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Coincidence measurements with X-ray synchrotron sources
Nuclear Instruments and Methods in Physics Research A297 (1990) 521-$2.5 North-Holland
Coincidence measurements with X-ray synchrotron sources K. Hiim~l~inen Department of Physics. University of Helsinki. Siltat,uorolpenger 201). SF-O0170 Helxmkl, Finland Received 9 April 1990 and in revised form 20 August 1990
The statistical accuracy in coincidence measurements is dependent on the amount of the random coincidences. The expression for the random coincidence count rate in the case of a pulsed synchrotron source is derived from the basic principles. A simple approximative relation between the random-to-true coincidence count rate ratio and the inelastic X-ray scattering cross scctions is presented. The different methods to improve the statistical accuracy and to optimize the coincidence measurements are discussed. It is shown that in the case of a synchrotron source the incident flux should sometimes be decreased offering a possibility to use larger detector solid angles, in this way a considerable improvement in the statistical accuracy can be achieved
I. Introduction
2. Estimation of random coincidem, es
The coincidence technique can be used to study the inelastic scattering from the tightly bound electrons. When a photon is inelastically scattered from an innershell electron it creates an electron hole which is immediately (in order of 10-15 s) filled and in the case of a radiative decay the resulting fluorescence photon can be detected. When the fluorescence photon is observed in coincidence with the inelastically scattered photon the innershell Compton profile can be separated from the dominating nearly free electron contribution. The simultaneity of the observations is determined by the experimental time resolution of the measuring system. A typical time resolution in the X-ray region is of the order of 10 ns if a solid state detector is used for good energy resolution. It is thus several orders of magnitude longer than the natural hole decay time. Therefore it is possible to detect two virtua21y coinciding photons which in fact are results Gf different processes and caused by different incident photons. The probability of these so-called random or chance coincidence increases when the average time between the incident photons becomes much shorter than the experimental time resolution. Their contribution is outstanding in the ca~.e of a pulsed synchrotron source [1-31 when, for examt~le, 10000 photons from a single electron bunch hit ~.he sample practically at the same time, i.e. during few hundred picoseconds. An expression for the random coincidence count rate in the case of a conventional and pulsed source is derived. It turns out that the time separation between the successive electron bunches in the synchrotron storage ring determines the effective time resolution.
In the case of a continuous source the probability to detect a photon during an infinitesimal time inte~'ai dt is ~ d t where ~ is the average count rate. Therefore the probability of not detecting a photon between time interval 0 - t decreases as e -a'. If we are measuring two independent processes (averag, • count rates ~1 and ~ , ) and we detect a photon in the first detector at t = 0. the probability to detect a random photon in the second detector increases as 1 - e-~:'. The count rate for random coincidences within time interval At corresponding to the experimental time resolution of the measuring system is therefore nrand = h i ( l -- e -~, ~') ----~!~2 At,
(i)
because in most cases h2 is chosen such that ~2 At << 1 due to the dead time limitation of the detector. In the case of a pulsed synchrotron source the incident photons hit the target m bunches within a time too short to be resolved and the time between the bunches, 1-, is often longer than the time resolution of the electronics. This increases the probability to get a random coincidence and also introduces a correlation between the different processes happening during the same bunch. The detection of a photon decreases the amoun, of the available incident photons in the same bunch and therefore decreases the probability to detect another photon. This complicates the subtraction of the random coincidences. The probability that a photon from a certain electron bunch hits the target in a given beam line is infinitesimal. This means that the number of the incident photons in a single bunch obeys Poisson statistics
0168-9002/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
522
K, HlJmllRlinen / Come'ideate measurements with X.ray SR sources
~hich can he verified by simple statistical argument, l'hu,~ the probability P ( N ) to have N incident photons from a single bunch is (2)
~" ~,, P(N)=~ N! e ,
where Nc~-~~it)r is the average number of incident photons in a single bunch and Ro the average incident flux. Typically No varies from a few up to several thousands incident photons per bunch. It is assumed that if a multihunch mode is used the electron hunches are identical which is justified in most cases, if the probabilities per incident photon to detect a photon in the first detector and the second detector are p~ and P2, respectively, the probability PI(NI) to detect NI photons in the first detector is given by the binomial probability N)
N-NI
Nt r,~'(l-t,~)
/'~(NI)-
(3)
The detection of a photon in the first detector decreases the probability to detect a photon in the second detector and these probabilities are therefore correlated. The probability to detect at least two coinciding photons in a single bunch is Pr~nd(N) =
N( N )
Z 81 PN~(1 --Pl NI ,, 1 ×
)N--N,
= ~ ? p l P.,( 1 - p~) e- ~"' "' '"~ ' " " ' --2
--- N(, P, Pz'
"(
(7)
The measured count rates with the individual detectors are nl..~ = N . P i , J r and the total random coincidence COLInt rate
",,,,d =
Prand ~. = ~ P l P : = ~ i l ~ : .
