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A high intensity apparatus for accurate compton profile measurements
NUCLEAR I N S T R U M E N T S AND METHODS
155 ( 1 9 7 8 )
1 1 5 - 1 1 9 :, (~) N O R T H - H O L L A N D PUBLISHING CO.
A HIGH INTENSITY APPARATUS FOR ACCURATE COMPTON PROFILE MEASUREMENTS S. MANNINEN and T. PAAKKARI
Department of Physics, University of Helsinki, SF-O0170 Helsinki 17, Finland Received 23 March 1978 A construction for a collimating system to measure the energy spectrum of Compton scattered photons is presented. This system is based on a 5 Ci annular 241Am source and on conical collimators. Using a scattering angle of 165° Compton profiles of liquids and polycrystalline solids can be measured with high statistical accuracy. Greatest scattering angle of the apparatus is 175° which can be used for anisotropy measurements. The energy and angular dependence of the resolution function is also studied at the energies commonly used for Compton scattering experiments.
1. Introduction The rapid growth of interest in Compton profile measurements over the last years is largely due to the following two reasons: 1) Mono-energetic gamma rays as a primary radiation instead of lower energy X-rays and the use of solid state detectors, which analyse the energy of each photon detected, have essentially shortened the desired measuring times and despite of that improved the statistical accuracy of measurements. 2) The present computer technology makes it possible to calculate wave functions and hence Compton profiles for solids, liquids and gases accurately enough for detailed comparison with experimental results. In 1970 Felsteiner, Fox and Kahane 1) performed first quantitative Compton profile measurement with gamma rays. They used a 300mCi 24~Am source, which emits 59.537 keV gamma rays, to measure the Compton profile o.f graphite. The other sources mainly used after that have been 123roTe (159 keV) and 198Au (412 keV). The source strengths have been up to 1 Ci for ~4~Am and t23mTe and up to 70 Ci for ~98Au. Recently Weyrich 2) pointed out that an annular source geometry, instead of disc shaped sources used previously, should give larger intensity because of the advantages in the beam collimation and also because the self-absorption in the source is smaller. In the present work two collimation systems based on a 5 Ci annular 24~Am source have been constructed. The first one, in which the scattering angle is 165°, is designed to measure isotropic electron properties and it also gives larger intensity: a statistical accuracy of about 1 ~ can be at-
tained at the Compton peak in one day. The second one, in which the scattering angle is 175°, is made to measure the anisotropies in the Compton profile; a high scattering angle is needed to fix the direction of the scattering vector. The effect of the beam divergences on the measured energy spectrum (Compton profile) has been calculated for both geometries using a computer simulation. The variation of the resolution function of the detector along the energy spectrum has been measured and the total effect (divergence+resolution) on the Compton profile has been estimated.
2. The apparatus The general description of a Compton profile measurement and data processing is given elsewhere3). The present scattering arrangement was based on following requirements: 1) In order to minimize the background intensity, it is necessary that no parts of the vacuum scattering chamber illuminated by the source, are seen by the detector. 2) The source-sample and sample--detector distances should be small to maximize the intensity, but the scattering angle, however, must still remain well defined. 3) The measurements of the anisotropies of the Compton profiles are useful only, if the direction of the scattering vector is fixed. In the case of an annular source this is not possible unless the scattering angle is close to 180°. Because of the fixed diameter of the annular source, it follows from requirements 1) and 2) that the scattering angle should be < 170° which is contrary to the requirement 3). It was therefore decided to construct two alternative geometries:
116
S. M A N N I N E N
A N D T.
