each othertt the remaining expansion is unambiguous:
4e, ,e,) =
c Al,,,,, mcoswz(c0s
"I,"2
0,)
b,(e,,e2)= C pmn,n,Pnm,(~~~ e,)C2(c0se,) nl,w
(m = 0, 1,2, ...),
W)
(m = 1,2,. ..>.
PC)
t The norm given by Biedenharn ‘) could not be confirmed. tt This point is not correctly given in Rose’s paper ‘).
TRIPLE
CORRELATION
481
FUNCTIONS
TABLE 1 Zeros 13‘ of the associated Legendre polynomials
P,m(cos 0) for n 5 4 (symmetrical at 90”)
n
6
0 1 2 3 4 1 2 3 4 2 3 4 3 4 4
90 54.6 39.2, 90 30.6, 70.1 0 0, 90 0, 63.4 0, 49.1, 90 0 0, 90 0, 67.8 0 0, 90 0
(de&
(Therefore, experimentalists will prefer to measure at angles ei which are zeros of the associated Legendre polynomials (cf. table 1) and at q2 = O”, -&go”, +45”, _+135”, . . .: This simplifies the determination of the coefficients CC,,,,,,,and /I,,,.,,,,.) Having determined the coefficients u,,,,, and j&,,n,n2,one easily finds the coefficients u~~~,.~.For no + nI + n, even, one has 1
=A,C-
a n0n1nz
2A,A, n 1+&o
(n,+m)!(n,+m)!
A,(no
(n,-m)!(n,-m)!
3 n1,
%)4m,n,
and for no+nI +n, odd: a “O”ln2
(~l+~)!(~2+~)!Am(n0,n1, ~2Mnnln2
L!L~
2%fi2 m (no-m)!(n,-m)!
W) The factors B,,,(n,, n,, nZ) follow from the recurrence relation B m+t =
-(n,+m)(n,-m++)(nl+m)(n,-m+l)B,_,
-[n&,+1)-n,(n,
+l)-n&z,+-1)+2m’]B,,
with
Bo(no ni , Q) = Ao(no n, n,), 31 A2 3
B,(no
3
7
n, , ~2)=
3
482
A. LINDNER
5. Application of triple correlation functions in polarization experiments In another paper “) we have explicitly shown the usefulness of triple correlation functions for complete three-body scattering experiments. This will now be done for polarization experiments. We rely on the results of an earlier paper “) where the particles were allowed to have spin s > 1. Their polarization was given in terms of Satchler’s s) spin tensors sv’ or in terms of density tensors
WI These complex quantities are connected with Lakin’s ‘) polarization recommended by the Madison convention lo):
tensors t,, ,
W) which are used here. The polarization produced by a magnetic field possesses axial symmetry around the field direction Q. In this case one has “) with real tl;’ tl”’ = tf’Cl”‘(i2).
(15)
It is shown in subsect. 5.1 that the differential cross section of a reaction initiated
by particles with this polarization can be expanded into triple correlation functions. The polarization of particles (with spin s > 3) produced in a reaction does not necessarily possess axial symmetry. Therefore, this polarization cannot be described by only one direction. Nevertheless it is possible to use triple correlation functions: In subsect. 5.2 we express the component of the polarization along some arbitrary direction a, t’“‘(R)
= c tS”‘Cyyi2) Y
= t(“)*(Lq,
(16)
in terms of triple correlation functions. The last equation defines a rotational invariant projection of the tensor of rank 12onto the unit vector in the direction $2 = (0, cp): If the tensor has a symmetry axis Sz,, i.e. if eq. (15) holds, then t’“‘(SZ)= tb”‘P,(cos e,>, with e1 being the angle between Q, and Q. Therefore, a scalar has t”‘(Q) = rho’ and a vector t’1’(8) = tc) cos d1 = ( t, cos cp+ t, sin cp)sin 8 + t, cos 6. Thus, for tensors of rank n s 1, the scalar t’“‘(Q) has the usual meaning of a component, not to be confused with the spherical component of tensors. Eq. (16) is a seemingly straightforward extension to tensors of higher rank. (Obviously, the component t’“‘(Q) is nothing more nor less than the spherical component tg’ referred to a coordinate system with z-axis along the direction a.)