(8)
Comparison with eq, (1) shows that the interpulse period time ~ determines the effective time resolution which is independent of the time resolution of the electronics as far as At < r, Furthermore, the random coincidence count rate is proportional to the square of the average flux, The rate of the random coincidences can be measured by delaying the output of the other detector by one interpulse period time r. If two photons are detected at the same time in this case they are truly uncorrelated because they are resulting from the interaction processes with the incident photons from the different bunches. The probability for a random coincidence is er:,na =
~
e-g"NPl( 1 - p t ) N-I
N~I
× ~
p~:(l_p~),.V_N,_N:
1
E N-N, ,v:- I N~
(6)
(4) I-yen if detectors with large solid angles are used the probabilities I'~.~.<< I and the chance to get multiple counts in a single detector is negligible. Furthermore. multiptc count~ in an individual detector during a single bunch would lead to a multiple energy which is not seen in the energy window set for the fluorescence. The count rate of the detector measuring the energy spectrum should be limited so that the probability to get multiple counts is negligible since this would distort the energy spectrum. Therefore the sums in the previous equation can be replaced by their first terms Prand(N) = NPl(l - P l ) A t - | ( N - l ) p 2 ( l - P2 )N-2
~
e ~ o M p ~ ( i - p : , ) ~t-t
M= I
=
~ P l t'~ e - N " ~ P ' + P ~
-" ~u'z't :'~.
(9) (lO)
The relative difference with the case of the correlated
random coincidence probability (6) is Pr~nd
-- 1 - (l - P t ) e v":"e: = P , << I.
(ll)
Thus the amount of the correlated random coincidences equals within a good accuracy the number of the uncorrelated coincidences. The random coincidence count rate can therefore be measured by delaying the output of the other detector by an interpulse period time even if the average number of incident photons in a single bunch N0 is of the order of unity.
(5)
Taking into account the Poisson distribution of the incident photons the total probability to get a random coincidence during a single bunch is I
Prand '=- ~,~ ~NoN e-~oN( N - l) N = 1
×plpz(l-pl)
'v-l(1 _p2) Iv-:
= e-N-pip2(1 - p l ) N o 2
.'v-2
(N
-
2)!
3. Statistical error and scattering probabilities Since the role of the random coincidences is so outstanding in the coincidence measurements it is useful to study whether the statistical accuracy can be improved with a higher incident flux obtainable from a synchrotron source. In the conventional scattering experiments the statistical accuracy is dependent on the total number of photoi~s hitting the detector and therefore the alternative way to improve the statistical accuracy is to increase the solid angle of the detector. However, in the case of the coincidence measurement.,
K tlilmiil~men
/
Comctdt.nc¢
the statistical error depends on the incident flux and the solid angles in a more complicated way. in order to study the statistical accuracy in the coincidence measurements some numerical estimations are done in the case of a fluorescence coincidence detection in the inelastic scattering studies from the K-shell ek.~:trons. In the case o f an inelastic scattering from the K-shell the total count rate in the coincidence m-ode is el,a,, = n tme + n .m,I " noPK + R 2oPr P,: r,
(12)
where PK is the probability per incident photon to detc,x:t an indastically scattered photon from the K-shell in coincidence with the fluorescence photon resulting from the scattering process and Pr and p~ are the corresponding probabilities to detect an uncorrelated fluorescence photon and inelastically scattered photon. respectively. The true coincidence count rate is obtained by subtracting the random coincidences, n t,~, = n , . t n,a,d, and since the statistical errors for the total and random coincidence measurements are independent the relative statistical accuracy is
o,,.,
VN, o, + N,,,.<, = , / 1 + 2...<,/,,,,.,,,
= N, rue 1 +
=
N'rue
V
(13)
n trueT
2"horPrpJp K ~oTpK "
(14)
where T is the measurement time for both the total and the random coincidences. The extra term in the numerator is due to the random coincidences and naturally vanishes when the time resolution (interpulse period time) approaches zero. In order to optimize the measurement a numerical estimation of this extra term should be made. This requires the evaluation of the scattering and fluorescence probabilities. The scattering and fluorescence probabilities are related to the corresponding cross sections. If the thickness of the sample is !, the density is p. the incident angle is a and the take-off angle flr.~ < 90 °. both given relative to the plane normal to the sample, the probability to detect a photon is
pl Pf.s =cf,scos
(do) a . d-~
.f,s
l-e p'J A~2r's P.r,! " ....
