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UNITED STATES ATOMIC ENERGY COMMISSION
AECU-715
(LADC-745)
ON THE STATISTICS OF LUMINESCENT
COUNTER SYSTEMS
By
Frederick Seitz
D. W. Mueller
Approved for Release: February 13, 1950
Los Alamos Scientific Laboratory
IL Technical Information Division, ORE, Oak Ridge, Tennessee
* 4.1,
Styled, retyped, and reproduced from copy
as submitted to this office.
Work performed under Contract No. W-7405-eng-36
PRINTED IN U.S.A.
PRICE 10 CENTS
ON THE STATISTICS OF LUMINESCENT COUNTER SYSTEMS
By Frederick Seitz and D. W. Mueller
1. INTRODUCTION
The type of crystal counter which depends upon the combination of luminescent crystals and a
photomultiplier tube shows promise of being of great service in the detection of radiations both be-
cause of its high sensitivity and speed of registry and recovery. This device has been developed by
a large number of individuals, almost too numerous to mention; however, the origin of the system
appears to rest with Coltman and Marshall,' who employed powdered luminescent materials of the
type used in previous commercial luminescent systems, and with Broser and Kallmann,2 who first
appreciated the advantages of employing large, transparent, luminescent crystals and introduced
organic materials.
The purpose of this paper is to analyze some of the factors which influence the statistical be-
havior of luminescent counter systems, in order to evaluate the limits within which a counter may be
used in making a particular type of measurement. The problems of interest range over a wide spec-
trum of possibilities. However, the problem on which attention is focused for immediate purposes in
order to provide a practical objective is the following:
A crystal-counter system is employed to count the gamma rays emitted from a source in time T.
If N gamma rays are emitted, what is the most probable number that will be counted and what is the
range of variation to be expected ? An attempt is made to examine this problem in a sufficiently gen-
eral way that the results will have value for a much broader group of problems.
It is interesting to consider the component parts of this problem in order to be able to examine
the sources of statistical variations. The components are as follows:
1. The source, even if constant in the sense that it remains unchanged during the time T, will
contribute to the statistical variation since the gamma rays are usually emitted at random. For
simplicity, it is assumed that the time T is sufficiently short that variations in the source strength
can be neglected and that the statistical variations in emission of gamma rays can be treated on the
basis of a Poisson distribution.
2. Unless the source is completely surrounded by the luminescent material, some of the gamma
rays will not pass through this material and hence will certainly fail to be registered. The average
fraction which passed through the material is designated by f, so that the average number of gamma
rays which pass through the detecting system, if N are emitted from the source, is
v = fN (1)
If the source is isotropic, f will be determined simply by the solid angle subtended by the crystal
system; otherwise a somewhat more involved calculation is needed to determine f.
3. A given gamma ray may or may not produce an ionizing pulse within the luminescent crystal.
The possible mechanisms for producing such a pulse are the photoelectric effect, the Compton ef-
fect, and pair production. In the first and third cases the gamma ray transmits all its energy to the
crystal provided the energetic electrons produced by the gamma ray do not escape from the crystal.
A greater statistical variation is possible when the range of gamma-ray energy and the atomic
number are such that the Compton effect predominates. This would be the case, for example, if the
luminescent material were one of the organic types such as naphthalene or anthracene and if the
gamma rays had an energy in the neighborhood of 2 Mev.
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The probability of a Compton encounter may be described in terms of the mean free path A for
the process, namely
A = 1/ne Oc (2)
where ne is the density of electrons in the luminescent material and 0c is the Compton cross section
per electron. If d is the thickness of the luminescent material in the direction in which the incident
gamma rays are traveling, the probability that a given gamma ray will pass through the system
without producing a Compton electron is e-Y where
a = d/A (3)
The initial energy ko of the gamma ray and the energy k after the collision are related by the
equation
k 1
k 1 + y(l cos 0) (
where 0 is the angle between the incident and scattered quantum and y is the energy of the incident
gamma ray expressed in units of the rest mass of the electron (507 kev). The energy gained by the
electron is e = k, k. From Eq. 4 the relation
d(cos 0) dk (5)
may be readily derived, connecting the differentials of cos 0 k. The differential cross section d
scattering into solid angle dD is
dp q- + k- sofe) (6)
in which ro is the classical electron radius e2/mc2. If the relation dQ = 2v sin 0 dO is used and do is
replaced by dk with the use of Eq. 5 the following equation is obtained.
