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Gamma-ray linear-polarization and angular-distribution formulas for mixtures of three multipoles
NIJCLEAR INSTRUMENTS
AND
METHODS
II 4
(1974)
333-34o;
©
NORTH-HOLLAND
PUBLISHING
CO.
GAMMA-RAY L I N E A R - P O L A R I Z A T I O N AND A N G U L A R - D I S T R I B U T I O N F O R M U L A S F O R M I X T U R E S OF T H R E E M U L T I P O L E S * R. V A I L L A N C O U R T and P. T A R A S
Laboratoire de Physique Nucldaire, University o f Montreal, Montreal, Canada Received 4 June 1973
G a m m a - r a y linear-polarization a n d angular-distribution form u l a s for mixtures o f three multipoles are presented in terms o f the phase-defined reduced matrix elements o f Rose a n d Brink. These f o r m u l a s are applicable to the simple a n g u l a r distribution as well as to the p a r t i c l e - g a m m a angular correlation k n o w n as
M e t h o d [I o f Litherland and Ferguson. In addition, a " r e c i p e " has been worked out for transforming an angular distribution formula into a linear polarization f o r m u l a in the general case o f a y radiation consisting of an arbitrary n u m b e r of multipoles.
1. Introduction
used to extract the sign and the magnitude of the mixing ratio, both of which can then be compared with the values predicted by a nuclear model, it is worthwhile at this stage to reproduce the phase conventions adopted by Rose and Brink. In their development, these authors have chosen the phase of their wavefunctions so that
The properties of the 3162 keV level in 35C1 and of the 1612 keV level in 3 7 A r require further analysis. Both of these levels are quoted in the literature 1) as having J~ - 7 - and a rather long mean lifetime, of the order of ns. I f indeed the 3162 keV level has J -~- , then a possible E1 transition to the 1763 keV, ~+ level is inhibited by a factor larger than 107 over the single particle estimate, while, if the 1612 keV level turned out to have J~ = ~ - , then again the E1 transition to the ~+ ground state would have a rather large inhibition, of the order of 106. The 7 decay of these two levels has been analyzed assuming mixtures of two multipoles only, i.e. E l M 2 or M2E3. It is solely on that basis that the values of J~ were assigned. In view of the rather large (possible in the case of 3 7 A r ) E1 inhibition, a further analysis of the y decay of these two levels in terms of a mixture of three multipoles, i.e. E1M2E3 is imperative. The assignment of J~ = 2s - would present an interesting challenge to the shell model. To our knowledge, general formulas for angular distribution and especially linear polarization of y rays consisting of mixtures of three multipoles have never been published. We do so in this paper. The formulas will be presented in terms of the phase-defined reduced matrix elements of Rose and Brink2), thus allowing a direct comparison of experimental and theoretical values of the sign and magnitude of the mixing ratio. These formulas should be very useful, as mixtures of three multipoles do seem to occur in nature3). 2. General
Since the formulas presented in this paper can be
0 [ ~ J M ) = ( - l ) J-M I ~ J - M ) ,
(1)
where 0 is the time-reversal operator and [ ~ J M ) is a discrete eigenstate of a time-reversal invariant hamiltonian H, which has only angular momentum degeneracy. This choice has the advantage of being invariant with respect to angular momentum coupling. In addition, the phases of the vector-coupling coefficients and of the spherical harmonics conform to the convention of Condon and Shortley4). Rose and Brink also fix the phase of their interaction multipole operators TLM such that OTLMO -1 = ( - - l) L - M T L - M .
(2)
This particular choice of phase ensures that the matrix elements of TLM are real if the wavefunctions transform according to eq. (1). In terms of the effective multipole operators c:.<~> ~LM the interaction multipole operators are given by the relation "~LM ,
(3)
where ~ = 0 for electric-interaction multipole operators and the superscript ( ~ = 0 ) or ( 0 ) simply means "electric", while rc-- 1 for magnetic-interaction multipole operators and the superscript ( r e = 1) or ( 1 ) simply means "magnetic". The coefficients cd~.> are
* Partially supported by a grant from the National Research
Council of Canada.
c~ = [(ik)L/(2L - 1)! !] [ ( L + 1)/(2L)] ~ ,
333
(4a)
334
R. V A I L L A N C O U K T
and m~ ~L
--ic~
(4b)
for electric and magnetic radiation, respectively. The relation between the reduced matrix element defined by Bohr and Mottelson 5) and that defined by Rose and Brink z) is then
A N D P. T A R A S
In eq. (7), q = 1 corresponds to right-handed and q=-1 to left handed-circular polarization with respect to ez,=k/lkl. The probability amplitude A~M2(k) is given by the relation
x ~ q~(Jj M I IT~2IJ2M2) D~q(R). LM~ (9)
(J1 H,/W(nL)HJ2)BM =
FL-( 2 L + l ) ( 2 J , +
1)'1k (JalIG2.>IIJz)RB,
(5)
where J~ and J2 are the spins of the initial and final states, respectively. The corresponding relation given by Rose and Brink is in error. It should be noted that in the formalism of Rose and Brink, the initial state of a transition is always written on the left of a matrix element and the final state on the right. The factor [(2L + 1)/(470] ~ in eq. (5) arises from the different definitions of the transition operators used by Bohr and Mottelson and Rose and Brink. These two groups of authors also utilize different versions of the Wigner-Eckart theorem; this gives rise to the factor (2J~ +1) ÷ in eq. (5). In the formalism of Rose and Brink, the Wigner-Eckart theorem has the form
D~tq(R) is the usual rotation matrix and the rotation R = (qs, 0, £2) is the rotation with Euler angles ~, 0, £2 taking the z axis to the direction of k, i.e. to the direction of the z' axis. We shall use eqs. (8) and (9) as our starting point to demonstrate the general formulas for 7-ray angular distribution and linear polarization and to obtain a "recipe" which can be utilized to transform an angulardistribution formula into a linear-polarization formula.
3. Angular-distributionformulas A polarization insensitive measurement requires an incoherent sum over q, thus in cases where the polarization is not observed the total 7-ray transition probability is given by the relation
(J1M1 ITLMIJ2ME)
P(k) = ~
w(M1)PMI(k),
(10)
M1
=(--1)2L(jzLM2MIJ~Ma)(J~IITLIIJ2),
(6)
where PM, (k) = ~ I-IAM~ o + 1 2 + ]Aq~ 7u12(k)12]. M2(k)[ =
where the phase factor ( - 1)2L can be dropped since L can only take integer values in the treatment of electromagnetic radiation. Using the above phase conventions, Rose and Brink obtain, after some straightforward but rather lengthy calculations, a formula for the total probability for a transition from a state ]J1MI> to a state [J2M2 > with emission of a circularly polarized photon along the direction of k. This formula is
Making use of eqs. (6) and (9), eq. (11) becomes PM~(k) = M2MLL' ~ ,~,~'( 2 ~ h ) (J2LM2MIJI M1) x
x ( J 2 E M 2 M I J 1 M I ) ( J 1 I[TL<=>I[J2) x x (JIIIT{,~'>IIJz)ED~+ I DM L'*+ a +
+ ( _ 1)~+.'
Pq(k) = ~w(M1)Pq,(k) = ~ w ( M , ) ~" IAq,M2(k)l 2 , M~
M1
M2
(7) where the population parameters w (Ml) of the substate M~ are normalized such that ~w(M1)---- 1,
-JI<_MI<_J1.
(8)
M1
This formula is valid if the radiating system is in a cylindrically symmetric environment so that Ma is a constant of the motion if the z axis is chosen along the symmetry axis. In a nuclear reaction the z axis would correspond to the beam direction.