(15)
where ~r,.~= # i / c ° s a + #r.JCOS fir..,, /t, and ttr.., are the linear absorption coefficients for the i,cident and scattered photons, Aflr., is the solid angle of the detector and cf,~ is the detection efficiency. Subscrip's f and s refer to the fluorescence and scattered photons, respectively. The cross section is given in cm2/g. Most of the detected fluorescence photons are resulting from the photoelectric absorption. The fluorescence radiation is isotropic to a good appro,~imation and the differential
ts
~'llll
X.rar
SR st~n't'~
cross section can be expre.ssed v,lth the total K-~h¢il photoabsorption cro~s section = .-~.
do
~IT..) ],f
I Io)
T.-: %h t,-
where ff~, is the K-fluorc.~cent'c ,,~eld l'hc mcl,r,t,c ncatterm 8 cro~ ~ t | o n ~s a I'um:twm o1" ttte ~'-attcrmg angle and the scattered photon energy and thus the scattering probability varies ,.n the measured .spectrum. The probability to detect a true coincidence, Le, both the inelastically scatte~'d and the fluore~ence photon ,s
pk = (r( . Pl (
d:o
~:o.--q-d~ d~Z, d.q,
)
A~')r -''q"
I - e ~'' ~1 "
(17)
where p~ ~. p, + ~ r / c o s Of. It" we assume that the radiative decay rate is independent of the creation process of the hole 14] the cross section can be written as
dIZ ad.q,
=K
~
,4--~"
(18)
where K < 1 gives the K-shell contribution to the total inelastic scattering cross section and is a function of the incident and scattered photon energy and especially equals zero at the energy region where the inelastic scattering process from the K-shell is not energetically possible. The interesting energies of the scattered photons are normally much higher than the fluorescence energies and because the absorption increases strongly in the low energy region, la r >-- p, and h __hr. Therefore q =
ranu
n t,.,,,,
PrP, pi avh K 1 - e " J = fi0r = hl.r . . . . . . . . . . . . . . . . . . . . PK COS a h fi,!
PiOph.K
" ~i,,r K cos ,~
( fi'l "~< 1 ).
(lq)
(20)
[hi:, expression can be used to e~tlmate the relative amount of the random coincidences. By measuring the count rates in the individual detectors eq. (8) together with eq. (20) gives a possibility to estimate the absolute true coincidence count rate. in a v-ray experiment n in eq. (19) can be noticeably less than unity [51 but in the case of a synchrotron source with a long interpulse period time it can be of the order of 10 4 [1.3]. The best way to optimize the measuring system depends on the parameter n. Only few parameters determining the relative statistical accuracy are adjustable in the experimental setup. Since the solid angle of the detector measuring the energy spectrum is determined by the desired angular resolution and the size of the detector crystal, the only ways to improve the statistical accuracy are to increase the measuring time. to increase the solid angle of the fluorescence aetector and to increase the incident intensity, if possible. However, with synchrotron sources measuring times of the order of v,e~'ks possible in the "y-ray experiments [6] are out of the: question. This gaves rise to the question whether the statistical accuracy
524
K. HIJm~lllllnen / Coincidence measurements with X.ray SR sources
I0: ~ . . . . . . . . . . . . . . . . .
I
.........!. . . . . . . . . . . . . . . . . . . . . . . . . .
~lcl
q>l
i
., * ~,~l. "0'l
Z°io'r"~io'~
to"
to°
Io~
to 2
_.
should be as large as possible without saturating the fluore~ence detector. If ~ ~ 1 the incident flux should be decreased and the solid angle should be increased as much as possible. It can be seen in fig. ! that decreasing the relative incident flux from 10 "~ by two orders of magnitude and increasing the solid angle respectively by a factor of 100, the fluorescence count rate is unchanged and one order of magnitude improvement in the relative statistical accuracy can be achieved. The fact that at the same time the inelastic scattering count rate has decreased has no effect on the relative statistical accuracy,
to'
Fig, 1, The relativ¢ 8tatlstlcal In'or as a function oi' the incident
flux i~0 relative to its optimum value n,~,t, The tumlng point of the curves corresponds the optimum 11- ttra,u/ntrue m 1, The different gurves refer to the different relative solid angles AOt, The best relative statistical accuracy can be reached in the case of ~ < 1 by increasing the incident flux to correspond to ~ ~, I and then increasing the solid angle of the fluorescence detector until the maximum tolerable count rate is obtained, in the case > ! the incident flux can be decreased without any significant effect on the relative statistical accuracy. This gives an opportunity to increa~ the solid angle which, on the contrary, has a crucial effect on the statistical accuracy,
4, Measuring time In the previous analysis it was a~umed that the measuring time for the coincidence and the random
coincidence modes were equal. It could be expected, however, that this does not necessarily guarantee the best possible statistical accuracy. If T is the total measurement time available and tT is used in the random coincidence mode there is time (1 - t ) T remaining for
the coincidence mode. The relative statistical error after the subtraction of the measured random coincidences can be calculated using eq. (13) and is