The short geometry, in which the mean scattering angle is about 165°, is shown in fig. la. Gamma rays from a 5 Ci annular 24~Am source (outer diameter 46.6 mm, inner diameter 26.2 ram, thickness 8.5 ram) are collimated using a lead cone and a heavy alloy tube (95% W, 5% Fe-Ni, p--18 g/cm3). Because the absorption coefficient for the heavy alloy is larger than for lead (in the case of 60 keV, for example, the linear absorption coefficient is 55 cm-1 for lead and 65 cm-~ for the heavy alloy), the tube also prevents effectively gamma rays to penetrate straight through the inner part of the collimator. The shape of the lead collimator is designed in such a way that the portion of the fluorescence and scattered radiation which can reach the sample or the inner side of the tube is minimized. A hole in the lead part surrounding the detector is made for a low activity 24, Am point source which is used for the daily energy calibration of the counting chain. The vacuum scattering chamber is about 30 cm long containing a window made of aluminized mylar foil. The long geometry, in which the mean scattering angle is about 175° , is shown in fig. lb. The short geometry can be easily converted into the long one by adding an extra lead block and by changing the positions of the sample and the heavy alloy tube. Because of the annular shape of the source, the scattering angle of 175° means that the direction of the scattering vector is fixed within ___2.5° which is same order than the geometrical diver(a) I I
A HO~E FOR SMALL CAL 'SRATION SO{JRCE
~.~ 5Cl ANNULAR Am SOUI~CE / ~
PAAKKARI
gences of the collimation system. One should also remember the small variation of the scattering vector along the Compton profile due to the change in the length of the scattered wave vector4).
3. Resolution of the apparatus The instrumental resolution function of a Compton spectrometer consists of two parts: 1) the energy resolution of the detector, R(a)), and 2) the geometrical resolution function of the collimator system, G(oJ). The total instrumental resolution function, /(co2), is the convolution of these two functions 1(o92) =
G(m) R ( ~ 2 - o ) d~.
(1)
-oo
The shape and the fwhm (Full Width at Half Maximum) of R(o~) depends on the properties of the detector crystal and Of the counting chain. The energy resolution (fwhm) of the Ge detector (Princeton Gamma-Tech. model IG 107, active size 12 r a m × 7 ram) used in the present system is given in fig. 2. The slope of the curve corresponds a Fano factor of 0.07 if 2.96 eV is taken as the mean energy required for the creation of an electron-hole pair in the present detector crystal. In fig. 3 the R(t~) is given at 59.537 keV. This measurement was made using an 24'Am point source with a holder covered by lead to prevent scattering from the holder to enter the detector. The fwhm of R(og) changes by - 2 5 e V from the energy of the unmodified line to the energy of the peak of the Compton modified line and varies from 320eV to 351eV over the range of interest in
DETECTOR RESOLUTION
i
rDleV]=33.96 ~
-9888
~
J _
HEAVY ALLOYrUSE
._. L...I
>
59.54ke~
(b) 2oo
I 144 keV
SAMPLE h
L~ iO0
/
~7Co
4 Fig. 1. T h e drawing o f the apparatus: a) T h e short g e o m e t r y (0= 165°), b) the long geometry ( 0 = 175°). Parts used for collimation and radiation shielding are m a d e of lead and of .heavy alloy.
' 2636key
~m
~lAm
~
~o
CENTER OF THE COMPTON L I N E
7 VE/keV"
Fig. 2. Resolution o f the Ge detector (Princeton G a m m a - T e c h . , active size 1 2 m m x 7 m m ) . T h e slope of the curvc gives a Fano factor of 0.07. Center of the C o m p t o n line corresponds with an energy of 48.440eV at 0 = 165 ° .
ACCURATE i J
10"-
i
i
1
COMPTON
PROFILE
0.8
i
DETECTOR RESOLUTION FUNCTION AT 59.537 keV
5
Z < T U 10 4
i
II
i
i
i
i
,,
:°-'m,
t
o.61
\ ~'~"x~"~ (FWHM)
T
n-hi n
1"!7
MEASUREMENTS
h
&j 0.-'
~I0 3
102
GeK ESCAPE PEAKS E)O
I I L 610 E/key Fig. 3. Resolution function o f the Ge detector at 59.537 eV measured using an 241 Am point source. The contribution of the tail is about 1%.
Compton profile measurements (from P2 = - 10 au to Pz = l0 au*). The geometrical resolution function, G(09), originates from the fact that the energy corresponding to the peak of the Compton profile, 09o, depends on the scattering angle through the well known relationship O91--0)0 =
2o91 too sin 2 0/2 mc 2
,
(2)
where 091 is the energy of the incoming photon and m c 2 =511 keV. Using this relationship and the general expression between the electron momentume p: and the energy of the scattered photon 092 ~°l -o92 - 2o91 c°2 sin 2 ~ p, = -mc.
