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Linear polarization measurements
NUCLEAR INSTRUMENTS AND METHODS 6t
(1968) 328-332; ©
LINEAR POLARIZATION
NORTH-HOLLAND PUBLISHING CO.
MEASUREMENTS
F. D. LEE* and D. D. WATSON Aerospace Research Laboratoriest, Wright-Patterson Air Force Base, Ohio. U.S.A. Received 2 January 1968 The use of the Compton polarimeter is discussed for the measuremerit of linear polarizations of gamma rays from aligned nuclear excited states. A set of formulas is given for calculating the polarization-correlation of any member of a multiple gamma-ray cascade and for calculating the Compton scattering asymmetry expected from a polarization measurement. Corrections for the finite geometry of the polarimeter are discussed and simple
approximate formulas are given for calculating the geometrical correction coefficients. The results of exact numerical computer calculations for one sample geometry are compared with the results of the approximate formulas. The mechanical construction and electronic circuitry for a typical Compton polarimeter and the shape of the pulse height spectrum obtained by summing the pulses from the scattering and absorbing detectors are discussed.
1. Polarization measurements
No/Ngo. In this case No is the counting rate measured for g a m m a rays f r o m the scattering crystal which are scattered in a direction parallel to the reaction plane, and Ng0 is the c o u n t i n g rate for g a m m a rays scattered p e r p e n d i c u l a r to the reaction plane. (The reaction plane is defined here as t h a t p l a n e which c o n t a i n s the direction of the incident particle beam and the direction o f the nuclear g a m m a ray which is incident on the scattering crystal.) F o r reasons which will become evident later, the a s y m m e t r y ratio ( N o o - N o ) / ( N g o + No) is, in many cases, a m o r e convenient n u m b e r to quote. We define two a d d i t i o n a l quantities as follows: The ratio R = W(O,yH)/W(0,yz) is the ratio o f the n u m b e r of g a m m a rays emitted at an angle 0 with respect to the incident b e a m and with the electric vector parallel to the reaction plane, to the n u m b e r o f g a m m a rays emitted with the electric vector p e r p e n d i c u l a r to the reaction plane. The q u a n t i t y S = Itl I l l is the ratio o f the intensity of C o m p t o n scattered r a d i a t i o n in a direction parallel to the electric vector o f the incident r a d i a t i o n to the intensity of the r a d i a t i o n scattered in a
A m e t h o d for m e a s u r i n g the linear p o l a r i z a t i o n of r e a c t i o n - i n d u c e d nuclear g a m m a rays is desirable since this q u a n t i t y is related to the change in parity o f the nucleus e m i t t i n g the radiation. In a d d i t i o n , for certain cases, a unique assignment o f the m u l t i p o l e mixing ratio of the emitted r a d i a t i o n can be o b t a i n e d by p o l a r i z a t i o n m e a s u r e m e n t s where o t h e r m e t h o d s fail. The intent o f this article is to discuss the C o m p t o n p o l a r i m e t e r as an i n t r u m e n t for m a k i n g such measurements and to present a consistent set of definitions and e q u a t i o n s with which p o l a r i z a t i o n d a t a may be analyzed. A discussion of the electronics and various corrections for finite d e t e c t o r size (geometry) is also included. The C o m p t o n p o l a r i m e t e r consists o f a central scattering crystal and one or more a b s o r b i n g crystals m o u n t e d to the side in such a way t h a t a coincidence between the scattering crystal and one o f the a b s o r b i n g crystals will indicate a C o m p t o n - s c a t t e r e d g a m m a ray. In our p a r t i c u l a r case, the central scattering element was a 2" x 2" NaI(T1) crystal and the side detectors were two 5" x 5" NaI(TI) crystals which were m o u n t e d with their central axes at 90'-' with respect to each other and at 60 ~ with respect to the central axis o f the scattering crystal. A lead c o l l i m a t o r was used to shield the side crystals f r o m direct radiation. Figs. 1 and 2 illustrate the c o n s t r u c t i o n of the polarimeter. F o r most p o l a r i z a t i o n m e a s u r e m e n t s , the p o l a r i m e t e r is m o u n t e d directly a b o v e the target, and hence, the central scattering elem e n t is at an angle o f 90 ° with respect to the incident p r o t o n beam. The q u a n t i t y which is directly measured with the C o m p t o n p o l a r i m e t e r is the c o u n t i n g rate r a t i o * NR.C.-O.A.R.-Post
~TUBE| I I
/.,ABSORBING / / / ' " . / XTAL // /
L
Doct. Assoc.
t An Element of the Office of Aerospace Research, U.S. Air Force. 328
Fig. I. Shows a front view of the lead shield and collimator as well as the positions of the scattering detector and one of the absorbing detectors. The other absorbing detector would be either behind or in front of the scattering detector.