d-= 7+ k sin2 0 dk (7)
This relation is to be employed in the range of k varying from ko to k1/(1 + 2y) corresponding to the
range of 0 from 0 to i. The quantity in parenthesis in Eq. 7 has the following values when 0 takes
the values 0, i/2, and n:
0=0 2
1+Y+Y2
0= /2
1 + (8)
1 + 2Y + 2y2
9=i 2
1+2Y
For values of y not larger than about 4, this variation is sufficiently small that it is reasonably
good to assume that k has equal probability of falling in any part of the allowed range, or that the
knocked-on electron has equal probabilities of receiving any energy in the range from 0 to 2y/(1 + 2y;
in units of k1. For very large values of y, the sin2 0 term in parenthesis in Eq. 7 may be neglected for
the most interesting collisions. It is then clear from the remaining terms in parenthesis that collisions
in which k is small compared with k. are preferred over those in which k is near ko.
The degree of preference is not exceedingly great for values of y in the normal radioactive range,
and it was assumed that the probability per unit energy range is constant within the allowed limits.
This gives the maximum statistical variation to be expected in a given Compton process.
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The gamma ray may conceivably make a number of Compton encounters in passing through the
crystal. There are two interesting extreme cases to consider which are referred to as the "thin"
and "thick" approximations. In the thin case, which corresponds to values of a appreciably less
than unity, the gamma ray has much smaller probability of making two collisions than one collision.
In this case it is assumed that a may be chosen to be a constant for each successive collision as if
its energy were not greatly affected by successive Compton encounters. In this event the probability
that the gamma ray will make n encounters may be described by the distribution function
ane-a
Pn n!
for the range of n of practical interest.
In the thick approximation the gamma ray transfers all its energy to the luminescent material
in a succession of encounters once it has made the first encounter. Thus this case is equivalent to
that in which the gamma ray transfers its energy by means of the photoelectric effect or pair pro-
duction, provided the electrons produced do not escape. These last two cases differ from the thick
approximation only with respect to the geometrical distribution of points within the crystal at which
the electrons are released-a difference which is not considered here.
The thin approximation is best achieved by employing a very thin crystal so that a is small
compared with unity and also employing soft gamma rays, for which y is 1 or less, which lose rela-
tively little energy in a Compton encounter. It is probably not a case which would be met in practice
but is interesting as one statistical extreme. It should be remarked that this limit cannot be achieved
by going to very soft radiation, for such radiation is scattered almost isotropically. The distance
which the scattered photon must traverse is usually different from that which the original photon
would have had to travel to pass through the crystal because it is traveling in a different direction.
Thus a is not a constant in this limit even though the energy of the photon is not greatly altered by
a Compton collision. The thick case can evidently be achieved by using a thick crystal and is the one
that will be met more commonly in practice.
4. The number of luminescent quanta which the crystal emits can vary even when the energy
transmitted to the crystal is fixed because of statistical fluctuations in the manner in which the
exciting radiation is distributed among the different excited states of the medium. This type of sta-
tistical fluctuation is partly responsible for the straggling in range of heavy ionizing particles as
they pass through matter. In order of magnitude the fractional variation in the number of light
quanta is l/V1" where I is the average number. Since in this paper cases in which rn is 1000 or
larger are of interest, corresponding to Compton encounters in which the knocked-on electron gains
several hundred key of energy, this source of statistical variation will be neglected. It could be
significant in cases in which the particle being detected produces very few light quanta, as for very
soft beta rays or x rays.
5. Only a fraction of the light quanta produced in the luminescent crystal will reach the photo-
electric surface. The fraction which does is determined primarily by geometrical factors involving
the angular distribution of emitted light and the angle subtended by the photosurface relative to the
luminescent material. 9 will be in the neighborhood of 0.5 for a relatively thin layer of luminescent
material which is immediately adjacent to the photosurface but may be considerably smaller if the
luminescent crystal is somewhat farther away. It may be enhanced by placing a reflecting backing
on the luminescent material or by employing other devices which cause the light to be "funneled"
toward the photocathode.