(I1)
M2
L _ 1 DM L'*_ 1]. DM
(12)
Using the properties of rotation matrices and of Clebsch-Gordan coefficients6) we obtain the angulardistribution formula for the emission of photons from a state J1 leading to a state Jz: P(k) =
k ~ BK(J 0 RK(LEJ , J:) 2 zrh LL"~" K X
x
(J~ HT<'>I[J2) (J1 IITI[J2) ( 2 L + 1) w2 ( 2 E + 1) 1/2
X
x [1 + ( - l) ~+''+L+L'-K] PK(COS 0),
(13)
FORMULAS
FOR
MIXTURES
where the "statistical tensor" B K( J l ) is defined as
B~(JI) = ~ (-- I)S'-M'(2J, + I) '/z X MI
x (dl Jt MI - M11K0) w(M1),
(14)
and the R coefficient, introduced by Rose and Brink, is
Rj~(LEJ1J2)
= ( - - 1) 1 +Jl-Jz+L'-L-K
X
x [(2J a + 1)(2L+ 1)(2E + 1)] l/a x
x (LEI - I[K0) W(Ja J1LE; K J2).
(15) It is convenient to write eq. (13) in terms of ratios of reduced matrix elements, i.e. in terms of mixing ratios defined as (5 < " >
6(L, n / E # )
-
_ (J, I[ T<">H Jz)/(2L+ l ) I / 2 m (Jr IIT~>IiJ:)/(2E+ 1) I/2
--
[r~(L, ~z)] '/~ [F~(E,~)] 1/2'
(16) where L, ff stand for the lowest order multipolarity occurring in the transition J~ ~ Jz and F~ is the partial y-width of the state of spin J~. Eq. (13) then becomes
P(k)-
k I(J~l[T~")llJ2)12x 2nh (2E+I)
OF THREE
335
MULTIPOLES
tion being that the initial state must have a definite value of J1. In the present work only initial and final states of definite parity will be considered, hence eq. (20) can be simplified further. Since the electromagnetic interaction conserves parity, the sum in this equation no longer runs independently over all possible LL'nn' values because there is now a fixed relationship between the L and n values. That is: a) If J~ and J2 have the same parity, then in combination with electric (n = 0) radiation only even values of L can occur whereas in combination with magnetic (n = 1) radiation only odd values of L can occur i.e. only M1, E2, M3, etc., radiations are allowed. b) Similarly, if J1 and J2 have opposite parity, then only El, M2, E3, etc., radiations are allowed. Hence all terms appearing in eq. (20) have L + L ' + +n+n'=even. The factor [1 + ( - 1 ) =+='+L+L'-K] is zero if K is odd and 2 if K is even. Thus if circular polarization is not observed and if all nuclear states have definite parity, then eq. (20) contains only terms with even K, the terms with odd values of K being zero. Thus eq. (20) becomes
w(o) :=
E
[ BK(J1) RK(LLtJ1J2)(~Ln}(~L~'')PK(cOSO)]
K even
L
(21)
x
y~ BK(J,)[I+(--1y+~'+L+L'-K]X
KLL'7tn"
x Rr(LL'J 1J2)6 6 PK(cos 0).
(17)
Experimentally the angular distribution is obtained in the form
W(O) = ~aKPK(cos O) = P(k)/f, K
(18)
where the normalization constant N is chosen such that
[W(O)]K:O = a0 = 1.0.
(19)
Thus, finally the angular distribution of 7 rays in terms of mixing ratios, when polarization is not observed, is given by the relation
W(O) =
~ BK(J1)[l + ( - l) '~+''+L+L'-K] x KLL"nn' x R K ( L E J 1J 2 )
~Or> 6 < n ' >L'L
X
x [2 ~ la<">12] -' PK(cos 0).
Ln
(20)
This equation is completely general, its sole restric-
1) Eq. (21) is restricted to the case that J1 and J2 have definite parity. 2) The sum over (Ln) is over all multipoles consistent with conservation of angular momentum and parity. The allowed values of L and L ' are
]J1-J2I
L,E¢O.
3) The sum is taken over even values of K only. Even if the initial state is polarized and not just aligned (i.e. BK(J1):~O when K is odd), odd order K terms cannot appear in eq. (21) unless the circular polarization is observed, but then eq. (21) is no longer valid.
4. Linear-polarization formulas In this section the development of the formulas will be presented in greater detail to facilitate the comprehension of the linear polarization process. By convention, the direction of the polarization vector t of a photon is chosen parallel to the direction of the electric vector E. I f the z' axis of a system of coordinates x', y', z' is chosen along the direction of
336
R. V A I L L A N C O U R T
propagation k of the photon, then it is quite natural to choose the x' axis to be either parallel to the E vector or perpendicular to it. Any other choice would lead to the same result but the intermediate steps will be algebraically more cumbersome. In this paper we shall take the x' axis along the direction of the E vector. With this choice, the probability amplitude for a linearly polarized photon takes the form '--
q=-I
q=+l
/,¢
A e~,, M2(k) = ½~j2EAM1M2(k)--AM, M2( )].
(22)
This expression arises from the fact that the probability amplitudes A~IM: (k) are the components of a spherical tensor of rank one and these must transform like the components of the position unit vector. The total 7-ray transition probability for the emission of a linearly polarized photon from a state ]J1M1) to a state ]J2M2) is then given by a relation equivalent to eq. (10)
pe(k ) = Z w ( M , ) P Me, ( k ) ,
(23)
M1
where e k ) = ½ ~ IA~MI:(k) --AM,M:( "~=+l"k"z PM,( )1
(24)
M2
AND
P. T A R A S
the z axis to the z' axis i.e. the direction k, using the rotation R=(q~, 0, (2). The angle 0 represents the angle between the z axis and the direction of emission of the photon while O is the angle between the direction of the E vector and the plane of the reaction defined by the z and z' axes. Since we are dealing with states of definite parity, as in section 3, [1 "b(--1) '~+n'+L+L'-K]= 2 and [ e x p ( i 2 ( 2 ) + ( - 1)"+''+L+L'-K exp(-i2f2)] = 2cos2Q. Then, using the relation between the product of three Clebsch-Gordan coefficients and a Racah coefficient, eqs. (14) and (15), and the definition of the coefficient
KK(LE)= _ ~ ( K - 2 ) ! - ] 1/2 (LE 1 ILK2) , L(K +2)!3 (LL' 1 -ILK0)
(26)
and substituting the result in eq. (23), we obtain
pe(k ) -- k BK(J1) RK(LE JI J2) × 2 nh KLL',~,~' >< (Jx I/T<~>HJ2) (J1 IIf[IJ2> [PK(cos 0)+ ( 2 L + 1) 1/2 ( 2 E + 1)1/2
Making use of eqs. (6) and (9), eq. (24) becomes + ( - 1)" KK (LE) (cos 2(2) p~Z)(cos 0)]. PM~(
)=½
E ~ M2MLL' nn'
(J;LM2MIJ1M1)x
Introducing the mixing ratios O defined by eq. (16), we get
x(J2EM2MIJ1 MI)(J~ lir~>ll J2)
pe(k) - - -k i(Jx ]lT{')ilJ2)i 2 x 2nh (2 E,+ 1)
x ( J , l[ TLIIJ2) x xE(_l)~ DLu - , _ DL+ 1] [(-- 1)~' DM-, L'* -- DM+ L'* 1] •
x
Using the properties of rotation matrices and of Clebsch-Gordan coefficients we get
PM,( )
S
~
B~(J,) RK(LL'JtJ2)bL<'>6{7 '> [-PK(COS0)+
KLL'nn'
+ ( - 1 ) ~' KK(LE)(cos 2f2) P~2)(cos 0)].
x (J2EM2M]J1M~) (Ja HTL<'>[]J2) (J1 UTL<'~'>HJ2) × × ( -- 1)M +~ (LE M - M[ KO)(LL' 1 - 1[K0) ×
F(K-2)!] ~
--]~)L ~ J
[W(0, Q)]K=o = [W(0)]K=o = 1.0,
(29)
(30)
i.e. it will correspond to the normalization of the angular distribution, as given by eq. (19). Thus, finally the linear-polarization distribution of 7 rays is given by the relation
[exp(+i2Y2)+
+(-1)'~+'e+L+L'-Kexp(--i2Y2)]P~Z)(cos 0)}.