LINEAR
POLARIZATION
329
MEASUREMENTS
and hence P(O) is discussed by Watson and Harris4). The linear polarization distribution in a directionpolarization measurement on a primary gamma ray, can be written as: W(01,~)~ W(01) ~__2 PmZCOS(2~b)BK,,P~(cosO'),
(2)
where Pm is the value of the population parameter for the + m and - m magnetic substates added together. W(O0 is simply the angular distribution of the primary radiation and is given by:
W(O,) = ~, P,, Z AK'PK(COS01)" m
K
The coefficients BK,. and AKIn are defined as
TA RG E
Br,n = ~ {~'/(1 +6~)~tI~(L1L,)EKo(JIL1L1Jzm), 2, , 0 , (3)
T)~
LIL' 1 4Z 0
r
AK,,= ~ { ~ P ' / ( l + f i z ) } ( 2 K + l ) EKO(J1LIL,J2m), (4) LIL'I
Fig. 2. An isometric drawing showing the spatial orientation of the three detectors of the polarimeter. direction perpendicular to the electric vector. It follows as a direct consequence of the above definitions that (N9o - No)/(N9o + No) = =
+s)}
{(R-I)/(R+
I)},
and hence
(R-I)/(R+I) = = {(1 + S ) / ( I - S ) } {(N9o-No)/(Ngo+No) }.
(I)
The left hand side of eq. (1) is identical with the so called degree of linear polarization P(O) as defined by Hoogenboom~). That is
P(O) = {W(0,90) - W(0,0)} / { W(0,90) + W(0,0)} = -= (R - I)/(R + 1). It should be noted that the left hand side ofeq. (l) is a function of the nuclear parameters alone, such as spin, parity change, and multipole mixing ratios, whereas the right hand side consists of the measured counting rates and associated errors, as well as the polarization sensitivity S together with its uncertainties. The evaluation of W(O,~) and hence the ratio R has been discussed by several authors 1-3). We note that in the literature there is no consistent agreement on the specification of the polarization angle ~b. The convention adopted here is to specify the polarization angle q5 as the angle between the electric vector of the radiation and the reaction plane. Because this definition is not universal, confusion may arise when comparing formulas from different authors. The evaluation of W(O,~b)
where the coefficient E°o(J~L1L'IJ2 m) is a function of the spins Jl and J2 of the states involved and of the multipolarity L, L' of the emitted radiation. These coefficients have been discussed and tabulated elsewhere4). The multipole mixing parameter 6 is defined as the ratio of the reduced matrix elements of the multipole orders present in the radiation. Specifically 6 = (J2 []L' ]lJ1 )/(J2 IIL1JJ1). In eq. (2), the plus sign is used if there is no parity change in the transition and the minus sign is used if there is a parity change. The coefficients q,.(LL') are products of the factors (2v +1) ~ with the >:~(LL') coefficients tabulated by Fagg and Hanna2), together with appropriate sign changes necessary to conform with the present notation. The values of the qv coefficients appear in table 1 for dipolequadrupole and quadrapole-octupole mixings. The linear polarization distribution of a secondary gamma radiation with unobserved primaries is given by eq. (2) in which K is replaced by M, 01 by 02, and :
=
Z
LIL'I ...LiL'i...LeL'e
(1 + J , ) 0
(i
×
0 t X EoM(JxLtLtJ2rn) x (5)
•..ItM(JiLiJj)...qM(LeL')h~4(J~LeL'Jf),
AMm ~
.~Pl .~pi Pe ~1 . . . . i ""(~e
E
(1 +
X
+
× ( 2 M + 1)~EOM(J1LILIJ2m)... o
•..ttM(JiLiJj)...hM(JeLeL'Jf).
(6)
The notation for the case of secondary radiation, as well as the values of the coefficients pM(J~LiJ~) and
330
F. D. LEE A N D D. D. W A T S O N
TABLE 1 Coefficients for use in calculating the polarization of a gamma ray between two nuclear energy levels as a function of spin, multipolarity mixing, and parity. Values of the 0 2
v
LL" 11 12 22 23 33
0 0 0 0 0
~I(LL')coefficients. 4
6
(l-S)l(l
-0.25000 +0.05000 -- 1.00000
+0.12019
P(O) = { W(0,90) - W(0,0)} / {W(0,90) + W(0,0)} = = + {~PmZB~mp2(COSO)}/{ZPmZavmPv(cosO)}, v
m
f
(8)
dO J4) = 0/3(0, q~)o'(0, ~ )
If one makes the assumption that the detection efficiency is approximately constant over the azimuthal spread of the detector, and if one replaces the cose~b dependence from the Compton cross section by the average value of the cosZq5 around the interval in question, one obtains the following result for (1 - S ) / ( 1 + S): ( 1 - S ) / ( I + S ) = (1 - 2 f i ) x
v
2. Finite geometry corrections In the case of a point scattering crystal and point detectors, the quantity S can be deduced directly from the ratio of the C o m p t o n scattering cross sections as given by the Klein-Nishina formula. Explicitly:
S = a(/£,O,q~= 0)/a(/£,O,(o = 90), where a ( ~ , 0 , 4 ) = (/£%g) [{(~g +/£2) //£/£o} - 2 cos 2 ~ sin 2 03. The ratio /£//£0 = {1 +~0(1 - c o s 0 ) } - ' , where /£o is the wave number of the incident photon, and/£ is the wave number of the scattered photon. In order to find the value of S in the actual case with side detectors of finite size, one must integrate these cross sections over the azimuthal spread Aq5 and polar spread AO of the detectors. Thus one obtains
f
f~b : 90
~(0,~)~7(0,q~) dO +
(7)
where the plus or minus sign is chosen as in eq. (2). The A vm coefficients are given by eq. (4) or eq. (6) and the B ~ are given by eq. (3) or eq. (5) according to whether they refer to the primary or secondary radiation.
