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Calculation of the detection efficiency of a cylindrical counter for X-rays from an internal source
NUCLEAR
INSTRUMENTS
AND METHODS
35 (I965) I 2 5 - I 2 9 ;
~
NORTH-HOLLAND
PUBLISHING
CO.
CALCULATION O F T H E D E T E C T I O N EFFICIENCY OF A CYLINDRICAL C O U N T E R FOR X-RAYS F R O M AN INTERNAL S O U R C E J. B Y R N E
Department of Natural Philosophy, University of Edinburgh, Edinburgh, Scotland Received 25 January 1965
This paper describes detailed calculations of the efficiencyof a long cylindrical proportional counter for detection of X-rays from an isotropic point source placed centrally in the wall of the
counter. The results are applicable to counters of varied length and radius for a range of X-ray energies and are presented in both analytic and graphical f o r m s .
1. Introduction The conventional apparatus used in nuclear spectroscopy for purposes of identifying and estimating the intensities of electromagnetic radiations, emitted after radio-active decay, consists of the NaI(TI) scintillation counter coupled to a photo-multiplier and pulseamplifier. For ~,-rays and X-rays in the energy range of 50 keV and less, however, the scintillation counter is not suitable for accurate work since the energy resolution is poor and the output pulses tend to lie in that part of the pulse-spectrum where photomultiplier noise begins to be important. A further disadvantage of the method arises from the fact that radiation of low energy is highly absorbed in the protective metal films with which NaI crystals are normally covered. These problems may be largely overcome if the X-rays are detected in a proportional counter with a 2re countingangle where the source of the radiation is deposited on a thin film placed in the wall of the counter in direct contact with the counter gas. The kind of experiment envisaged here might be for example a study of the L-shell X-rays emitted following internal conversion of the E2 ~-rays of about 40 keV energy which are characteristic of a whole class of even-A nuclei of heavy alpha-emitting elements such as plutonium and uranium. A further application exists in the study of electron-capture in nuclei for which gaseous sources are not easily manufactured. The technique for these and similar measurements was initiated by Curran and his collaborators more than a decade ago 1'2) and considerable development has taken place since then. Energy resolution attained by this method is far superior to that available using scintillation counters. The principal difficulty in proportional counter X-ray spectroscopy is associated with the measurement of absolute intensity; for all X-rays except those of the lowest energy the counter gas is to a large extent transparent and the probability of detecting the X-ray quantum becomes critically dependent, not only on the absorption coefficient, but also on the direction of
emission. Since the absorption law is exponential the problem of calculating the absolute detection efficiency for X-rays of a given energy can be formidable even for the very simplest geometry. In this paper are presented calculations of the detection efficiency of a long cylindrical counter for X-rays emitted isotropically from a point source placed centrally in the wall of the counter. By a "long" counter is understood one for which the half-length is considerably greater than the diameter. It turns out that the resUlts for an infinitely long counter are expressible in closed form in terms of products of hyperbolic Bessel functions. For counters of finite length there is in addition a series in inverse powers of the half-length expressed in units of the diameter. In the type of counter normally used the half-length is at least twice the diameter in which circumstances the series converges so rapidly that for computation purposes only the first few terms in the series need be used. As a rough guide one may take it that each additional term contributes a significant figure in the final result. 2. The efficiency function q Let the counter be of radius a and length I and let/t be the absorption coefficient of the counter gas. The f
j,
1 I
ft
I I
' I
Fig. 1. These diagrams show the geometrical arrangement of source and counter and indicate the relevant angles.
125
126
J. BYRNE
source of the radiation is assumed to be located at a point in the cylindrical boundary a distance l from either end of the counter. Thus only those X-rays emitted into the solid angle 2n on the counter side of the source have the possibility of being detected. The geometry of the arrangement is shown in fig. 1. The probability that an X-ray emitted on the counter side of the source will be detected in the counter is obtained by integrating the exponential absorption law over all directions of emission giving an expression for q in the form
1I
t=r,
Iz[ I. (1)
t
q = 1 -2--n~ _ e-~td~2;
In order to reduce the integral in (1) to a form suitable for calculation, it is first necessary to express the element of solid angle dO in terms of convenient variables. Writing dzds for an element of area at the point (r, z) on the cylindrical surface we have r2
dO = dz ds cos ~ cos ~k,
where the angles ~, ~Oare as indicated in fig. 1. From the same figure we see that d s c o s ~ =pdq~,
dzcos~b = dz(p/r),
r dp
.dpdz =
p2dpdz
r2
(0~)
p2,
r
(r 2
r _ p2)i"
r 2 {(r 2
- -
p2)(4a2
_
2_f2. /tdo
/I(P)
f l p2 +L2)½ e -2at dE r2(r 2 _ p2)~
=
and I
f
(p) =
~o
e -~'(r2-~2)-~ dr
(/02+.~2)½ r2(r2 __ 'p2)~"
In order to reduce the integral Ia(p) we make a change of variable and write r = p cosh t,
(p2 _~_ ~2)½ : p cosh co.