W(O, Q) = Pe(k)/N',
where the normalization constant N' is chosen such that
{[1 + ( - - l) ~+'c+L+L'-K] PK(COS 0)--(-- 1)"' ×
(LL'llIK2) x (LE 1
(28)
Experimentally the linear-polarization distribution is obtained in the form
(J2LMzMIJ1M1)×
M2M,LL'7tn'K
X
(27)
(25)
W(O, Q) = The angles 0 and ~2 appear in the Euler rotation taking
~ (Ln)(L' n') K even
BK(J1) RK(LEJ1J2) x
337
F O R M U L A S FOR MIXTURES OF THREE M U L T I P O L E S TABLE 1
× 6~
+ ( - 1)" Kx(LE)(cos 2f2) PkZ)(cos 0)3.
(31)
Possible values of L, L', 6L and 6L' in cases of mixtures of three multipoles.
Comparing eqs. (21) and (31), both of which are quite general formulas, it is evident that a general method of obtaining a linear-polarization distribution formula from an angular-distribution formula is to make the substitution
P,; (cos 0) ~ PK (cos 0) + + ( - 1)"' KK(LE)(cos 20) P~2)(cos 0),
(32)
where, as has been stated in sec. 2, r r ' = 0 for electric and 7r'= 1 for magnetic radiation. The "recipe" [eq. (32)] had previously been quoted by Fagg and Hanna 7) for a mixture of two multipoles; in this paper, the recipe has been shown to be valid for a mixture of any number of multipoles. In the process, it is hoped that the reader has gained a deeper insight into the phenomenon of y-ray linear polarization.
E
E
l
l
E E
L+I L+2
1 I
61 ?Je
t-}-I
L
(~1
1
L+I L+ 1 L+2 [+2
L+I L+2 L E+l
61 61 ?J~ ?~2
61 6z 1 ~
L+2
L+2
62
62
The possible values of L, L', 6~~> and fi~,~'>are given in table l for mixtures of three multipoles. For ease of notation, we redefine L = L, L + 1 =- E, L + 2 = E' ,
whence the mixing ratios 61 and 6z can be expressed as
5. Mixtures of three multipo~:s 5.1. ANGULAR:D1STRIBU'I-IONFORMULAS In general it can be stated that if an initial state of definite parity lJ1 M1 > is formed in a nuclear reaction and if neither the beam of the incident particles nor the target nuclei are polarized then there will be alignment with respect to the beam axfs. We speak of an alignment of the initial state J1 if
w ( - M O = w(Ml),
(33)
and call this the alignment condition. In this case the statistical tensors BK(J1) which describe the nuclear alignment can be written as BK(J1) = ~ ( - -
c51_--
(38) 62 -=
(Jl iIT~">IIJD/(2L+ l) k
W(O) = E B K ( J 1 ) [RK(LLJ 1J 2 ) q- 261 Rr(LI' J1 J 2 ) + K
+62 Rr(L' L' J1 J2) +26162 Rr(L' E' J1J2)+ + 262 RK(LE'J1Jz) + 62 RK(L'L"JI J2)] x x PK(COS 0)/(1 +62+62),
x (J1 J1 M1 - M I I K 0 ) w(M1),
(34)
and the population parameters w(M~) as X
K X (J1 J1 M 1 - M1 [K0) BK(J1) ,
(35)
where Ew(M1)
( J l I]TLU'>IIJ2)/(2E' + 1)k
Substitution of the values of table 1 in eq. (21) gwes the y-ray angular distribution for a mixture of three multipoles:
MI
MI
(Jl IITL<,='>IIJ2)/(2L' + 1)½ (dl I]TL<~>I[Jz)/( 2 L + 1) ~: '
I)J~-M'(2J 1 + 1) 1/2 x
w(MO = (2-6M,, o) ~ ( - 1)s' -M'(2J1 + 1)- 1/2
=
1,
and Bo(J 0 = 1.
(37)
where use has been made of the symmetric property in L and L' of the Rr(LL'JaJ2) coefficient. Eq. (39) is applicable to the mixture of any three multipoles, e.g. dipole-quadrupole-octupole, or quadrupole-octupole-hexadecapole mixtures. In practice only E1M2E3 mixtures are expected to occur. In that case L = 1, L' = 2 , L " = 3 and
(36)
In the summations we now have M I > 0 and the population of the negative magnetic substates has been folded into the population of the positive magnetic substates, i.e. w(M1) - w ( M l ) + w ( - M t ) .
(39)
6~ = 6(M2/EI) =
/3 (JI lIT(M2)IIJ2), V 5 (J1 I[T(E1)IIJz)
and
(40)
~/~(JI"T(E3)"J2) 6 2 - fi(E3/E1) = (J~ I[T(EI)IIJ:>
338
R. V A I L L A N C O U R T
In addition, eq. (39) describes a simple 7-ray angular distribution following a particle in-particle out reaction or a particle 7 capture reaction as well as the particle-7 angular correlation known as Method I I of Litherland and Ferguson8). Physically, these reactions differ from each other in the way the initial state is aligned, i.e. in the number and magnitude of the population parameters w(M~) of the initial state. This is discussed in greater detail by Rose and Brink 2) and by Tarasg), who also discuss the use of the attenuation factors QK due to the finite size of the counters. Eq. (39) gives the angular distribution in terms of mixing ratios of a 7 ray originating from the decay of a state of spin J1 whose alignment is described by the tensor BK(J1). Quite frequently the decay of the state of spin J1 proceeds through a gamma cascade to the ground state (fig. 1) and it is desirable to obtain a distribution formula for any one of the subsequent 7-rays in terms of the population parameters w(M~) of the magnetic substates of the initial state of spin J1. In fig. 1, the observed 7 ray is the one without subscripts and is emitted in a transition from the state of spin J, to a state of spin J,+ ~, with mixing ratios 61 and ~ 2 ; the other 7 rays are assumed r o b e unobserved. By analogy with eq. (39), the angular distribution of the observed Y ray is
W(O)=~BK(Jn)[RK(LLJnJn+ l)+ 261 RK(LE J.J.+1)+ K
A N D P. T A R A S
by the relation 1°)
BK(J,) = BK(J1) UK(J1J2) UK(J2J3)..- UK(Jn-1J,), (42) where
-2
UK(J, 1J,) =
¢) Ln - t L.- I
UK(L,_,j,_Ij,)/, ~ (~L.-2 1 Ln- 1
(43) and
UK(Ln_~Jn_1Jn)
=
W(J. _ 1J. Jn- 1J. ; L._ 1K)
W(J,-1J, J,-tJ,; L,±I 0)
,
(44)
so that Uo (L, and
1 Jn-
1 Jn)
=
1,
U K ( L . _ I J . _ L J . ) = UK(L._IJ.J._a).