s=
+S) =
f ~o=90g(O'(]))O'(O'~o)dQ-f 4,=oe(O'qS)a(O'4)dQ
+ 1.11803 --0.37268 + 1.11803 +0.55902 -0.74536
hM(JeLeL'eJc) have been discussed and tabulated elsewhere4). Finally, the degree of linear polarization P(O) is defined by
m
center of the scattering crystal and as measured from the reaction plane. The quantity of interest, (1 - S)/(1 + S) can be written in terms of the detector efficiency and Compton cross sections as
f
where the factor e(0,qS) is the detector efficiency and includes information about the geometrical shape of the detector. The subscripts q5 = 0 and q5 = 90 on the integrals indicate that the integral is to be performed about ~b = 0 ° and 4~ = 90 ° respectively; where q~ is the azimuthal angle of the position vector with origin at the
x .
.
f
.
(K2/K )sin 2 0 (0)d0 .
.
.
. , f (/£2//£2) [{(/£2 .~_N2)//£KO} __ sin 2 0 ] g ( 0 ) d 0
(9)
where/3 is equal to the value of the cosZq5 averaged about 90 ° over an interval equal to the azimuthal extent of the crystal. Explicitly: fl --
cos2qSdqS,with 1 = ½7t-½W/(rsinOo),
(10)
where W is equal to the width of the crystal, r is the distance from the center of the scattering crystal to the center of the side detector, and 0 o is the median polar scattering angle determined by the position of the front face of the side detector, and the lower limit for acceptable scattering angles. The value offl was found to be approximately equal to 0.060 for our polarimeter. If one assumes that 5(0) is a constant over the polar spread of the crystal, then the integrals which remain in eq. (9) can be integrated in closed form. However, the functional form of the solution is quite complicated and in this respect not particularly suited for computation. It was found that the energy dependence of the ratio of the two integrals in eq. (9) could be closely approximated by simply the ratio & t h e integrands evaluated at a suitable choice of 0. (1-S)/(I +S) ~ (1-2fl) × x (sin 2 0m) / {(/C0/K) + (K/h'0) -- sin 2 0m}, where = 1 + (G/(mc2)}0--cOS0m).
(1 1)
LINEAR
POLARIZATION
0.4
MEASUREMENTS
331
duced by such an approximation was largest for g a m m a rays o f energy less than a b o u t 2 MeV.
z
o 0.3
3. Polarimeter construction
I-w rY
02 o cr"
gn o.~
- - - . . ~
_
I
I
I
2
I
I
4 Eo
I
I
6
I
8
I
°
I
I
10
(MeV)
Fig. 3. The solid line is the ratio of the exact integrals over the polar angle 0 from eq. (9). The points are calculated from the ratio of the integrands evaluated at an angle 0m = 45° as in eq. (11). This shows that the energy dependence of the correction factor obtained from eq. (11) is nearly the same as from the exact eq. (9). I n order to find the proper value o f 0 m to use in this approximation, one can either carry out the integration for a fixed energy and equate the ratio of the integrands to the resulting number, thus yielding a value for 0m or one can experimentally calibrate the polarimeter by measuring the polarization o f a k n o w n g a m m a ray such as the 3.16 MeV g a m m a ray from the 1214 keV resonance in the 34S(p,7)35Cl reaction and hence determine the ratio of the integrals and thereby the value o f 0 m to be used in the approximation. Shown in fig. 3 is the value of the ratio of these integrals ploted as a function o f energy as c o m p u t e d analytically, and as c o m p u t e d f r o m the ratio of the integrands with a value of 0m = 45 °. A numerical integration was performed in order to examine the importance o f the finite size of the scattering crystal. The results o f this integration showed that for scattering crystals of the order o f 1.5"x 1.5" or smaller, the error introduced by ignoring the finite size was on the order of 3% or smaller. The error intro-
Since the quantity to be measured with the C o m p t o n polarimeter is the asymmetry of scattering at 0 ° and 90 ° and since this asymmetry is often quite small, it is of p a r a m o u n t importance that no asymmetry be introduced by the mechanical or the electronic system of the polarimeter. It is desirable to have a system which rigidly constrains the target and incident particle beam spot to lie accurately on the axis of the scattering crystal and to have the lead collimator open far enough that the complete area of the beam spot will lie inside the acceptance cone of the collimator. The collimator in front of the scattering crystal is necessary to shield the absorbing crystals f r o m direct radiation. The solid angles are generally chosen to he as large as possible without seriously reducing the measured asymmetries. The one exception to