=
2 - 2~(pa+).2)½ + e p2(p2 + 22)~ +__ P
cosht cosh
dt.
(5)
o
An analogous transformation to a new variable u in /2(0) given by
(2)
p2))½ "
In performing the integral in (1), the variable p twice traverses the range from zero to 2a. Thus, noting that both ends of the counter must be included, the substitution of (2) into (1) leads to the equation for r/, =1
where
and a further integration by parts yields a similar result for h(P),
p2 dp dr =
(4)
u = r(r 2 - p2)-~
leading to the final expression d~
P = f ' P~Zd--p-P {I,(P) + I2(p)}, ,/0( 1 _pZ)~
r(4a2_p2)~ "
The variable z may now be eliminated using the relations
Z2
The primes may be immediately omitted on the variables of integration p' and r' and the problem is seen as calculation of two double integrals each of which requires different treatment. Thus we have
ii(p )
= P 2 d~
r
p'=p/2a, r'=r/2a; 2=1/2a, e = # a , f l = # l = 2 e 2 .
One integration by parts leads to the result
which gives the results r2dO= pd~b.dz-
and since it proves convenient to operate with dimensionless quantities we make the substitutions
p~ d p [ f(P2+'e)fexp(-#r) dr + {4a2_p2} ½[.]p rZ{r2_p2} ~
+ f (~+,2)½exp {-l~rl(r~-l~)-f}]
r2(r2p2)~
j.
(3)
We now introduce the quantity P through the relation t/ = 1 - (2/~z)P,
12(p) = e -p /92
2 e- 2a(p2+~2)½ _ p2(p2 ..~ ,~2)~-
flf(11+p~/~)' e-PU du pZ u
(6)
When eqs. (4), (5) and (6) are combined, the resulting expression for P is composed of two parts; a part independent of 2 which we recognise as the value of P appropriate to a counter of infinite length, together with a part which depends on 2 and describes the increase in P arising from the escape of X-rays over the ends of the counter. Thus we may write --
THE
DETECTION
EFFICIENCY
OF
t p dp O(l_p2~. x
f ×{f/e-""°°"'(cosht j/'(t +o2/z2)½e-,U u
2a
C(fl,2) = f~
{e-'-dp do(1 _p2)~
-2c
f+
/-s /
= ¼n {Io(O0Kl(a) - I,(e)Ko(a )}.
(7)
(
')I
coshi dt
(12)
Combining the results (11) and (12) we obtain finally Poo(a) = ½7ra { [Io(~)Kl(~) - I,(c0Ko(a)] + + 2~[Io(a)Ko(a) +
d.-
o,e-2~P¢°*ht c o s h t
127
COUNTER
f ~ c o s 0 Kl(2a cos 0 ) dO
cos~)dt},
and
CYLINDRICAL
differentiating* the resulting relation with respect to the variable z we find
where
P~(a) =
A
. (8)
I,(a)Kl(a)] -
21.
(13)
Q8
In the next section we proceed to calculate P®(e) in closed form. 3. Calculation of P~o(a) In the first of the two double integrals in (7), the integration with respect to the variable t reduces immediately to a special case of the integral representation of the hyperbolic Bessel function K.(z), ref. 3)
K.(z)
=
f+
e-~°°Shtcosh
nt dt.
O.E
8 c" 0.4
0
The same formula may be applied to the second double integral in (7) after one integration by parts with respect to the variable p has been carried out. Finally the substitution p = cos 0 yields the result Poo(a) = 2a
0.2
cos 0 Kx(2a cos 0) dO + 0
+ (2a) 2
sin z 0 Ko(2a cos 0) dO - n~.