6. Only a fraction p of the photons striking the photosurface will eject electrons from it. This
parameter appears4 to be about 0.03 for the type of photosurface in which the photons penetrate the
photosurface and electrons are ejected from the back side, and it appears to be about 0.05 for the
type of photosurface for which electrons are ejected from the front surface. (Morton and Mitchell'
have shown that the pulse-height distribution is broader than that expected on the basis of a Poisson
distribution of electrons at each stage.)
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7. The electrons ejected from the photocathode will give rise to pulses of various size, de-
pending upon the accidents which befall the primary photoelectron and the secondaries which it emits
from the multiplying surfaces. Actually there are two problems associated with an analysis of the
pulse distribution: first, the problem of determining the probability that the photoelectron will
actually create a measurable pulse and second, the problem of determining distribution of pulse
sizes when pulses are generated. If the secondary emission ratio is s, the probability that the
photoelectron will not eject a secondary from the first stage of the multiplier is e-s, provided it is
assumed that the emission of secondaries is random. This probability is of the order of a few per
cent for normal values of s (between 3 and 5) and is essentially equal to the probability that the
primary electron will not generate a pulse. Since the percentage of uncertainty in p is at least as
large as this, this factor may be combined with p in the following discussion, and the assumption
may be made that a measurable pulse is produced whenever an electron is ejected. The distribu-
tion of pulse sizes has been measured by Engstrom4 using a light source. Later, his results will
be approximated with an appropriate mathematical function. Evidently the pulse distribution is not
important if the luminescent counter is employed simply as a counter of events and if a pulse of
arbitrary size can be employed as the signal for a significant count. Knowledge of the distribution
becomes important, however, if a pulse discriminator is employed so that only pulses larger than
a certain size are counted (as when a noise background is eliminated) or if the integrated current
of the photomultiplier is recorded. The first of these cases may be treated by redefining the
parameter p as the probability that an observable pulse is measured when a photon strikes the
cathode and introducing measured values of this quantity. The second case is discussed in detail.
2. THE GENERATING FUNCTION AND ITS APPLICATIONS TO THE PROBLEMs
The generating function was introduced into probability theory very early in its development,
and some of its properties are described in textbooks, for example, the book by Uspensky.s The
authors have benefitted by reading a mimeographed survey of the subject by O. R. Frisch. An
account of some of the relations employed here is given by Jorgenson.5
The aggregate contribution of the various unit parts of the photomultiplier system to the sta-
tistical variation of the system can be determined most simply with the use of generating functions
appropriate to each stage. If pn is the probability that a given observation shall yield n events, for
example, that the source in the problem will emit n gamma rays in time T, the generating function
G(e) for the process of observation is defined by the series
G(e) = poEo + pREl + pe + ... + pne + ... (10)
The generating function is readily found to possess the following properties
G(0) = Po G(1) = 1 (11)
The mean value m of a series of observations, namely
m = E pn (12)
n
is readily seen to satisfy the relation
m = (13)
Similarly the variance of a sequence of observations, defined by the relation
v = E (n m)p = E n'- m' (14)
n n
is readily found to be related to the generating function by the equation
[G[dG dG dG = l d + m -m" (15)
v d2-d-\de-J=l d C-, +m
It
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In the case of the Poisson distribution
an
Pn = e-a (16)
G(E) is readily found to be
G() = ea(e-1) (17)
whence
m = a v =a (18)
In the following calculations the ratio j7-/m is employed to provide a measure of the fractional
deviation from the mean or the fractional deviation. In the case of the Poisson distribution this
quantity is the familiar ratio 1/Ya.
Although the generating function is interesting and useful because of the properties already
outlined, its real service appears when the following two additional properties are considered:
I. Suppose that, instead of making one observation of the number of events of interest (such as
the number of gamma rays emitted from the source in time T), two observations are made (e.g., for
two time intervals T) and ask for the probability that n events are observed en toto is asked for.
The probability for this is the sum
PnPo + Pn- Pi + Pn-P2 2+ *+ PoPn
which is the coefficient of tn in the expansion of G2(W). This is a special case of the more general
theorem: The generating function governing the probability distribution of the sum of r identical
observations is Gr(e) if G(E) is the generating function for a single observation.