this rule is that the absorbing crystal must be biased above 0.51 MeV so that the polarimeter will reject annihilation q u a n t a from pair formation in the scattering crystal. Thus a g a m m a ray of energy, say 3 MeV, will not be detected if it is C o m p t o n scattered at an angle of greater than 70 ° since the scattered q u a n t a would then have an energy which is below the bias on the absorbing crystals. In our polarimeter a forward scattering angle cutoff of a b o u t 30 ° was selected as a reasonable compromise over a wide energy range. The azimuthal half-angle for this polarimeter is 25 °. This angle results in an azimuthal correction factor ( 1 - 2 / / ) = 0.88. The mechanical construction of the polarimeter is illustrated in figs. I and 2. The unusual orientation of the absorbing crystal was chosen so that the forward scattering angle would be limited by the fiat front faces of the crystals rather than the edges of the cylindrical crystals. This resulted in a well defined forward angle cutoff, which was desirable because the sensitivity o f the polarimeter is quite
ABSORBING
CRYSTAL
M,x
INPUT
c,,,-. L2 CRYSTALI Fig. 4. Shows the electronic arrangement used for the polarimeter. The two absorbing crystals are actually set at 0° and 90° with respect to the reaction plane.
332
600
V. D. LEE A N D D. D. W A T S O N
O°
--
90°
Fig. 5. Shows the results from a polarization measurement at the E~= 1214keV resonance in the 34S(p,y)35C1 reaction. The scattering asymmetries of the two strongest gamma rays are opposite as can be seen in the pulse height spectrum. sensitive to the position of the forward angle cutoff. The amount of energy deposited in one of the crystals by a Compton scattered gamma ray depends on the angle through which it was scattered. It is therefore useful to electronically sum the coincident pulses from the scattering and absorbing crystals. This results in a pulse height proportional to the total incident gamma-ray energy and independent of its scattering angle. A satisfactory electronic arrangement which we have used is illustrated in fig. 4. The two absorbing crystals are a matched pair. Both crystals are connected to a common high voltage power supply and pulse height discrimination for both absorbing crystals is performed by the same differential discriminator. This technique is used to maintain electronic symmetry between the two absorbing crystals at 0 ~ and 90 °. Fig. 5 shows the result from a polarization measurement at the Ep = 1214 keV resonance i n t h e 348(p,y)35Cl reaction. This resonance decays predominantly by a two-step cascade from R - - , 3 . 1 6 ~ 0 giving a 4.38 and 3.16 MeV gamma ray. A weak 2.64 MeV gamma ray can also be seen in the spectrum.This gamma ray results from the branch 3.16-. 2.64~ 0. The line shapes of the
individual gamma rays are indicated. The line shapes from monoenergetic gamma rays incident upon the Compton polarimeter can be understood qualitatively in the following way. Suppose a gamma ray of energy Eo is scattered at a definite angle qS~ towards the absorbing crystal. The scattered gamma ray will now have an energy E~ which depends on q~ and is considerably less than E 0. This scattered gamma ray of energy E~ will interact with the absorbing crystal by photocapture, Compton scatter, or pair formation if E~ is sufficiently high. The resulting pulse will be added to the pulse from the scattering crystal so that for all gamma rays scattered at an angle ~D~we should expect to see a photopeak at energy Eo along with the usual complement of Compton tail, single and double escape peaks. However, the Compton edge and tail and the escape peaks will be characteristic of a gamma ray of energy E~. Since the actual line shape is a composite of gamma rays scattered over a wide range of angles the resulting line shape will have a photopeak corresponding to Eo, escape peaks characteristic of gamma rays of much lower energy than Eo, and a Compton shape which is smeared out by the large acceptance angle of the absorbing crystals. In the example shown in fig. 5 the escape peaks are not visible and the strange looking humps below the full energy peaks are the integrated Compton edges. References 1) M. Suffert, P. M. Endt and A. M. H o o g e n b o o m , Physica 25 (1959) 965.
e) L. W. Fagg and S. S. Hanna, Rev. Mod. Phys. 31 (1959) 711. a) H. E. Gore and A. E. Litherland, Nuclear Spectroscopy, Part A (ed. by F. Ajzenberg-Selove;Academic Press, 1960). 4) D. D. Watson and G. I. Harris, Nuclear Data, section A, 3, no. 1 (1967).