should be improved by increasing the incident flux ~; or the solid angle of the fluorescence detector A9 t. According to eqs. (13) and (15) the relative statistical error ~, is proportional to ~/i
+ 2~I
(21)
where 110c ;io¢ according to eq. (19), The relative statistical error as a function of the relative incident flux is presented in fig, I in the case of different solid angles, The dependence on the flux can be divided into two separate regions: 1) -q < 1, The role of the random coincidences i~ unimportant and the improvement of the time reso .~tion is irrelevant, By increasing the incident flux :,~e relative statistical error decreases as 1/(~ o AfJr) I/" es long as the detector is not saturated. The same improvemeat can be achieved b~, increasing the solid angle of the fluorescence detector by the same amount. 2) ~1> 1. The role of the random coincidences is dominant. No practical improvement is ach, cved with a higher flux. The relative statistical error is proportional to (T/ARf)t/2 and can be decreased with a better time resolution in the case of a continuous source or with a shorter interpulse period time in the case of a synchrotron source as far as At < ~-. The statistical accuracy can also be improved by increasing the solid angle of the fluorescence deter:tar. As a general rule the solid angle
i
i
l-× a
/
i T
x
(22)
+7 '
where ate , is the total count rate (12). The relative statistical error e has its minimum when the derivative dF./dt ffi 0 which gives
1 t., o = I + I/v
1
<
(23)
where X = nrand/nlot gives the relative amount of the random coincidences. Therefore the ideal division of the measuring time depends on the amount of the random coincidences and the time spent for the random coincidence mode should always be less than a half of the total m~ ~suring time. The relative statistical accuracy as a function of the random coincidence measurement time is shown in fig. 2. In the synchrotron experiments with high random coincidence count rates X approaches to unity and trot n "* 1., which is the most commonly adapted choice for the measuring time division. In the previous -~-ray experiment [5] X ffi 0.25 and tmi,, = ~, However, improvement of the statistical accuracy by decreasing t is not significant until X "~ 1, It can also be seen in fig, 2 how the statistical error greatly increases with the random coincidence count rate which supports the minimization of the random coincidences instead of just increasing the measuring time or the incident flux.
K. H a m ~ i ~
/ Cma~admce m m s w e m e m s ,,+'.h X.ray gR m~a~:¢s
I0)
~ 0 . ~ W
10:
/
I0
t 0,1 0,2 0,3 0.4 03 0~6 T~ITw Fill, 2, The relative suitisti¢ll ¢ n e r i s a |uneUoa of gqtliv¢
time I - Tme/Tto, N~eat for the random coinciden~ measu~,ment in the cue of different random coincidence ratios X nr.~/a~, ., The dashed line shows the optimum value which approaches ~ when ~ .-, I, 5. Results and
The role of the random coincidences is important in the coincidence measurement and becomes pronounced in the case of a pulsed synchrotron source. It is necessary to be make an estimate of the relative amount of the random coincidences before the experiment in order to optimize the experimental setup and to be able to decide whether a reasonable statistical accuracy can be vbtained during the available beam time. Therefore a simple expression based on the scattering cross sections has been derived to make these considerations possible. Also the different division of the measuring time has been discussed as a method to improve the statistical accuracy. According to calculations the pulsed synchro-
tron soerc'e is nm necessarily ideal for the X-ra)* ~ n ¢ : . dence measunmaents. It is shown that in the c a ~ of : high incident flux and a long interpulse peciggl time the relative statistical accuracy is dominated by the random coincidem.-es and cannot be impro~,ed wuh the mcreased flux. On ~ ~ o m r ~ u ~ ,ho*n that the incidem flux and instead using ~ solid ~ in the f l u o o ~ ' e n ~ detector a ¢ o n s i d ~ a b k i m ~ t in the stads~al aceura~ can be IPdned, Furthermore, it is shown that the interpulse period time determiMs the effective time resolution with a syachrotroa soerce and the use of a multiple bunch mode can esscnda!~ improve the statistical ac~ur~'y. An ideal sy~ch~t~rofl source for the coincidence measurements would he a storage ring with an interpulse period time coml~rable to the experimental time resolution, The radio-frequcno~ signal obtainable from the storage ring .gives an excellent opportunity to adjust the timing electronics and to optimize the time resolution [3l. However. the X-ray coincidence measurements seem to be one of the exceplions where no significant advantage can be achi¢ved by using the synchrotron source in comparison with the conventional sources.
R d ~ [I] V. Mm'chctti and C. Franck. Roy. Sol. Instr 59 (]98g) 407 [21 V. Marchem and C. Franck. Phys. Rev. [.ell. 59 (Ig8?) 1557. [3] K. H~a~il~imen. S. Manmnen and J.R. Schneider. th:s issue. Nucl. Instr. and Moth. A293 (1990) 526. [4] V. Marchcu| and C. Franck. Phys. Ro,. A3q (l~gg) 647 [5[ S. Manmnen. K. H~imMiiinen and J. Graeffe. Ph~,s. R¢~ 1541 (1990) 1224. [61 S. Manninen. K. H~imiil~inen. T Paakkari and P. Suorm. J. Physique C9 (1987) 823.