2
2
mc
(3)
(o9~+o92 - 2o91 o92 cos0) ½
=o,~o"
tion of the Compton peak (092 = 09°, Pz = 0). As shown in fig. 5, no essential improvement can be obtained in resolution by utilizing energies above 200 keV, say. In fact, the situation may even get worse if wider collimation or/and lower scattering angles are used as demonstrated in fig. 4. On the other hand, the cross-section for photoelectric absorption will become smaller at higher photon energies, making measurements on heavy atoms possible. To be more accurate, the energy dependence of d 0 9 /d O should be taken into account at each point 092 of the profile. At large O (cos 0= - 1 ) one obtains do92
do92///p, o92 ) d0/Ikm--c~ + 1 .
(5)
It is found that the effect of the correction term in parentheses is always less than 10% to the fwhm of G(09) and can thus be omitted.
(4)
In fig. 4 the fwhm of G ( 0 9 ) is drawn in momentum units corresponding the angular fwhm of two degrees in 0 for scattering angles of 150°, 165° and 175° at 092 : w ° as a function of the energy of the incoming photon 09). The energies typically used for Compton experiments are indicated by arrows. In fig. 5 the fwhm of 1(09) is given in momentum units assuming Gaussian line shapes for the R(09) of the present detector and for the G(09) with the fwhm of two degrees (cf. fig. 4). The given values of the fwhm of 1 ( 0 9 ) refer to the posi* In atomic units h = m = e = I, c = 137.
100 200 300 400 500 t~/keV Fig. 4, Line broadening due to the beam divergence (Fo) at the C o m p t o n peak in m o m e n t u m units as a function o f the incoming photon energy. Solid lines correspond to three scattering angles typically used in C o m p t o n experiments. The broken curve (FD) gives the energy resolution of the detector.
dO =
one obtains for the shift of the Compton profile in momentum units due to the change of the scattering angle oJ, =~,o = \ d o 9 2 , ] \ d 0 / , o ~
L~
o,-,O.g 1 3 /
,
,
2 2 ½ r _iFoore ),
re = 2" (FWHM)
~ 0.6
L 0.3[4t
l
, t ,t tit Lt , , 100 200 300 400 500 ~/keV Fig. 5. Total resolution of a Complon spectrometer / = (FZo + .r'#)t at the Compton peak as a Function of the incoming photon energy. Two degrees are taken as the fwhm o f G(O) and Gaussian line shapes are assumed.
11"8
S. M A N N I N E N AND T. P A A K K A R I
of G(co): The angular distribution of the scattered power, G(0), was determined using a computer simulation. To save computing time the calculation was made only in two dimensions by projecting the source, sample and detector on a plane. This omits the effects of vertical divergence. However, the change of the scattering angle, 7, due to the vertical divergence (deviation from the scattering plane) ~p, obtained from Calculation
0 q~2
7 = - ½ tan ~-
,
(6)
is well of second order at an scattering angle of 165° (short geometry). The resulting curve for the short geometry is given in fig. 6 after conversion into an energy scale according eq. (2). The derivate - co2 sin 0
(_~02)
'02=O'20
( 2co1
0 )2 sin 2 ~-+ 1 mc 2
(7)
has a value of 21 eV/degree at 0= 165° and it is seen from fig. 6 that the fwhm of G(co) corresponds to about 5°. The total instrumental function at 165° was calculated using eq. (1) and the fwhm of 1(¢o) was found to be 355 eV which was very close to the result of 353 eV obtained using the sum rule for the fwhm of Gaussian type functions. For final instrumental resolution function for the short geometry experiments one can therefore use the detector resolution function R(w) measured at 59.537 keV because its fwhm deviates only little from the calculated I(09) at the Compton peak, 09° = 48.440 keV. Thus for the present system calculation of I(o9) for each measurement is not necessary and small changes in the resolution of the detector are taken automatically into
1.0
~
/
110eV
-2.50 0 250 ~E/eV Fig. 6. Detector resolution function (solid curve) and geometrical resolution function (dashed curve) of the present Compton spectrometer.