(1968) 328-332; ©
LINEAR POLARIZATION
NORTH-HOLLAND PUBLISHING CO.
MEASUREMENTS
F. D. LEE* and D. D. WATSON Aerospace Research Laboratoriest, Wright-Patterson Air Force Base, Ohio. U.S.A. Received 2 January 1968 The use of the Compton polarimeter is discussed for the measuremerit of linear polarizations of gamma rays from aligned nuclear excited states. A set of formulas is given for calculating the polarization-correlation of any member of a multiple gamma-ray cascade and for calculating the Compton scattering asymmetry expected from a polarization measurement. Corrections for the finite geometry of the polarimeter are discussed and simple
approximate formulas are given for calculating the geometrical correction coefficients. The results of exact numerical computer calculations for one sample geometry are compared with the results of the approximate formulas. The mechanical construction and electronic circuitry for a typical Compton polarimeter and the shape of the pulse height spectrum obtained by summing the pulses from the scattering and absorbing detectors are discussed.
1. Polarization measurements
No/Ngo. In this case No is the counting rate measured for g a m m a rays f r o m the scattering crystal which are scattered in a direction parallel to the reaction plane, and Ng0 is the c o u n t i n g rate for g a m m a rays scattered p e r p e n d i c u l a r to the reaction plane. (The reaction plane is defined here as t h a t p l a n e which c o n t a i n s the direction of the incident particle beam and the direction o f the nuclear g a m m a ray which is incident on the scattering crystal.) F o r reasons which will become evident later, the a s y m m e t r y ratio ( N o o - N o ) / ( N g o + No) is, in many cases, a m o r e convenient n u m b e r to quote. We define two a d d i t i o n a l quantities as follows: The ratio R = W(O,yH)/W(0,yz) is the ratio o f the n u m b e r of g a m m a rays emitted at an angle 0 with respect to the incident b e a m and with the electric vector parallel to the reaction plane, to the n u m b e r o f g a m m a rays emitted with the electric vector p e r p e n d i c u l a r to the reaction plane. The q u a n t i t y S = Itl I l l is the ratio o f the intensity of C o m p t o n scattered r a d i a t i o n in a direction parallel to the electric vector o f the incident r a d i a t i o n to the intensity of the r a d i a t i o n scattered in a
A m e t h o d for m e a s u r i n g the linear p o l a r i z a t i o n of r e a c t i o n - i n d u c e d nuclear g a m m a rays is desirable since this q u a n t i t y is related to the change in parity o f the nucleus e m i t t i n g the radiation. In a d d i t i o n , for certain cases, a unique assignment o f the m u l t i p o l e mixing ratio of the emitted r a d i a t i o n can be o b t a i n e d by p o l a r i z a t i o n m e a s u r e m e n t s where o t h e r m e t h o d s fail. The intent o f this article is to discuss the C o m p t o n p o l a r i m e t e r as an i n t r u m e n t for m a k i n g such measurements and to present a consistent set of definitions and e q u a t i o n s with which p o l a r i z a t i o n d a t a may be analyzed. A discussion of the electronics and various corrections for finite d e t e c t o r size (geometry) is also included. The C o m p t o n p o l a r i m e t e r consists o f a central scattering crystal and one or more a b s o r b i n g crystals m o u n t e d to the side in such a way t h a t a coincidence between the scattering crystal and one o f the a b s o r b i n g crystals will indicate a C o m p t o n - s c a t t e r e d g a m m a ray. In our p a r t i c u l a r case, the central scattering element was a 2" x 2" NaI(T1) crystal and the side detectors were two 5" x 5" NaI(TI) crystals which were m o u n t e d with their central axes at 90'-' with respect to each other and at 60 ~ with respect to the central axis o f the scattering crystal. A lead c o l l i m a t o r was used to shield the side crystals f r o m direct radiation. Figs. 1 and 2 illustrate the c o n s t r u c t i o n of the polarimeter. F o r most p o l a r i z a t i o n m e a s u r e m e n t s , the p o l a r i m e t e r is m o u n t e d directly a b o v e the target, and hence, the central scattering elem e n t is at an angle o f 90 ° with respect to the incident p r o t o n beam. The q u a n t i t y which is directly measured with the C o m p t o n p o l a r i m e t e r is the c o u n t i n g rate r a t i o * NR.C.-O.A.R.-Post
~TUBE| I I
/.,ABSORBING / / / ' " . / XTAL // /
L
Doct. Assoc.
t An Element of the Office of Aerospace Research, U.S. Air Force. 328
Fig. I. Shows a front view of the lead shield and collimator as well as the positions of the scattering detector and one of the absorbing detectors. The other absorbing detector would be either behind or in front of the scattering detector.
