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A multidetector apparatus for linear polarisation correlation measurements
NUCLEAR
INSTRUMENTS
AND
57 0967)
METHODS
A MULTIDETECTOR
295-312;
APPARATUS
CORRELATION
© NORTH-HOLLAND
PUBLISHING
CO.
FOR LINEAR POLARISATION
MEASUREMENTS
P. WEIGT, H. HOBEL, P. GOTTEL, P. HERZOG and E. BODENSTEDT lnstitut fiir Strahlen- und Kernphysik der Universitiit Bonn Received 29 August 1967 A nine detector apparatus for the measurement of linear polarisation correlations of gamma-gamma-cascades is described. The high statistical accuracy which can be obtained with this apparatus requires a reinvestigation of the solid angle correction. A modified approximation of the solid angle correction is de-
veloped and its performance is checked by careful calibration measurements with the isotopes l°6Pd, 46Sc and ~°Co. The theory of the attenuation of linear polarisation correlations by internal fields is developed.
1. Introduction
p l a n e o f p o l a r i s a t i o n o f the initial g a m m a - r a y and the scattering plane (fig. 1).
Usually the C o m p t o n effect is used for the measurem e n t o f the linear p o l a r i s a t i o n o f low energy g a m m a rays. Its sensitivity to the linear p o l a r i s a t i o n can be seen from the K l e i n - N i s h i n a 1) f o r m u l a for the differential cross section d a / d O for C o m p t o n scattering dtr d O -- ½r2°{(hv')/(hv)}2" • {(hv')/(hv)+(hv)/(hv')-2sin2Ocos2¢#}.
(1)
The energy hv' o f the scattered g a m m a - r a y is related to the energy hv o f the initial g a m m a - r a y by 2) hv' = h v / [ l + { ( h v ) / ( m o c 2 ) } ( 1 - c o s 0 ) ] . 9 is the scattering angle. • is the angle between the
Fig. 1. Schematic illustration of a basic element for a gamma-ray polarimeter2). F o r 9 = const, the differential cross section da/df2 shows ° a m a x i m u m for 4 ~ = 9 0 and a m i n i m u m for = 0 . T h e a s y m m e t r y ratio
90°-
da Ro= ~(~=90
,~ d a )/~(~=0
,, dago d a o )= ~-/-j~-
(2)
80q
70%
3~.... ~o~ ] I
~ ,~ ~ "
~
e
p
is a m e a s u r e o f the sensitivity o f the C o m p t o n effect to t h e linear p o l a r i s a t i o n . F o r a given energy hv o f the initial g a m m a - r a y the e n d s on the angle O F o r
60°
50 °I~ h Y" [ M e V ]
Fig. 2. Plot of Compton scattering angles @which yield a maximum for the asymmetry ratio R v~ gamma-ray energy. 295
296
P. WEIGT
et al. No(O) oc Io(O) dd°'° ~-
+ 19°(0) da9° dQ '
g9o(O) oc Io(O) da9° d~-
/
/
da° d~'~
N°(O)
/
I0(0 ) da° +
Ngo(0) - [
dO
190(0 ) da90 dO } /
= {n(o)+no}/{P(O)'no+ tO~- 0 . . . . Ro becomes a maximum, tOmaxis plotted for the energy range 0 < hv < 5 MeV in fig. 2. The first description of an apparatus for the measurement of linear polarisation correlations with a polarimeter based on the Compton effect was given by Metzger and Deutsch3). A simple three detector arrangement is shown schematically in fig. 3. If for instance the degree of polarisation of the ','2 radiation of a 7 t - 7z cascade is to be determined, the l't radiation must be detected by El and the '/2 radiation by Z. Thus the positions of the detectors E1 and Z define the angle 0 between the propagation vectors of the two gammarays. A coincidence event is registered only if the radiation is scattered by the Compton effect from Z into the third detector A 1 where its remaining energy must be totally absorbed. In order to measure the linear polarisation correlation the triple coincidence counting rates are recorded at r/-- 0 ° and r! = 90 ° for different values of 0. In this way one obtains the counting rates No(O) and U9o(O). The linear polarisation P(O) is defined by
P(O) = 10(0)/190(0 ). I0(0 ) and •90(0) are the intensities of the components the 1'2 radiation polarised parallel and perpendicular
of to the 0 plane, respectively. The theory of the linear polarisation correlation delivers for Io(0) and 19o(0) the expressions
Io(0 ) : W(O)+[B2p2(cosO)+ B,,pI(cosO)],
(3)
['V(O)--[B 2P2z(cOs0) + B,p42(cos 0)],
(4)
=
(6)
/ {io(o) da9o + 19o(0) ?-ff dao j
Fig. 3. Gamma-ray triple coincidence polarimeter, schematicS).
190(0)
"
Alternatively,
E1 ~
-1- 1 9 0 ( 0 )
(5)
where W(O) = l + A2P2(cos 0) + A~P,(cos 0) is the directional correlation and B2 and B~ are the linear polarisation coefficients. The polarisation P(O) can be derived from the measured quantities No(O) and Ngo(0) with the help of the asymmetry ratio R o in the following way :
1}.
Finally one obtains for the linear polarisation
P(O) =
In 0-
{No(O)/N90(0)} ]/[no{No(O)/N90(O)} -
1].
(7) This formula gives the right polarisation only for an ideal geometry. Therefore one has to correct for the finite solid angles of the detectors. Linear polarisation measurements with the three detector arrangement described, are restricted to a few especially favourable cases. This restriction is caused by insufficient triple coincidence counting rates. In comparison with normal gamma-gamma-directional correlation measurements the counting rates are reduced by an additional solid angle, i.e. the effective angle of the detector for the scattered radiation. Because of this difficulty, only a few linear polarisation correlation measurements have been published up to now, in spite of the fact that the method has been known since 1950. On the other hand measurements of linear polarisation correlations are very valuable for nuclear spectroscopy as they represent the only known method to distinguish absolutely between electric and magnetic multipole radiation. In this work we describe a multidetector apparatus in which the counting rates are increased by a factor of 16 as compared to the simple three detector arrangement. 2. Description of the apparatus We use four NaI(T1) scintillators El, E2, E3, and E4 (referred to as detectors E in the following) for the registration of the radiation ~/1. The polarimeter for the detection of the radiation 7'2 consists of the scatterer Z surrounded by four additional NaI(TI) scintillators A 1, A2, BI, and B2 (referred to in the following as detectors A and B) for the measurement of the scattered radiation. The geometrical arrangement of the detectors is shown in fig. 4. lfthe energy of the radiation 72 is above 200 keV one can use a NaI(TI) scintillator also for the scatterer Z. Only for low energies, plastic scintillators
A
MULTIDETECTOR
297
APPARATUS
ion
Fig. 4. Apparatus for the measurement of the linear polarisation correlations. have to be used in order to keep the probability of photo absorption small enough. All nine detectors are cylindrical (lk" dia. x 2" height). The NaI(T1) scintillators were mounted integrally with 153 AVP photo tubes with energy resolutions between 7.0% and 8.1C}/o fwhm for the 662 keV line of 13VCs. We used a distance of 74 mm between the axes of the scatterer Z and the detectors A and B. The distances between the source and the detectors E and Z were 77 mm. The size of the detectors and the solid angles were so chosen as to yield a maximum of triple coincidence counting rates at a still fairly high asymmetry ratio. The calculations of Metzger and Deutsch 3) were helpful when we tried to find the best compromise solution for gamma-radiations of medium energies. As shown in fig. 2, the best value for the scattering angle 0 .... is about 90 °. It decreases slowly for higher energies of the radiation Y2. Our detector arrangement (fig. 4) allows a rather large range of scattering angles. It is possible, however, to choose a specific range of scattering angles by selecting the energy of the Compton electrons through pulse height discrimination of the pulses produced in the scatterer Z. The four detectors E are mounted on a revolving platform. The measurements are performed at two different positions of the platform which differ by 12.9 °. In this way one measures P(O)at eight equidistant angles between 90 ° and 180° taking advantage of the symmetry of the linear polarisation correlation around 0 = 180°. No correction for the different detection
efficiencies of the four detectors E is necessary because in the measured quantity No/Ngo these efficiencies cancel out. In order to avoid corrections for the different detection efficiencies of the polarimeter detectors A and B, the polarimeter is rotated around the axis of Z in steps of 90 °. In this way the detectors A and B exchange their position several times during one cycle of the measurement. After four steps the direction of rotation is reversed in order to prevent a twisting of the cables from the polarimeter detectors. The sequence of the different positions of the platform and the polarimeter during one complete cycle of the measurement is shown in table 1. The rotations of the TABLE 1 The sixteen detector positions of one complete cycle of the polarisation correlation measurement.
Position hr.
1 (9)
2 (10) 3 (11) 4 (12) 5 (13) 6 (14) 7 (15) 8 (16)
Position Position of the Position of the polarimeter detectors platform of the relative to the 0-plane with the polarimeter detectors E detector A detector B 1 1 1 1 1 1 1 1
(2) (2) (2) (2) (2) (2) (2) (2)
1.
II
LL
L
±
II
II
±
II
3_
.L
II
2_
I1
P.WEIGT et al.
298
platform and the polarimeter are performed by small electromotors controlled automatically. We did not mount any lead shielding between the detectors E. A Compton scattering of gamma-radiation from one E-detector into another does not disturb the measurement, because coincidences between the Edetectors are not measured. The polarimeter detectors A and B must be shielded against direct radiation from the source and scattered radiation from the detectors E; but the space is limited and the lead shield alone is not sufficient if high energy gamma-lines are present. We were able, however, to keep background coincidences negligibly small by electronic discrimination even in rather unfavourable cases. A fast triple coincidence is registered only if the following three conditions are fulfilled: 1. The energy of the 71 radiation has to be totally absorbed in one of the detectors E. Four single channel analysers are used which accept the photo peaks of the 71 radiation. 2. A given fraction of the energy of the ?2 radiation
must be lost in the detector Z by a Compton scattering. A single channel analyzer selects the right energy interval of the Compton electron. 3. The scattered 72 radiation must be totally absorbed in one of the detectors A or B. The pulses from the scatterer Z and from each of the polarimeter detectors A or B are added. Four single channel analyzers are set on the photo lines of the 72 radiation in the four sum spectra. A part of the fast-slowcoincidence circuits is shown in the block diagram fig. 5. Automatic peak stabilisation is used for all nine detectors in order to keep the overall pulse height amplification constant. The detectors E and Z can be stabilised on a photo line of the isotope under investigation. Extra sources are needed for the detectors A and B. We have chosen the monoenergetic radiation of ~aVcs. The four necessary 137Cs sources were mounted inside the lead shield in front of the detectors. The adder ( A D D in fig. 5) are followed by a gate which opens only when an event is detected in the respective polarimeter detector. Without this gate the sum spectrum would be masked
.,:(
?
SCATLE R r¢ O
coL; -I
Y FAST L
Fig. 5. Block diagram of the electronic equipment of one triple coincidence circuit.