TRIPLE CORRELATLON FUNCTIONS 5.1. REACTIONS Of: THE TYPE ikb
483
-+ c+d
Let us consider the case of particles a polarized along 0, and impinging along Szi on particles b, particles c being detected in the direction Q, (in the c.m. system). We have (17a) (According to the Madison convention 51, should fix the z-direction and Sz, the X-Z plane such that k, x k, is along y: We can use P,,, *,[& B,, cp,).) Defining the transition matrix 2’ as Newton ‘I), namely through S = 1-2ziT, one has in the channel spin representation “)
and in the j-r~presentatiou
“>
Parity conservation requires n, -l-n to be even *)_ The sum n&n-i-n, may be odd: Vector polarized particles (or, more generally, particles with n, odd) give rise to a well-known right-left asymmetry (cf. eq. (5b)). If only a single transition matrix element contributes (resonance), the sum nl -f-n + n, must be even (otherwise the coefficients in eq. (17) would not be real): In that case only polarization tensors of even degree contribute, which is a well-known fact.
484
A. LINDNER
If the polarization direction Q, is perpendicular to the reaction plane (0, = 90”, (Pi = +90”)a n d parity is conserved (8,-i-n even), one only needs
C,W& , 90°, t- 90°) (*
a
even)
(n, odd).
(18)
In this case (often applied!) the expansion into triple correlation functions seems to be superfluous since associated Legendre polynomials suffice. However, it should be noticed that the expansion into associated Legendre polynomials is only unambiguous if m is fixed - that is for pure vector-polarized spin-$ particles. In general, the data at d, = 90”, qo, = 490” do not fix unambiguously the expansion coefficients and then the triple correlation functions are needed at other angles too. 5.2. REACTIONS
OF THE TYPE a+b -+:-i-d
Another example of a scattering problem with three distinguished directions is a two-body reaction a+ b -+ -d+ d, where the polarization of the reaction product c is to be measured: Besides the directions of relative motion 8i and Or we take some direction 42, and ask for the component of the spin expectation value (the polarization) in this direction. (Vector polarization is directed along + ki x k, if parity is conserved.) The component of the polarization tensor of rank n along 52, is given by eq. (16). Using the results of the paper mentioned “) it is easy to show that t(“E)(.CJ,)can be expressed through triple correlation functions:
(According to the Madison convention the direction !2r should fix z and 52; the x-z plane such that ki x k, gives y_ Adopting this convention results in the triple correlation function PEtnn,(Oi, B,, cp,).) Using eq. (14b), we find “) for the expansion coefficients
X
(E, I(S, s&I TJlEi I(SaSb)S)*(Ef
F(S,Sd)S’( &JEi E’(S, S,)s>t
Wb)
TRIPLE CORRELATION
in the channel spin representation,
FUNCTIONS
485
and
(19c)
X
in the j-representation (if the transition matrix is defined i ‘) through S = l -2niT). Parity conservation requires E,+n to be even ‘). In this case all components t(“E)(S2,)with n, odd must vanish in the reaction plane (cp, = 0, cf. eq. (5b)). Therefore, vector polarization is directed perpendicular to the reaction plane (along + ki x k,). In this case the special triple correlation function of eq. (18) suffices to calculate tg’, since this is now equal to t (I) ( 90”, 90’). Following the Madison convention one would call this quantity ~%t,,. This result can be generalized: The direction ki x kf is a main axis in the following sense. Taking this axis as quantization axis (z-axis) parity conservation results - because of eq. (9) - in triple correlation functions Pfil,,,=(90”, CJ+;90”, 4oi;Q,) depending on the direction 52, through the spherical harmonics YenFc)(s2Jwith only even v,. Particles of spin 1 are usually described by the quantities t2,,, Re t,, and Re tz2 besides itI, given above. These quantities refer to the coordinate system recommended by the Madison convention. They can be calculated from the components tf2’(R,) along suitable directions, e.g. from tc2) (90”, 900), tc2) (90”, 0”) and tc2) (4.5”, 00): t 20 = - tC2)(900,O”)- t(2)(90”, 907, Re t 21 = -&[2t’2’(45”,
O’)+ tC2)(900,9071,
Re t 22 = &[tC2)(90”, 00)- t(2)(900, 90”)]. 6. Conclusion Biedenharn’s triple correlation functions ‘), real and of high symmetry, form a complete orthogonal and rotational invariant basis for all problems depending on three directions in space. We have shown how they can easily be calculated and how experiments should be planned to obtain the relevant expansion coefficients. In addition to the well-known case of sequential decay with well-defined intermediate states 1,2), triple correlation functions can be applied in complete three-body experiments “) and in those polarization experiments where three directions are relevant.