TABLE 1 Counting rates for various samples at the scattering angle of" 165% Sample
Thickness
H20 H20 AI Ni Nb
1 mm 3 mm 3 mm 1 mm 1 mm
85 320 395 146 56
W
1 mm
59
Background
c/s c/s (38.5-58.5 keV) at peak channel
0.4
3.0 10.5 10.3 2.6 1.0 0.9
10- 3
account by measuring R(oJ) at 59.537 keV before and after the measurement of the Compton profile. In the case of the long geometry, scattering angle 0 = 175°, the effect of the geometrical resolution function, G(oJ), is negligible. The value of (doo2/dO)~o2=o3 ° is equal to 7eV/degree at 0 = 175° which makes the fwhm of G(~o)less than 35 eV because of tighter collimation. Assuming Gaussian form, the fwhm of/(6o) is changed only about 0.5% in convolution which is, for example, much less than the variation of R(to) over the Compton profile. Using long geometry the total instrumental function can be thus obtained by narrowing the R ( w ) measured at 59.537 keV by approximately 25 eV in the value of fwhm. 4. P e r f o r m a n c e
of t h e a p p a r a t u s Some characteristic counting rates for the short geometry are given in table 1. Data have been collected in the energy region of 38.5-58.5keV (-17.2+14.5 atomic units of momentum). Also given in table 1 is the counting rate at the peak channel in each case (the channel width was about 63eV). Compared with the 24~Am and ~23mTe systems used so far the intensity is better by more than one order of magnitude in the present apparatus. As can be seen in table 1 a statistical accuracy of about 1%o can be obtained in one day for light materials. In the case of the long geometry, the intensity is lower by about one order of magnitude compared with the results given in table 1. For example, in order to have a statistical accuracy of i% in anisotropy measurements of AI and Ni crystals, measuring times of about 3 h and 10 h are needed,
A C C U R A T E COMPTON PROFILE M E A S U R E M E N T S
but the corresponding times are 10 d for AI and 40 d for Ni if an accuracy of 1%o is required. References i) j. Felsteiner, R. Fox and S. Kahane, Phys. Lett. 33A (1970)
119
442. 2) W. Weyrich, Ber. Bunsen. Ges. Phys. Chem. 79 (1975) 1085. 3) R. J. Weiss, W. A. Reed and P. Pattison, Compton Scattering (ed. B. Williams; McGraw-llill, London, 1977) p. 43. 4) p. Eisenberger and W. A. Reed, Phys. Rev. B9 (1974) 3237.
155 ( 1 9 7 8 )
1 1 5 - 1 1 9 :, (~) N O R T H - H O L L A N D PUBLISHING CO.
A HIGH INTENSITY APPARATUS FOR ACCURATE COMPTON PROFILE MEASUREMENTS S. MANNINEN and T. PAAKKARI
Department of Physics, University of Helsinki, SF-O0170 Helsinki 17, Finland Received 23 March 1978 A construction for a collimating system to measure the energy spectrum of Compton scattered photons is presented. This system is based on a 5 Ci annular 241Am source and on conical collimators. Using a scattering angle of 165° Compton profiles of liquids and polycrystalline solids can be measured with high statistical accuracy. Greatest scattering angle of the apparatus is 175° which can be used for anisotropy measurements. The energy and angular dependence of the resolution function is also studied at the energies commonly used for Compton scattering experiments.
1. Introduction The rapid growth of interest in Compton profile measurements over the last years is largely due to the following two reasons: 1) Mono-energetic gamma rays as a primary radiation instead of lower energy X-rays and the use of solid state detectors, which analyse the energy of each photon detected, have essentially shortened the desired measuring times and despite of that improved the statistical accuracy of measurements. 2) The present computer technology makes it possible to calculate wave functions and hence Compton profiles for solids, liquids and gases accurately enough for detailed comparison with experimental results. In 1970 Felsteiner, Fox and Kahane 1) performed first quantitative Compton profile measurement with gamma rays. They used a 300mCi 24~Am source, which emits 59.537 keV gamma rays, to measure the Compton profile o.f graphite. The other sources mainly used after that have been 123roTe (159 keV) and 198Au (412 keV). The source strengths have been up to 1 Ci for ~4~Am and t23mTe and up to 70 Ci for ~98Au. Recently Weyrich 2) pointed out that an annular source geometry, instead of disc shaped sources used previously, should give larger intensity because of the advantages in the beam collimation and also because the self-absorption in the source is smaller. In the present work two collimation systems based on a 5 Ci annular 24~Am source have been constructed. The first one, in which the scattering angle is 165°, is designed to measure isotropic electron properties and it also gives larger intensity: a statistical accuracy of about 1 ~ can be at-
tained at the Compton peak in one day. The second one, in which the scattering angle is 175°, is made to measure the anisotropies in the Compton profile; a high scattering angle is needed to fix the direction of the scattering vector. The effect of the beam divergences on the measured energy spectrum (Compton profile) has been calculated for both geometries using a computer simulation. The variation of the resolution function of the detector along the energy spectrum has been measured and the total effect (divergence+resolution) on the Compton profile has been estimated.
