Polarizational-directional correlation from oriented ... - Science Direct

7 downloads 0 Views 639KB Size Report
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