LINEAR
POLARIZATION
329
MEASUREMENTS
and hence P(O) is discussed by Watson and Harris4). The linear polarization distribution in a directionpolarization measurement on a primary gamma ray, can be written as: W(01,~)~ W(01) ~__2 PmZCOS(2~b)BK,,P~(cosO'),
(2)
where Pm is the value of the population parameter for the + m and - m magnetic substates added together. W(O0 is simply the angular distribution of the primary radiation and is given by:
W(O,) = ~, P,, Z AK'PK(COS01)" m
K
The coefficients BK,. and AKIn are defined as
TA RG E
Br,n = ~ {~'/(1 +6~)~tI~(L1L,)EKo(JIL1L1Jzm), 2, , 0 , (3)
T)~
LIL' 1 4Z 0
r
AK,,= ~ { ~ P ' / ( l + f i z ) } ( 2 K + l ) EKO(J1LIL,J2m), (4) LIL'I
Fig. 2. An isometric drawing showing the spatial orientation of the three detectors of the polarimeter. direction perpendicular to the electric vector. It follows as a direct consequence of the above definitions that (N9o - No)/(N9o + No) = =
+s)}
{(R-I)/(R+
I)},
and hence
(R-I)/(R+I) = = {(1 + S ) / ( I - S ) } {(N9o-No)/(Ngo+No) }.
(I)
The left hand side of eq. (1) is identical with the so called degree of linear polarization P(O) as defined by Hoogenboom~). That is
P(O) = {W(0,90) - W(0,0)} / { W(0,90) + W(0,0)} = -= (R - I)/(R + 1). It should be noted that the left hand side ofeq. (l) is a function of the nuclear parameters alone, such as spin, parity change, and multipole mixing ratios, whereas the right hand side consists of the measured counting rates and associated errors, as well as the polarization sensitivity S together with its uncertainties. The evaluation of W(O,~) and hence the ratio R has been discussed by several authors 1-3). We note that in the literature there is no consistent agreement on the specification of the polarization angle ~b. The convention adopted here is to specify the polarization angle q5 as the angle between the electric vector of the radiation and the reaction plane. Because this definition is not universal, confusion may arise when comparing formulas from different authors. The evaluation of W(O,~b)
where the coefficient E°o(J~L1L'IJ2 m) is a function of the spins Jl and J2 of the states involved and of the multipolarity L, L' of the emitted radiation. These coefficients have been discussed and tabulated elsewhere4). The multipole mixing parameter 6 is defined as the ratio of the reduced matrix elements of the multipole orders present in the radiation. Specifically 6 = (J2 []L' ]lJ1 )/(J2 IIL1JJ1). In eq. (2), the plus sign is used if there is no parity change in the transition and the minus sign is used if there is a parity change. The coefficients q,.(LL') are products of the factors (2v +1) ~ with the >:~(LL') coefficients tabulated by Fagg and Hanna2), together with appropriate sign changes necessary to conform with the present notation. The values of the qv coefficients appear in table 1 for dipolequadrupole and quadrapole-octupole mixings. The linear polarization distribution of a secondary gamma radiation with unobserved primaries is given by eq. (2) in which K is replaced by M, 01 by 02, and :
=
Z
LIL'I ...LiL'i...LeL'e
(1 + J , ) 0
(i
×
0 t X EoM(JxLtLtJ2rn) x (5)
•..ItM(JiLiJj)...qM(LeL')h~4(J~LeL'Jf),
AMm ~
.~Pl .~pi Pe ~1 . . . . i ""(~e
E
(1 +
X
+
× ( 2 M + 1)~EOM(J1LILIJ2m)... o
•..ttM(JiLiJj)...hM(JeLeL'Jf).
(6)
The notation for the case of secondary radiation, as well as the values of the coefficients pM(J~LiJ~) and
330
F. D. LEE A N D D. D. W A T S O N
TABLE 1 Coefficients for use in calculating the polarization of a gamma ray between two nuclear energy levels as a function of spin, multipolarity mixing, and parity. Values of the 0 2
v
LL" 11 12 22 23 33
0 0 0 0 0
~I(LL')coefficients. 4
6
(l-S)l(l
-0.25000 +0.05000 -- 1.00000
+0.12019
P(O) = { W(0,90) - W(0,0)} / {W(0,90) + W(0,0)} = = + {~PmZB~mp2(COSO)}/{ZPmZavmPv(cosO)}, v
m
f
(8)
dO J4) = 0/3(0, q~)o'(0, ~ )
If one makes the assumption that the detection efficiency is approximately constant over the azimuthal spread of the detector, and if one replaces the cose~b dependence from the Compton cross section by the average value of the cosZq5 around the interval in question, one obtains the following result for (1 - S ) / ( 1 + S): ( 1 - S ) / ( I + S ) = (1 - 2 f i ) x
v
2. Finite geometry corrections In the case of a point scattering crystal and point detectors, the quantity S can be deduced directly from the ratio of the C o m p t o n scattering cross sections as given by the Klein-Nishina formula. Explicitly:
S = a(/£,O,q~= 0)/a(/£,O,(o = 90), where a ( ~ , 0 , 4 ) = (/£%g) [{(~g +/£2) //£/£o} - 2 cos 2 ~ sin 2 03. The ratio /£//£0 = {1 +~0(1 - c o s 0 ) } - ' , where /£o is the wave number of the incident photon, and/£ is the wave number of the scattered photon. In order to find the value of S in the actual case with side detectors of finite size, one must integrate these cross sections over the azimuthal spread Aq5 and polar spread AO of the detectors. Thus one obtains
f
f~b : 90
~(0,~)~7(0,q~) dO +
(7)
where the plus or minus sign is chosen as in eq. (2). The A vm coefficients are given by eq. (4) or eq. (6) and the B ~ are given by eq. (3) or eq. (5) according to whether they refer to the primary or secondary radiation.