The coefficients 6L,_ * are the mixing ratios of the unobserved 7 rays emitted between the states of spin J . - 1 and J,. Since these ? rays are unobserved there are no interference terms between the different multipolarities L,_ 1, L~_ 1 and L~_ ~ in eq. (43). Thus for unobserved 7 rays consisting of mixtures of three multipoles
UK(J,- 1J,) =
[UK(Ln_IJn aJn) + ( 6-21 ) n - - 1 U K ( I Jt' n - I J n - ' J n ) + -~- ( ~ 22) n -- 1 ( L n,,-
~
(45)
Jn - 1 s.)] [1 + 0 ~ ) . _ ~ + ( ~ ) . _ I ] - L
+ 52 RK(E E J.J.+ l) + 26~ 62 RK(E E' J,,J.+ l) +
+232RK(LE, J,j,+l)+ Oz -2 RK(L.... 1£ JnJn+ l)] x x PK(cos 0)/(1 +621 +62),
(41)
where the statistical tensors BK(J,) of the state of spin J, can be related to the statistical tensors BK(J1)
5.2. LINEAR-POLARIZATION FORMULAS In order to transform the angular-distribution formula into a linear-polarization distribution formula with the "recipe" given by eq. (32) it is convenient to express eqs. (39) and (41) in a different form such that
W(O) = B~kL(O)+26 , WLL,(O)+6~[4k'L'(O)+ +26132 WL,L,,(O)+232 WLL"(O)+6~WL"L"(O),
w(M I ) L
j
,L I ,L'I' dl i
n
d2
I' L 2 ' L 2 ' L 2
d3 I
I I I '1
i
(46)
where in the case of the observed Y ray emitted in the transition from a state of spin J1 to a state of spin J 2
WLL'(O) = EBK(J1) RK(LEJ, J 2 ) x K
X(I ,
,din- J
Lrri,Ln-i,l'n-I
L,L',L"
Jn dn+l
Fig. 1, Example o f a cascade of unobserved 7 rays occurring before the observed transition L, L', L " . The angular-distribution formula can be expressed in terms of the statistical tensors BK(J1) or BK(Jn).
,'2 +61"2 +02)
-1
PK(COS0),
(47)
and in the case of ( n - 1) unobserved y rays preceding the observed y-ray transition between the states of spin J, and J, +1 WLL,(O ) =
~Br(J.)
RK(LE
J.J.+
1) x
K
x (l + 62 + 62)-1 PK(COS 0),
(48)
F O R M U L A S F O R M I X T U R E S OF T H R E E M U L T I P O L E S
the statistical tensor Br(J,) being given by eq. (42). The formula for the linear-polarization distribution of a ~ ray consisting of a mixture of three multipoles can then be written down immediately. It is
However, for the sake of completeness we shall present the relation that exists between the measured quantities and the linear-polarization parameterp. This parameter is usually defined as
W(O, 0) = WLz(O, f2)+ 260 WLL,(O,0)+6~ WL,L,(O, 0)+ -]-26162 WL,L,,(O, 0 ) + 2 6 2 WLL"(O,0)+62 WZ,,L"(O,0), (49)
p =
Jo ']'90
:
W(O,0 = 0 °)
(53)
W(O,O : 90 °)
According to this definition p (E 1M2E3) = [p (M 1E2M 3)]- 1,
where WLL,(0, O) : Z B , , ( J O R K ( L L ' J ~ J g ( 1 K
+ 6 1 ~+ 6 2 ) ~ - ' X
(50) or
WCL.(O,n) = ~BK(Jn)RK(LEJ, J,+I)(I +62 +(~2)-~ x K
x [PK (COS0) + ( -- 1)~' K K(LE) (cos 20) P~2)(cos 0)], (51) depending on whether unobserved transitions precede the observed transition [eq. (51)] or not [eq. (50)]. Formulas for mixtures of only two multipoles can be: obtained immediately by putting fi2 = 0 in the preceding equations. In cases where the population parameters of the initial state of spin J1 are not known it may become necessary to express the linear-polarization distribution in terms of the measured angular-distribution (or correlation) coefficients aJao. This can be done by comparing eqs. (18), (39), (49) and (50). Eq. (49) still applies but eq. (50) [or eq. (51)] is transformed according te the substitution: 1 + 62 + 62) - 1 ~ -ak - IRK(EL J1J2) 7tao
+26x RK(LE J1Jz) + 62 Rr(E E Ja J2)+
(54)
Usually the experiment consists of measuring the intensity of y rays, Compton scattered in the reaction plane (No) and perpendicular to it (N9o). These quantities are related to the linear-polarization parameter p through the Klein-Nishina cross section da~ such that No = Jodao=o-l-Jgodo'q,:90, and
(55) N9o = Jo da~ = 90 + J9o da~o= o,
where (p is the angle between the E vector and the scattering plane defined by the propagation vectors of the incident and scattered y rays. For a Compton polarimeter made up of point detectors the asymmetry ratio is given by R=da¢=9o/da~,=o and if we let N = N9o/N o, we obtain from eq. (55)
N=(pR+I)/(p+R),
orp=(1-NR)/(N-R).
(56)
The value of the asymmetry ratio taking into account the finite size of the counters of a real Compton polarimeter has been discussed by Taras and Matas11). Some authors use a different definition of p given by the relation p' = ( J o - Jgo)/(Jo + Jgo)" 6. Conclusions
+ 26162 RK(EE'Ja J2) + 26z Rr(LE'J1J2)+ + 02 RK(E,E,j, j2)] -1.
and p (E2M3E4) = [p (M2E3 M4)]- ~.
x [PK(COS 0 ) + ( - - I ) ~ ' K r ( L E ) ( c o s 20) P(rZ)(cos O)],
BK(J0(
339
(52)
This equation is valid regardless of possible unobserved y-rays preceding the observed transition between states of spin J1 and J2 (or J, and J,+ t). It should be noted however that there may exist a linear polarization even though the angular distribution is isotropic, thus great care must be exercised in the use of eq. (52). Measurements of linear polarization and applications of the above formulas to specific cases have been discussed extensively in a review article by Tarasg).
In this paper we have demonstrated a "recipe" which allows the transformation of an angular-distribution formula into a linear-polarization distribution formula for a ~ ray consisting of a mixture of any number of multipoles. The formulas for mixtures of any three multipoles were then presented. These were further reduced to the only case expected in nature, namely E1M2E3 radiation. These formulas describe the processes following a particle in - particle out reaction or a particle-~ capture reaction as well as the (particle-~/) angular correlation known as Method II of Litherland and Ferguson. In this last case, the polarization should
340
R. V A I L L A N C O U R T
be measured in coincidence with the particle detector placed along the symmetry axis. In this way the statistical tensors will be the same in the polarization and in the correlation measurements, and the two measurements can then be analyzed simultaneously. Such a simultaneous analysis should normally lead to a unique determination of the spin and parity of the initial state as well as of the mixing ratio of its 7-ray decay. References 1) p. M. Endt and C. Van der Leun, Nucl. Phys. A105 (1967) 1. e) H. J. Rose and D. M. Brink, Rev. Mod. Phys. 39 (1967) 306.
A N D P. T A R A S a) A. G. Schmidt, E. G. Funk, and J. W. Mihelich, Bull. Am. Phys. Soc. 18 (1973) 582. 4) E. U. Condon and G. H. Shortley, Theor7 o f atomic spectra (Cambridge University Press, London, 1935). 5) A. Bohr and B. R. Mottelson, Kgl. Danske. Videnskab. Selskab Mat.-F:ys. Medd. 27 (1953) 1. 6) A. J. Ferguson, Angular correlation methods in gamma-ray spectroscopy (North-Holland Publ. Co., Amsterdam, 1965). 7) L. W. Fagg and S. S. Hanna, Rev. Mod. Phys. 31 (1959) 711. 8) A. E. Litherland and A. J. Ferguson, Can. J. Phys. 39 (1961) 788. 9) p. Taras, Can. J. Phys. 49 (1971) 328. 10) A. R. Poletti and E. K. Warburton, Phys. Rev. 137 (1965) B595. 11) p. Taras and J. Matas, Nucl. Instr. and Meth. 61 (1968) 317.