2. The apparatus The general description of a Compton profile measurement and data processing is given elsewhere3). The present scattering arrangement was based on following requirements: 1) In order to minimize the background intensity, it is necessary that no parts of the vacuum scattering chamber illuminated by the source, are seen by the detector. 2) The source-sample and sample--detector distances should be small to maximize the intensity, but the scattering angle, however, must still remain well defined. 3) The measurements of the anisotropies of the Compton profiles are useful only, if the direction of the scattering vector is fixed. In the case of an annular source this is not possible unless the scattering angle is close to 180°. Because of the fixed diameter of the annular source, it follows from requirements 1) and 2) that the scattering angle should be < 170° which is contrary to the requirement 3). It was therefore decided to construct two alternative geometries:
116
S. M A N N I N E N
A N D T.
The short geometry, in which the mean scattering angle is about 165°, is shown in fig. la. Gamma rays from a 5 Ci annular 24~Am source (outer diameter 46.6 mm, inner diameter 26.2 ram, thickness 8.5 ram) are collimated using a lead cone and a heavy alloy tube (95% W, 5% Fe-Ni, p--18 g/cm3). Because the absorption coefficient for the heavy alloy is larger than for lead (in the case of 60 keV, for example, the linear absorption coefficient is 55 cm-1 for lead and 65 cm-~ for the heavy alloy), the tube also prevents effectively gamma rays to penetrate straight through the inner part of the collimator. The shape of the lead collimator is designed in such a way that the portion of the fluorescence and scattered radiation which can reach the sample or the inner side of the tube is minimized. A hole in the lead part surrounding the detector is made for a low activity 24, Am point source which is used for the daily energy calibration of the counting chain. The vacuum scattering chamber is about 30 cm long containing a window made of aluminized mylar foil. The long geometry, in which the mean scattering angle is about 175° , is shown in fig. lb. The short geometry can be easily converted into the long one by adding an extra lead block and by changing the positions of the sample and the heavy alloy tube. Because of the annular shape of the source, the scattering angle of 175° means that the direction of the scattering vector is fixed within ___2.5° which is same order than the geometrical diver(a) I I
A HO~E FOR SMALL CAL 'SRATION SO{JRCE
~.~ 5Cl ANNULAR Am SOUI~CE / ~
PAAKKARI
gences of the collimation system. One should also remember the small variation of the scattering vector along the Compton profile due to the change in the length of the scattered wave vector4).
3. Resolution of the apparatus The instrumental resolution function of a Compton spectrometer consists of two parts: 1) the energy resolution of the detector, R(a)), and 2) the geometrical resolution function of the collimator system, G(oJ). The total instrumental resolution function, /(co2), is the convolution of these two functions 1(o92) =
G(m) R ( ~ 2 - o ) d~.
(1)
-oo
The shape and the fwhm (Full Width at Half Maximum) of R(o~) depends on the properties of the detector crystal and Of the counting chain. The energy resolution (fwhm) of the Ge detector (Princeton Gamma-Tech. model IG 107, active size 12 r a m × 7 ram) used in the present system is given in fig. 2. The slope of the curve corresponds a Fano factor of 0.07 if 2.96 eV is taken as the mean energy required for the creation of an electron-hole pair in the present detector crystal. In fig. 3 the R(t~) is given at 59.537 keV. This measurement was made using an 24'Am point source with a holder covered by lead to prevent scattering from the holder to enter the detector. The fwhm of R(og) changes by - 2 5 e V from the energy of the unmodified line to the energy of the peak of the Compton modified line and varies from 320eV to 351eV over the range of interest in
DETECTOR RESOLUTION
i
rDleV]=33.96 ~
-9888
~
J _
HEAVY ALLOYrUSE
._. L...I
>
59.54ke~
(b) 2oo
I 144 keV
SAMPLE h
L~ iO0
/
~7Co
4 Fig. 1. T h e drawing o f the apparatus: a) T h e short g e o m e t r y (0= 165°), b) the long geometry ( 0 = 175°). Parts used for collimation and radiation shielding are m a d e of lead and of .heavy alloy.