I. Suppose next that a situation is being dealt with in which each member of a set of initial
events that are statistically distributed (e.g., gamma rays from a source) can give rise to a series
of events of possibly different type (e.g., production of Compton electrons) and the statistical distri-
bution of the second type of event is asked for. Let Gi(t) be the generating function for the first type
of event (e.g., the number of gamma rays emitted by the source in a given time for the example
under consideration) and G2(W) be the generating function for the number of events of the second type
associated with one primary event (e.g., the number of Compton recoils produced by a single gamma
ray). It Is readily shown that the generating function GII(E) for the number of events of the second
type when the statistical variation of the number of events of the first type is taken into account is
given by
Gn(c) = G[G,(e)] (19)
The validity of this theorem may readily be demonstrated by writing GI1 in the form
Gjn(e) = poG(e) + pG2((E) + p2G (e) + p3G(e) + ... + pnGn() + ... (20)
in which pn is the probability of n events of the first type, so that
G(e) = po +pi+ P't1 2 +...+pn + ...
The coefficient of Pn in Eq. 20 is the generating function for the total number of events of the second
type when it is known that n events of the first type have occurred, in accordance with theorem I.
This coefficient appears suitably weighted with the probability that n primary events will occur.
Using Eqs. 13 and 14, the mean and variance associated with the generating functions
GI(e) = Gr(e) Gn(s) = GJ[G,(<)] (21)
may readily be found which occur in the cases I and II described above. The results are, respectively
mI = rm vI = rv
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in which m and v are the mean and variance associated with a single observation in case I, and
mH = m'm2 v1 v +mi + v2m1 (23)
It is clear that, if case H were extended to that in which the second type of event can give rise
to a third type (e.g., if a Compton electron can give rise to ion pairs or to luminescent quanta) which
is statistically distributed in accordance with a generating function G,(e), the complete generating
function which takes account of the statistical variation in events of the three types is
Gm = GI[G2(e)] (24)
for which the mean and variance are, by analogy with Eq. 23
mm = mim3 v = vIIm + v.mi (25)
The appropriate form of generating functions to be employed in each of the constituent processes
described in Sec. 1 will now be examined.
1. Emission from source. Since the gamma rays are emitted at random, the appropriate gen-
erating function is of the Poisson type, Eq. 17, namely
Gi(e) = eN(&-l) (26)
in which N is the average number of gamma rays emitted in the time T.
2. Passage of gamma rays into system. A given gamma ray either does or does not enter the
crystal. If the probability that it does is f, the generating function for this event is simply
G2(e) = (1- f) + fe (27)
Using Eq. 19, it is readily found that the generating function G'(e), giving the distribution of proba-
bilities that the gamma rays emitted at random by the source enter the crystal, is
G,(e) = G,[G2(e)] = eNf(-) = ev(-1) (28)
where v = fN.
3. Generation of photons. As stated in the introduction, the assumption is that a fixed fraction of
the energy which the gamma photon gives up to the crystal is transformed into light quanta. If this
energy is E, the number of light quanta produced is then
7 = pE (29)
where p is a factor measuring the efficiency with which the luminescent crystal converts the ex-
citation energy it receives into light quanta. If hv is the average energy of the luminescent quanta
emitted, fhv is the fraction of the energy of excitation which appears in the form of luminescent
radiation. This may be as large as 0.20 for some of the most efficient materials but can easily
be much smaller. According to Broser, Kallman, and Martius' the efficiencies of energy conver-
sion in zinc sulfide activated with silver and in the organic materials naphthalene, diphenyl, and
phenanthrene are given in Table 1.
Table 1-Efficiency of Energy Conversion in Luminescent Materials
under Gamma-ray Excitation
(After Broser, Kallmann, and Martius. Values in fractions.)
Ohv
ZnS:Ag 0.135
Naphthalene 0.05
Diphenyl 0.075
Phenanthrene 0.11
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The investigators find somewhat different efficiencies for beta-ray excitation. Similarly, Colt-
man, Ebbighausen, and Altar7 have found the energy conversion in calcium tungstate to be 5.0 per
cent for x rays. The interest here is in detailed values of the efficiency in Sec. 3.