Coincidence measurements with X-ray synchrotron sources K. Hiim~l~inen Department of Physics. University of Helsinki. Siltat,uorolpenger 201). SF-O0170 Helxmkl, Finland Received 9 April 1990 and in revised form 20 August 1990
The statistical accuracy in coincidence measurements is dependent on the amount of the random coincidences. The expression for the random coincidence count rate in the case of a pulsed synchrotron source is derived from the basic principles. A simple approximative relation between the random-to-true coincidence count rate ratio and the inelastic X-ray scattering cross scctions is presented. The different methods to improve the statistical accuracy and to optimize the coincidence measurements are discussed. It is shown that in the case of a synchrotron source the incident flux should sometimes be decreased offering a possibility to use larger detector solid angles, in this way a considerable improvement in the statistical accuracy can be achieved
I. Introduction
2. Estimation of random coincidem, es
The coincidence technique can be used to study the inelastic scattering from the tightly bound electrons. When a photon is inelastically scattered from an innershell electron it creates an electron hole which is immediately (in order of 10-15 s) filled and in the case of a radiative decay the resulting fluorescence photon can be detected. When the fluorescence photon is observed in coincidence with the inelastically scattered photon the innershell Compton profile can be separated from the dominating nearly free electron contribution. The simultaneity of the observations is determined by the experimental time resolution of the measuring system. A typical time resolution in the X-ray region is of the order of 10 ns if a solid state detector is used for good energy resolution. It is thus several orders of magnitude longer than the natural hole decay time. Therefore it is possible to detect two virtua21y coinciding photons which in fact are results Gf different processes and caused by different incident photons. The probability of these so-called random or chance coincidence increases when the average time between the incident photons becomes much shorter than the experimental time resolution. Their contribution is outstanding in the ca~.e of a pulsed synchrotron source [1-31 when, for examt~le, 10000 photons from a single electron bunch hit ~.he sample practically at the same time, i.e. during few hundred picoseconds. An expression for the random coincidence count rate in the case of a conventional and pulsed source is derived. It turns out that the time separation between the successive electron bunches in the synchrotron storage ring determines the effective time resolution.
In the case of a continuous source the probability to detect a photon during an infinitesimal time inte~'ai dt is ~ d t where ~ is the average count rate. Therefore the probability of not detecting a photon between time interval 0 - t decreases as e -a'. If we are measuring two independent processes (averag, • count rates ~1 and ~ , ) and we detect a photon in the first detector at t = 0. the probability to detect a random photon in the second detector increases as 1 - e-~:'. The count rate for random coincidences within time interval At corresponding to the experimental time resolution of the measuring system is therefore nrand = h i ( l -- e -~, ~') ----~!~2 At,
(i)
because in most cases h2 is chosen such that ~2 At << 1 due to the dead time limitation of the detector. In the case of a pulsed synchrotron source the incident photons hit the target m bunches within a time too short to be resolved and the time between the bunches, 1-, is often longer than the time resolution of the electronics. This increases the probability to get a random coincidence and also introduces a correlation between the different processes happening during the same bunch. The detection of a photon decreases the amoun, of the available incident photons in the same bunch and therefore decreases the probability to detect another photon. This complicates the subtraction of the random coincidences. The probability that a photon from a certain electron bunch hits the target in a given beam line is infinitesimal. This means that the number of the incident photons in a single bunch obeys Poisson statistics
0168-9002/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
522
K, HlJmllRlinen / Come'ideate measurements with X.ray SR sources
~hich can he verified by simple statistical argument, l'hu,~ the probability P ( N ) to have N incident photons from a single bunch is (2)
~" ~,, P(N)=~ N! e ,
where Nc~-~~it)r is the average number of incident photons in a single bunch and Ro the average incident flux. Typically No varies from a few up to several thousands incident photons per bunch. It is assumed that if a multihunch mode is used the electron hunches are identical which is justified in most cases, if the probabilities per incident photon to detect a photon in the first detector and the second detector are p~ and P2, respectively, the probability PI(NI) to detect NI photons in the first detector is given by the binomial probability N)
N-NI
Nt r,~'(l-t,~)
/'~(NI)-
(3)
The detection of a photon in the first detector decreases the probability to detect a photon in the second detector and these probabilities are therefore correlated. The probability to detect at least two coinciding photons in a single bunch is Pr~nd(N) =
N( N )
Z 81 PN~(1 --Pl NI ,, 1 ×
)N--N,
= ~ ? p l P.,( 1 - p~) e- ~"' "' '"~ ' " " ' --2
--- N(, P, Pz'
"(
(7)
The measured count rates with the individual detectors are nl..~ = N . P i , J r and the total random coincidence COLInt rate
",,,,d =
Prand ~. = ~ P l P : = ~ i l ~ : .