' 2636key
~m
~lAm
~
~o
CENTER OF THE COMPTON L I N E
7 VE/keV"
Fig. 2. Resolution o f the Ge detector (Princeton G a m m a - T e c h . , active size 1 2 m m x 7 m m ) . T h e slope of the curvc gives a Fano factor of 0.07. Center of the C o m p t o n line corresponds with an energy of 48.440eV at 0 = 165 ° .
ACCURATE i J
10"-
i
i
1
COMPTON
PROFILE
0.8
i
DETECTOR RESOLUTION FUNCTION AT 59.537 keV
5
Z < T U 10 4
i
II
i
i
i
i
,,
:°-'m,
t
o.61
\ ~'~"x~"~ (FWHM)
T
n-hi n
1"!7
MEASUREMENTS
h
&j 0.-'
~I0 3
102
GeK ESCAPE PEAKS E)O
I I L 610 E/key Fig. 3. Resolution function o f the Ge detector at 59.537 eV measured using an 241 Am point source. The contribution of the tail is about 1%.
Compton profile measurements (from P2 = - 10 au to Pz = l0 au*). The geometrical resolution function, G(09), originates from the fact that the energy corresponding to the peak of the Compton profile, 09o, depends on the scattering angle through the well known relationship O91--0)0 =
2o91 too sin 2 0/2 mc 2
,
(2)
where 091 is the energy of the incoming photon and m c 2 =511 keV. Using this relationship and the general expression between the electron momentume p: and the energy of the scattered photon 092 ~°l -o92 - 2o91 c°2 sin 2 ~ p, = -mc.
2
2
mc
(3)
(o9~+o92 - 2o91 o92 cos0) ½
=o,~o"
tion of the Compton peak (092 = 09°, Pz = 0). As shown in fig. 5, no essential improvement can be obtained in resolution by utilizing energies above 200 keV, say. In fact, the situation may even get worse if wider collimation or/and lower scattering angles are used as demonstrated in fig. 4. On the other hand, the cross-section for photoelectric absorption will become smaller at higher photon energies, making measurements on heavy atoms possible. To be more accurate, the energy dependence of d 0 9 /d O should be taken into account at each point 092 of the profile. At large O (cos 0= - 1 ) one obtains do92
do92///p, o92 ) d0/Ikm--c~ + 1 .
(5)
It is found that the effect of the correction term in parentheses is always less than 10% to the fwhm of G(09) and can thus be omitted.
(4)
In fig. 4 the fwhm of G ( 0 9 ) is drawn in momentum units corresponding the angular fwhm of two degrees in 0 for scattering angles of 150°, 165° and 175° at 092 : w ° as a function of the energy of the incoming photon 09). The energies typically used for Compton experiments are indicated by arrows. In fig. 5 the fwhm of 1(09) is given in momentum units assuming Gaussian line shapes for the R(09) of the present detector and for the G(09) with the fwhm of two degrees (cf. fig. 4). The given values of the fwhm of 1 ( 0 9 ) refer to the posi* In atomic units h = m = e = I, c = 137.
100 200 300 400 500 t~/keV Fig. 4, Line broadening due to the beam divergence (Fo) at the C o m p t o n peak in m o m e n t u m units as a function o f the incoming photon energy. Solid lines correspond to three scattering angles typically used in C o m p t o n experiments. The broken curve (FD) gives the energy resolution of the detector.
dO =
one obtains for the shift of the Compton profile in momentum units due to the change of the scattering angle oJ, =~,o = \ d o 9 2 , ] \ d 0 / , o ~
L~
o,-,O.g 1 3 /
,
,
2 2 ½ r _iFoore ),
re = 2" (FWHM)
~ 0.6
L 0.3[4t
l
, t ,t tit Lt , , 100 200 300 400 500 ~/keV Fig. 5. Total resolution of a Complon spectrometer / = (FZo + .r'#)t at the Compton peak as a Function of the incoming photon energy. Two degrees are taken as the fwhm o f G(O) and Gaussian line shapes are assumed.