AND
METHODS
II 4
(1974)
333-34o;
©
NORTH-HOLLAND
PUBLISHING
CO.
GAMMA-RAY L I N E A R - P O L A R I Z A T I O N AND A N G U L A R - D I S T R I B U T I O N F O R M U L A S F O R M I X T U R E S OF T H R E E M U L T I P O L E S * R. V A I L L A N C O U R T and P. T A R A S
Laboratoire de Physique Nucldaire, University o f Montreal, Montreal, Canada Received 4 June 1973
G a m m a - r a y linear-polarization a n d angular-distribution form u l a s for mixtures o f three multipoles are presented in terms o f the phase-defined reduced matrix elements o f Rose a n d Brink. These f o r m u l a s are applicable to the simple a n g u l a r distribution as well as to the p a r t i c l e - g a m m a angular correlation k n o w n as
M e t h o d [I o f Litherland and Ferguson. In addition, a " r e c i p e " has been worked out for transforming an angular distribution formula into a linear polarization f o r m u l a in the general case o f a y radiation consisting of an arbitrary n u m b e r of multipoles.
1. Introduction
used to extract the sign and the magnitude of the mixing ratio, both of which can then be compared with the values predicted by a nuclear model, it is worthwhile at this stage to reproduce the phase conventions adopted by Rose and Brink. In their development, these authors have chosen the phase of their wavefunctions so that
The properties of the 3162 keV level in 35C1 and of the 1612 keV level in 3 7 A r require further analysis. Both of these levels are quoted in the literature 1) as having J~ - 7 - and a rather long mean lifetime, of the order of ns. I f indeed the 3162 keV level has J -~- , then a possible E1 transition to the 1763 keV, ~+ level is inhibited by a factor larger than 107 over the single particle estimate, while, if the 1612 keV level turned out to have J~ = ~ - , then again the E1 transition to the ~+ ground state would have a rather large inhibition, of the order of 106. The 7 decay of these two levels has been analyzed assuming mixtures of two multipoles only, i.e. E l M 2 or M2E3. It is solely on that basis that the values of J~ were assigned. In view of the rather large (possible in the case of 3 7 A r ) E1 inhibition, a further analysis of the y decay of these two levels in terms of a mixture of three multipoles, i.e. E1M2E3 is imperative. The assignment of J~ = 2s - would present an interesting challenge to the shell model. To our knowledge, general formulas for angular distribution and especially linear polarization of y rays consisting of mixtures of three multipoles have never been published. We do so in this paper. The formulas will be presented in terms of the phase-defined reduced matrix elements of Rose and Brink2), thus allowing a direct comparison of experimental and theoretical values of the sign and magnitude of the mixing ratio. These formulas should be very useful, as mixtures of three multipoles do seem to occur in nature3). 2. General
Since the formulas presented in this paper can be
0 [ ~ J M ) = ( - l ) J-M I ~ J - M ) ,
(1)
where 0 is the time-reversal operator and [ ~ J M ) is a discrete eigenstate of a time-reversal invariant hamiltonian H, which has only angular momentum degeneracy. This choice has the advantage of being invariant with respect to angular momentum coupling. In addition, the phases of the vector-coupling coefficients and of the spherical harmonics conform to the convention of Condon and Shortley4). Rose and Brink also fix the phase of their interaction multipole operators TLM such that OTLMO -1 = ( - - l) L - M T L - M .
(2)
This particular choice of phase ensures that the matrix elements of TLM are real if the wavefunctions transform according to eq. (1). In terms of the effective multipole operators c:.<~> ~LM the interaction multipole operators are given by the relation "~LM ,
(3)
where ~ = 0 for electric-interaction multipole operators and the superscript ( ~ = 0 ) or ( 0 ) simply means "electric", while rc-- 1 for magnetic-interaction multipole operators and the superscript ( r e = 1) or ( 1 ) simply means "magnetic". The coefficients cd~.> are
* Partially supported by a grant from the National Research
Council of Canada.
c~ = [(ik)L/(2L - 1)! !] [ ( L + 1)/(2L)] ~ ,
333
(4a)
334
R. V A I L L A N C O U K T
and m~ ~L
--ic~
(4b)
for electric and magnetic radiation, respectively. The relation between the reduced matrix element defined by Bohr and Mottelson 5) and that defined by Rose and Brink z) is then
A N D P. T A R A S
In eq. (7), q = 1 corresponds to right-handed and q=-1 to left handed-circular polarization with respect to ez,=k/lkl. The probability amplitude A~M2(k) is given by the relation
x ~ q~(Jj M I IT~2IJ2M2) D~q(R). LM~ (9)
(J1 H,/W(nL)HJ2)BM =
FL-( 2 L + l ) ( 2 J , +
1)'1k (JalIG2.>IIJz)RB,
(5)
where J~ and J2 are the spins of the initial and final states, respectively. The corresponding relation given by Rose and Brink is in error. It should be noted that in the formalism of Rose and Brink, the initial state of a transition is always written on the left of a matrix element and the final state on the right. The factor [(2L + 1)/(470] ~ in eq. (5) arises from the different definitions of the transition operators used by Bohr and Mottelson and Rose and Brink. These two groups of authors also utilize different versions of the Wigner-Eckart theorem; this gives rise to the factor (2J~ +1) ÷ in eq. (5). In the formalism of Rose and Brink, the Wigner-Eckart theorem has the form
D~tq(R) is the usual rotation matrix and the rotation R = (qs, 0, £2) is the rotation with Euler angles ~, 0, £2 taking the z axis to the direction of k, i.e. to the direction of the z' axis. We shall use eqs. (8) and (9) as our starting point to demonstrate the general formulas for 7-ray angular distribution and linear polarization and to obtain a "recipe" which can be utilized to transform an angulardistribution formula into a linear-polarization formula.
3. Angular-distributionformulas A polarization insensitive measurement requires an incoherent sum over q, thus in cases where the polarization is not observed the total 7-ray transition probability is given by the relation
(J1M1 ITLMIJ2ME)
P(k) = ~
w(M1)PMI(k),
(10)
M1
=(--1)2L(jzLM2MIJ~Ma)(J~IITLIIJ2),
(6)
where PM, (k) = ~ I-IAM~ o + 1 2 + ]Aq~ 7u12(k)12]. M2(k)[ =
where the phase factor ( - 1)2L can be dropped since L can only take integer values in the treatment of electromagnetic radiation. Using the above phase conventions, Rose and Brink obtain, after some straightforward but rather lengthy calculations, a formula for the total probability for a transition from a state ]J1MI> to a state [J2M2 > with emission of a circularly polarized photon along the direction of k. This formula is
Making use of eqs. (6) and (9), eq. (11) becomes PM~(k) = M2MLL' ~ ,~,~'( 2 ~ h ) (J2LM2MIJI M1) x
x ( J 2 E M 2 M I J 1 M I ) ( J 1 I[TL<=>I[J2) x x (JIIIT{,~'>IIJz)ED~+ I DM L'*+ a +
+ ( _ 1)~+.'
Pq(k) = ~w(M1)Pq,(k) = ~ w ( M , ) ~" IAq,M2(k)l 2 , M~
M1
M2
(7) where the population parameters w (Ml) of the substate M~ are normalized such that ~w(M1)---- 1,
-JI<_MI<_J1.
(8)
M1
This formula is valid if the radiating system is in a cylindrically symmetric environment so that Ma is a constant of the motion if the z axis is chosen along the symmetry axis. In a nuclear reaction the z axis would correspond to the beam direction.
