Abstract. A general formula is given for correlation between two polarized gamma rays (y, and -yZ) emitted in a cascade from an oriented (for example. due to a ...
Nuclear
Instruments
and Methods
in Physics Research A 378
( 1996)
5 18-525
NUCLEAR INSTRUMENTS & METHODS IN PllVSlCS RESEARCH
I -\ M
SectIon A
ELSWIER
PDCO: Polarizational-directional Ch. Droste”‘*, S.G. Rohoziiiski”,
correlation
from oriented nuclei
K. Starosta”, T. Morek”. J. Srebrny”, P. Magierskib
Received
29 December
1995
Abstract A general formula is given for correlation between two polarized gamma rays (y, and -yZ)emitted in a cascade from an oriented (for example. due to a heavy ion reaction) nucleus. It allows us or one to calculate the angular correlation between: (a) Linear polarizations of y, and y?. (b) Polarization of y, and direction of yZ or vice versa. (c) Directions of y, and y2 (DCO). The formula, discussed in detail for the case (b). can be used in the analysis of data coming from the modern multidetector gamma ray spectrometers that contain new generation detectors (e.g. CLOVER) sensitive to the polarization. The analysis of polarization together with DC0 ratio can lead to a unique spin/parity assignment and a mixing ratio determination.
1. Introduction Multidetector y-ray spectrometers located on the heavy ion beam facilities have recently become the standard instruments in nuclear spectroscopy. Large amounts of experimental data and their complexity call for an effective method of spin and parity assignment. Usually the information about the spin of excited states is obtained from a study of angular correlation between y-rays emitted from aligned nuclei (DC0 method [I] ). This method does not give information about parities, because the DCO-function does not depend on the electric or magnetic character of the radiation. To determine spin and parity one should combine the results of DC0 analysis with measurements of the linear polarization of the y-transitions. In large multidetector arrays the segmented Ge detectors (e.g. CLOVER) are used [2,3]. A construction of these detectors and their position in the array (perpendicular to the ion beam axis) make them suitable for y-polarization measurements [4]. The in-beam polarization measurements without coincidences (in singles) are limited to the most intense and well separated y-lines following nuclear reactions. Nowadays, the large total efficiency of multidetector arrays allows one or us to carry out coincidence measurements between the y-ray polarimeters and the remaining detectors. Thus .the polarization measurements of weak transitions or complex y-spectra become possible. In Section 2 we present a general formula that allows us to calculate the angular correlation between two polarized gamma quanta (y, and y?yz)emitted in a cascade from an
oriented nucleus. The orientation can be the result of a nuclear reaction. In the derivation of the formula we follow a general theory of angular correlation of y-rays [5]. In Section 3 we discuss in detail a particular case of the formula when the correlation between polarization of y, and direction of emission of y2 (or vice versa) is studied. The conclusions are presented in Section 4. Theoretical and experimental aspects of angular correlation and polarization of y-rays are widely discussed in the literature, see for example Refs. i 1.S- I S].
2. Angular correlation of two polarized gamma rays emitted in cascade from an oriented nuclear state Let an initial nuclear state I*,,) be deexcited via the successive emission of two photons, y, and yZ of momenta k, and k2 leading, through an intermediate state (9,) to a tinal state I*,). Energies, spins and parities of the states are denoted E,, I, and n-,, respectively, for j = 0. I ,2 (Fig.
1). The initial state is supposed to be oriented. The orientation is given by n.,,$M~,). the population of magnetic substates with the angular momentum projections M,, on a initial state
IYOuo>
inrmn&te
1yl > _
YI
0168.9002/96/$15.00 P/I
SO I68-9002(
Fig.
author. Copyright 96 )00426-3
0
1996 Elsevier
Science B.V. All rights reserved
---
El ,I1,n1
Y2 finalstate
*Corresponding
.%J,~O,~O
IYz>
1. Emission
--
I
~729127x2
of two successive
photons.
C. llroste
et al. I NW/.
Z- axis e,_, this axis is called the orientation
laboratory
519
Instr. trnd Meth. in Phvs. Rrs. A 378 (1996) S/8-.53
axis.