11"8
S. M A N N I N E N AND T. P A A K K A R I
of G(co): The angular distribution of the scattered power, G(0), was determined using a computer simulation. To save computing time the calculation was made only in two dimensions by projecting the source, sample and detector on a plane. This omits the effects of vertical divergence. However, the change of the scattering angle, 7, due to the vertical divergence (deviation from the scattering plane) ~p, obtained from Calculation
0 q~2
7 = - ½ tan ~-
,
(6)
is well of second order at an scattering angle of 165° (short geometry). The resulting curve for the short geometry is given in fig. 6 after conversion into an energy scale according eq. (2). The derivate - co2 sin 0
(_~02)
'02=O'20
( 2co1
0 )2 sin 2 ~-+ 1 mc 2
(7)
has a value of 21 eV/degree at 0= 165° and it is seen from fig. 6 that the fwhm of G(co) corresponds to about 5°. The total instrumental function at 165° was calculated using eq. (1) and the fwhm of 1(¢o) was found to be 355 eV which was very close to the result of 353 eV obtained using the sum rule for the fwhm of Gaussian type functions. For final instrumental resolution function for the short geometry experiments one can therefore use the detector resolution function R(w) measured at 59.537 keV because its fwhm deviates only little from the calculated I(09) at the Compton peak, 09° = 48.440 keV. Thus for the present system calculation of I(o9) for each measurement is not necessary and small changes in the resolution of the detector are taken automatically into
1.0
~
/
110eV
-2.50 0 250 ~E/eV Fig. 6. Detector resolution function (solid curve) and geometrical resolution function (dashed curve) of the present Compton spectrometer.
TABLE 1 Counting rates for various samples at the scattering angle of" 165% Sample
Thickness
H20 H20 AI Ni Nb
1 mm 3 mm 3 mm 1 mm 1 mm
85 320 395 146 56
W
1 mm
59
Background
c/s c/s (38.5-58.5 keV) at peak channel
0.4
3.0 10.5 10.3 2.6 1.0 0.9
10- 3
account by measuring R(oJ) at 59.537 keV before and after the measurement of the Compton profile. In the case of the long geometry, scattering angle 0 = 175°, the effect of the geometrical resolution function, G(oJ), is negligible. The value of (doo2/dO)~o2=o3 ° is equal to 7eV/degree at 0 = 175° which makes the fwhm of G(~o)less than 35 eV because of tighter collimation. Assuming Gaussian form, the fwhm of/(6o) is changed only about 0.5% in convolution which is, for example, much less than the variation of R(to) over the Compton profile. Using long geometry the total instrumental function can be thus obtained by narrowing the R ( w ) measured at 59.537 keV by approximately 25 eV in the value of fwhm. 4. P e r f o r m a n c e
of t h e a p p a r a t u s Some characteristic counting rates for the short geometry are given in table 1. Data have been collected in the energy region of 38.5-58.5keV (-17.2+14.5 atomic units of momentum). Also given in table 1 is the counting rate at the peak channel in each case (the channel width was about 63eV). Compared with the 24~Am and ~23mTe systems used so far the intensity is better by more than one order of magnitude in the present apparatus. As can be seen in table 1 a statistical accuracy of about 1%o can be obtained in one day for light materials. In the case of the long geometry, the intensity is lower by about one order of magnitude compared with the results given in table 1. For example, in order to have a statistical accuracy of i% in anisotropy measurements of AI and Ni crystals, measuring times of about 3 h and 10 h are needed,
A C C U R A T E COMPTON PROFILE M E A S U R E M E N T S
but the corresponding times are 10 d for AI and 40 d for Ni if an accuracy of 1%o is required. References i) j. Felsteiner, R. Fox and S. Kahane, Phys. Lett. 33A (1970)
119
442. 2) W. Weyrich, Ber. Bunsen. Ges. Phys. Chem. 79 (1975) 1085. 3) R. J. Weiss, W. A. Reed and P. Pattison, Compton Scattering (ed. B. Williams; McGraw-llill, London, 1977) p. 43. 4) p. Eisenberger and W. A. Reed, Phys. Rev. B9 (1974) 3237.