(8)
Comparison with eq, (1) shows that the interpulse period time ~ determines the effective time resolution which is independent of the time resolution of the electronics as far as At < r, Furthermore, the random coincidence count rate is proportional to the square of the average flux, The rate of the random coincidences can be measured by delaying the output of the other detector by one interpulse period time r. If two photons are detected at the same time in this case they are truly uncorrelated because they are resulting from the interaction processes with the incident photons from the different bunches. The probability for a random coincidence is er:,na =
~
e-g"NPl( 1 - p t ) N-I
N~I
× ~
p~:(l_p~),.V_N,_N:
1
E N-N, ,v:- I N~
(6)
(4) I-yen if detectors with large solid angles are used the probabilities I'~.~.<< I and the chance to get multiple counts in a single detector is negligible. Furthermore. multiptc count~ in an individual detector during a single bunch would lead to a multiple energy which is not seen in the energy window set for the fluorescence. The count rate of the detector measuring the energy spectrum should be limited so that the probability to get multiple counts is negligible since this would distort the energy spectrum. Therefore the sums in the previous equation can be replaced by their first terms Prand(N) = NPl(l - P l ) A t - | ( N - l ) p 2 ( l - P2 )N-2
~
e ~ o M p ~ ( i - p : , ) ~t-t
M= I
=
~ P l t'~ e - N " ~ P ' + P ~
-" ~u'z't :'~.
(9) (lO)
The relative difference with the case of the correlated
random coincidence probability (6) is Pr~nd
-- 1 - (l - P t ) e v":"e: = P , << I.
(ll)
Thus the amount of the correlated random coincidences equals within a good accuracy the number of the uncorrelated coincidences. The random coincidence count rate can therefore be measured by delaying the output of the other detector by an interpulse period time even if the average number of incident photons in a single bunch N0 is of the order of unity.
(5)
Taking into account the Poisson distribution of the incident photons the total probability to get a random coincidence during a single bunch is I
Prand '=- ~,~ ~NoN e-~oN( N - l) N = 1
×plpz(l-pl)
'v-l(1 _p2) Iv-:
= e-N-pip2(1 - p l ) N o 2
.'v-2
(N
-
2)!
3. Statistical error and scattering probabilities Since the role of the random coincidences is so outstanding in the coincidence measurements it is useful to study whether the statistical accuracy can be improved with a higher incident flux obtainable from a synchrotron source. In the conventional scattering experiments the statistical accuracy is dependent on the total number of photoi~s hitting the detector and therefore the alternative way to improve the statistical accuracy is to increase the solid angle of the detector. However, in the case of the coincidence measurement.,
K tlilmiil~men
/
Comctdt.nc¢
the statistical error depends on the incident flux and the solid angles in a more complicated way. in order to study the statistical accuracy in the coincidence measurements some numerical estimations are done in the case of a fluorescence coincidence detection in the inelastic scattering studies from the K-shell ek.~:trons. In the case o f an inelastic scattering from the K-shell the total count rate in the coincidence m-ode is el,a,, = n tme + n .m,I " noPK + R 2oPr P,: r,
(12)
where PK is the probability per incident photon to detc,x:t an indastically scattered photon from the K-shell in coincidence with the fluorescence photon resulting from the scattering process and Pr and p~ are the corresponding probabilities to detect an uncorrelated fluorescence photon and inelastically scattered photon. respectively. The true coincidence count rate is obtained by subtracting the random coincidences, n t,~, = n , . t n,a,d, and since the statistical errors for the total and random coincidence measurements are independent the relative statistical accuracy is
o,,.,
VN, o, + N,,,.<, = , / 1 + 2...<,/,,,,.,,,
= N, rue 1 +
=
N'rue
V
(13)
n trueT
2"horPrpJp K ~oTpK "
(14)
where T is the measurement time for both the total and the random coincidences. The extra term in the numerator is due to the random coincidences and naturally vanishes when the time resolution (interpulse period time) approaches zero. In order to optimize the measurement a numerical estimation of this extra term should be made. This requires the evaluation of the scattering and fluorescence probabilities. The scattering and fluorescence probabilities are related to the corresponding cross sections. If the thickness of the sample is !, the density is p. the incident angle is a and the take-off angle flr.~ < 90 °. both given relative to the plane normal to the sample, the probability to detect a photon is
pl Pf.s =cf,scos
(do) a . d-~
.f,s
l-e p'J A~2r's P.r,! " ....