1
A MULTIDETECTOR APPARATUS
299
Fig. 6. View of the apparatus for the measurement of the linear polarisation correlation.
totally by the single spectrum of Z. The measurement of the pure sum spectrum is necessary in order to set the single channel analysers correctly on the )'2 photolines. The signals Z + A 1 and Z + A 2 of the sum spectra are fed to the fast and slow coincidence units via an OR-gate. This can be done because the signals contribute the same type of information. This procedure reduces the necessary number of fast-slow coincidence circuits to eight. The entire electronic equipment of the described apparatus is rather bulky but completely transistorised circuits make it possible to reach a very high reliability and to keep all settings stable for long periods of continuous measurements. A view of the whole experimental set up 4' 5) is shown in fig. 6. 3. Determination of the solid angle correction
The asymmetry ratio R 0 for ideal geometry can be
calculated immediately from the Klein-Nishina formula. Fig. 7 shows R 0 as a function of the energy hr. The actual polarisation sensitivity of the apparatus is smaller, however, because of the finite solid angles of the detectors. The exact theory of the solid angle correction has not yet been worked out. The usual method which has been applied so far takes into account all solid angle effects by using formula (7) with a reduced value of R instead of R o. This procedure is only an approximation. The exact correction for the measured polarisation at the angle 0 depends also on the linear polarisation correlation coefficients. This is caused by the fact that the detectors E and Z average the linear polarisation over a finite range of 0. We developed an approximate theory of the solid angle correction which takes into account this last effect. For real geometries, formulae (5) and (6) must be written in the form:
300
P. WEIGT
rA(8) =
No(O)
T6 ('E,'z,rA(.))J l o ( 1 9 o ) = = differential cross section for Compton scattering from rz to rAta) of a radiation which is polarised parallel (perpendicular) to the plane rE,rz, e(rE,rZ,rA) = detection probability for 71 at the point r E, and that for '/2 by Compton scattering at the point r z and 7; at the point r A. The exact evaluation of the solid angle correction could, in principle, be done according to formulas (8) and (9) by the Monte Carlo technique. This procedure has to be carried out, however, for each special pair of 7-ray energies hv I and hv 2. This does not seem to be feasible because it would imply a very long execution time even for very fast computers. We used instead the following approximation. We split the efficiency function e into two factors
+ flgo(O')13(rE,rZ,rA)" (8) and
(O)oc f ([o' ( 0 )13 rE,rz,rB).
"4-
90
point of absorption ofT; in one of the detectors A(B),
]
"[~(l"E,rz,rA)]lodZrEd3rzd3rA"F
N9o
et al.
(o') 13( rE,rz,r B) • (9)
8(rE,rz,r A) = sl(rE,rz). ~2(rE,rZ,r A) ,
where: 0' arccos[{rE'rz}/{IrEl'lrzl}], = point of absorption of 71 in one of the detectors r E E, = point of Compton scattering of 72 in the der Z tectorZ,
where el(rE,rz)
=
probability for detection of 71 at the point r E and that for 72 by Compton scattering in the point r z without absorption of 7; within the detector Z, e2(rE,rz,rA) = probability of detection of 7; by absorpsion in the point rA, and rewrote equations (8) and (9) in the following approximation : =
No(O) oc f lo(O')131(rE,rz)d3rE d3rz • 12
" fe2(rE,rz,rA) [dd~ (rE,rz,ra)] tod3rEd3rz d3rA + 10 ¸
+ fZgo(O')e,(r,rz)d3rE d3rz ' "f132(rE,rz,ra)[~(rE,rz,rA)llgod3rEd3rzd3rA, (10)
0'5
1'0
115
2'0
2'5
3'0
3'5
4'0
415 b hvo [MeV]
Fig. 7. Plot of the asymmetry ratio R0 as a function of the gamma-ray energy hr.
50
A MULTIDETECTOR APPARATUS /. N9o(0 ) oc I
JIo(O')~drE,rz)d3rE darz • f e2(rE,rz,ra)[d~(rE,rZ,rB)] d3rEd3rzd3rB+
The evaluation of the integrals of eqs. (14) and (15) has been performed in the following way: By inserting the expressions (3) and (4) for Io(0') and 19o(0') into eqs. (14) and (15) we obtain:
J to
ud~,~
]o(0) -~/
fl9o(O')e,(rE,rz) dare darz •
+ Q
(11)
+B4fP~(cosO')~tdarEd3rz],
We believe that this splitting of the integrals of equations (8) and (9) is a good approximation because we obtained it by the following consideration: We separated the integrands of equations (8) and (9) into two parts, the first of which depends mainly on the angle 0' and the second depends mainly on the other angles. By dividing No(O)by N9o(0 ) we obtain"
No(O) N9o(0)
= [{1o(0)/I9o(0)}+ R(O)]/[{lo(O)/t9o(O))R(O)+1],
R(O)="
[do
e2(rE'rz'r^) ~
(rE'rz'rA)
]
(12)
darEdarzd3rA I,o , (13)
and
/
Io(0) = lo(O)el(rE,rz)d2rEdarz, I9o(0) =
-
f
~9o(0')~,(rE,r~) d3rE darz.
W(0')gl d3rE darz +
+ [a2f P~(cosO')~,d3rEd3rz +
fe2(rE,rz,ra) IL-d-~ d a ,[rE'rz'rB)Jl9o ,] --3 0 rEO--3r Z d3r B"
with :
301
(16)
I90(0 ) = f W(0')~, darE d a r z -
- [B2f P~(cosO')~,d3,'Ed3,'z+ +B4fP~(cosO')~,darEd3rz].
(,7)
For the evaluation of the first integrals we used the well known "Rose correction" 0) which is usually applied for the solid angle correction of gamma-gamma-directional correlations. The first step for the evaluation of the remaining integrals is the determination of the efficiency coefficient 51. We applied the Monte Carlo method which took into account the probability that 72 has no further interaction in the detector Z on its path to A. Fig. 8 shows, as an example, the result of a Monte Carlo calculation with 10000 events. The execution time for the IBM 7090 was about two minutes, e~ is not normalised in this figure. This calculation was done for
(14)
(15)
In this evaluation we made use of the identity of the integrals:
i I
f e2(rE,rz,rA(a))[ ~ (rE,rz,rAta))]todarEd3rzdarAtB~ = f e2(rE,rz,rBtA))[~(ra,rz,ra(A,)ll~oda rEd3rzda ra(Ar I
It should be pointed out that the improvement of this approximation compared to the usual method consists of the fact that in our definition, R depends only on the geometry of the apparatus and the two gamma-ray energies and does not depend on the linear polarisation correlation coefficients.
./
.30 o
_20 o
_l'Oo
OO
I'0 o
2'0 o
3'{]~ 0 O'
Fig. 8. Probability distribution of 0' as a result of Monte Carlo calculation for J°6Pd energies hva= 513 keV, h)'2= 624 keV.
P. WEIGT et al.
302
,oi 8-
,
p2 2 P2
0
, 100 °
, 120 °
, 140 °
. 160 °
180 °
(3 -2
-6
-s 1 Fig. 9. Plot of the Legendre polynomials Pz2(cosO), PJ(cos0) and the modified functions ~2(0), P4e(0). the geometry of our apparatus and the energies: hvl = 513 keV and hv2 = 624 keV (l°6pd). el is plotted as a function of ( 0 - 0 ' ) . In the second step the remaining integrals were solved numerically. Fig. 9 shows the result for p2(0) =
f
P~(cos0')~l d3rEd3rz
(18)
RIO)T30
and
P](O) =
JP1(cos0') l d3vE d3rz,
The evaluation of R(O), formula (13), has been programmed for a Monte Carlo calculation. The result for the special case of the gamma-gamma-cascade of 106pd (hv I = 513 keV, hv2 = 624 keV) is shown in fig. 10. The asymmetry ratio R decreases slightly with 0. This is expected because the range of angles between the 0 plane and the 0' plane increases with 0. Finally at 0 = 180 <~the 0 plane is no longer defined and the asymmetry ratio reduces to the value R = 1. This calculation of R has two disadvantages: 1. The execution time for the computer is rather long (The calculation for the example of fig. 10 took two hours on an IBM 7090); 2. The systematic error which is produced by the different approximations is unknown and its estimation is difficult. Therefore we have performed calibration measurements with our apparatus for a number of simple cases where the linear polarisation coefficients and angular correlation coefficients can be predicted exactly from the known spins and multipolarities. We selected for this purpose the 624 keV-513 keV cascade of ~°6pd, the 887 keV-1119 keV cascade of 46Sc, and the 1173 keV1332 keV cascade of 6°Co. In all three cases the linear polarisation of both gamma-radiations has been measured. Figs. 11, 12 and 13 show the decay schemes, the single pulse height spectra, and the sum spectra together with the window settings which were used for the linear polarisation correlation measurements. Figs. 14, 15 and 16 show the measured counting rates No/N9o for the 624 keV gamma-transition of a°6pd, the 1332 keV gamma-transition of 6°Co, and the 887 keV
et
(19)
for the example mentioned above. Thus we have reduced the expressions for 10(0) and 190(0) to the form:
2.5 ¸
I0(0 ) = l + A2{J2/Jo}P2(cos O) + A4{J4/Jo}n4(cos O) +
2.0
+ [Bzp2(o) + B4p2(o)],
(20)
I90(0 ) = 1 + Az{Jz/Jo}P2(cos O) + A4{JJJo}P4(cos O) -
[B2p2(o) + B,p2(o)],
(21)
90 °
180 ° II, 0
where J2/Jo and J4/Jo are the " R o s e correction" 6) factors.
Fig. 10. Asymmetry ratio R(O) as a result of Monte Carlo calculation for l°6pd energies hva = 513 keV, hv2 = 624 keV.
303
A MULTIDETECTOR APPARATUS
Fig. 11. Decay scheme, single pulse height spectrum, and sum spectrum together with window settings of the single channel analysers which were used for the linear polarisation correlation measurements in ]°6Pd.
513 oo p-
Z 0 U
/
I ~
1o6 45
;
, b
\ ' c
~
'd
106 Rh 61
,o, EiMeY)
I
•
° *
1 1 0 1 . ~ %~
I, :IT
1.56
2
*
1.137
0
•
1.125
+
0 513 2 ,,
7 8 % ~' -II ~ - A
0
0 *
p d I06 46
60
~' PULSE HEIGHT 887 key 1119
kev
Fig. 12. Decay scheme, single pulse height spectrum, and sum spectrum together with window settings of the single channel analysers which were used for the linear polarisation correlation measurements in 4%c.
U3 o,--
Z
Sc 46
Tl/2= 84d
0 o
T
SINGLE
SPECTRUM
El.key]
I I~
2006
4 ...
B87
2 *
0
0 +
o i
b
IE
2 SUM S P E C T R U M
I
stab[¢
I e
PULSE HEIGHT
Ti 46
P. W E I G T et al.
304
1173 MeV l
1 332 MeV
Lf) O 104-
E[kev]
I
2505
~.