s=
+S) =
f ~o=90g(O'(]))O'(O'~o)dQ-f 4,=oe(O'qS)a(O'4)dQ
+ 1.11803 --0.37268 + 1.11803 +0.55902 -0.74536
hM(JeLeL'eJc) have been discussed and tabulated elsewhere4). Finally, the degree of linear polarization P(O) is defined by
m
center of the scattering crystal and as measured from the reaction plane. The quantity of interest, (1 - S)/(1 + S) can be written in terms of the detector efficiency and Compton cross sections as
f
where the factor e(0,qS) is the detector efficiency and includes information about the geometrical shape of the detector. The subscripts q5 = 0 and q5 = 90 on the integrals indicate that the integral is to be performed about ~b = 0 ° and 4~ = 90 ° respectively; where q~ is the azimuthal angle of the position vector with origin at the
x .
.
f
.
(K2/K )sin 2 0 (0)d0 .
.
.
. , f (/£2//£2) [{(/£2 .~_N2)//£KO} __ sin 2 0 ] g ( 0 ) d 0
(9)
where/3 is equal to the value of the cosZq5 averaged about 90 ° over an interval equal to the azimuthal extent of the crystal. Explicitly: fl --
cos2qSdqS,with 1 = ½7t-½W/(rsinOo),
(10)
where W is equal to the width of the crystal, r is the distance from the center of the scattering crystal to the center of the side detector, and 0 o is the median polar scattering angle determined by the position of the front face of the side detector, and the lower limit for acceptable scattering angles. The value offl was found to be approximately equal to 0.060 for our polarimeter. If one assumes that 5(0) is a constant over the polar spread of the crystal, then the integrals which remain in eq. (9) can be integrated in closed form. However, the functional form of the solution is quite complicated and in this respect not particularly suited for computation. It was found that the energy dependence of the ratio of the two integrals in eq. (9) could be closely approximated by simply the ratio & t h e integrands evaluated at a suitable choice of 0. (1-S)/(I +S) ~ (1-2fl) × x (sin 2 0m) / {(/C0/K) + (K/h'0) -- sin 2 0m}, where = 1 + (G/(mc2)}0--cOS0m).
(1 1)
LINEAR
POLARIZATION
0.4
MEASUREMENTS
331
duced by such an approximation was largest for g a m m a rays o f energy less than a b o u t 2 MeV.
z
o 0.3
3. Polarimeter construction
I-w rY
02 o cr"
gn o.~
- - - . . ~
_
I
I
I
2
I
I
4 Eo
I
I
6
I
8
I
°
I
I
10
(MeV)
Fig. 3. The solid line is the ratio of the exact integrals over the polar angle 0 from eq. (9). The points are calculated from the ratio of the integrands evaluated at an angle 0m = 45° as in eq. (11). This shows that the energy dependence of the correction factor obtained from eq. (11) is nearly the same as from the exact eq. (9). I n order to find the proper value o f 0 m to use in this approximation, one can either carry out the integration for a fixed energy and equate the ratio of the integrands to the resulting number, thus yielding a value for 0m or one can experimentally calibrate the polarimeter by measuring the polarization o f a k n o w n g a m m a ray such as the 3.16 MeV g a m m a ray from the 1214 keV resonance in the 34S(p,7)35Cl reaction and hence determine the ratio of the integrals and thereby the value o f 0 m to be used in the approximation. Shown in fig. 3 is the value of the ratio of these integrals ploted as a function o f energy as c o m p u t e d analytically, and as c o m p u t e d f r o m the ratio of the integrands with a value of 0m = 45 °. A numerical integration was performed in order to examine the importance o f the finite size of the scattering crystal. The results o f this integration showed that for scattering crystals of the order o f 1.5"x 1.5" or smaller, the error introduced by ignoring the finite size was on the order of 3% or smaller. The error intro-
Since the quantity to be measured with the C o m p t o n polarimeter is the asymmetry of scattering at 0 ° and 90 ° and since this asymmetry is often quite small, it is of p a r a m o u n t importance that no asymmetry be introduced by the mechanical or the electronic system of the polarimeter. It is desirable to have a system which rigidly constrains the target and incident particle beam spot to lie accurately on the axis of the scattering crystal and to have the lead collimator open far enough that the complete area of the beam spot will lie inside the acceptance cone of the collimator. The collimator in front of the scattering crystal is necessary to shield the absorbing crystals f r o m direct radiation. The solid angles are generally chosen to he as large as possible without seriously reducing the measured asymmetries. The one exception to