As mentioned in the introduction there are two extreme approximations that are of interest,
namely, those designated as "thin" and "thick." In the second case, all the energy of the gamma
ray is transmitted to the crystal once a first collision has occurred. The number of quanta emitted
is then equal to r7o, the value of ,i when ko is the energy of the gamma ray. If c is the probability
that such a collision occurs, the generating function for the number of quanta is evidently
S3(e) = (1 c) + ceLo (30)
If this is combined with Eq. 28, the complete generating function for the production of luminescent
quanta in the thick case is
S3 = exp [v c( 0o )] (31)
In the thin case there are two sources of statistical variation, for both the number of Compton
encounters and the energy transferred to the counter per collision may vary. The first of these
quantities is distributed in accordance with the Poisson law, Eq. 17, in the ideal thin case, for
which the generating function is
H3 = ea(-l)
where a is the ratio (Eq. 3). The energy which the Compton electron receives is randomly distributed
between 0 and the maximum value 2ke1y/(l + 2y) in the approximation described in paragraph 3 of the
introduction. This means that the number of quanta generated will vary between 0 and a maximum
rim, where
2koY
m 2k= Y (32)
1 + 2(
in which [ is the efficiency factor appearing in Eq. 29. A generating function for this random dis-
tribution is readily constructed by treating 7 as a continuous variable and is
K,(e) = f edrT = exp (77m log ) 1 (33)
m 0 77m log
for which the mean and variance are ,rm/2 and n2/12, respectively. The complete generating func-
tion for the number of quanta associated with a single gamma ray is H,[K3(e)].
4. Emission of photoelectrons. A given light quantum either does or does not emit a photoelectron
from the photosurface of the multiplier. The probability that it does is fp, so that the generating
function for this process is
G(e) = [(I Qp) + Dpe] (34)
As stated in paragraph 7 of the introduction, it is assumed that a measurable pulse is associated
with each photoelectron ejected from the cathode of the multiplier.
5. Generating function for pulse distribution. Engstrom4 has measured the pulse-height distri-
bution of a typical multiplier tube. His empirical distribution is represented by the analytical
function
f(h) = Ah2 exp (-h/p) (35)
in which h is the pulse height on an arbitrary scale, p is a constant measuring the width of the dis-
tribution, and A is a normalization factor 1/2p3. A generating function
G,(e) = 1/(1 p log E)3 (36)
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may readily be constructed for this distribution. The mean and variance are
m = 3p v = 3p2 (37)
It will be seen later in this paper that it is not necessary to know p in order to determine the frac-
tional deviation of interest here.
3. CORRELATION BETWEEN GAMMA RAYS FROM SOURCE AND PULSES IN MULTIPLIERS
The probability that a gamma ray from the source will produce a pulse in the multiplier is dis-
cussed in this section. The thick and thin cases are discussed separately.
A. Thick Case
In the thick case, a gamma ray passing through the crystal has probability c of making an en-
counter, in which case it generates no photons. The distribution of such encounters is random, being
governed by the Poisson distribution. The average number is Nfc = vc, and the deviation is vc. Thus,
as far as luminescent pulses are concerned, the effective strength of the source is vc.
The probability that n of the ro light quanta will eject photoelectrons from the multiplier is given
by the generating function
[G,()]"0 = [(1 -p) +.pE]'o (38)
The probability that none will eject electrons is (1 -Qp)7o, so that the probability of observing a
pulse, if one electron is sufficient to produce an observable pulse, is
p = 1 (1 ap) o (39)
Since 1ro is usually large compared with unity, this may be approximated by
P= 1-exp (- i0Qp) (40)
in which the quantity
%o P (41)
is the average number of photoelectrons emitted from the cathode. With the use of the rule, Eq. 25,
for determining the mean and variance of a chain of events, ;he mean number of counts is
M = Nfc[l exp (- inop)] = NfcP (42)
whereas the variance is
V = NfcP (43)
The fractional variance is
S(44)
M NfcP
B. Thin Case
There are four statistical processes in the chain extending from the passage of gamma rays into
the crystal to the ejection of electrons from the photocathode, namely, those described by the gen-
erating functions G2, H3, K,, and G4 of the preceding section. The number of electrons ejected from
the cathode when a single gamma ray passes through the crystal is governed by the generating
function
E(c) = Ha KI[G4(E)] (45)
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The probability that no electron will be ejected and hence that no pulse will be recorded is E(0),
so that the probability of a pulse is 1 E(0), and the generating function for pulses is
{E(0) + [1 -E(0)],} (46)
Hence the mean and variance in the number of pulses are
M = V = Nf[l E(0)j (47)
Since Qp is of the order of 3 per cent even when Q is unity, it is readily found that K,[G4(0)] can be
approximated by the expression
K[G4(0)] [1 exp (-~mOp)] (8)
K,[G O) P (48)
A simple examination of E(0) shows that it approaches e-a when lm.Qp is large compared with unity
and approaches exp (-a7lmQp/2) when nmlp is small. It may be concluded that, in both the thick and
thin cases, the counts are governed by a Poisson distribution and that it is desirable to have the
quantities c mnd 1 e-o as near unity as possible and the quantities 7o9p andimS2p somewhat larger
than unity, although there probably is little advantage to having them as large as 10.