1332
2•
:o 3_ e
f
0
0.
sta~¢ N, g0
P PULSE HEIGHT Fig. 13. Decay scheme, single pulse height spectrum, and sum spectrum together with window settings of the single channel analysers which were used {'or the linear polarisation correlation measurements in 6°Co.
I" 1.7- NO/ N90
1.61.51.41.312 11 i
i
i
~
I
"
i
140"
I
i
160 °
i
I
I@0"
e
Q9 080.70.60.50.4-
Fig. 14. R a t i o o f triple coincidence c o u n t i n g rates
No(O)/Ngo(O)for
l°sPd energies h~'l = 513 keY, h~'2 = 624 keV.
A MULTIDETECTOR
305
APPARATUS
N ~90
-'~ e l 00-
~
900
. . . .
lo'0o
11~
f
r 12oo
1
"'i
, oo
098-
096-
09~
/
Sc 46
0.B87MeV
092]
0881 Fig. ! 5. R a t i o o f triple coincidence counting rates
No(O)/N,o(O) for
46Sc energies/ml --- 11 t9 keV,/n,2 = 887 keV.
---.~ 1,190 100
?
90 °
I
100°
T
1'1
'
~
Y o
o
~
170 °
O L
1,B0°
0 98-
0.96-
09/."
Co60
1.332 MeV
0.92"
Fig. 16. Ratio of triple coincidence counting rates
No(O)/Ngo(O)for
gamma-transition of 46Sc. The linear polarisation of the three other transitions has been measured with the same statistical accuracy. The evaluation of R(O) from the measured ratio of counting r a t e s No(O)/Ngo(O ) is made by use of eq. (12). One obtains for R(O) the expression:
R(O) = [{Io(0)/19o(0)}- {No(O)/N9o(O)}]/ /[{io(0)/I9o(0)} {No(O)~Ngo(O)}- 13.
(22
~°Co energies h~,] = 1173 keV,/n,2 = 1332 keV.
For To(0) and/9o(0) the expressions of eqs. (20) and (21) are inserted. For 6°Co and 46Sc we used the theoretical coefficients for a 4 + ( E 2 ) 2 + ( E 2 ) 0 + cascade. In the case of ~°6pd the measurement of the 0 + ( E 2 ) 2 + ( E 2 ) 0 + cascade contains a ( 1 0 + 3 ) % contribution of the two 2 + ( E 2 , M I ) 2 + ( E 2 ) 0 + cascades starting from the 1125 keV and 1560 keV levels. The E2,M1 mixing ratios were taken from the work of Miinich et al.7). The
P. WEIGT et al.
306 R'
2.0-
2.0
/0
2.0-
10 9'0o
' --~.
1.0
9'0o
~80 o
,
'180o
90 o
e
(3 R ~
2.0.
1.0 ' @
30-
20-
20-
180o P
F i g . 17. A s y m m e t r y ratio
R j
30-
10 9'0o
180o I,
,
i
10
i
90 °
180 °
g0 o
IBO o
0
R(O)in
~°Co a) b) 4"Sc c) d) l " ~ P d e) f)
o u r definition as a result o f c a l i b r a t i o n m e a s u r e m e n t s in
hvl = 1173 k e V , hvl = 1 3 3 2 k e V , by1 = 8 8 7 k e V , by1 = 1 1 1 9 k e V , hJ'l = 6 2 4 k e V , h~'l = 513 k e V ,
evaluation of R(O) is very sensitive to these admixtures and mixing parameters. To get more precise information a reinvestigation on the isotope l°6Pd is in progress. The results for the six measured cases are plotted in fig. 17. All six curves show the expected slight decrease of R with increasing 0. It is satisfactory to see that the shapes of the curves are very similar in spite of the fact that the linear polarisation coefficients B2 and B, and the angular correlation coefficients Az and A 4 are very different for 0 - 2 - 0 and 4 - 2 - 0 cascades. This shows that our approximation in the definition of R is quite effective in making R independent of the correlation coefficients. When we applied, however, the usual definition of R instead of ours we obtained the curves of fig. 18. These curves, indeed, exhibit a strong dependence of the shape R(O) on the correlation coefficients. This dem-
hv2 hvz hv2 hv2
= 1332 k e V , = 1173 k e V ; = 1119 k e V , = 887keV; hv2 = 513 k e V , hv2 = 6 2 4 k e V .
onstrates that especially for large coefficients the usual method of solid angle correction is rather inadequate. A comparison of the empirical values for R(O) of fig. 17 with the "Monto Carlo" calculation plotted in fig. 10 shows a satisfactory agreement. This proves that the systematic errors which are introduced by our approximation cannot be very large. The experimental determination of the absolute linear polarisation correlation coefficients B2 and B, of gamma-gamma-cascades can thus be performed in the following way. 1. R(O) in our definition is derived from the described calibration measurements by an extrapolation to the gamma-energies of the special case. The extrapolation can be performed with high precision by application of the described "Monte Carlo" calculation of R(O). 2. By use of eq. (12) the ratio !o(0)/190(0) is derived
A MULTIDETECTOR
RA
2.0-
R,
R'
2.0
20"
1.0
10
9'0o
10
9'o*
180g
R~
@
180°
LI @ r
]
180°
30-
•
20-
20.
r
1.0
10 90 °
•
90 °
R ~
R d
3.0-
20'
307
APPARATUS
180o
10 90 °
180 °
180°
90 ° G
Fig. 18. A s y m m e t r y r a t i o
R(O)in 6°Co a) b) 46Sc4c) d) L°aPd e) f)
the u s u a l d e f i n i t i o n as a r e s u l t o f c a l i b r a t i o n m e a s u r e m e n t s in h*'l by1 hl, i h~t hvl hvt
= 1173 keV, = 1332keV, = 887 keV, = lll9keV, = 624 keV, = 513 k e V ,
from the measured ratio of coincidence counting rates
hi'2 = 1332 keV, h~'2 = 1173 k e V ; hi'2 = l 1 1 9 k e V , h*'z = 887 k e V ; hvz = 513 keV, hi,2 = 624 keV.
4. By dividing eq. (20) by eq. (21) one gets the ratio
lo(0)/i9o(0 ) expressed by the coefficients A2, A4, Bz and
Xo(O)/Ugo(O):
B4, and one obtains by trivial algebraic computation.~:
Io(0)/19o(0) = J R ( 0 ) - {No(O)/N9o(0)}] / /[R(O){No(O)/Ngo(O)}-I].
(23)
3. The functions p2(O) and p2(O) are calculated for" the special gamma-ray energies by the " M o n t e Carlo" technique and J2/Jo and J,~/Jo are calculated by the Rose method. In principle, one could now derive both, the linear polarisation correlation coefficients B2 and B4 as well as the directional correlation coefficients A2 and A4 from eqs. (20) and (21) by a least-squares fit. The accuracy obtained for B2 and B~ can be increased, however, if ,4, and A~ are determined independently in a directional correlation measurement.
B 2 P2(O) + B4P24(O) = = [1
+ A 2{J2/Jo}Pz(cOs O) + Aa{J4/Jo}Pa(cos 0 ) ]
x[{Io(0)/I9o(0)}-1]/[{Io(0)/I9o(0)}+1].
x
(24)
5. The absolute coefficients B2 and B 4 are determined from eq. (24) by a least-squares fit. 4. Dependence of the asymmetry R on the range of Compton scattering angles In order to optimise the admitted range of Compton scattering angles A,9 we investigated experimentally the dependence of the asymmetry ratio R on this quantity.
P. WE[GT et al.
308
(.9
W(k, k 2 t ) =
~
Z
(malp(k,)[m'a>(m'b[p(k2)lmb> •
roam' a
mhm%
'(mblA(t)lma>*. .25
./
25-
By inserting the expressions for the matrix elements [eqs. (66) and (67)in 8)] one obtains: 0 (.J UJ ¢..)
Z
-200
(25)
W(klk2t
) =
=(--1) r'+r~
u z O
t%/t
( - 1 ) -L''-L'2-Rz-N'-s2n~'N2:'~ ~Jklk2 ~tj'
E E
LtL'I kiNitl
L2L'2 k2N2r2
,,k2 t
"[LiL2Ii] [L2L'2IfJ" l
-150
2.0
t
t
*
•< ILlrci[li>
,
t
t
,
zlflL2r¢21I>
•
• Ck,r,(L'~ L 1)cLr~(L 2L[)D%' ,, (Z ~ k,)D~m(Z ~ k2), i
10 °
w
i
i
20 °
30 °
t. 0 o i,
AO Fig. 19. Plot o f R(90 °) a n d triple coincidence c o u n t i n g rates Ne versus /I as a result o f linear polarisation m e a s u r e m e n t s in l°6Pd (hvl = 624 keV, h~2 = 513 keV).
The range of admitted Compton scattering angles is selected in our apparatus by the single channel analyzer of the detector Z which measures the energy of the recoil electron. This energy is connected with the scattering angle by the well known equation:
where ~,~.N,N~:,a are the general perturbation factors of k l k 2 ~.'1 the theory of perturbed angular correlations [ref. 8), eq. (209)] and ck(LL') are the radiation parameters [ref. 8), eq. (63)]. For z~ = r2 = 0, eq. (26) changes into the normal directional correlation function [ref. 8), eq. (208)3• If the linear polarisation of the second radiation is measured, eq. (26) describes the linear polarisation correlation by summing over z~ = 0 and r2 = 0 , + 2 and - 2; one then obtains:
W(ktk2t)=(-1)
h+t~ E A'k,(LIL;IilN~)" NtN2 klk2
T~ = h v - hv'
t t kl •Ak2(L2LzlflN2)DN,o(Z
= hv[{hv/(moc2)} (1 - cosg)] / / [l +
(26)
Stmax - A o <-- St <-- St . . . .
+ AO.
The result for R(90 °) is plotted in fig. 19 vs AO. This shows that R decreases only slightly with increasing ASt. On the other hand, the triple coincidence counting rates increase very fast with increasing ASt. Apparently a good compromise is the choice ASt= 30°; this has been used for all our linear polarisation correlation measurements so far. 5. Theory of attenuation of linear polarisation correlation by extranuclear fields In this section the theory of perturbed gammagamma-angular correlations is extended to the attenuation of linear polarisation correlation. We start from the general expression of the correlation function [ref. 8), eq. (20)] :
k2* ~ N N2 kl)DN~o(Z k2)G,,k2(t)+
+ A'k,(L,L' 1lilN,)" B'[~(L2L'21dN2 ) •
(i - cos O)].