(9) 0
A complete reduction of these single integrals is now made possible by means of a formula 3) which links the Bessel functions Is(z) and K,(z),
Im(z)K,(z) =2 (-1)m f i"K,_m(2z cosO)cos(v+ m)OdO, IRe(v-m) l < l .
(10)
Direct application of this relation leads immediately to a value for the second integral in (9), thus we find
I 05
i 1.0 Gt=]J cl
=
{Io(
)Ko(a) +
(n)
Equation (10) cannot be used directly to give the Value of the first integral in (9) since the relation is not valid for v - m = 1. Instead by setting v = m = 0 in (10) and
] 2.0
I 2.5
Fig. 2. The efficiencyfunction t/oo(e)for an infinitely long counter shown as a function of ~, the ratio of the counter radius to the mean free path of the radiation. In fig. 2 the function qoo(a) based on (13) is plotted over a range of a from 0 to 2.5. For larger values of (counter radius much greater than the mean free path of the radiation in the gas) use may be made of the asymptotic form P/~(~) ,~ 1 - 1/4~,
f ~sin2 OKo(2a cosO)dO
I 1.5
a ~ 0%
(14)
a result which easily follows from (13) when the Bessel functions are replaced by their asymptotic expansions. * Differentiation under the integral sign is a valid operation since the integrand possesses a Riemann integral with respect to 0 and the partial derivative of the integrand with respect to z is a continuous function of 0 and z, ref. 4).
128
J. B Y R N E
It is perhaps worth noting that, even for a value of as small as 2, the asymptotic form of q~o(~) gives a result which is correct to 1%. For small values of ~ the complete expression for r/oo(~) may conveniently be replaced by the expansion q~o(e) "~ 2~ + ~2[log ½e + 1 - 37], ~,~1,
),=2
r
(15)
where y--0.577216.., is Euler's constant. At g = 0.2, this form of rt oo(~) gives an error of about 6 %, reflecting the importance of the geometrical arrangement when the mean free path of the radiation is great.
L) c~l~ ×
A=3
%
4. Correction for finite length
The contribution to the total escape probability arising from the finite length of the counter is contained in the function C(fl,2) given in (8). In this equation the integral over t, which represents essentially a sum of incomplete Bessel functions, can be reduced to a more useful form through the substitution u =
(p/).)
cosh t.
A simple manipulation of the remaining terms then leads to the result C(fl,2)
=
f l o(1 ---o2)~flck(fl,x),,. dp
(16)
where
0.2
0.5
__e-P"[(u 2 - x) ½- 1] du d (1 +x)½ U
(17)
n
=
(18) tl=O
f ~Oe-Zt dt
J t
=
t--.
They are closely related to the confluent hypergeometric functions 4, 5) and satisfy the recurrence formulae:
E.(z)
The function ¢p(fl,x) is here expressed in convenient form for expansion in a Taylor series
2.5
straightforward. In these formulae the functions E,(fl) are the generalized exponential integral functions defined by the relation
and 0 < x < 1.
2.0
counter diameter.
E.(z)
_|~or
1.5
~-2P
Fig. 3. The correction term (2/~)C(fl, ,~) expressed as a function o f fl for four values o f 2, the ratio o f counter half-length to
(u - 1)du ~(t~,x) = :f°°e-P" ,
x = (p/2) z,
1.0
1 n-1
= - -
{ e -= -
zE._
co e - t
E,(z) = - Ei(-
z) = f • ~ / - - ,
,(z) },
dt
n >
Eo(Z) =
1,
e-°/z.
The integral over p in (16) is easily carried out with the aid of (18) and the formula
since the conditions* permitting differentiation under the integral sign hold. Thus we find
f
i pendp 1"3"5...(2n--1) 0( 1 __p2)~r = 2.4.6...2n "2-"
~(0) = 0 (~t(0)
=
½E2(fl)
(19)
We can finally collect these results together to give C(fl,2) in the form
~"(0) = ¼ { E , ( # ) - e -p } C(#,,~)
~b"(0) = } (3E6(fl) + fle -p + e-P}. The calculation of the higher derivatives is equally
=
½n ~_, a.(#)/2 n=0
where
2",
(20)
THE DETECTION EFFICIENCY OF A CYLINDRICAL COUNTER
References
ao=O a,
=
129
!
a2 = ~r~b"(O)/2!