(I1)
M2
L _ 1 DM L'*_ 1]. DM
(12)
Using the properties of rotation matrices and of Clebsch-Gordan coefficients6) we obtain the angulardistribution formula for the emission of photons from a state J1 leading to a state Jz: P(k) =
k ~ BK(J 0 RK(LEJ , J:) 2 zrh LL"~" K X
x
(J~ HT<'>I[J2) (J1 IIT
X
x [1 + ( - l) ~+''+L+L'-K] PK(COS 0),
(13)
FORMULAS
FOR
MIXTURES
where the "statistical tensor" B K( J l ) is defined as
B~(JI) = ~ (-- I)S'-M'(2J, + I) '/z X MI
x (dl Jt MI - M11K0) w(M1),
(14)
and the R coefficient, introduced by Rose and Brink, is
Rj~(LEJ1J2)
= ( - - 1) 1 +Jl-Jz+L'-L-K
X
x [(2J a + 1)(2L+ 1)(2E + 1)] l/a x
x (LEI - I[K0) W(Ja J1LE; K J2).
(15) It is convenient to write eq. (13) in terms of ratios of reduced matrix elements, i.e. in terms of mixing ratios defined as (5 < " >
6(L, n / E # )
-
_ (J, I[ T<">H Jz)/(2L+ l ) I / 2 m (Jr IIT~>IiJ:)/(2E+ 1) I/2
--
[r~(L, ~z)] '/~ [F~(E,~)] 1/2'
(16) where L, ff stand for the lowest order multipolarity occurring in the transition J~ ~ Jz and F~ is the partial y-width of the state of spin J~. Eq. (13) then becomes
P(k)-
k I(J~l[T~")llJ2)12x 2nh (2E+I)
OF THREE
335
MULTIPOLES
tion being that the initial state must have a definite value of J1. In the present work only initial and final states of definite parity will be considered, hence eq. (20) can be simplified further. Since the electromagnetic interaction conserves parity, the sum in this equation no longer runs independently over all possible LL'nn' values because there is now a fixed relationship between the L and n values. That is: a) If J~ and J2 have the same parity, then in combination with electric (n = 0) radiation only even values of L can occur whereas in combination with magnetic (n = 1) radiation only odd values of L can occur i.e. only M1, E2, M3, etc., radiations are allowed. b) Similarly, if J1 and J2 have opposite parity, then only El, M2, E3, etc., radiations are allowed. Hence all terms appearing in eq. (20) have L + L ' + +n+n'=even. The factor [1 + ( - 1 ) =+='+L+L'-K] is zero if K is odd and 2 if K is even. Thus if circular polarization is not observed and if all nuclear states have definite parity, then eq. (20) contains only terms with even K, the terms with odd values of K being zero. Thus eq. (20) becomes
w(o) :=
E
[ BK(J1) RK(LLtJ1J2)(~Ln}(~L~'')PK(cOSO)]
K even
L
(21)
x
y~ BK(J,)[I+(--1y+~'+L+L'-K]X
KLL'7tn"
x Rr(LL'J 1J2)6
(17)
Experimentally the angular distribution is obtained in the form
W(O) = ~aKPK(cos O) = P(k)/f, K
(18)
where the normalization constant N is chosen such that
[W(O)]K:O = a0 = 1.0.
(19)
Thus, finally the angular distribution of 7 rays in terms of mixing ratios, when polarization is not observed, is given by the relation
W(O) =
~ BK(J1)[l + ( - l) '~+''+L+L'-K] x KLL"nn' x R K ( L E J 1J 2 )
~Or> 6 < n ' >L'L
X
x [2 ~ la<">12] -' PK(cos 0).
Ln
(20)
This equation is completely general, its sole restric-
1) Eq. (21) is restricted to the case that J1 and J2 have definite parity. 2) The sum over (Ln) is over all multipoles consistent with conservation of angular momentum and parity. The allowed values of L and L ' are
]J1-J2I
L,E¢O.
3) The sum is taken over even values of K only. Even if the initial state is polarized and not just aligned (i.e. BK(J1):~O when K is odd), odd order K terms cannot appear in eq. (21) unless the circular polarization is observed, but then eq. (21) is no longer valid.
4. Linear-polarization formulas In this section the development of the formulas will be presented in greater detail to facilitate the comprehension of the linear polarization process. By convention, the direction of the polarization vector t of a photon is chosen parallel to the direction of the electric vector E. I f the z' axis of a system of coordinates x', y', z' is chosen along the direction of
336
R. V A I L L A N C O U R T
propagation k of the photon, then it is quite natural to choose the x' axis to be either parallel to the E vector or perpendicular to it. Any other choice would lead to the same result but the intermediate steps will be algebraically more cumbersome. In this paper we shall take the x' axis along the direction of the E vector. With this choice, the probability amplitude for a linearly polarized photon takes the form '--
q=-I
q=+l
/,¢
A e~,, M2(k) = ½~j2EAM1M2(k)--AM, M2( )].
(22)
This expression arises from the fact that the probability amplitudes A~IM: (k) are the components of a spherical tensor of rank one and these must transform like the components of the position unit vector. The total 7-ray transition probability for the emission of a linearly polarized photon from a state ]J1M1) to a state ]J2M2) is then given by a relation equivalent to eq. (10)
pe(k ) = Z w ( M , ) P Me, ( k ) ,
(23)
M1
where e k ) = ½ ~ IA~MI:(k) --AM,M:( "~=+l"k"z PM,( )1
(24)
M2
AND
P. T A R A S
the z axis to the z' axis i.e. the direction k, using the rotation R=(q~, 0, (2). The angle 0 represents the angle between the z axis and the direction of emission of the photon while O is the angle between the direction of the E vector and the plane of the reaction defined by the z and z' axes. Since we are dealing with states of definite parity, as in section 3, [1 "b(--1) '~+n'+L+L'-K]= 2 and [ e x p ( i 2 ( 2 ) + ( - 1)"+''+L+L'-K exp(-i2f2)] = 2cos2Q. Then, using the relation between the product of three Clebsch-Gordan coefficients and a Racah coefficient, eqs. (14) and (15), and the definition of the coefficient
KK(LE)= _ ~ ( K - 2 ) ! - ] 1/2 (LE 1 ILK2) , L(K +2)!3 (LL' 1 -ILK0)
(26)
and substituting the result in eq. (23), we obtain
pe(k ) -- k BK(J1) RK(LE JI J2) × 2 nh KLL',~,~' >< (Jx I/T<~>HJ2) (J1 IIf[IJ2> [PK(cos 0)+ ( 2 L + 1) 1/2 ( 2 E + 1)1/2
Making use of eqs. (6) and (9), eq. (24) becomes + ( - 1)" KK (LE) (cos 2(2) p~Z)(cos 0)]. PM~(
)=½
E ~ M2MLL' nn'
(J;LM2MIJ1M1)x
Introducing the mixing ratios O defined by eq. (16), we get
x(J2EM2MIJ1 MI)(J~ lir~>ll J2)
pe(k) - - -k i(Jx ]lT{')ilJ2)i 2 x 2nh (2 E,+ 1)
x ( J , l[ TL
x
Using the properties of rotation matrices and of Clebsch-Gordan coefficients we get
PM,( )
S
~
B~(J,) RK(LL'JtJ2)bL<'>6{7 '> [-PK(COS0)+
KLL'nn'
+ ( - 1 ) ~' KK(LE)(cos 2f2) P~2)(cos 0)].
x (J2EM2M]J1M~) (Ja HTL<'>[]J2) (J1 UTL<'~'>HJ2) × × ( -- 1)M +~ (LE M - M[ KO)(LL' 1 - 1[K0) ×
F(K-2)!] ~
--]~)L ~ J
[W(0, Q)]K=o = [W(0)]K=o = 1.0,
(29)
(30)
i.e. it will correspond to the normalization of the angular distribution, as given by eq. (19). Thus, finally the linear-polarization distribution of 7 rays is given by the relation
[exp(+i2Y2)+
+(-1)'~+'e+L+L'-Kexp(--i2Y2)]P~Z)(cos 0)}.