486
A. LINDNER
References 1) L. C. Biedenharn, in Nuclear spectroscopy B, ed. F. Ajzenberg-Selove (Academic Press, New York, 1960) 2) M. E. Rose, Triple correlations, ORNL-2516 (1958) 3) A. Lindner, Nucl. Phys. A199 (1973) 110 4) A. Lindner, Nucl. Phys. Al55 (1970) 145 5) L. C. Biedenharn and M. E. Rose, Rev. Mod. Phys. 25 (1953) 729 6) S. Devons and L. J. B. Goldfarb, in Handbuch der Physik, vol. 42 (Springer, Berlin, 1953) 7) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton Univ. Press, 1957) 8) G. R. Satchler, Nucl. Phys. 21 (1960) 116 9) W. Lakin, Phys. Rev. 98 (1955) 139 10) H. H. Barschall and W. Haeberli, ed., Polarization phenomena in nuclear reactions, Proc. Third Int. Symp. Madison 1970 (Univ. Wisconsin Press, 1971) 11) R. G. Newton, Scattering theory of waves and particles (McGraw-Hill, New York, 1966)
Nuclear Physics A230 (1974) 477-486; Not to be reproduced
by photoprint
@
or microfilm
TRIPLE CORRELATION FOR DESCRIBING
North-Holland without
Co.. Amsterdam
permission
from the publisher
FUNCTIONS
THREE-BODY
AND POLARIZATION
Publishing
written
REACTIONS
EXPERIMENTS
ALBRECHT LINDNER 1. Znstitut fiir Theoretische Physik der Unioersitiit Hamburg Received 21 January 1974 (Revised 11 March 1974) Abstract: The triple correlation functions P.o.l,Z (Q,, 9, , Q,) form a rotational invariant basis for all experiments depending on three directions. Their usefulness for the description of threebody reactions and polarization experiments is demonstrated. In the latter case the sum nofn1+n2 may be odd, pertaining to the well-known right-left asymmetries. Numerical methods are given to calculate these functions and the relevant expansion coefficients. The directions are tabulated which are best suited for experiments.
1. Applicabiiities of triple correlation functions Every physical problem depending on three directions in space, e.g. Q,, R, and Q,, can be described in terms of Biedenharn’s ‘) triple correlation functions, ~flOnlnl(Qo 2 Q2,3a,>
since these functions form a complete orthogonal system, as shown in sect. 4. (Here, -__ fi = J2n+l.) Until now, triple correlation functions have been considered in some detail only for even no, n, and n2 [refs. r* 2)], pertaining to the sequential decay of intermediate states with sharp parities. Triple correlation functions can be applied in a much wider range. They can be used in the description of all complete three-body scattering experiments “) (where parity conservation still requires the sum no + nl + n2 to be even) as well as in all those polarization experiments where three directions are relevant (ki, k, and the polarization direction). In the latter case, as shown implicitly in an earlier paper 4), the sum n,+n, +n2 may be odd, e.g. in the case of right-left asymmetries. This is detailed in sect. 5. 477
478
A. LINDNER
2. Symmetries and other triple correlation functions
Biedenharn’s triple correlation functions are real and very symmetrical, as follows from eq. (lb):
If one of the indices n,, n, or n2 is zero, one obtains a Legendre polynomial:
In the literature functions with different norm and phase are called triple correlation functions too. Yet, they are of lower symmetry and are not real, if n,+n, +n, is odd: Biederharn and Rose “) use Anon,n2(Q05Q, 7 Q,) = ~-“0-“‘+“2(479-*fio fi, fiZ,P”,,,“,(%, 0, Q2); 3
Devons and Goldfarb “) use ~“On,n2(%5 Q, >52,) = ~no+n’-“2&l fh fiZP”on,n2(Qo 9 Q, 3 Q,); Rose “) uses
3. The triple correlation functions P,_,nln@l, 02, q2) Until now we did not fix the coordinate system. The triple correlation functions are spherical invariants ‘). Choosing the z-direction along 8, and the x-z plane of our right-handed coordinate system to go through 52,, we retain the angles 8,, 0, and (p2 as (the only physically relevant) variables:
x
P;(COS
e,)qcos
8,)
exp
(hcp,).