Suppose one is dealing with gamma rays in the vicinity of 1.5 Mev, to provide a concrete example.
In this case the mean free path for the Compton effect in a material such as naphthalene is of the order
of 15 cm. Hence if the crystal is a cube 5 cm on an edge, the factor e-a is 0.72. The Compton elec-
tron will have an average energy of the order of 0.7 Mev, so that the average number of luminescent
quanta produced is 10,000 if the energy efficiency is taken to be 0.05. Choosing p to be 0.03, it is
found that mflp/2 is 300 9. Hence Q should be at least 10-2 if each Compton electron is expected to
register with reasonable faithfulness. If it is assumed that the photosurface of the multiplier has an
active area of about 15 cm2 and that this surface is 5 cm from the center of the crystal, the factor
should be as large as 0.G5 even if the photons are isotropically distributed, which would guarantee
faithful counting of Compton encounters. The same photosurface would be more nearly borderline if
the crystal were chosen to be a 10-cm cube and the surface were placed 10 cm from its center, for
then 9 would be about 0.01, which is very close to the limit set above. In fact, those Compton en-
counters which take place at points within the crystal which are most distant from the surface may
fail to register if the photon distribution is isotropic. In this event it may prove profitable to employ
a method of light funneling, for example, by covering all surfaces of the crystal except that opposite
the multiplier with a reflecting metallic covering.
4. PHOTOMULTIPLIER CURRENT
Consider next the current in the photomultiplier, or rather the charge which arrives at the
anode end when N gamma rays are emitted from the source. The "thick" and "thin "cases are dis-
cussed separately once again.
A. Thick Case
In this case the distribution of charge in the photomultiplier may be regarded as if compounded
of the three statistical processes which are described by the generating functions S3, G4, and G5 of
Sec. 2. The first of these functions gives the distribution of photons in the crystal associated with the
N gamma rays, th, second function gives the distribution of the photoelectrons from the cathode of
the multiplier, and the third function gives the distribution of pulses in the multiplier. The mean and
variance of these distributions are as follows:
Mean Variance
S; VCylo vcrO p
G4 Qp Qp- (Qp)2 Opp
Gs 3p 3p2
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The quantity (Op)2 may be neglected in comparison withDp since the latter is at most a few per cent.
By compounding these statistical quantities in accordance with the rule, Eq. 25, the following
values of the mean and variance for the charge in the multiplier are obtained:
M = 3pvc 70Qp
V = 3p2 vc oQp (4 + 3roQp) (49)
The fractional variance is
M :-[ 3v c = + o0 p J (50)
This quantity is independent of p, as pointed out previously. Moreover, it becomes independent of
the quantity x = o Dp when this quantity is large compared with unity. The condition placed upon x
for this limit to be valid is somewhat more stringent than the condition required for faithful counting
of luminescent pulses. That is, x must be larger than 5 for this approximation to be precise.