Apart from the momenta k, and k2, the linear polarization of one or both photons is observed whereas the orientation of
the tinal
emission
nuclear
state is not. The
geometry
of the
process is shown in Fig. 2. When the orientation
is due to a heavy
ion reaction
the axis e; coincides
with
that of the ion beam. The plane spanned on vectors k, and e; is called the emission 8, between
plane of the photon Y,. The angle
these vectors
angle. The angle between Y, and
Yr is
4,
The
i\
referred
angle
plane(i.e.
the plane in which
field
electromagnetic
of
polarimeter
detecting
to as the emission
the emission
planes containing
between
the
the intensity
wave
photon
is
polarization
of the electric
measured)
of
the
Y, and the corresponding
emission plane is denoted by $,. We consider the following four
cases of angular
directional
correlation
from an oriented tional
correlation
photons:
W,,,, of two gamma quanta emitted
nucleus (DCO).
correlations
W*,,*, of two
W,,,
and
q,,
the polarizational-direc(PDCO)
and
W, ,
correlation
Thus, if the linear polarization
of the photon y, (j = I, 2) is
y, is set equal to
it is
nuclear
first-order system
following
c) Integer
perturbation with
formula
theory
of interaction
the electromagnetic
for the angular
field
correlation
gives
of the emission
and the detection
of two
of a
p from 0 to min( A, .A).
Thus, only even values of A,, enter in Eq. of whether the oriented Quantities
equal to 0. The
Geometry
c.
(PPCO).
I, otherwise
L
the
polarizational-polarizational measured a parameter
Fig.
polarized gamma quanta. The ion beam coincides with the :-axis
initial
of the right-hand
( 1) regardless
state is aligned or polarized. side of Eq.
C1) are defined as
follows:
the
[ lh,l7]: (2) the orientation
parameter
of the initial
state.
and
A:;-(y)
“T%(TL.y
= &
)U’(T’L’,Y)
.ZZ,(-1)
x H,(LL’jF:‘““(LL’Z,l,), the generalized
directional
(4)
and polarizational
coefficients,
respectively
for photon
distribution
coefficients
are expressed
absolute
transition
(electric for T=E
amplitudes
for
distribution
y (= Y, or yZ). The in
terms
radiations
or magnetic for T=M)
of
of
the
type
T
and multipolarity
L:
x ~~lIJw~)II~) Summations a) Even where [I]
in Eq. (I)
are over:
A,, and A, up to 2[1,,] and 2[1, I. respectively means the integer
b) Integer
A from
IA,, - A,
part of 1.
1to A,, + A,.
and the total transition cu~y)=C14TL,y)Iz. ,..T
(5)
probability (6)
520
C. Droste et al. I Nucl. Instr. and Meth. in Phys. Res. A 378 (1996) 518-525
Indices i and f stand for initial and tinal states, respectively. The symbolic Kronecker delta S,, is equal to 1 for T=M and 0 for T=E. Algebraic coefficients in Eq. (3) and Eq. (4) are: the generalized F-coefficient [8] connected with Fano’s 9j-symbol [18] in the following way
polarization plane of the polarimeter detecting emission plane of -yZ.Then we have:
y, and the
Wy,,2(sz.lCl,,,~~.,.r,~~) =
(2-q,)(2-q,)
F:‘“^(LL’I,J,)
32~’ x
c
o,~,(r,,)~{~(n,,OA,OlhO)A::,*“(y,
h(,,,i,- cvcn
J
x Lf$,(y,W,:,:(f$I - q,A’:::crz)d,::(B,)cos2~~1
(7)
- 3q,(A,,OA,21A2)A:1*“(y,) and the ratio of Clebsch-Gordan
coefficients X [Al::,j(y,)d:(:(eZ)c0~2~,*
(L-l
HA(LL’) =
L’-
I)#-2)
(8)
(L _ 1 L’]IAO)
- ;q&f;(X)