(21)
aa = ?~q~"(0)/3! G r a p h s of the correction function (2/n) C(fl, 2), computed using eqs. (19), (20), (21) and published tables 6) o f the functions E,(z) are shown in fig. 3 for four values of the p a r a m e t e r 2.
1) Curran, Angus and Cockroft, Phil. Mag. 40 (1949) 36. 2) Curran, Cockroft and Insch, Proc. Phys. Sot., A 63 (1950) 845. 3) G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, 1952). 4) Whittaker and Watson, Modern Analysis (Cambridge 1952). 5) A. Erd61yi, Higher Transcendental Functions, Bateman Manuscript Project, Vol. 1 (McGraw Hill, 1953). 6) G. Placzek, Declassified Canadian Report MT-I (NRC-1547) (1947).
INSTRUMENTS
AND METHODS
35 (I965) I 2 5 - I 2 9 ;
~
NORTH-HOLLAND
PUBLISHING
CO.
CALCULATION O F T H E D E T E C T I O N EFFICIENCY OF A CYLINDRICAL C O U N T E R FOR X-RAYS F R O M AN INTERNAL S O U R C E J. B Y R N E
Department of Natural Philosophy, University of Edinburgh, Edinburgh, Scotland Received 25 January 1965
This paper describes detailed calculations of the efficiencyof a long cylindrical proportional counter for detection of X-rays from an isotropic point source placed centrally in the wall of the
counter. The results are applicable to counters of varied length and radius for a range of X-ray energies and are presented in both analytic and graphical f o r m s .
1. Introduction The conventional apparatus used in nuclear spectroscopy for purposes of identifying and estimating the intensities of electromagnetic radiations, emitted after radio-active decay, consists of the NaI(TI) scintillation counter coupled to a photo-multiplier and pulseamplifier. For ~,-rays and X-rays in the energy range of 50 keV and less, however, the scintillation counter is not suitable for accurate work since the energy resolution is poor and the output pulses tend to lie in that part of the pulse-spectrum where photomultiplier noise begins to be important. A further disadvantage of the method arises from the fact that radiation of low energy is highly absorbed in the protective metal films with which NaI crystals are normally covered. These problems may be largely overcome if the X-rays are detected in a proportional counter with a 2re countingangle where the source of the radiation is deposited on a thin film placed in the wall of the counter in direct contact with the counter gas. The kind of experiment envisaged here might be for example a study of the L-shell X-rays emitted following internal conversion of the E2 ~-rays of about 40 keV energy which are characteristic of a whole class of even-A nuclei of heavy alpha-emitting elements such as plutonium and uranium. A further application exists in the study of electron-capture in nuclei for which gaseous sources are not easily manufactured. The technique for these and similar measurements was initiated by Curran and his collaborators more than a decade ago 1'2) and considerable development has taken place since then. Energy resolution attained by this method is far superior to that available using scintillation counters. The principal difficulty in proportional counter X-ray spectroscopy is associated with the measurement of absolute intensity; for all X-rays except those of the lowest energy the counter gas is to a large extent transparent and the probability of detecting the X-ray quantum becomes critically dependent, not only on the absorption coefficient, but also on the direction of
emission. Since the absorption law is exponential the problem of calculating the absolute detection efficiency for X-rays of a given energy can be formidable even for the very simplest geometry. In this paper are presented calculations of the detection efficiency of a long cylindrical counter for X-rays emitted isotropically from a point source placed centrally in the wall of the counter. By a "long" counter is understood one for which the half-length is considerably greater than the diameter. It turns out that the resUlts for an infinitely long counter are expressible in closed form in terms of products of hyperbolic Bessel functions. For counters of finite length there is in addition a series in inverse powers of the half-length expressed in units of the diameter. In the type of counter normally used the half-length is at least twice the diameter in which circumstances the series converges so rapidly that for computation purposes only the first few terms in the series need be used. As a rough guide one may take it that each additional term contributes a significant figure in the final result. 2. The efficiency function q Let the counter be of radius a and length I and let/t be the absorption coefficient of the counter gas. The f
j,
1 I
ft
I I
' I
Fig. 1. These diagrams show the geometrical arrangement of source and counter and indicate the relevant angles.