We performed the measurement with the 513 keV gamma-ray of ~°6pd for different ranges
__~
kt NIN2 • Du,o(Z--+ k,)Gk,k~ (t),
(27)
with the abbreviations:
Ilk,1 E (--l) -L''-N't L1L'I I Li L fl i] • ( l l L l r t l l l i > ( l l L l n ' 'I 1I~>* ck,o(L,L,), ,
Ak,(LIL'tIiINI)=
' 'Ii Ak~(L2L2 r N2)= ~ (-
(28)
(L2L2Id
L2L'2
t
t
.(IflL2~EII>(IflL2~21I>
*
*
1
t
%o(L2L2),
(29)
" ' I If N2) = ~, :t , - 1 ) [-L'2-N~-k2, llk,2 1 Bk~(L2L2 L2L'2 (L2L2Ir] • ' ,
c
•
t
L L D
k2*
Z~k
*C k*2 0 ( L 2 L 2 ) . +c k* L L P Dk2*
CR2°(LzL2)
Z~k (30)
For the evaluation of the radiation parameters ck~ we
A MULTIDETECTOR
309
APPARATUS
Lu
yb172
/~fiTd
key
I n K 9~//I
I~* ~)
2287
Wl i
2194
/
2075 0,,4~
gg~
,g~
1"/50 1701
}
4t+
1
{4÷
3)
:
(4.
3) 3
,
,..~ ~--e-.,~_ ma~
II04
' I
~
i
1663
1510
172
-
,~
I ~
.
|
i
i
6(+) 3
1376 12~:, _ _ 11
i
r
~
I1
~I,
- -
3(+} 3 1112. 9.7 ns (2*)
6÷
I
--!
-
~
-
0
4*
0
2..
0
O-
0
Fig. 20. Decay scheme of t72Lu. use the expressions for (oalLp,~) and ( a f e l a ' ) = 6~o' for the case of a polarisation insensitive detector 8) and e =½(_ 11- 1 ~) for a detector sensitive to photons linearly polarised in the x-direction. We obtain:
cko(LE) = ½(-1) L-' (2k+ 1)½(2L+ I)½(2L' + 1)+' (i
-L', o k ) {t + ( - 0"''+~'''(-')~+~'+~}'
Ca2(LE)= ¼(-1 )"-' +~v)(2k + I)'I-(2L + 1)}"
1
1
2
'
ck_2(LC) = (-I)'~L"+~L'( - I)L+L'+%dLC), where s ( L ' ) = 1, s(L)= I for L', L denoting electric multipole radiation and s ( L ' ) = 0, s(L)= 0 for L', L denoting magnetic multipole radiation. The factor
(_ 1),~',+~,.(_ 1)~+~'+~ must always be positive, otherwise
Cao(LL') and
P. WE|GT et aL
310
therefore the coefficients _4k will vanish. With this assumption and the expressions for the radiation parameters Ck~ eq. (30) may be written in the form:
B'k'2(L2L'2IfIN2)=
~. ( - - I ) - L % - k 2 - N 2 [ I l k s I" L2I.'2 [L2LzlfJ
• (IflL21~2iI>* ck20(L2L2)" • {ck*2(LsL'2)/Ck*o(L2L'2) } [ ( - I),(L2)+*(L'=). --
DN2_z(Z--~ks)4-D,%2(Z
ks)]
t t k2* , = B,=(L2LslflN=) [ON2 _s(Z--+ ks) 4- D kN2 *, s ( Z ~ k s ) ], (31)
with
Bk2(LsL2IJN2) = 2
W(k, k2t) = ( - 1 ) t'+'~ Z A'k,(l)'A'k2(2)Dk',o(Z-*k,)" kl~2
N1N2 k2*
NjN2
• o,,,o(Z-~ k s ) ~ , ~ (0 + ! ! kI __+ +Ak,(1)Bk2(2)Ou,o(Z k,)" k2* k2 NjN2 • [o~,_~(z-~ ks)+ o~2+s(Z-~k,)]c,,~, (0.
(32) In the special case of liquid sources one has randomly fluctuating interactions. It is convenient then to choose the propagation vector of the first radiation as the quantisation axis. This restricts N~ and N2 to the value Nl = N2 = 0 , and k~ = k s = k . We can now make use of the relation
( - - l ) -L'2-k2-Nl'i Ilk't2 I"
(LsLsIfJ * • ( l f l L s x 2 I [ ) ( l f l L 2 x 2 I17 "
L2L'2
t ¢
• L ~L~){ ' c *,,s(L~Ls)/q~o(L~Ls)}. ' * ' • C,~o(
By inserting the eqs. (28), (29) and (31)into eq. (27) one obtains:
k k2* k2* Doo(Z -+ kl) [Do2(Z--* k2) 4- D O+ 2(Z--* k2)] =
= o~sao(k~-+k,) = 2 { ( k - 2)!/(k 4- 2)~?P~(cos O)cos2~ and reduce eq. (32) to:
W(k, k2 t) = ~. Ak(L1L' I l_i])Ak(L2L'2lfI ) • k
• G°Ok(t)ek(COS O) + 108-
W,°
+ Ak(L 1L; I,I)B,(L2L'21d)" • G°k°(t)P~(cos O) COS2r/.
107
Ak(L1L'llil ) is the normalised directional correlation coefficient for the first gamma-radiation, A k ( L 2 L ; l d ) is the normalised directional correlation coefficient for the second gamma-radiation, Bk(L2L'zld ) is the normalised linear polarisation coefficient for the second gamma-radiation.
1,06
1.05"
The theory of perturbed angular correlations derives for the pertubation factor for randomly fluctuating interactionsS):
104
a°°(t) = e- ~'.
103
102
101
100
go o
180 °
e ogg-
Fig. 21. Ratio of triple coincidence counting rates No(O)/Ngo(O)for 172Yb(hvl = 79 keY, hvz = 1094 keY).
A
MULTIDETECTOR
Our result for the perturbed linear polarisation correlation in liquid sources is in agreement with the formula which Davies and Hamilton used for the attenuated correlation of their linear polarisation correlation measurementg). There may be a few other cases in which the factor Z ok,,o(Z~ k,)
k2*
*,*
311
APPARATUS
0 0O8 0006-
~exp
000~ 0002 00
.
tQ=
02
014
0'6
0'8
62 _ _ 1+6 2
NIN2 ¢~.NIN2[t' ~ • Uklk2 k'l,
can be reduced to only one rotation function multiplied by an attenuation factor. This is true for example for a powder source and for correlations in a transverse magnetic field.
6. Measurement of the linear polarisation correlation of the 1094 keV-79 keV gamma-gamma-cascade in the decay of 172Lu The measurement of the linear polarisation correlation of the 1094 keV-79 keV gamma-gamma-cascade in the decay of 172Lu was important for the determination of the multipolarity of the K-forbidden 1094 keV transition. In a recent publication ~°) we have described this measureme at together with a preliminary analysis of the data. In the meanwhile we have completed the final analysis and represent the result in the following as an example for the experimental determination of absolute values of B2 and B4. The decay scheme is shown in fig. 20. The linear polarisation of the 1094 keV line was measured. The observed ratio of the triple coincidence counting rates No(O)/N9o(O) is shown in fig. 21. The background of coincidences of high energy gamma-lines in the 79 keV window and its contribution to the linear polarisation correlation has been determined experimentally by shifting the low energy window above the 79 keV photo peak. The admixtures of the 91 keV-1094 keV cascade and the 1082 k e V 79 keV cascade have been taken into account by use of theoretical coefficients for A2, A4, B2 and 94. The result of the evaluation of B2 and B4 is:
B2 = - 0 . 0 4 3
+ 0.008,
B4 = 0.0050 + 0.0025. These coefficients have to be corrected for the attenuation by internal fields. We used the attenuation parameters : 22 = (0.025 +__0.010) x 109/see,
k4=O, which were determined by differential directional corre-
132 001 -I, Q : _ _b2
0.0
1 *62
-0.02
- O.0t. B2ex p - 0.06
-008
-0.10
- 0.12
0.I/. Fig. 22. Comparison of experimental values for B2 and B4 for the cascade by1 = 79 keV, hv2 = 1094 keV in 172Yb with the theoretical coefficients for a 3(MI,E2)2(2)0 cascade. lation measurements 1 ~). The final corrected coefficients are: B2 = - 0 . 0 4 5 + 0.008, B4 =
0.0050 + 0.0025.
The theory of linear polarisation correlations gives for the Bk values of mixed transitions:
Bk = Ak(I)'Bk(2), where Ak(1) are the directional correlation coefficients of the gamma-line measured in the detectors E and Bk(2) are the linear polarisation correlation coefficients of the gamma-line which is measured by the polarimeter. Bk(2) is given by the expression: Bk(2 ) = (1 + 62)-1 [( _ 1)S(L2)×k(L2Lz)Fk(LE Lzld) + +(_1)J(L'~)×k(L~L~) "62" z Fk(L2L , flrl) , +
+ 2(-
312
P. WEIGT et aL
References
where J(L)
/0 if the
2 L p o l e r a d i a t i o n is electric, if the 2 L pole r a d i a t i o n is magnetic,
t(k+2)!
1 I
-1
0
"
I n fig. 22 t h e c o m p a r i s o n o f o u r e x p e r i m e n t a l values for B2 a n d //4 with t h e t h e o r e t i c a l coefficients for a 3(M 1,E2)2(2)0 cascade is given. T h e o n l y s o l u t i o n is Q = 0.84 + 0.04, which is in a g r e e m e n t with the analysis o f t h e g a m m a g a m m a - a n g u l a r c o r r e l a t i o n measurements~°). T h e a u t h o r s are i n d e b t e d to the B u n d e s m i n i s t e r i u m ftir wissenschaftliche F o r s c h u n g for generous s u p p o r t o f this investigation. T h e n u m e r i c a l calculations have been d o n e with the I B M 7090 c o m p u t e r o f the Rheinisch-Westf/ilisches I n s t i t u t f a r l n s t r u m e n t e l l e M a t h e matik.
1) O. Klein and Y. Nishina, Z. Physik 52 (•929) 835. 2) R. D. Evans, The atomic nucleus (Mc Graw Hill, New York, 1955). 3) F. Metzger and M. Deutsch, Phys. Rev. 79 (1950) 551. 4) p. G6ttel, Diplomarbeit (Universit/it Bonn, 1967) unpublished. 5) H. Htibel, Bundesministerium fiir wissenschaftliche Forschung, Forschungsbericht K66-37, Bonn (1966). 6) M. E. Rose, Phys. Rev. 91 (1953) 610. 7) F. Mfinich, K. Fricke, J. Koch and J. Nitschke, Tagung des Fachausschusses "Kernphysik", Freudenstadt (March, 1966). 8) H. Frauenfelder and R, M. Steffen in Alpha-, beta- and gamma-ray spectroscopy, 2 (ed. K. Siegbahn; North-Holland Publishing Company, Amsterdam). 9) K. E. Davies and W. D. Hamilton, Nucl, Physics A96 (1967) 65. 10) H. Blumberg, K.-H. Speidel, H. Schlenz, P. Weigt, H. Hiibel, P. G6ttel, H. F. Wagner and E. Bodenstedt, Nucl. Physics A90 (1967) 65. ~1) C. Giinther, W. Engels and E. Bodenstedt, Phys. Letters 10 (1964) 77.