The definitions and notation of Bohr and Mottelson [ 191 for the electromagnetic transition operators. reduced matrix elements and Clebsch-Gordan coefficients are used. The emission angle dependence is given through the Wigner functions dLV [20]. Possible multipolarities TL in Eq. (5) take values such that max( I ,[I, -I, 1)5 L ‘1, + I, and r, rr,( - I )” = I - 2S,,. The two lowest multipolarities L,, = max( I& -I, 1) and L, = L,, + I are usually taken into account in Eq. (3), Eq. (4) and Eq. (6) [17]. This introduces an additional restriction for A. namely h12L,. In most cases L,, = 1, L, = 2, what means the restriction to the dipole and the quadrupole transitions only. Then the summation in Eq. (I ) is over A54. The angular correlation function Wy,1,2 (Eq. (I )) is invariant under the following transformations: a) ~,+(v+P)T+(-)“+w 6 and I,!+(-Y‘k+m,n and v=O,l,m,=O,~l fori=l and ++)I’+, wherep=O,l, 2; transformation should be done for both detectors b) e,-+n-o! and +,+-k and 4+++rr, for i= 1 or 2; transformation should be done for one detector, only. When either one of the emission angles H,. 0, or both are equal to zero the angle 4 is not defined and the angular correlation of two polarized photons takes a simpler form. Namely, when 8, =& =0 it depends only on the angle += I/L- $? between the polarization planes of the polarimeters and reads:
~~[~(~,~0~,0lh0)A:j,“~~(y,)A:,:,(y2) + $y, qz(A,,0A,21A2)A:~““(y,
)4;;(y,)cos2+
1 .
(9)
When only B, =0 but t$#O the emission plane of y, is not defined and the angular correlation depends, apart from the angles ez and &, on the angle (c1,> between the
- 2&) + d;Lz(8,)COs(2cCI12
X(d;;vl,)cos(2+,,
(10)
+ 216-,))I}. Similarly. when 8, f0
and eZ=O, the angular correlation
is: w~,q?(e,.rcII,~~,.Y,‘Y~) = (2 - q, )(2 - qz) 32~’
X [A:;,“” (Y, )d,:,,UI, ) - q,A:;+Yy,
)d:?(& )cos2~,41
- ~q~(A,,OA,2lA2)A~:lcy,) X [A::,““( y, )d,,V, - +q,A:;?y, X(&B,
)cos2~~;,
)
)cos(2+, - 2JI,, ) + (-
+ 24, ))I).
1Y&@,
)coNW, (II)
where I)*;,is the angle between the polarization plane for y2 and the emission plane of ‘y,. When photon ‘yZis not observed Eq. (I ) gives the known formula [I I.131 for the linear polarization angular distribution of photon ‘y, from the oriented nucleus:
C. Drnste
et al. I Nucl. Instr.
and Meth.
It means that PDCO integrated over the entire solid angle 47r of the emission direction of yZ is equivalent to the uncorrelated (observed in singles) polarizational angular distribution of y,. However, when the photon -y, is not observed Eq. (I ) gives the formula for the polarizational angular distribution of photon yZ but emitted from the state (‘u,) of a different orientation than that of state I*,,). The corresponding distribution reads:
in Phxs. Res. A 378 (1996)
= (2 - 4,)(2 - 4,,) 32~’
521
Sl8-53
x
,+,).*,-c””
B,IIU,,)
where (14)
is the de-orientation
coefficient
and where 4 is the angle between the emission planes of y and
is the de-orientation factor. A change in the spin orientation after successive gamma emissions drives one to consider a more general case. Let a with quanta momenta gamma cascade of n a sek,,kz. .k,, . *km,,. . . k,, deexcite successively states I*,J, oriented nuclear quence of I*,), . $P,:), . . .I*,,,), . . .I*,,‘,>(Fig. 3A). When two polarized photons, say, x and y,,, are detected the corresponding formula for the correlation reads:
B-2
Y,“. It is well known that the gamma quanta emitted in a stretched gamma cascade of multipolarity L = I all have the same angular distribution [5] provided that the influence of extranuclear electric and magnetic fields and sidefeedings can be neglected. The angular correlation between two photons both emitted in the same stretched cascade exhibits a similar property. Namely, the correlation between the first observed photon yt in the cascade and all the following photons is the same i.e., it does not depend on the consecutive number m of the second observed photon. This is because of the following relation between the distribution coefficients: L’,,c~+,,.