125
126
J. BYRNE
source of the radiation is assumed to be located at a point in the cylindrical boundary a distance l from either end of the counter. Thus only those X-rays emitted into the solid angle 2n on the counter side of the source have the possibility of being detected. The geometry of the arrangement is shown in fig. 1. The probability that an X-ray emitted on the counter side of the source will be detected in the counter is obtained by integrating the exponential absorption law over all directions of emission giving an expression for q in the form
1I
t=r,
Iz[
t
q = 1 -2--n~ _ e-~td~2;
In order to reduce the integral in (1) to a form suitable for calculation, it is first necessary to express the element of solid angle dO in terms of convenient variables. Writing dzds for an element of area at the point (r, z) on the cylindrical surface we have r2
dO = dz ds cos ~ cos ~k,
where the angles ~, ~Oare as indicated in fig. 1. From the same figure we see that d s c o s ~ =pdq~,
dzcos~b = dz(p/r),
r dp
.dpdz =
p2dpdz
r2
(0~)
p2,
r
(r 2
r _ p2)i"
r 2 {(r 2
- -
p2)(4a2
_
2_f2. /tdo
/I(P)
f l p2 +L2)½ e -2at dE r2(r 2 _ p2)~
=
and I
f
(p) =
~o
e -~'(r2-~2)-~ dr
(/02+.~2)½ r2(r2 __ 'p2)~"
In order to reduce the integral Ia(p) we make a change of variable and write r = p cosh t,
(p2 _~_ ~2)½ : p cosh co.
=
2 - 2~(pa+).2)½ + e p2(p2 + 22)~ +__ P
cosht cosh
dt.
(5)
o
An analogous transformation to a new variable u in /2(0) given by
(2)
p2))½ "
In performing the integral in (1), the variable p twice traverses the range from zero to 2a. Thus, noting that both ends of the counter must be included, the substitution of (2) into (1) leads to the equation for r/, =1
where
and a further integration by parts yields a similar result for h(P),
p2 dp dr =
(4)
u = r(r 2 - p2)-~
leading to the final expression d~
P = f ' P~Zd--p-P {I,(P) + I2(p)}, ,/0( 1 _pZ)~
r(4a2_p2)~ "
The variable z may now be eliminated using the relations
Z2
The primes may be immediately omitted on the variables of integration p' and r' and the problem is seen as calculation of two double integrals each of which requires different treatment. Thus we have
ii(p )
= P 2 d~
r
p'=p/2a, r'=r/2a; 2=1/2a, e = # a , f l = # l = 2 e 2 .
One integration by parts leads to the result
which gives the results r2dO= pd~b.dz-
and since it proves convenient to operate with dimensionless quantities we make the substitutions
p~ d p [ f(P2+'e)fexp(-#r) dr + {4a2_p2} ½[.]p rZ{r2_p2} ~
+ f (~+,2)½exp {-l~rl(r~-l~)-f}]
r2(r2p2)~
j.
(3)
We now introduce the quantity P through the relation t/ = 1 - (2/~z)P,
12(p) = e -p /92
2 e- 2a(p2+~2)½ _ p2(p2 ..~ ,~2)~-
flf(11+p~/~)' e-PU du pZ u
(6)
When eqs. (4), (5) and (6) are combined, the resulting expression for P is composed of two parts; a part independent of 2 which we recognise as the value of P appropriate to a counter of infinite length, together with a part which depends on 2 and describes the increase in P arising from the escape of X-rays over the ends of the counter. Thus we may write --
THE
DETECTION
EFFICIENCY
OF
t p dp O(l_p2~. x
f ×{f/e-""°°"'(cosht j/'(t +o2/z2)½e-,U u
2a
C(fl,2) = f~
{e-'-dp do(1 _p2)~
-2c
f+
/-s /
= ¼n {Io(O0Kl(a) - I,(e)Ko(a )}.
(7)
(
')I
coshi dt
(12)
Combining the results (11) and (12) we obtain finally Poo(a) = ½7ra { [Io(~)Kl(~) - I,(c0Ko(a)] + + 2~[Io(a)Ko(a) +
d.-
o,e-2~P¢°*ht c o s h t
127
COUNTER
f ~ c o s 0 Kl(2a cos 0 ) dO
cos~)dt},
and
CYLINDRICAL
differentiating* the resulting relation with respect to the variable z we find
where
P~(a) =
A
. (8)
I,(a)Kl(a)] -
21.