this rule is that the absorbing crystal must be biased above 0.51 MeV so that the polarimeter will reject annihilation q u a n t a from pair formation in the scattering crystal. Thus a g a m m a ray of energy, say 3 MeV, will not be detected if it is C o m p t o n scattered at an angle of greater than 70 ° since the scattered q u a n t a would then have an energy which is below the bias on the absorbing crystals. In our polarimeter a forward scattering angle cutoff of a b o u t 30 ° was selected as a reasonable compromise over a wide energy range. The azimuthal half-angle for this polarimeter is 25 °. This angle results in an azimuthal correction factor ( 1 - 2 / / ) = 0.88. The mechanical construction of the polarimeter is illustrated in figs. I and 2. The unusual orientation of the absorbing crystal was chosen so that the forward scattering angle would be limited by the fiat front faces of the crystals rather than the edges of the cylindrical crystals. This resulted in a well defined forward angle cutoff, which was desirable because the sensitivity o f the polarimeter is quite
ABSORBING
CRYSTAL
M,x
INPUT
c,,,-. L2 CRYSTALI Fig. 4. Shows the electronic arrangement used for the polarimeter. The two absorbing crystals are actually set at 0° and 90° with respect to the reaction plane.
332
600
V. D. LEE A N D D. D. W A T S O N
O°
--
90°
Fig. 5. Shows the results from a polarization measurement at the E~= 1214keV resonance in the 34S(p,y)35C1 reaction. The scattering asymmetries of the two strongest gamma rays are opposite as can be seen in the pulse height spectrum. sensitive to the position of the forward angle cutoff. The amount of energy deposited in one of the crystals by a Compton scattered gamma ray depends on the angle through which it was scattered. It is therefore useful to electronically sum the coincident pulses from the scattering and absorbing crystals. This results in a pulse height proportional to the total incident gamma-ray energy and independent of its scattering angle. A satisfactory electronic arrangement which we have used is illustrated in fig. 4. The two absorbing crystals are a matched pair. Both crystals are connected to a common high voltage power supply and pulse height discrimination for both absorbing crystals is performed by the same differential discriminator. This technique is used to maintain electronic symmetry between the two absorbing crystals at 0 ~ and 90 °. Fig. 5 shows the result from a polarization measurement at the Ep = 1214 keV resonance i n t h e 348(p,y)35Cl reaction. This resonance decays predominantly by a two-step cascade from R - - , 3 . 1 6 ~ 0 giving a 4.38 and 3.16 MeV gamma ray. A weak 2.64 MeV gamma ray can also be seen in the spectrum.This gamma ray results from the branch 3.16-. 2.64~ 0. The line shapes of the
individual gamma rays are indicated. The line shapes from monoenergetic gamma rays incident upon the Compton polarimeter can be understood qualitatively in the following way. Suppose a gamma ray of energy Eo is scattered at a definite angle qS~ towards the absorbing crystal. The scattered gamma ray will now have an energy E~ which depends on q~ and is considerably less than E 0. This scattered gamma ray of energy E~ will interact with the absorbing crystal by photocapture, Compton scatter, or pair formation if E~ is sufficiently high. The resulting pulse will be added to the pulse from the scattering crystal so that for all gamma rays scattered at an angle ~D~we should expect to see a photopeak at energy Eo along with the usual complement of Compton tail, single and double escape peaks. However, the Compton edge and tail and the escape peaks will be characteristic of a gamma ray of energy E~. Since the actual line shape is a composite of gamma rays scattered over a wide range of angles the resulting line shape will have a photopeak corresponding to Eo, escape peaks characteristic of gamma rays of much lower energy than Eo, and a Compton shape which is smeared out by the large acceptance angle of the absorbing crystals. In the example shown in fig. 5 the escape peaks are not visible and the strange looking humps below the full energy peaks are the integrated Compton edges. References 1) M. Suffert, P. M. Endt and A. M. H o o g e n b o o m , Physica 25 (1959) 965.
e) L. W. Fagg and S. S. Hanna, Rev. Mod. Phys. 31 (1959) 711. a) H. E. Gore and A. E. Litherland, Nuclear Spectroscopy, Part A (ed. by F. Ajzenberg-Selove;Academic Press, 1960). 4) D. D. Watson and G. I. Harris, Nuclear Data, section A, 3, no. 1 (1967).