(4
These triple correlation functions possess the symmetries
Pnond4 e2,(~~1 = cde2, el, q2) = (-)no+nl +~~~no.ln2(e1 , e2, - ~7~). 7
Pa> (w
TRIPLE CORRELATION
Evidently,
if no +n, +n,
479
FUNCTIONS
is even,
min (nl, 14
and, if n,+n, Konl”*(&
+n, is odd,
343 cpz) =
The real expansion
coefficients
A In+1 =
with the starting
min (III, nt) mz, 4n(~o 9 n,> %)c@
-
&I sin mcpz-
w
follow from the useful recurrence relation “)
A,-,+[no(no+l)-nI(nI+l)-n,(n,+1)+2mZ]A, (n,-m)(n,+m+l)(n,-m)(n,+m+l)
(7)
’
values
A
=
_
1
if no-i-n, +n,
4PXcos
n,(n,+l)-n,(n,+l)-n,(n,+l)~
02
is even, or
A, = 0, A, = J.,(.,+:).,(.,+l)
II:
= _ J(no+n,+nz+l)(-no+n,
“I’
4
+n,)(n,-nn,+n,)(no+n,-nn,+l) n,(n1+
l)u,(u*
+ 1) X
no n, 0
0
n,--I 0
1’ (8’3) 1’
if no+n,+n, is odd. The associated Legendre polynomials PT(cos 0) can be calculated by recurrence relations too, which is a well-known procedure ‘): Especially suited for eq. (6) is the recurrence relation ‘) p::+'(cose) = 2 m cot ep,m(cose) -(n
+ m)(n - m 4- l)P;-‘(~0s
but only for analytical expressions, since the relation 8 z 0. Using computers one should apply ‘)
cxx) =
is numerically
1 [(2n-l)xP,“_,(x)-(n+m-1)P,“_z(x)], n- m
e),
unstable
for sin
480
A. LINDNER
with Pz(cos 0) = (2m - l)! ! sinm0,
P;+ 1(~~~0) = (2m + l)!! cos 0 sinm 8. Often useful is the value at 8 = 90”: i”-”
e(o)
=
i
()
(n+m-l)!! -__ (n-m)!!
if
n-m
= 0,2,4,.
if
n-m=1,3,5
..
,....
(9
In this case the sums in eq. (6) run only over even or only over odd m. This is applied in eq. (18). 4. Expansion into triple correlation functions In their range of definition, the triple correlation functions form a complete orthogonal system’ :
=
8n
‘tel -e;) s(e2-e;) sin 8,
sin
e
2
(The integers no, nl and n2 must of course satisfy the triangular conditions.) Therefore, every real, square integrable function f(e,, 8,, (p2) in the range 0 5 Bi $ R, 0 5 40, 6 27~can be expanded according to (11)
In order to evaluate the coefficients, it is useful to start with a Fourier series in q2:
f(e, , e2, cpz) = C (“$:i mL0
“)COS mcp, + b,(@i ,0,) sin mq,) .
(12a)
PIlO
Since the associated Legendre polynomials of given order m are orthogonal to each othertt the remaining expansion is unambiguous:
4e, ,e,) =
c Al,,,,, mcoswz(c0s
"I,"2
0,)
b,(e,,e2)= C pmn,n,Pnm,(~~~ e,)C2(c0se,) nl,w
(m = 0, 1,2, ...),
W)
(m = 1,2,. ..>.
PC)
t The norm given by Biedenharn ‘) could not be confirmed. tt This point is not correctly given in Rose’s paper ‘).
TRIPLE
CORRELATION
481
FUNCTIONS
TABLE 1 Zeros 13‘ of the associated Legendre polynomials
P,m(cos 0) for n 5 4 (symmetrical at 90”)
n
6
0 1 2 3 4 1 2 3 4 2 3 4 3 4 4
90 54.6 39.2, 90 30.6, 70.1 0 0, 90 0, 63.4 0, 49.1, 90 0 0, 90 0, 67.8 0 0, 90 0
(de&
(Therefore, experimentalists will prefer to measure at angles ei which are zeros of the associated Legendre polynomials (cf. table 1) and at q2 = O”, -&go”, +45”, _+135”, . . .: This simplifies the determination of the coefficients CC,,,,,,,and /I,,,.,,,,.) Having determined the coefficients u,,,,, and j&,,n,n2,one easily finds the coefficients u~~~,.~.For no + nI + n, even, one has 1
=A,C-
a n0n1nz
2A,A, n 1+&o
(n,+m)!(n,+m)!