B. Thin Case
In this approximation the distribution of pulses is governed by a generating function that is
compounded of the generating functions G2, H3, K3, G4, and Gs. Respectively, these correspond to
the distribution of gamma rays in the crystal, the distribution of Compton encounters, the distribu-
tion of luminescent quanta produced in the crystal, the distribution of photoelectrons from the
cathode, and the distribution of pulses in the multiplier. The corresponding means and variances
are as follow:
Mean Variance
H3 a a
K, nm/2 l /12
G4 Qp 2p
Gs 3p 3p2
The mean and variance for the distribution of pulses is found to be
3
M = p v a mQp
(51)
V = jp2 amQp 4+imQp (4+3a)
Once again it is noticed that p drops out of the fractional variance. Whenever the quantity y =1nmgp
is very small compared with unity, the fractional variance may be approximated by the expression
M-V ( 8/3 '2 (52)
M an p)
In the opposite extreme, in which y is very large compared with unity, the fractional variance is
(4 + 3aty2
M \ 3=va ) (53)
which approaches 2/ faiv if a is small compared with unity and approaches 1/-v if a is very
large. The latter case, in which a is large, is in contradiction with the assumptions of the thin
approximation; however, it is of mathematical interest.
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5. FLUCTUATIONS IN CHARGE ON CONDENSOR
When dealing with a high-intensity source, it is frequently convenient to feed the current pulses
from the photomultiplier into a condensor which is shunted with a high resistance and measure the
voltage across the condenser in order to provide a measure of the average current which arrives
at the condenser. This voltage exhibits fluctuations because the pulses are distributed statistically
both in magnitude and in time. The influence of the distribution in time has been investigated by
Schiff and Evans8 for the case in which the pulses are equal in magnitude. The generalization of
their results when the pulses vary in size is of interest here.
If the capacity of the condenser is C and the shunting resistance is R, the decay time for the
shunted capacity is T = RC. A charge which is fed into the condensor at time t' will have decayed by
a factor exp [-(t t')/rT by the later time t.
The assumption is that the charge associated with each pulse of the multiplier arrives in a time
that is short compared with the decay time of the condensor. It is also assumed that the pulses are
distributed in time in accordance with the distribution law governing the frequency with which gamma
rays enter the luminescent crystal, that is, in accordance with the generating function G (e) of Sec. 2
(see Eq. 28). Since interest is in specific intervals of time t, v in Eq. 28 is replaced by nt, where n
is the average number of gamma rays entering the crystal per unit time. Those gamma rays which
do not excite the crystal will give rise to pulses of 0 size. For the purposes of this section, the
generating function is designated for the pulse in the photomultiplier associated with the passage of
a single gamma ray into the crystal by G(E). The pulse size will be assumed to be expressed in units
of charge. G(e) will differ in the soft and hard approximations but may be left arbitrary for the
moment.
Consider the gamma rays which arrive in the time interval dt' between t' and t' + dt'. The
generating function associated with the current they contribute to the condensor at the time t' is
1 + ndt'[G(E) 1] (54)
which is the expansion of G~I[G(E)] in terms of dt' when v is replaced by ndt'. The mean value of the
charge associated with this generating function is
ndt'G'(1) (55)
This mean contribution will have decayed by a factor exp [(t' t)/T] by the time t. Thus the mean
charge at time t resulting from the accumulation for all previous times is
nG'(1) exp [(t'- t)/T]dt' = nG'(1) (56)
G'(1) evidently is the mean charge pulse Q in the photomultiplier associated with the entrance of a
single gamma ray.
Similarly, the variance in the charge on the condensor at time t is the integral of the variance of
Eq. 54 from t' = -oo to t' = t with a weighting coefficient exp [2(t' t)/T] since the decay constant for
the square of the charge is twice as large as that for the charge. The result is
n [G"(1) + G'(1)] (57)
2
The quantity G"(1) + G'(1) is the mean of the square of the charge pulse associated with a single
gamma ray, which we shall designate as Q2. This is also equal to the variance of the charge pulse
associated with a single gamma ray plus Q2.
The fractional variance of the charge on the condensor is
V-V (i2:2 (58)
M k T Q2)n
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The coefficient (1/2Tn)y represents the result obtained by Schiff and Evans for pulses of constant
amplitude. The coefficient Q2/Q2 for the thick and thin cases may now be investigated.
A. Thick Case
In this case the generating function G(e) is SafG4[Gs(,)]J. The means and variances of G, and Gs
were tabulated in the previous section. The corresponding quantities for G, are %oc and I2~c(1 c).