(15)
where ~r,.~= # i / c ° s a + #r.JCOS fir..,, /t, and ttr.., are the linear absorption coefficients for the i,cident and scattered photons, Aflr., is the solid angle of the detector and cf,~ is the detection efficiency. Subscrip's f and s refer to the fluorescence and scattered photons, respectively. The cross section is given in cm2/g. Most of the detected fluorescence photons are resulting from the photoelectric absorption. The fluorescence radiation is isotropic to a good appro,~imation and the differential
ts
~'llll
X.rar
SR st~n't'~
cross section can be expre.ssed v,lth the total K-~h¢il photoabsorption cro~s section = .-~.
do
~IT..) ],f
I Io)
T.-: %h t,-
where ff~, is the K-fluorc.~cent'c ,,~eld l'hc mcl,r,t,c ncatterm 8 cro~ ~ t | o n ~s a I'um:twm o1" ttte ~'-attcrmg angle and the scattered photon energy and thus the scattering probability varies ,.n the measured .spectrum. The probability to detect a true coincidence, Le, both the inelastically scatte~'d and the fluore~ence photon ,s
pk = (r( . Pl (
d:o
~:o.--q-d~ d~Z, d.q,
)
A~')r -''q"
I - e ~'' ~1 "
(17)
where p~ ~. p, + ~ r / c o s Of. It" we assume that the radiative decay rate is independent of the creation process of the hole 14] the cross section can be written as
dIZ ad.q,
=K
~
,4--~"
(18)
where K < 1 gives the K-shell contribution to the total inelastic scattering cross section and is a function of the incident and scattered photon energy and especially equals zero at the energy region where the inelastic scattering process from the K-shell is not energetically possible. The interesting energies of the scattered photons are normally much higher than the fluorescence energies and because the absorption increases strongly in the low energy region, la r >-- p, and h __hr. Therefore q =
ranu
n t,.,,,,
PrP, pi avh K 1 - e " J = fi0r = hl.r . . . . . . . . . . . . . . . . . . . . PK COS a h fi,!
PiOph.K
" ~i,,r K cos ,~
( fi'l "~< 1 ).
(lq)
(20)
[hi:, expression can be used to e~tlmate the relative amount of the random coincidences. By measuring the count rates in the individual detectors eq. (8) together with eq. (20) gives a possibility to estimate the absolute true coincidence count rate. in a v-ray experiment n in eq. (19) can be noticeably less than unity [51 but in the case of a synchrotron source with a long interpulse period time it can be of the order of 10 4 [1.3]. The best way to optimize the measuring system depends on the parameter n. Only few parameters determining the relative statistical accuracy are adjustable in the experimental setup. Since the solid angle of the detector measuring the energy spectrum is determined by the desired angular resolution and the size of the detector crystal, the only ways to improve the statistical accuracy are to increase the measuring time. to increase the solid angle of the fluorescence aetector and to increase the incident intensity, if possible. However, with synchrotron sources measuring times of the order of v,e~'ks possible in the "y-ray experiments [6] are out of the: question. This gaves rise to the question whether the statistical accuracy
524
K. HIJm~lllllnen / Coincidence measurements with X.ray SR sources
I0: ~ . . . . . . . . . . . . . . . . .
I
.........!. . . . . . . . . . . . . . . . . . . . . . . . . .
~lcl
q>l
i
., * ~,~l. "0'l
Z°io'r"~io'~
to"
to°
Io~
to 2
_.
should be as large as possible without saturating the fluore~ence detector. If ~ ~ 1 the incident flux should be decreased and the solid angle should be increased as much as possible. It can be seen in fig. ! that decreasing the relative incident flux from 10 "~ by two orders of magnitude and increasing the solid angle respectively by a factor of 100, the fluorescence count rate is unchanged and one order of magnitude improvement in the relative statistical accuracy can be achieved. The fact that at the same time the inelastic scattering count rate has decreased has no effect on the relative statistical accuracy,
to'
Fig, 1, The relativ¢ 8tatlstlcal In'or as a function oi' the incident
flux i~0 relative to its optimum value n,~,t, The tumlng point of the curves corresponds the optimum 11- ttra,u/ntrue m 1, The different gurves refer to the different relative solid angles AOt, The best relative statistical accuracy can be reached in the case of ~ < 1 by increasing the incident flux to correspond to ~ ~, I and then increasing the solid angle of the fluorescence detector until the maximum tolerable count rate is obtained, in the case > ! the incident flux can be decreased without any significant effect on the relative statistical accuracy. This gives an opportunity to increa~ the solid angle which, on the contrary, has a crucial effect on the statistical accuracy,
4, Measuring time In the previous analysis it was a~umed that the measuring time for the coincidence and the random
coincidence modes were equal. It could be expected, however, that this does not necessarily guarantee the best possible statistical accuracy. If T is the total measurement time available and tT is used in the random coincidence mode there is time (1 - t ) T remaining for
the coincidence mode. The relative statistical error after the subtraction of the measured random coincidences can be calculated using eq. (13) and is