A,(no
(n,-m)!(n,-m)!
3 n1,
%)4m,n,
and for no+nI +n, odd: a “O”ln2
(~l+~)!(~2+~)!Am(n0,n1, ~2Mnnln2
L!L~
2%fi2 m (no-m)!(n,-m)!
W) The factors B,,,(n,, n,, nZ) follow from the recurrence relation B m+t =
-(n,+m)(n,-m++)(nl+m)(n,-m+l)B,_,
-[n&,+1)-n,(n,
+l)-n&z,+-1)+2m’]B,,
with
Bo(no ni , Q) = Ao(no n, n,), 31 A2 3
B,(no
3
7
n, , ~2)=
3
482
A. LINDNER
5. Application of triple correlation functions in polarization experiments In another paper “) we have explicitly shown the usefulness of triple correlation functions for complete three-body scattering experiments. This will now be done for polarization experiments. We rely on the results of an earlier paper “) where the particles were allowed to have spin s > 1. Their polarization was given in terms of Satchler’s s) spin tensors sv’ or in terms of density tensors
WI These complex quantities are connected with Lakin’s ‘) polarization recommended by the Madison convention lo):
tensors t,, ,
W) which are used here. The polarization produced by a magnetic field possesses axial symmetry around the field direction Q. In this case one has “) with real tl;’ tl”’ = tf’Cl”‘(i2).
(15)
It is shown in subsect. 5.1 that the differential cross section of a reaction initiated
by particles with this polarization can be expanded into triple correlation functions. The polarization of particles (with spin s > 3) produced in a reaction does not necessarily possess axial symmetry. Therefore, this polarization cannot be described by only one direction. Nevertheless it is possible to use triple correlation functions: In subsect. 5.2 we express the component of the polarization along some arbitrary direction a, t’“‘(R)
= c tS”‘Cyyi2) Y
= t(“)*(Lq,
(16)
in terms of triple correlation functions. The last equation defines a rotational invariant projection of the tensor of rank 12onto the unit vector in the direction $2 = (0, cp): If the tensor has a symmetry axis Sz,, i.e. if eq. (15) holds, then t’“‘(SZ)= tb”‘P,(cos e,>, with e1 being the angle between Q, and Q. Therefore, a scalar has t”‘(Q) = rho’ and a vector t’1’(8) = tc) cos d1 = ( t, cos cp+ t, sin cp)sin 8 + t, cos 6. Thus, for tensors of rank n s 1, the scalar t’“‘(Q) has the usual meaning of a component, not to be confused with the spherical component of tensors. Eq. (16) is a seemingly straightforward extension to tensors of higher rank. (Obviously, the component t’“‘(Q) is nothing more nor less than the spherical component tg’ referred to a coordinate system with z-axis along the direction a.)
TRIPLE CORRELATLON FUNCTIONS 5.1. REACTIONS Of: THE TYPE ikb
483
-+ c+d
Let us consider the case of particles a polarized along 0, and impinging along Szi on particles b, particles c being detected in the direction Q, (in the c.m. system). We have (17a) (According to the Madison convention 51, should fix the z-direction and Sz, the X-Z plane such that k, x k, is along y: We can use P,,, *,[& B,, cp,).) Defining the transition matrix 2’ as Newton ‘I), namely through S = 1-2ziT, one has in the channel spin representation “)
and in the j-r~presentatiou
“>
Parity conservation requires n, -l-n to be even *)_ The sum n&n-i-n, may be odd: Vector polarized particles (or, more generally, particles with n, odd) give rise to a well-known right-left asymmetry (cf. eq. (5b)). If only a single transition matrix element contributes (resonance), the sum nl -f-n + n, must be even (otherwise the coefficients in eq. (17) would not be real): In that case only polarization tensors of even degree contribute, which is a well-known fact.
484
A. LINDNER
If the polarization direction Q, is perpendicular to the reaction plane (0, = 90”, (Pi = +90”)a n d parity is conserved (8,-i-n even), one only needs
C,W& , 90°, t- 90°) (*
a
even)
(n, odd).