By combining the means and variances
M = Q = 3p ro0,cp V = 3p2 r70oCSp[4 + 3r,.Q p(l c)] (59)
are obtained. Moreover,
Q2 = V + M2 = 3p2 oi7,cp(4 + 3noQp) (60)
so that
42 +31oc.Qp
(\2 3mcQp ) (61)
As should be expected, this approaches 1/ Y whenio9Qp becomes sufficiently large, for the pulses
then approach the constant size and the only source of statistical variation is in the random pro-
duction of luminescent bursts.
B. Thin Case
In this case G(e) is H3(K3[G4[GGs()]1) whose averages were tabulated in the previous section.
3
M = Q = p a mp V = 3p2 a rQp (2 +nmS2p) (62)
Q2 = 3p2 a nmpP[2 + inmlp l + a)]
(Q)2 4 2+ rm9Qp (1 +%a) 1/2
Q/ 13 a 7m-p
In this case Q2/12 approaches (4/3 + a)/a when imr p becomes sufficiently large.
6. CONCLUSIONS
1. The statistical variations in a counting system which consists of a source, a luminescent
crystal, and a photomultiplier are examined. It is assumed that the source is constant for a fixed
period of time, although it emits particles at random. For definiteness and to provide a maximum
degree of statistical variation, it is assumed that the source is a gamma emitter and that only a
fraction of the gamma rays fall on the luminescent crystal. The method of generating functions is
employed to treat the chain of events which the particles emitted from the source engender. Two
oppositely extreme cases are considered, namely, that in which all the energy of a gamma ray which
enters the crystal is transferred to the electrons and that in which the gamma ray transfers only a
portion of its energy in a manner that depends upon the Compton encounters it makes. The two ap-
proximations are referred to as the "thick" and "thin" approximations. The first can be realized
by using a crystal which is sufficiently thick that the gamma ray is completely absorbed. The second
case can be approximated by using a very thin specimen and using gamma ray energies for which the
Compton process predominates.
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2. As might be expected the results show that the effectiveness of the system depends upon the
ability of the crystal to receive energy from the crystal. They also show that a measure of the ef-
fectiveness of the remainder of the system is provided by the quantity
x = 77p
Here ?7 is equal to the number of light quanta, Y, produced per gamma ray in the luminescent crystal
in the thick case and is, im, the maximum number which can be produced per Compton encounter in
the thin case. 9 is the probability that a light quantum emitted from the crystal will strike the photo-
surface of the multiplier, and p is the probability that a photoelectron will be emitted from the
cathode. The system will be a faithful counter of those gamma rays which transfer energy to the
crystal provided x is of the order of 5 or larger. The statistical fluctuations are then determined
primarily by the Poisson distribution of encounters in the crystal. If, on the other hand, the current
from the multiplier is measured instead of the rate of counts, the contribution of the photomultiplier
to the statistical error is appreciable until x is considerably larger than 5, although this error can
be reduced to that corresponding to the Poisson distribution of encounters in the crystal when x is
increased.
The statistics of the case, in which the pulses are fed into a capacitor with a time constant and
the voltage of the capacitor is measured, are treated from a standpoint somewhat more general
than that considered by Schiff and Evans.
REFERENCES
1. 1. W. Coltman and F. Marshall, Phys. Rev., 72: 528 (1947); F. Marshall, J. Applied Phys., 18: 512
(1947).
2. I. Broser and H. Kallmann, Z. Naturforsch., 2a: 439 (1937); 642 (1947); I. Broser, L. Herforth,
H. Kallmann, and U. Martius, ibid., 3a: 6 (1948).
3. W. Heitler, The Quantum Theory of Radiation, Oxford University Press, 1936.
4. P. W. Engstrom, J. Optical Soc. Am., 37: 420 (1947); G. A. Morton and J. A. Mitchell, R C A Rev.,
9: 632 (1948).
5. 1. V. Uspensky, Introduction to Mathematical Probability, McGraw-Hill Book Company, Inc.,
New York, 1937; T. Jorgenson, Am. J. Phys., 16: 285 (1948).
6. L. Herforth and H. Kallmann, Ann. Physik, 4: 231 (1949).
7. J. W. Coltman, E. G. Ebbighausen, and W. Altar, I. Applied Phys., 18: 530 (1947).
8. L. I. Schiff and R. D. Evans, Rev. Sci. Instruments, 7: 456 (1937); L. I. Schiff, Phys. Rev., 50: 88
(1936).
END OF DOCUMENT
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