should be improved by increasing the incident flux ~; or the solid angle of the fluorescence detector A9 t. According to eqs. (13) and (15) the relative statistical error ~, is proportional to ~/i
+ 2~I
(21)
where 110c ;io¢ according to eq. (19), The relative statistical error as a function of the relative incident flux is presented in fig, I in the case of different solid angles, The dependence on the flux can be divided into two separate regions: 1) -q < 1, The role of the random coincidences i~ unimportant and the improvement of the time reso .~tion is irrelevant, By increasing the incident flux :,~e relative statistical error decreases as 1/(~ o AfJr) I/" es long as the detector is not saturated. The same improvemeat can be achieved b~, increasing the solid angle of the fluorescence detector by the same amount. 2) ~1> 1. The role of the random coincidences is dominant. No practical improvement is ach, cved with a higher flux. The relative statistical error is proportional to (T/ARf)t/2 and can be decreased with a better time resolution in the case of a continuous source or with a shorter interpulse period time in the case of a synchrotron source as far as At < ~-. The statistical accuracy can also be improved by increasing the solid angle of the fluorescence deter:tar. As a general rule the solid angle
i
i
l-× a
/
i T
x
(22)
+7 '
where ate , is the total count rate (12). The relative statistical error e has its minimum when the derivative dF./dt ffi 0 which gives
1 t., o = I + I/v
1
<
(23)
where X = nrand/nlot gives the relative amount of the random coincidences. Therefore the ideal division of the measuring time depends on the amount of the random coincidences and the time spent for the random coincidence mode should always be less than a half of the total m~ ~suring time. The relative statistical accuracy as a function of the random coincidence measurement time is shown in fig. 2. In the synchrotron experiments with high random coincidence count rates X approaches to unity and trot n "* 1., which is the most commonly adapted choice for the measuring time division. In the previous -~-ray experiment [5] X ffi 0.25 and tmi,, = ~, However, improvement of the statistical accuracy by decreasing t is not significant until X "~ 1, It can also be seen in fig, 2 how the statistical error greatly increases with the random coincidence count rate which supports the minimization of the random coincidences instead of just increasing the measuring time or the incident flux.
K. H a m ~ i ~
/ Cma~admce m m s w e m e m s ,,+'.h X.ray gR m~a~:¢s
I0)
~ 0 . ~ W
10:
/
I0
t 0,1 0,2 0,3 0.4 03 0~6 T~ITw Fill, 2, The relative suitisti¢ll ¢ n e r i s a |uneUoa of gqtliv¢
time I - Tme/Tto, N~eat for the random coinciden~ measu~,ment in the cue of different random coincidence ratios X nr.~/a~, ., The dashed line shows the optimum value which approaches ~ when ~ .-, I, 5. Results and
The role of the random coincidences is important in the coincidence measurement and becomes pronounced in the case of a pulsed synchrotron source. It is necessary to be make an estimate of the relative amount of the random coincidences before the experiment in order to optimize the experimental setup and to be able to decide whether a reasonable statistical accuracy can be vbtained during the available beam time. Therefore a simple expression based on the scattering cross sections has been derived to make these considerations possible. Also the different division of the measuring time has been discussed as a method to improve the statistical accuracy. According to calculations the pulsed synchro-
tron soerc'e is nm necessarily ideal for the X-ra)* ~ n ¢ : . dence measunmaents. It is shown that in the c a ~ of : high incident flux and a long interpulse peciggl time the relative statistical accuracy is dominated by the random coincidem.-es and cannot be impro~,ed wuh the mcreased flux. On ~ ~ o m r ~ u ~ ,ho*n that the incidem flux and instead using ~ solid ~ in the f l u o o ~ ' e n ~ detector a ¢ o n s i d ~ a b k i m ~ t in the stads~al aceura~ can be IPdned, Furthermore, it is shown that the interpulse period time determiMs the effective time resolution with a syachrotroa soerce and the use of a multiple bunch mode can esscnda!~ improve the statistical ac~ur~'y. An ideal sy~ch~t~rofl source for the coincidence measurements would he a storage ring with an interpulse period time coml~rable to the experimental time resolution, The radio-frequcno~ signal obtainable from the storage ring .gives an excellent opportunity to adjust the timing electronics and to optimize the time resolution [3l. However. the X-ray coincidence measurements seem to be one of the exceplions where no significant advantage can be achi¢ved by using the synchrotron source in comparison with the conventional sources.
R d ~ [I] V. Mm'chctti and C. Franck. Roy. Sol. Instr 59 (]98g) 407 [21 V. Marchem and C. Franck. Phys. Rev. [.ell. 59 (Ig8?) 1557. [3] K. H~a~il~imen. S. Manmnen and J.R. Schneider. th:s issue. Nucl. Instr. and Moth. A293 (1990) 526. [4] V. Marchcu| and C. Franck. Phys. Ro,. A3q (l~gg) 647 [5[ S. Manmnen. K. H~imMiiinen and J. Graeffe. Ph~,s. R¢~ 1541 (1990) 1224. [61 S. Manninen. K. H~imiil~inen. T Paakkari and P. Suorm. J. Physique C9 (1987) 823.