INSTRUMENTS
AND
57 0967)
METHODS
A MULTIDETECTOR
295-312;
APPARATUS
CORRELATION
© NORTH-HOLLAND
PUBLISHING
CO.
FOR LINEAR POLARISATION
MEASUREMENTS
P. WEIGT, H. HOBEL, P. GOTTEL, P. HERZOG and E. BODENSTEDT lnstitut fiir Strahlen- und Kernphysik der Universitiit Bonn Received 29 August 1967 A nine detector apparatus for the measurement of linear polarisation correlations of gamma-gamma-cascades is described. The high statistical accuracy which can be obtained with this apparatus requires a reinvestigation of the solid angle correction. A modified approximation of the solid angle correction is de-
veloped and its performance is checked by careful calibration measurements with the isotopes l°6Pd, 46Sc and ~°Co. The theory of the attenuation of linear polarisation correlations by internal fields is developed.
1. Introduction
p l a n e o f p o l a r i s a t i o n o f the initial g a m m a - r a y and the scattering plane (fig. 1).
Usually the C o m p t o n effect is used for the measurem e n t o f the linear p o l a r i s a t i o n o f low energy g a m m a rays. Its sensitivity to the linear p o l a r i s a t i o n can be seen from the K l e i n - N i s h i n a 1) f o r m u l a for the differential cross section d a / d O for C o m p t o n scattering dtr d O -- ½r2°{(hv')/(hv)}2" • {(hv')/(hv)+(hv)/(hv')-2sin2Ocos2¢#}.
(1)
The energy hv' o f the scattered g a m m a - r a y is related to the energy hv o f the initial g a m m a - r a y by 2) hv' = h v / [ l + { ( h v ) / ( m o c 2 ) } ( 1 - c o s 0 ) ] . 9 is the scattering angle. • is the angle between the
Fig. 1. Schematic illustration of a basic element for a gamma-ray polarimeter2). F o r 9 = const, the differential cross section da/df2 shows ° a m a x i m u m for 4 ~ = 9 0 and a m i n i m u m for = 0 . T h e a s y m m e t r y ratio
90°-
da Ro= ~(~=90
,~ d a )/~(~=0
,, dago d a o )= ~-/-j~-
(2)
80q
70%
3~.... ~o~ ] I
~ ,~ ~ "
~
e
p
is a m e a s u r e o f the sensitivity o f the C o m p t o n effect to t h e linear p o l a r i s a t i o n . F o r a given energy hv o f the initial g a m m a - r a y the e n d s on the angle O F o r
60°
50 °I~ h Y" [ M e V ]
Fig. 2. Plot of Compton scattering angles @which yield a maximum for the asymmetry ratio R v~ gamma-ray energy. 295
296
P. WEIGT
et al. No(O) oc Io(O) dd°'° ~-
+ 19°(0) da9° dQ '
g9o(O) oc Io(O) da9° d~-
/
/
da° d~'~
N°(O)
/
I0(0 ) da° +
Ngo(0) - [
dO
190(0 ) da90 dO } /
= {n(o)+no}/{P(O)'no+ tO~- 0 . . . . Ro becomes a maximum, tOmaxis plotted for the energy range 0 < hv < 5 MeV in fig. 2. The first description of an apparatus for the measurement of linear polarisation correlations with a polarimeter based on the Compton effect was given by Metzger and Deutsch3). A simple three detector arrangement is shown schematically in fig. 3. If for instance the degree of polarisation of the ','2 radiation of a 7 t - 7z cascade is to be determined, the l't radiation must be detected by El and the '/2 radiation by Z. Thus the positions of the detectors E1 and Z define the angle 0 between the propagation vectors of the two gammarays. A coincidence event is registered only if the radiation is scattered by the Compton effect from Z into the third detector A 1 where its remaining energy must be totally absorbed. In order to measure the linear polarisation correlation the triple coincidence counting rates are recorded at r/-- 0 ° and r! = 90 ° for different values of 0. In this way one obtains the counting rates No(O) and U9o(O). The linear polarisation P(O) is defined by
P(O) = 10(0)/190(0 ). I0(0 ) and •90(0) are the intensities of the components the 1'2 radiation polarised parallel and perpendicular
of to the 0 plane, respectively. The theory of the linear polarisation correlation delivers for Io(0) and 19o(0) the expressions
Io(0 ) : W(O)+[B2p2(cosO)+ B,,pI(cosO)],
(3)
['V(O)--[B 2P2z(cOs0) + B,p42(cos 0)],
(4)
=
(6)
/ {io(o) da9o + 19o(0) ?-ff dao j
Fig. 3. Gamma-ray triple coincidence polarimeter, schematicS).
190(0)
"
Alternatively,
E1 ~
-1- 1 9 0 ( 0 )
(5)
where W(O) = l + A2P2(cos 0) + A~P,(cos 0) is the directional correlation and B2 and B~ are the linear polarisation coefficients. The polarisation P(O) can be derived from the measured quantities No(O) and Ngo(0) with the help of the asymmetry ratio R o in the following way :
1}.
Finally one obtains for the linear polarisation
P(O) =
In 0-
{No(O)/N90(0)} ]/[no{No(O)/N90(O)} -
1].
(7) This formula gives the right polarisation only for an ideal geometry. Therefore one has to correct for the finite solid angles of the detectors. Linear polarisation measurements with the three detector arrangement described, are restricted to a few especially favourable cases. This restriction is caused by insufficient triple coincidence counting rates. In comparison with normal gamma-gamma-directional correlation measurements the counting rates are reduced by an additional solid angle, i.e. the effective angle of the detector for the scattered radiation. Because of this difficulty, only a few linear polarisation correlation measurements have been published up to now, in spite of the fact that the method has been known since 1950. On the other hand measurements of linear polarisation correlations are very valuable for nuclear spectroscopy as they represent the only known method to distinguish absolutely between electric and magnetic multipole radiation. In this work we describe a multidetector apparatus in which the counting rates are increased by a factor of 16 as compared to the simple three detector arrangement. 2. Description of the apparatus We use four NaI(T1) scintillators El, E2, E3, and E4 (referred to as detectors E in the following) for the registration of the radiation ~/1. The polarimeter for the detection of the radiation 7'2 consists of the scatterer Z surrounded by four additional NaI(TI) scintillators A 1, A2, BI, and B2 (referred to in the following as detectors A and B) for the measurement of the scattered radiation. The geometrical arrangement of the detectors is shown in fig. 4. lfthe energy of the radiation 72 is above 200 keV one can use a NaI(TI) scintillator also for the scatterer Z. Only for low energies, plastic scintillators
A
MULTIDETECTOR
297
APPARATUS
ion
Fig. 4. Apparatus for the measurement of the linear polarisation correlations. have to be used in order to keep the probability of photo absorption small enough. All nine detectors are cylindrical (lk" dia. x 2" height). The NaI(T1) scintillators were mounted integrally with 153 AVP photo tubes with energy resolutions between 7.0% and 8.1C}/o fwhm for the 662 keV line of 13VCs. We used a distance of 74 mm between the axes of the scatterer Z and the detectors A and B. The distances between the source and the detectors E and Z were 77 mm. The size of the detectors and the solid angles were so chosen as to yield a maximum of triple coincidence counting rates at a still fairly high asymmetry ratio. The calculations of Metzger and Deutsch 3) were helpful when we tried to find the best compromise solution for gamma-radiations of medium energies. As shown in fig. 2, the best value for the scattering angle 0 .... is about 90 °. It decreases slowly for higher energies of the radiation Y2. Our detector arrangement (fig. 4) allows a rather large range of scattering angles. It is possible, however, to choose a specific range of scattering angles by selecting the energy of the Compton electrons through pulse height discrimination of the pulses produced in the scatterer Z. The four detectors E are mounted on a revolving platform. The measurements are performed at two different positions of the platform which differ by 12.9 °. In this way one measures P(O)at eight equidistant angles between 90 ° and 180° taking advantage of the symmetry of the linear polarisation correlation around 0 = 180°. No correction for the different detection
efficiencies of the four detectors E is necessary because in the measured quantity No/Ngo these efficiencies cancel out. In order to avoid corrections for the different detection efficiencies of the polarimeter detectors A and B, the polarimeter is rotated around the axis of Z in steps of 90 °. In this way the detectors A and B exchange their position several times during one cycle of the measurement. After four steps the direction of rotation is reversed in order to prevent a twisting of the cables from the polarimeter detectors. The sequence of the different positions of the platform and the polarimeter during one complete cycle of the measurement is shown in table 1. The rotations of the TABLE 1 The sixteen detector positions of one complete cycle of the polarisation correlation measurement.
Position hr.
1 (9)
2 (10) 3 (11) 4 (12) 5 (13) 6 (14) 7 (15) 8 (16)
Position Position of the Position of the polarimeter detectors platform of the relative to the 0-plane with the polarimeter detectors E detector A detector B 1 1 1 1 1 1 1 1
(2) (2) (2) (2) (2) (2) (2) (2)
1.
II
LL
L
±
II
II
±
II
3_
.L
II
2_
I1
P.WEIGT et al.
298
platform and the polarimeter are performed by small electromotors controlled automatically. We did not mount any lead shielding between the detectors E. A Compton scattering of gamma-radiation from one E-detector into another does not disturb the measurement, because coincidences between the Edetectors are not measured. The polarimeter detectors A and B must be shielded against direct radiation from the source and scattered radiation from the detectors E; but the space is limited and the lead shield alone is not sufficient if high energy gamma-lines are present. We were able, however, to keep background coincidences negligibly small by electronic discrimination even in rather unfavourable cases. A fast triple coincidence is registered only if the following three conditions are fulfilled: 1. The energy of the 71 radiation has to be totally absorbed in one of the detectors E. Four single channel analysers are used which accept the photo peaks of the 71 radiation. 2. A given fraction of the energy of the ?2 radiation
must be lost in the detector Z by a Compton scattering. A single channel analyzer selects the right energy interval of the Compton electron. 3. The scattered 72 radiation must be totally absorbed in one of the detectors A or B. The pulses from the scatterer Z and from each of the polarimeter detectors A or B are added. Four single channel analyzers are set on the photo lines of the 72 radiation in the four sum spectra. A part of the fast-slowcoincidence circuits is shown in the block diagram fig. 5. Automatic peak stabilisation is used for all nine detectors in order to keep the overall pulse height amplification constant. The detectors E and Z can be stabilised on a photo line of the isotope under investigation. Extra sources are needed for the detectors A and B. We have chosen the monoenergetic radiation of ~aVcs. The four necessary 137Cs sources were mounted inside the lead shield in front of the detectors. The adder ( A D D in fig. 5) are followed by a gate which opens only when an event is detected in the respective polarimeter detector. Without this gate the sum spectrum would be masked
.,:(
?