(18)
In this case (often applied!) the expansion into triple correlation functions seems to be superfluous since associated Legendre polynomials suffice. However, it should be noticed that the expansion into associated Legendre polynomials is only unambiguous if m is fixed - that is for pure vector-polarized spin-$ particles. In general, the data at d, = 90”, qo, = 490” do not fix unambiguously the expansion coefficients and then the triple correlation functions are needed at other angles too. 5.2. REACTIONS
OF THE TYPE a+b -+:-i-d
Another example of a scattering problem with three distinguished directions is a two-body reaction a+ b -+ -d+ d, where the polarization of the reaction product c is to be measured: Besides the directions of relative motion 8i and Or we take some direction 42, and ask for the component of the spin expectation value (the polarization) in this direction. (Vector polarization is directed along + ki x k, if parity is conserved.) The component of the polarization tensor of rank n along 52, is given by eq. (16). Using the results of the paper mentioned “) it is easy to show that t(“E)(.CJ,)can be expressed through triple correlation functions:
(According to the Madison convention the direction !2r should fix z and 52; the x-z plane such that ki x k, gives y_ Adopting this convention results in the triple correlation function PEtnn,(Oi, B,, cp,).) Using eq. (14b), we find “) for the expansion coefficients
X
(E, I(S, s&I TJlEi I(SaSb)S)*(Ef
F(S,Sd)S’( &JEi E’(S, S,)s>t
Wb)
TRIPLE CORRELATION
in the channel spin representation,
FUNCTIONS
485
and
(19c)
X
in the j-representation (if the transition matrix is defined i ‘) through S = l -2niT). Parity conservation requires E,+n to be even ‘). In this case all components t(“E)(S2,)with n, odd must vanish in the reaction plane (cp, = 0, cf. eq. (5b)). Therefore, vector polarization is directed perpendicular to the reaction plane (along + ki x k,). In this case the special triple correlation function of eq. (18) suffices to calculate tg’, since this is now equal to t (I) ( 90”, 90’). Following the Madison convention one would call this quantity ~%t,,. This result can be generalized: The direction ki x kf is a main axis in the following sense. Taking this axis as quantization axis (z-axis) parity conservation results - because of eq. (9) - in triple correlation functions Pfil,,,=(90”, CJ+;90”, 4oi;Q,) depending on the direction 52, through the spherical harmonics YenFc)(s2Jwith only even v,. Particles of spin 1 are usually described by the quantities t2,,, Re t,, and Re tz2 besides itI, given above. These quantities refer to the coordinate system recommended by the Madison convention. They can be calculated from the components tf2’(R,) along suitable directions, e.g. from tc2) (90”, 900), tc2) (90”, 0”) and tc2) (4.5”, 00): t 20 = - tC2)(900,O”)- t(2)(90”, 907, Re t 21 = -&[2t’2’(45”,
O’)+ tC2)(900,9071,
Re t 22 = &[tC2)(90”, 00)- t(2)(900, 90”)]. 6. Conclusion Biedenharn’s triple correlation functions ‘), real and of high symmetry, form a complete orthogonal and rotational invariant basis for all problems depending on three directions in space. We have shown how they can easily be calculated and how experiments should be planned to obtain the relevant expansion coefficients. In addition to the well-known case of sequential decay with well-defined intermediate states 1,2), triple correlation functions can be applied in complete three-body experiments “) and in those polarization experiments where three directions are relevant.
486
A. LINDNER
References 1) L. C. Biedenharn, in Nuclear spectroscopy B, ed. F. Ajzenberg-Selove (Academic Press, New York, 1960) 2) M. E. Rose, Triple correlations, ORNL-2516 (1958) 3) A. Lindner, Nucl. Phys. A199 (1973) 110 4) A. Lindner, Nucl. Phys. Al55 (1970) 145 5) L. C. Biedenharn and M. E. Rose, Rev. Mod. Phys. 25 (1953) 729 6) S. Devons and L. J. B. Goldfarb, in Handbuch der Physik, vol. 42 (Springer, Berlin, 1953) 7) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton Univ. Press, 1957) 8) G. R. Satchler, Nucl. Phys. 21 (1960) 116 9) W. Lakin, Phys. Rev. 98 (1955) 139 10) H. H. Barschall and W. Haeberli, ed., Polarization phenomena in nuclear reactions, Proc. Third Int. Symp. Madison 1970 (Univ. Wisconsin Press, 1971) 11) R. G. Newton, Scattering theory of waves and particles (McGraw-Hill, New York, 1966)