.U,,(Y,.
,)A:::(%,,)=A~:~(y,+,)
(17)
for u=O,2. Likewise, the correlation does not depend on the consecutive number I of the first observed photon in the cascade, because of the following identity:
=A
A
B
Fig. 3. Cascades of gamma quanta. (A) idealized for the theoretical considerations. (9)
observed in an experiment.
:f”(y,)u,,(x)
”
U+(x)
(18)
for v=O,2. In other words one obtains the same double correlation also for different consecutive number I of the first detected photon in the stretched cascade. The FORTRAN program that calculates the correlation
522
C. Draw
et al. I Nucl. lnstr. und Meth. in Phy.
function for PDCO according to the formula of Eq. (I ). Eq. ( 16) was developed and applied in a number of examples. The source codes are available from the authors.
3. Discussion In the present paper we limit our discussion to the polarizational-directional correlation of two quanta emitted in the cascade from an aligned nucleus. We assume that the nucleus is excited during a heavy ion reaction. Then, according to [Zl] the population of the magnetic substates may be approximated by the Gaussian distribution
C exp(-M”l2cT’) M’:.,
(19)
where the parameter tr describes the width of distribution. In the experiment, the y-radiation accompanying the nuclear reactions is studied using an array of detectors. For the sake of simplicity we consider only two of them (see Fig. 2). namely a polarimeter placed at @o, relatively to the beam direction and a directional (usual) detector located at 0,,,,. One can consider a case in which polarization and direction of y, and only direction of ‘yZare measured. Then e poL = 0, and %,, = &. This type of measurement we shall call further as the POL-DIR mode. Then the linear polarization of ‘y, (being in coincidence with y2) is defined as follows: P Po,.-~D,R = [W,,,(& - W,,,(G,
= 0”) = 90”)11]W,,,(@, = 0”)
+ W,,,($, = 9O”)l.
this is a sensitive function of spin alignment it can be used to estimate the parameter u. Applying the PDCO formulae we have found that: I. polarization depends strongly on spin sequence, character of transitions, mixing ratios and emission angles. The latter dependence is shown in Fig. 4 where polarization P is plotted vs or,,, and r#~for fixed position of the polarimeter (#r,,,,. =90”). In some cases we have observed even much stronger anisotropy than that presented in Fig. 4. 2. the following symmetries are fulfilled: P PO,. -I,IR(%IK = PPO,.L,,,K(
I w,(M) = exp(-M’l2c+‘)
Res. A 378 (1996) 5/S-.Q5
II P po,.
180” - &,,*
.,,(4,,,~~80”
$1
- 4)
= P PO,. ,,,R(%m’~~o”
+ 4)
= Pp,,,.~,,,K(e,,,R,360°
- 4)
for I&,, = 90”. P POLL-UIKv%IK’ S,l) = Pm
4,,,(180”
- 4x,,
9,:)
and P P0L.GDIK(%IK’ $4z)~cos(2+,2) for 4,,
= O”, 180”.
Similar symmetries hold for the DIR-POL mode. 3. in general if polarimeter is placed at 8,,, Z90” (such situation occurs for the CLOVER detectors in the EUROGAM II array) then P(O,,,, qb) is not symmetric with respect to the plane perpendicular to the beam axis, but the following symmetries are fulfilled:
(20) P ~o~--D,~(‘L,x~
where W,,, is given by Eq. (I); for a definition of polarization see also [7]. All the arguments but (CI,of W,, are suppressed in Eq. (20) to make the formula clear. Similarly, for the DIR-POL mode (correlation between direction of 7, and polarization of yZ) one has:
+ W,,($: = 9O”)l.
4)
4) = P,,,-r,1R(@01~~360° - b) =P P”,,-I,,K(1800 - e,,..lgo”-
= P p