W(O, Q) = Pe(k)/N',
where the normalization constant N' is chosen such that
{[1 + ( - - l) ~+'c+L+L'-K] PK(COS 0)--(-- 1)"' ×
(LL'llIK2) x (LE 1
(28)
Experimentally the linear-polarization distribution is obtained in the form
(J2LMzMIJ1M1)×
M2M,LL'7tn'K
X
(27)
(25)
W(O, Q) = The angles 0 and ~2 appear in the Euler rotation taking
~ (Ln)(L' n') K even
BK(J1) RK(LEJ1J2) x
337
F O R M U L A S FOR MIXTURES OF THREE M U L T I P O L E S TABLE 1
× 6~
+ ( - 1)" Kx(LE)(cos 2f2) PkZ)(cos 0)3.
(31)
Possible values of L, L', 6L and 6L' in cases of mixtures of three multipoles.
Comparing eqs. (21) and (31), both of which are quite general formulas, it is evident that a general method of obtaining a linear-polarization distribution formula from an angular-distribution formula is to make the substitution
P,; (cos 0) ~ PK (cos 0) + + ( - 1)"' KK(LE)(cos 20) P~2)(cos 0),
(32)
where, as has been stated in sec. 2, r r ' = 0 for electric and 7r'= 1 for magnetic radiation. The "recipe" [eq. (32)] had previously been quoted by Fagg and Hanna 7) for a mixture of two multipoles; in this paper, the recipe has been shown to be valid for a mixture of any number of multipoles. In the process, it is hoped that the reader has gained a deeper insight into the phenomenon of y-ray linear polarization.
E
E
l
l
E E
L+I L+2
1 I
61 ?Je
t-}-I
L
(~1
1
L+I L+ 1 L+2 [+2
L+I L+2 L E+l
61 61 ?J~ ?~2
61 6z 1 ~
L+2
L+2
62
62
The possible values of L, L', 6~~> and fi~,~'>are given in table l for mixtures of three multipoles. For ease of notation, we redefine L = L, L + 1 =- E, L + 2 = E' ,
whence the mixing ratios 61 and 6z can be expressed as
5. Mixtures of three multipo~:s 5.1. ANGULAR:D1STRIBU'I-IONFORMULAS In general it can be stated that if an initial state of definite parity lJ1 M1 > is formed in a nuclear reaction and if neither the beam of the incident particles nor the target nuclei are polarized then there will be alignment with respect to the beam axfs. We speak of an alignment of the initial state J1 if
w ( - M O = w(Ml),
(33)
and call this the alignment condition. In this case the statistical tensors BK(J1) which describe the nuclear alignment can be written as BK(J1) = ~ ( - -
c51_--
(38) 62 -=
(Jl iIT~">IIJD/(2L+ l) k
W(O) = E B K ( J 1 ) [RK(LLJ 1J 2 ) q- 261 Rr(LI' J1 J 2 ) + K
+62 Rr(L' L' J1 J2) +26162 Rr(L' E' J1J2)+ + 262 RK(LE'J1Jz) + 62 RK(L'L"JI J2)] x x PK(COS 0)/(1 +62+62),
x (J1 J1 M1 - M I I K 0 ) w(M1),
(34)
and the population parameters w(M~) as X
K X (J1 J1 M 1 - M1 [K0) BK(J1) ,
(35)
where Ew(M1)
( J l I]TLU'>IIJ2)/(2E' + 1)k
Substitution of the values of table 1 in eq. (21) gwes the y-ray angular distribution for a mixture of three multipoles:
MI
MI
(Jl IITL<,='>IIJ2)/(2L' + 1)½ (dl I]TL<~>I[Jz)/( 2 L + 1) ~: '
I)J~-M'(2J 1 + 1) 1/2 x
w(MO = (2-6M,, o) ~ ( - 1)s' -M'(2J1 + 1)- 1/2
=
1,
and Bo(J 0 = 1.
(37)
where use has been made of the symmetric property in L and L' of the Rr(LL'JaJ2) coefficient. Eq. (39) is applicable to the mixture of any three multipoles, e.g. dipole-quadrupole-octupole, or quadrupole-octupole-hexadecapole mixtures. In practice only E1M2E3 mixtures are expected to occur. In that case L = 1, L' = 2 , L " = 3 and
(36)
In the summations we now have M I > 0 and the population of the negative magnetic substates has been folded into the population of the positive magnetic substates, i.e. w(M1) - w ( M l ) + w ( - M t ) .
(39)
6~ = 6(M2/EI) =
/3 (JI lIT(M2)IIJ2), V 5 (J1 I[T(E1)IIJz)
and
(40)
~/~(JI"T(E3)"J2) 6 2 - fi(E3/E1) = (J~ I[T(EI)IIJ:>
338
R. V A I L L A N C O U R T
In addition, eq. (39) describes a simple 7-ray angular distribution following a particle in-particle out reaction or a particle 7 capture reaction as well as the particle-7 angular correlation known as Method I I of Litherland and Ferguson8). Physically, these reactions differ from each other in the way the initial state is aligned, i.e. in the number and magnitude of the population parameters w(M~) of the initial state. This is discussed in greater detail by Rose and Brink 2) and by Tarasg), who also discuss the use of the attenuation factors QK due to the finite size of the counters. Eq. (39) gives the angular distribution in terms of mixing ratios of a 7 ray originating from the decay of a state of spin J1 whose alignment is described by the tensor BK(J1). Quite frequently the decay of the state of spin J1 proceeds through a gamma cascade to the ground state (fig. 1) and it is desirable to obtain a distribution formula for any one of the subsequent 7-rays in terms of the population parameters w(M~) of the magnetic substates of the initial state of spin J1. In fig. 1, the observed 7 ray is the one without subscripts and is emitted in a transition from the state of spin J, to a state of spin J,+ ~, with mixing ratios 61 and ~ 2 ; the other 7 rays are assumed r o b e unobserved. By analogy with eq. (39), the angular distribution of the observed Y ray is
W(O)=~BK(Jn)[RK(LLJnJn+ l)+ 261 RK(LE J.J.+1)+ K
A N D P. T A R A S
by the relation 1°)
BK(J,) = BK(J1) UK(J1J2) UK(J2J3)..- UK(Jn-1J,), (42) where
-2
UK(J, 1J,) =
¢) Ln - t L.- I
UK(L,_,j,_Ij,)/, ~ (~L.-2 1 Ln- 1
(43) and
UK(Ln_~Jn_1Jn)
=
W(J. _ 1J. Jn- 1J. ; L._ 1K)
W(J,-1J, J,-tJ,; L,±I 0)
,
(44)
so that Uo (L, and
1 Jn-
1 Jn)
=
1,
U K ( L . _ I J . _ L J . ) = UK(L._IJ.J._a).