SCATLE R r¢ O
coL; -I
Y FAST L
Fig. 5. Block diagram of the electronic equipment of one triple coincidence circuit.
1
A MULTIDETECTOR APPARATUS
299
Fig. 6. View of the apparatus for the measurement of the linear polarisation correlation.
totally by the single spectrum of Z. The measurement of the pure sum spectrum is necessary in order to set the single channel analysers correctly on the )'2 photolines. The signals Z + A 1 and Z + A 2 of the sum spectra are fed to the fast and slow coincidence units via an OR-gate. This can be done because the signals contribute the same type of information. This procedure reduces the necessary number of fast-slow coincidence circuits to eight. The entire electronic equipment of the described apparatus is rather bulky but completely transistorised circuits make it possible to reach a very high reliability and to keep all settings stable for long periods of continuous measurements. A view of the whole experimental set up 4' 5) is shown in fig. 6. 3. Determination of the solid angle correction
The asymmetry ratio R 0 for ideal geometry can be
calculated immediately from the Klein-Nishina formula. Fig. 7 shows R 0 as a function of the energy hr. The actual polarisation sensitivity of the apparatus is smaller, however, because of the finite solid angles of the detectors. The exact theory of the solid angle correction has not yet been worked out. The usual method which has been applied so far takes into account all solid angle effects by using formula (7) with a reduced value of R instead of R o. This procedure is only an approximation. The exact correction for the measured polarisation at the angle 0 depends also on the linear polarisation correlation coefficients. This is caused by the fact that the detectors E and Z average the linear polarisation over a finite range of 0. We developed an approximate theory of the solid angle correction which takes into account this last effect. For real geometries, formulae (5) and (6) must be written in the form:
300
P. WEIGT
rA(8) =
No(O)
T6 ('E,'z,rA(.))J l o ( 1 9 o ) = = differential cross section for Compton scattering from rz to rAta) of a radiation which is polarised parallel (perpendicular) to the plane rE,rz, e(rE,rZ,rA) = detection probability for 71 at the point r E, and that for '/2 by Compton scattering at the point r z and 7; at the point r A. The exact evaluation of the solid angle correction could, in principle, be done according to formulas (8) and (9) by the Monte Carlo technique. This procedure has to be carried out, however, for each special pair of 7-ray energies hv I and hv 2. This does not seem to be feasible because it would imply a very long execution time even for very fast computers. We used instead the following approximation. We split the efficiency function e into two factors
+ flgo(O')13(rE,rZ,rA)" (8) and
(O)oc f ([o' ( 0 )13 rE,rz,rB).
"4-
90
point of absorption ofT; in one of the detectors A(B),
]
"[~(l"E,rz,rA)]lodZrEd3rzd3rA"F
N9o
et al.
(o') 13( rE,rz,r B) • (9)
8(rE,rz,r A) = sl(rE,rz). ~2(rE,rZ,r A) ,
where: 0' arccos[{rE'rz}/{IrEl'lrzl}], = point of absorption of 71 in one of the detectors r E E, = point of Compton scattering of 72 in the der Z tectorZ,
where el(rE,rz)
=
probability for detection of 71 at the point r E and that for 72 by Compton scattering in the point r z without absorption of 7; within the detector Z, e2(rE,rz,rA) = probability of detection of 7; by absorpsion in the point rA, and rewrote equations (8) and (9) in the following approximation : =
No(O) oc f lo(O')131(rE,rz)d3rE d3rz • 12
" fe2(rE,rz,rA) [dd~ (rE,rz,ra)] tod3rEd3rz d3rA + 10 ¸
+ fZgo(O')e,(r,rz)d3rE d3rz ' "f132(rE,rz,ra)[~(rE,rz,rA)llgod3rEd3rzd3rA, (10)
0'5
1'0
115
2'0
2'5
3'0
3'5
4'0
415 b hvo [MeV]
Fig. 7. Plot of the asymmetry ratio R0 as a function of the gamma-ray energy hr.
50
A MULTIDETECTOR APPARATUS /. N9o(0 ) oc I
JIo(O')~drE,rz)d3rE darz • f e2(rE,rz,ra)[d~(rE,rZ,rB)] d3rEd3rzd3rB+
The evaluation of the integrals of eqs. (14) and (15) has been performed in the following way: By inserting the expressions (3) and (4) for Io(0') and 19o(0') into eqs. (14) and (15) we obtain:
J to
ud~,~
]o(0) -~/
fl9o(O')e,(rE,rz) dare darz •
+ Q
(11)
+B4fP~(cosO')~tdarEd3rz],
We believe that this splitting of the integrals of equations (8) and (9) is a good approximation because we obtained it by the following consideration: We separated the integrands of equations (8) and (9) into two parts, the first of which depends mainly on the angle 0' and the second depends mainly on the other angles. By dividing No(O)by N9o(0 ) we obtain"
No(O) N9o(0)
= [{1o(0)/I9o(0)}+ R(O)]/[{lo(O)/t9o(O))R(O)+1],
R(O)="
[do
e2(rE'rz'r^) ~
(rE'rz'rA)
]
(12)
darEdarzd3rA I,o , (13)
and
/
Io(0) = lo(O)el(rE,rz)d2rEdarz, I9o(0) =
-
f
~9o(0')~,(rE,r~) d3rE darz.
W(0')gl d3rE darz +
+ [a2f P~(cosO')~,d3rEd3rz +
fe2(rE,rz,ra) IL-d-~ d a ,[rE'rz'rB)Jl9o ,] --3 0 rEO--3r Z d3r B"
with :
301
(16)
I90(0 ) = f W(0')~, darE d a r z -
- [B2f P~(cosO')~,d3,'Ed3,'z+ +B4fP~(cosO')~,darEd3rz].
(,7)
For the evaluation of the first integrals we used the well known "Rose correction" 0) which is usually applied for the solid angle correction of gamma-gamma-directional correlations. The first step for the evaluation of the remaining integrals is the determination of the efficiency coefficient 51. We applied the Monte Carlo method which took into account the probability that 72 has no further interaction in the detector Z on its path to A. Fig. 8 shows, as an example, the result of a Monte Carlo calculation with 10000 events. The execution time for the IBM 7090 was about two minutes, e~ is not normalised in this figure. This calculation was done for
(14)
(15)
In this evaluation we made use of the identity of the integrals:
i I
f e2(rE,rz,rA(a))[ ~ (rE,rz,rAta))]todarEd3rzdarAtB~ = f e2(rE,rz,rBtA))[~(ra,rz,ra(A,)ll~oda rEd3rzda ra(Ar I
It should be pointed out that the improvement of this approximation compared to the usual method consists of the fact that in our definition, R depends only on the geometry of the apparatus and the two gamma-ray energies and does not depend on the linear polarisation correlation coefficients.
./
.30 o
_20 o
_l'Oo
OO
I'0 o
2'0 o
3'{]~ 0 O'
Fig. 8. Probability distribution of 0' as a result of Monte Carlo calculation for J°6Pd energies hva= 513 keV, h)'2= 624 keV.
P. WEIGT et al.
302
,oi 8-
,
p2 2 P2
0
, 100 °
, 120 °
, 140 °
. 160 °
180 °
(3 -2
-6
-s 1 Fig. 9. Plot of the Legendre polynomials Pz2(cosO), PJ(cos0) and the modified functions ~2(0), P4e(0). the geometry of our apparatus and the energies: hvl = 513 keV and hv2 = 624 keV (l°6pd). el is plotted as a function of ( 0 - 0 ' ) . In the second step the remaining integrals were solved numerically. Fig. 9 shows the result for p2(0) =
f
P~(cos0')~l d3rEd3rz
(18)
RIO)T30
and
P](O) =
JP1(cos0') l d3vE d3rz,
The evaluation of R(O), formula (13), has been programmed for a Monte Carlo calculation. The result for the special case of the gamma-gamma-cascade of 106pd (hv I = 513 keV, hv2 = 624 keV) is shown in fig. 10. The asymmetry ratio R decreases slightly with 0. This is expected because the range of angles between the 0 plane and the 0' plane increases with 0. Finally at 0 = 180 <~the 0 plane is no longer defined and the asymmetry ratio reduces to the value R = 1. This calculation of R has two disadvantages: 1. The execution time for the computer is rather long (The calculation for the example of fig. 10 took two hours on an IBM 7090); 2. The systematic error which is produced by the different approximations is unknown and its estimation is difficult. Therefore we have performed calibration measurements with our apparatus for a number of simple cases where the linear polarisation coefficients and angular correlation coefficients can be predicted exactly from the known spins and multipolarities. We selected for this purpose the 624 keV-513 keV cascade of ~°6pd, the 887 keV-1119 keV cascade of 46Sc, and the 1173 keV1332 keV cascade of 6°Co. In all three cases the linear polarisation of both gamma-radiations has been measured. Figs. 11, 12 and 13 show the decay schemes, the single pulse height spectra, and the sum spectra together with the window settings which were used for the linear polarisation correlation measurements. Figs. 14, 15 and 16 show the measured counting rates No/N9o for the 624 keV gamma-transition of a°6pd, the 1332 keV gamma-transition of 6°Co, and the 887 keV
et
(19)
for the example mentioned above. Thus we have reduced the expressions for 10(0) and 190(0) to the form:
2.5 ¸
I0(0 ) = l + A2{J2/Jo}P2(cos O) + A4{J4/Jo}n4(cos O) +
2.0
+ [Bzp2(o) + B4p2(o)],
(20)
I90(0 ) = 1 + Az{Jz/Jo}P2(cos O) + A4{JJJo}P4(cos O) -
[B2p2(o) + B,p2(o)],
(21)
90 °
180 ° II, 0
where J2/Jo and J4/Jo are the " R o s e correction" 6) factors.
Fig. 10. Asymmetry ratio R(O) as a result of Monte Carlo calculation for l°6pd energies hva = 513 keV, hv2 = 624 keV.
303
A MULTIDETECTOR APPARATUS
Fig. 11. Decay scheme, single pulse height spectrum, and sum spectrum together with window settings of the single channel analysers which were used for the linear polarisation correlation measurements in ]°6Pd.
513 oo p-
Z 0 U
/
I ~
1o6 45
;
, b
\ ' c
~
'd
106 Rh 61
,o, EiMeY)
I
•
° *
1 1 0 1 . ~ %~
I, :IT
1.56
2
*
1.137
0
•
1.125
+
0 513 2 ,,
7 8 % ~' -II ~ - A
0
0 *
p d I06 46
60
~' PULSE HEIGHT 887 key 1119
kev
Fig. 12. Decay scheme, single pulse height spectrum, and sum spectrum together with window settings of the single channel analysers which were used for the linear polarisation correlation measurements in 4%c.
U3 o,--
Z
Sc 46
Tl/2= 84d
0 o
T
SINGLE
SPECTRUM
El.key]
I I~
2006
4 ...
B87
2 *
0
0 +
o i
b
IE
2 SUM S P E C T R U M
I
stab[¢
I e
PULSE HEIGHT
Ti 46
P. W E I G T et al.
304
1173 MeV l
1 332 MeV
Lf) O 104-
E[kev]
I
2505
~.
1332
2•
:o 3_ e
f
0
0.
sta~¢ N, g0
P PULSE HEIGHT Fig. 13. Decay scheme, single pulse height spectrum, and sum spectrum together with window settings of the single channel analysers which were used {'or the linear polarisation correlation measurements in 6°Co.
I" 1.7- NO/ N90
1.61.51.41.312 11 i
i
i
~
I
"
i
140"
I
i
160 °
i
I
I@0"
e
Q9 080.70.60.50.4-
Fig. 14. R a t i o o f triple coincidence c o u n t i n g rates
No(O)/Ngo(O)for
l°sPd energies h~'l = 513 keY, h~'2 = 624 keV.
A MULTIDETECTOR
305
APPARATUS
N ~90
-'~ e l 00-
~
900
. . . .
lo'0o
11~
f
r 12oo
1
"'i
, oo
098-
096-
09~
/
Sc 46
0.B87MeV
092]
0881 Fig. ! 5. R a t i o o f triple coincidence counting rates
No(O)/N,o(O) for
46Sc energies/ml --- 11 t9 keV,/n,2 = 887 keV.
---.~ 1,190 100
?
90 °
I
100°
T
1'1
'
~
Y o
o
~
170 °
O L
1,B0°
0 98-
0.96-
09/."
Co60
1.332 MeV
0.92"
Fig. 16. Ratio of triple coincidence counting rates
No(O)/Ngo(O)for
gamma-transition of 46Sc. The linear polarisation of the three other transitions has been measured with the same statistical accuracy. The evaluation of R(O) from the measured ratio of counting r a t e s No(O)/Ngo(O ) is made by use of eq. (12). One obtains for R(O) the expression:
R(O) = [{Io(0)/19o(0)}- {No(O)/N9o(O)}]/ /[{io(0)/I9o(0)} {No(O)~Ngo(O)}- 13.
(22
~°Co energies h~,] = 1173 keV,/n,2 = 1332 keV.
For To(0) and/9o(0) the expressions of eqs. (20) and (21) are inserted. For 6°Co and 46Sc we used the theoretical coefficients for a 4 + ( E 2 ) 2 + ( E 2 ) 0 + cascade. In the case of ~°6pd the measurement of the 0 + ( E 2 ) 2 + ( E 2 ) 0 + cascade contains a ( 1 0 + 3 ) % contribution of the two 2 + ( E 2 , M I ) 2 + ( E 2 ) 0 + cascades starting from the 1125 keV and 1560 keV levels. The E2,M1 mixing ratios were taken from the work of Miinich et al.7). The
P. WEIGT et al.
306 R'
2.0-
2.0
/0
2.0-
10 9'0o
' --~.
1.0
9'0o
~80 o
,
'180o
90 o
e
(3 R ~
2.0.
1.0 ' @
30-
20-
20-
180o P
F i g . 17. A s y m m e t r y ratio
R j
30-
10 9'0o
180o I,
,
i
10
i
90 °
180 °
g0 o
IBO o
0
R(O)in
~°Co a) b) 4"Sc c) d) l " ~ P d e) f)
o u r definition as a result o f c a l i b r a t i o n m e a s u r e m e n t s in
hvl = 1173 k e V , hvl = 1 3 3 2 k e V , by1 = 8 8 7 k e V , by1 = 1 1 1 9 k e V , hJ'l = 6 2 4 k e V , h~'l = 513 k e V ,
evaluation of R(O) is very sensitive to these admixtures and mixing parameters. To get more precise information a reinvestigation on the isotope l°6Pd is in progress. The results for the six measured cases are plotted in fig. 17. All six curves show the expected slight decrease of R with increasing 0. It is satisfactory to see that the shapes of the curves are very similar in spite of the fact that the linear polarisation coefficients B2 and B, and the angular correlation coefficients Az and A 4 are very different for 0 - 2 - 0 and 4 - 2 - 0 cascades. This shows that our approximation in the definition of R is quite effective in making R independent of the correlation coefficients. When we applied, however, the usual definition of R instead of ours we obtained the curves of fig. 18. These curves, indeed, exhibit a strong dependence of the shape R(O) on the correlation coefficients. This dem-
hv2 hvz hv2 hv2
= 1332 k e V , = 1173 k e V ; = 1119 k e V , = 887keV; hv2 = 513 k e V , hv2 = 6 2 4 k e V .
onstrates that especially for large coefficients the usual method of solid angle correction is rather inadequate. A comparison of the empirical values for R(O) of fig. 17 with the "Monto Carlo" calculation plotted in fig. 10 shows a satisfactory agreement. This proves that the systematic errors which are introduced by our approximation cannot be very large. The experimental determination of the absolute linear polarisation correlation coefficients B2 and B, of gamma-gamma-cascades can thus be performed in the following way. 1. R(O) in our definition is derived from the described calibration measurements by an extrapolation to the gamma-energies of the special case. The extrapolation can be performed with high precision by application of the described "Monte Carlo" calculation of R(O). 2. By use of eq. (12) the ratio !o(0)/190(0) is derived
A MULTIDETECTOR
RA
2.0-
R,
R'
2.0
20"
1.0
10
9'0o
10
9'o*
180g
R~
@
180°
LI @ r
]
180°
30-
•
20-
20.
r
1.0
10 90 °
•
90 °
R ~
R d
3.0-
20'
307
APPARATUS
180o
10 90 °
180 °
180°
90 ° G
Fig. 18. A s y m m e t r y r a t i o
R(O)in 6°Co a) b) 46Sc4c) d) L°aPd e) f)
the u s u a l d e f i n i t i o n as a r e s u l t o f c a l i b r a t i o n m e a s u r e m e n t s in h*'l by1 hl, i h~t hvl hvt
= 1173 keV, = 1332keV, = 887 keV, = lll9keV, = 624 keV, = 513 k e V ,
from the measured ratio of coincidence counting rates
hi'2 = 1332 keV, h~'2 = 1173 k e V ; hi'2 = l 1 1 9 k e V , h*'z = 887 k e V ; hvz = 513 keV, hi,2 = 624 keV.
4. By dividing eq. (20) by eq. (21) one gets the ratio
lo(0)/i9o(0 ) expressed by the coefficients A2, A4, Bz and
Xo(O)/Ugo(O):
B4, and one obtains by trivial algebraic computation.~:
Io(0)/19o(0) = J R ( 0 ) - {No(O)/N9o(0)}] / /[R(O){No(O)/Ngo(O)}-I].
(23)
3. The functions p2(O) and p2(O) are calculated for" the special gamma-ray energies by the " M o n t e Carlo" technique and J2/Jo and J,~/Jo are calculated by the Rose method. In principle, one could now derive both, the linear polarisation correlation coefficients B2 and B4 as well as the directional correlation coefficients A2 and A4 from eqs. (20) and (21) by a least-squares fit. The accuracy obtained for B2 and B~ can be increased, however, if ,4, and A~ are determined independently in a directional correlation measurement.
B 2 P2(O) + B4P24(O) = = [1
+ A 2{J2/Jo}Pz(cOs O) + Aa{J4/Jo}Pa(cos 0 ) ]
x[{Io(0)/I9o(0)}-1]/[{Io(0)/I9o(0)}+1].
x
(24)
5. The absolute coefficients B2 and B 4 are determined from eq. (24) by a least-squares fit. 4. Dependence of the asymmetry R on the range of Compton scattering angles In order to optimise the admitted range of Compton scattering angles A,9 we investigated experimentally the dependence of the asymmetry ratio R on this quantity.
P. WE[GT et al.
308
(.9
W(k, k 2 t ) =
~
Z
(malp(k,)[m'a>(m'b[p(k2)lmb> •
roam' a
mhm%
'(mblA(t)lma>
./
25-
By inserting the expressions for the matrix elements [eqs. (66) and (67)in 8)] one obtains: 0 (.J UJ ¢..)
Z
-200
(25)
W(klk2t
) =
=(--1) r'+r~
u z O
t%/t
( - 1 ) -L''-L'2-Rz-N'-s2n~'N2:'~ ~Jklk2 ~tj'
E E
LtL'I kiNitl
L2L'2 k2N2r2
,,k2 t
"[LiL2Ii] [L2L'2IfJ" l
-150
2.0
t
t
*
•< ILlrci[li>
,
t
t
,
•
• Ck,r,(L'~ L 1)cLr~(L 2L[)D%' ,, (Z ~ k,)D~m(Z ~ k2), i
10 °
w
i
i
20 °
30 °
t. 0 o i,
AO Fig. 19. Plot o f R(90 °) a n d triple coincidence c o u n t i n g rates Ne versus /I as a result o f linear polarisation m e a s u r e m e n t s in l°6Pd (hvl = 624 keV, h~2 = 513 keV).
The range of admitted Compton scattering angles is selected in our apparatus by the single channel analyzer of the detector Z which measures the energy of the recoil electron. This energy is connected with the scattering angle by the well known equation:
where ~,~.N,N~:,a are the general perturbation factors of k l k 2 ~.'1 the theory of perturbed angular correlations [ref. 8), eq. (209)] and ck(LL') are the radiation parameters [ref. 8), eq. (63)]. For z~ = r2 = 0, eq. (26) changes into the normal directional correlation function [ref. 8), eq. (208)3• If the linear polarisation of the second radiation is measured, eq. (26) describes the linear polarisation correlation by summing over z~ = 0 and r2 = 0 , + 2 and - 2; one then obtains:
W(ktk2t)=(-1)
h+t~ E A'k,(LIL;IilN~)" NtN2 klk2
T~ = h v - hv'
t t kl •Ak2(L2LzlflN2)DN,o(Z
= hv[{hv/(moc2)} (1 - cosg)] / / [l +
(26)
Stmax - A o <-- St <-- St . . . .
+ AO.
The result for R(90 °) is plotted in fig. 19 vs AO. This shows that R decreases only slightly with increasing ASt. On the other hand, the triple coincidence counting rates increase very fast with increasing ASt. Apparently a good compromise is the choice ASt= 30°; this has been used for all our linear polarisation correlation measurements so far. 5. Theory of attenuation of linear polarisation correlation by extranuclear fields In this section the theory of perturbed gammagamma-angular correlations is extended to the attenuation of linear polarisation correlation. We start from the general expression of the correlation function [ref. 8), eq. (20)] :
k2* ~ N N2 kl)DN~o(Z k2)G,,k2(t)+
+ A'k,(L,L' 1lilN,)" B'[~(L2L'21dN2 ) •
(i - cos O)].