The coefficients 6L,_ * are the mixing ratios of the unobserved 7 rays emitted between the states of spin J . - 1 and J,. Since these ? rays are unobserved there are no interference terms between the different multipolarities L,_ 1, L~_ 1 and L~_ ~ in eq. (43). Thus for unobserved 7 rays consisting of mixtures of three multipoles
UK(J,- 1J,) =
[UK(Ln_IJn aJn) + ( 6-21 ) n - - 1 U K ( I Jt' n - I J n - ' J n ) + -~- ( ~ 22) n -- 1 ( L n,,-
~
(45)
Jn - 1 s.)] [1 + 0 ~ ) . _ ~ + ( ~ ) . _ I ] - L
+ 52 RK(E E J.J.+ l) + 26~ 62 RK(E E' J,,J.+ l) +
+232RK(LE, J,j,+l)+ Oz -2 RK(L.... 1£ JnJn+ l)] x x PK(cos 0)/(1 +621 +62),
(41)
where the statistical tensors BK(J,) of the state of spin J, can be related to the statistical tensors BK(J1)
5.2. LINEAR-POLARIZATION FORMULAS In order to transform the angular-distribution formula into a linear-polarization distribution formula with the "recipe" given by eq. (32) it is convenient to express eqs. (39) and (41) in a different form such that
W(O) = B~kL(O)+26 , WLL,(O)+6~[4k'L'(O)+ +26132 WL,L,,(O)+232 WLL"(O)+6~WL"L"(O),
w(M I ) L
j
,L I ,L'I' dl i
n
d2
I' L 2 ' L 2 ' L 2
d3 I
I I I '1
i
(46)
where in the case of the observed Y ray emitted in the transition from a state of spin J1 to a state of spin J 2
WLL'(O) = EBK(J1) RK(LEJ, J 2 ) x K
X(I ,
,din- J
Lrri,Ln-i,l'n-I
L,L',L"
Jn dn+l
Fig. 1, Example o f a cascade of unobserved 7 rays occurring before the observed transition L, L', L " . The angular-distribution formula can be expressed in terms of the statistical tensors BK(J1) or BK(Jn).
,'2 +61"2 +02)
-1
PK(COS0),
(47)
and in the case of ( n - 1) unobserved y rays preceding the observed y-ray transition between the states of spin J, and J, +1 WLL,(O ) =
~Br(J.)
RK(LE
J.J.+
1) x
K
x (l + 62 + 62)-1 PK(COS 0),
(48)
F O R M U L A S F O R M I X T U R E S OF T H R E E M U L T I P O L E S
the statistical tensor Br(J,) being given by eq. (42). The formula for the linear-polarization distribution of a ~ ray consisting of a mixture of three multipoles can then be written down immediately. It is
However, for the sake of completeness we shall present the relation that exists between the measured quantities and the linear-polarization parameterp. This parameter is usually defined as
W(O, 0) = WLz(O, f2)+ 260 WLL,(O,0)+6~ WL,L,(O, 0)+ -]-26162 WL,L,,(O, 0 ) + 2 6 2 WLL"(O,0)+62 WZ,,L"(O,0), (49)
p =
Jo ']'90
:
W(O,0 = 0 °)
(53)
W(O,O : 90 °)
According to this definition p (E 1M2E3) = [p (M 1E2M 3)]- 1,
where WLL,(0, O) : Z B , , ( J O R K ( L L ' J ~ J g ( 1 K
+ 6 1 ~+ 6 2 ) ~ - ' X
(50) or
WCL.(O,n) = ~BK(Jn)RK(LEJ, J,+I)(I +62 +(~2)-~ x K
x [PK (COS0) + ( -- 1)~' K K(LE) (cos 20) P~2)(cos 0)], (51) depending on whether unobserved transitions precede the observed transition [eq. (51)] or not [eq. (50)]. Formulas for mixtures of only two multipoles can be: obtained immediately by putting fi2 = 0 in the preceding equations. In cases where the population parameters of the initial state of spin J1 are not known it may become necessary to express the linear-polarization distribution in terms of the measured angular-distribution (or correlation) coefficients aJao. This can be done by comparing eqs. (18), (39), (49) and (50). Eq. (49) still applies but eq. (50) [or eq. (51)] is transformed according te the substitution: 1 + 62 + 62) - 1 ~ -ak - IRK(EL J1J2) 7tao
+26x RK(LE J1Jz) + 62 Rr(E E Ja J2)+
(54)
Usually the experiment consists of measuring the intensity of y rays, Compton scattered in the reaction plane (No) and perpendicular to it (N9o). These quantities are related to the linear-polarization parameter p through the Klein-Nishina cross section da~ such that No = Jodao=o-l-Jgodo'q,:90, and
(55) N9o = Jo da~ = 90 + J9o da~o= o,
where (p is the angle between the E vector and the scattering plane defined by the propagation vectors of the incident and scattered y rays. For a Compton polarimeter made up of point detectors the asymmetry ratio is given by R=da¢=9o/da~,=o and if we let N = N9o/N o, we obtain from eq. (55)
N=(pR+I)/(p+R),
orp=(1-NR)/(N-R).
(56)
The value of the asymmetry ratio taking into account the finite size of the counters of a real Compton polarimeter has been discussed by Taras and Matas11). Some authors use a different definition of p given by the relation p' = ( J o - Jgo)/(Jo + Jgo)" 6. Conclusions
+ 26162 RK(EE'Ja J2) + 26z Rr(LE'J1J2)+ + 02 RK(E,E,j, j2)] -1.
and p (E2M3E4) = [p (M2E3 M4)]- ~.
x [PK(COS 0 ) + ( - - I ) ~ ' K r ( L E ) ( c o s 20) P(rZ)(cos O)],
BK(J0(
339
(52)
This equation is valid regardless of possible unobserved y-rays preceding the observed transition between states of spin J1 and J2 (or J, and J,+ t). It should be noted however that there may exist a linear polarization even though the angular distribution is isotropic, thus great care must be exercised in the use of eq. (52). Measurements of linear polarization and applications of the above formulas to specific cases have been discussed extensively in a review article by Tarasg).
In this paper we have demonstrated a "recipe" which allows the transformation of an angular-distribution formula into a linear-polarization distribution formula for a ~ ray consisting of a mixture of any number of multipoles. The formulas for mixtures of any three multipoles were then presented. These were further reduced to the only case expected in nature, namely E1M2E3 radiation. These formulas describe the processes following a particle in - particle out reaction or a particle-~ capture reaction as well as the (particle-~/) angular correlation known as Method II of Litherland and Ferguson. In this last case, the polarization should
340
R. V A I L L A N C O U R T
be measured in coincidence with the particle detector placed along the symmetry axis. In this way the statistical tensors will be the same in the polarization and in the correlation measurements, and the two measurements can then be analyzed simultaneously. Such a simultaneous analysis should normally lead to a unique determination of the spin and parity of the initial state as well as of the mixing ratio of its 7-ray decay. References 1) p. M. Endt and C. Van der Leun, Nucl. Phys. A105 (1967) 1. e) H. J. Rose and D. M. Brink, Rev. Mod. Phys. 39 (1967) 306.
A N D P. T A R A S a) A. G. Schmidt, E. G. Funk, and J. W. Mihelich, Bull. Am. Phys. Soc. 18 (1973) 582. 4) E. U. Condon and G. H. Shortley, Theor7 o f atomic spectra (Cambridge University Press, London, 1935). 5) A. Bohr and B. R. Mottelson, Kgl. Danske. Videnskab. Selskab Mat.-F:ys. Medd. 27 (1953) 1. 6) A. J. Ferguson, Angular correlation methods in gamma-ray spectroscopy (North-Holland Publ. Co., Amsterdam, 1965). 7) L. W. Fagg and S. S. Hanna, Rev. Mod. Phys. 31 (1959) 711. 8) A. E. Litherland and A. J. Ferguson, Can. J. Phys. 39 (1961) 788. 9) p. Taras, Can. J. Phys. 49 (1971) 328. 10) A. R. Poletti and E. K. Warburton, Phys. Rev. 137 (1965) B595. 11) p. Taras and J. Matas, Nucl. Instr. and Meth. 61 (1968) 317.