We performed the measurement with the 513 keV gamma-ray of ~°6pd for different ranges
__~
kt NIN2 • Du,o(Z--+ k,)Gk,k~ (t),
(27)
with the abbreviations:
Ilk,1 E (--l) -L''-N't L1L'I I Li L fl i] • ( l l L l r t l l l i > ( l l L l n ' 'I 1I~>* ck,o(L,L,), ,
Ak,(LIL'tIiINI)=
' 'Ii Ak~(L2L2 r N2)= ~ (-
(28)
(L2L2Id
L2L'2
t
t
.(IflL2~EII>(IflL2~21I>
*
*
1
t
%o(L2L2),
(29)
" ' I If N2) = ~, :t , - 1 ) [-L'2-N~-k2, llk,2 1 Bk~(L2L2 L2L'2 (L2L2Ir] •
c
•
t
L L D
k2*
Z~k
*C k*2 0 ( L 2 L 2 ) . +c k* L L P Dk2*
CR2°(LzL2)
Z~k (30)
For the evaluation of the radiation parameters ck~ we
A MULTIDETECTOR
309
APPARATUS
Lu
yb172
/~fiTd
key
I n K 9~//I
I~* ~)
2287
Wl i
2194
/
2075 0,,4~
gg~
,g~
1"/50 1701
}
4t+
1
{4÷
3)
:
(4.
3) 3
,
,..~ ~--e-.,~_ ma~
II04
' I
~
i
1663
1510
172
-
,~
I ~
.
|
i
i
6(+) 3
1376 12~:, _ _ 11
i
r
~
I1
~I,
- -
3(+} 3 1112. 9.7 ns (2*)
6÷
I
--!
-
~
-
0
4*
0
2..
0
O-
0
Fig. 20. Decay scheme of t72Lu. use the expressions for (oalLp,~) and ( a f e l a ' ) = 6~o' for the case of a polarisation insensitive detector 8) and e =½(_ 11- 1 ~) for a detector sensitive to photons linearly polarised in the x-direction. We obtain:
cko(LE) = ½(-1) L-' (2k+ 1)½(2L+ I)½(2L' + 1)+' (i
-L', o k ) {t + ( - 0"''+~'''(-')~+~'+~}'
Ca2(LE)= ¼(-1 )"-' +~v)(2k + I)'I-(2L + 1)}"
1
1
2
'
ck_2(LC) = (-I)'~L"+~L'( - I)L+L'+%dLC), where s ( L ' ) = 1, s(L)= I for L', L denoting electric multipole radiation and s ( L ' ) = 0, s(L)= 0 for L', L denoting magnetic multipole radiation. The factor
(_ 1),~',+~,.(_ 1)~+~'+~ must always be positive, otherwise
Cao(LL') and
P. WE|GT et aL
310
therefore the coefficients _4k will vanish. With this assumption and the expressions for the radiation parameters Ck~ eq. (30) may be written in the form:
B'k'2(L2L'2IfIN2)=
~. ( - - I ) - L % - k 2 - N 2 [ I l k s I" L2I.'2 [L2LzlfJ
• (IflL21~2iI>
DN2_z(Z--~ks)4-D,%2(Z
ks)]
t t k2* , = B,=(L2LslflN=) [ON2 _s(Z--+ ks) 4- D kN2 *, s ( Z ~ k s ) ], (31)
with
Bk2(LsL2IJN2) = 2
W(k, k2t) = ( - 1 ) t'+'~ Z A'k,(l)'A'k2(2)Dk',o(Z-*k,)" kl~2
N1N2 k2*
NjN2
• o,,,o(Z-~ k s ) ~ , ~ (0 + ! ! kI __+ +Ak,(1)Bk2(2)Ou,o(Z k,)" k2* k2 NjN2 • [o~,_~(z-~ ks)+ o~2+s(Z-~k,)]c,,~, (0.
(32) In the special case of liquid sources one has randomly fluctuating interactions. It is convenient then to choose the propagation vector of the first radiation as the quantisation axis. This restricts N~ and N2 to the value Nl = N2 = 0 , and k~ = k s = k . We can now make use of the relation
( - - l ) -L'2-k2-Nl'i Ilk't2 I"
(LsLsIfJ * • ( l f l L s x 2 I [ ) ( l f l L 2 x 2 I17 "
L2L'2
t ¢
• L ~L~){ ' c *,,s(L~Ls)/q~o(L~Ls)}. ' * ' • C,~o(
By inserting the eqs. (28), (29) and (31)into eq. (27) one obtains:
k k2* k2* Doo(Z -+ kl) [Do2(Z--* k2) 4- D O+ 2(Z--* k2)] =
= o~sao(k~-+k,) = 2 { ( k - 2)!/(k 4- 2)~?P~(cos O)cos2~ and reduce eq. (32) to:
W(k, k2 t) = ~. Ak(L1L' I l_i])Ak(L2L'2lfI ) • k
• G°Ok(t)ek(COS O) + 108-
W,°
+ Ak(L 1L; I,I)B,(L2L'21d)" • G°k°(t)P~(cos O) COS2r/.
107
Ak(L1L'llil ) is the normalised directional correlation coefficient for the first gamma-radiation, A k ( L 2 L ; l d ) is the normalised directional correlation coefficient for the second gamma-radiation, Bk(L2L'zld ) is the normalised linear polarisation coefficient for the second gamma-radiation.
1,06
1.05"
The theory of perturbed angular correlations derives for the pertubation factor for randomly fluctuating interactionsS):
104
a°°(t) = e- ~'.
103
102
101
100
go o
180 °
e ogg-
Fig. 21. Ratio of triple coincidence counting rates No(O)/Ngo(O)for 172Yb(hvl = 79 keY, hvz = 1094 keY).
A
MULTIDETECTOR
Our result for the perturbed linear polarisation correlation in liquid sources is in agreement with the formula which Davies and Hamilton used for the attenuated correlation of their linear polarisation correlation measurementg). There may be a few other cases in which the factor Z ok,,o(Z~ k,)
k2*
*,*
311
APPARATUS
0 0O8 0006-
~exp
000~ 0002 00
.
tQ=
02
014
0'6
0'8
62 _ _ 1+6 2
NIN2 ¢~.NIN2[t' ~ • Uklk2 k'l,
can be reduced to only one rotation function multiplied by an attenuation factor. This is true for example for a powder source and for correlations in a transverse magnetic field.
6. Measurement of the linear polarisation correlation of the 1094 keV-79 keV gamma-gamma-cascade in the decay of 172Lu The measurement of the linear polarisation correlation of the 1094 keV-79 keV gamma-gamma-cascade in the decay of 172Lu was important for the determination of the multipolarity of the K-forbidden 1094 keV transition. In a recent publication ~°) we have described this measureme at together with a preliminary analysis of the data. In the meanwhile we have completed the final analysis and represent the result in the following as an example for the experimental determination of absolute values of B2 and B4. The decay scheme is shown in fig. 20. The linear polarisation of the 1094 keV line was measured. The observed ratio of the triple coincidence counting rates No(O)/N9o(O) is shown in fig. 21. The background of coincidences of high energy gamma-lines in the 79 keV window and its contribution to the linear polarisation correlation has been determined experimentally by shifting the low energy window above the 79 keV photo peak. The admixtures of the 91 keV-1094 keV cascade and the 1082 k e V 79 keV cascade have been taken into account by use of theoretical coefficients for A2, A4, B2 and 94. The result of the evaluation of B2 and B4 is:
B2 = - 0 . 0 4 3
+ 0.008,
B4 = 0.0050 + 0.0025. These coefficients have to be corrected for the attenuation by internal fields. We used the attenuation parameters : 22 = (0.025 +__0.010) x 109/see,
k4=O, which were determined by differential directional corre-
132 001 -I, Q : _ _b2
0.0
1 *62
-0.02
- O.0t. B2ex p - 0.06
-008
-0.10
- 0.12
0.I/. Fig. 22. Comparison of experimental values for B2 and B4 for the cascade by1 = 79 keV, hv2 = 1094 keV in 172Yb with the theoretical coefficients for a 3(MI,E2)2(2)0 cascade. lation measurements 1 ~). The final corrected coefficients are: B2 = - 0 . 0 4 5 + 0.008, B4 =
0.0050 + 0.0025.
The theory of linear polarisation correlations gives for the Bk values of mixed transitions:
Bk = Ak(I)'Bk(2), where Ak(1) are the directional correlation coefficients of the gamma-line measured in the detectors E and Bk(2) are the linear polarisation correlation coefficients of the gamma-line which is measured by the polarimeter. Bk(2) is given by the expression: Bk(2 ) = (1 + 62)-1 [( _ 1)S(L2)×k(L2Lz)Fk(LE Lzld) + +(_1)J(L'~)×k(L~L~) "62" z Fk(L2L , flrl) , +
+ 2(-
312
P. WEIGT et aL
References
where J(L)
/0 if the
2 L p o l e r a d i a t i o n is electric, if the 2 L pole r a d i a t i o n is magnetic,
t(k+2)!
1 I
-1
0
"
I n fig. 22 t h e c o m p a r i s o n o f o u r e x p e r i m e n t a l values for B2 a n d //4 with t h e t h e o r e t i c a l coefficients for a 3(M 1,E2)2(2)0 cascade is given. T h e o n l y s o l u t i o n is Q = 0.84 + 0.04, which is in a g r e e m e n t with the analysis o f t h e g a m m a g a m m a - a n g u l a r c o r r e l a t i o n measurements~°). T h e a u t h o r s are i n d e b t e d to the B u n d e s m i n i s t e r i u m ftir wissenschaftliche F o r s c h u n g for generous s u p p o r t o f this investigation. T h e n u m e r i c a l calculations have been d o n e with the I B M 7090 c o m p u t e r o f the Rheinisch-Westf/ilisches I n s t i t u t f a r l n s t r u m e n t e l l e M a t h e matik.
1) O. Klein and Y. Nishina, Z. Physik 52 (•929) 835. 2) R. D. Evans, The atomic nucleus (Mc Graw Hill, New York, 1955). 3) F. Metzger and M. Deutsch, Phys. Rev. 79 (1950) 551. 4) p. G6ttel, Diplomarbeit (Universit/it Bonn, 1967) unpublished. 5) H. Htibel, Bundesministerium fiir wissenschaftliche Forschung, Forschungsbericht K66-37, Bonn (1966). 6) M. E. Rose, Phys. Rev. 91 (1953) 610. 7) F. Mfinich, K. Fricke, J. Koch and J. Nitschke, Tagung des Fachausschusses "Kernphysik", Freudenstadt (March, 1966). 8) H. Frauenfelder and R, M. Steffen in Alpha-, beta- and gamma-ray spectroscopy, 2 (ed. K. Siegbahn; North-Holland Publishing Company, Amsterdam). 9) K. E. Davies and W. D. Hamilton, Nucl, Physics A96 (1967) 65. 10) H. Blumberg, K.-H. Speidel, H. Schlenz, P. Weigt, H. Hiibel, P. G6ttel, H. F. Wagner and E. Bodenstedt, Nucl. Physics A90 (1967) 65. ~1) C. Giinther, W. Engels and E. Bodenstedt, Phys. Letters 10 (1964) 77.