(13)
Q8
In the next section we proceed to calculate P®(e) in closed form. 3. Calculation of P~o(a) In the first of the two double integrals in (7), the integration with respect to the variable t reduces immediately to a special case of the integral representation of the hyperbolic Bessel function K.(z), ref. 3)
K.(z)
=
f+
e-~°°Shtcosh
nt dt.
O.E
8 c" 0.4
0
The same formula may be applied to the second double integral in (7) after one integration by parts with respect to the variable p has been carried out. Finally the substitution p = cos 0 yields the result Poo(a) = 2a
0.2
cos 0 Kx(2a cos 0) dO + 0
+ (2a) 2
sin z 0 Ko(2a cos 0) dO - n~.
(9) 0
A complete reduction of these single integrals is now made possible by means of a formula 3) which links the Bessel functions Is(z) and K,(z),
Im(z)K,(z) =2 (-1)m f i"K,_m(2z cosO)cos(v+ m)OdO, IRe(v-m) l < l .
(10)
Direct application of this relation leads immediately to a value for the second integral in (9), thus we find
I 05
i 1.0 Gt=]J cl
=
{Io(
)Ko(a) +
(n)
Equation (10) cannot be used directly to give the Value of the first integral in (9) since the relation is not valid for v - m = 1. Instead by setting v = m = 0 in (10) and
] 2.0
I 2.5
Fig. 2. The efficiencyfunction t/oo(e)for an infinitely long counter shown as a function of ~, the ratio of the counter radius to the mean free path of the radiation. In fig. 2 the function qoo(a) based on (13) is plotted over a range of a from 0 to 2.5. For larger values of (counter radius much greater than the mean free path of the radiation in the gas) use may be made of the asymptotic form P/~(~) ,~ 1 - 1/4~,
f ~sin2 OKo(2a cosO)dO
I 1.5
a ~ 0%
(14)
a result which easily follows from (13) when the Bessel functions are replaced by their asymptotic expansions. * Differentiation under the integral sign is a valid operation since the integrand possesses a Riemann integral with respect to 0 and the partial derivative of the integrand with respect to z is a continuous function of 0 and z, ref. 4).
128
J. B Y R N E
It is perhaps worth noting that, even for a value of as small as 2, the asymptotic form of q~o(~) gives a result which is correct to 1%. For small values of ~ the complete expression for r/oo(~) may conveniently be replaced by the expansion q~o(e) "~ 2~ + ~2[log ½e + 1 - 37], ~,~1,
),=2
r
(15)
where y--0.577216.., is Euler's constant. At g = 0.2, this form of rt oo(~) gives an error of about 6 %, reflecting the importance of the geometrical arrangement when the mean free path of the radiation is great.
L) c~l~ ×
A=3
%
4. Correction for finite length
The contribution to the total escape probability arising from the finite length of the counter is contained in the function C(fl,2) given in (8). In this equation the integral over t, which represents essentially a sum of incomplete Bessel functions, can be reduced to a more useful form through the substitution u =
(p/).)
cosh t.
A simple manipulation of the remaining terms then leads to the result C(fl,2)
=
f l o(1 ---o2)~flck(fl,x),,. dp
(16)
where
0.2
0.5
__e-P"[(u 2 - x) ½- 1] du d (1 +x)½ U
(17)
n
=
(18) tl=O
f ~Oe-Zt dt
J t
=
t--.
They are closely related to the confluent hypergeometric functions 4, 5) and satisfy the recurrence formulae:
E.(z)
The function ¢p(fl,x) is here expressed in convenient form for expansion in a Taylor series
2.5
straightforward. In these formulae the functions E,(fl) are the generalized exponential integral functions defined by the relation
and 0 < x < 1.
2.0
counter diameter.
E.(z)
_|~or
1.5
~-2P
Fig. 3. The correction term (2/~)C(fl, ,~) expressed as a function o f fl for four values o f 2, the ratio o f counter half-length to
(u - 1)du ~(t~,x) = :f°°e-P" ,
x = (p/2) z,
1.0
1 n-1
= - -
{ e -= -
zE._
co e - t
E,(z) = - Ei(-
z) = f • ~ / - - ,
,(z) },
dt
n >
Eo(Z) =
1,
e-°/z.
The integral over p in (16) is easily carried out with the aid of (18) and the formula
since the conditions* permitting differentiation under the integral sign hold. Thus we find
f
i pendp 1"3"5...(2n--1) 0( 1 __p2)~r = 2.4.6...2n "2-"
~(0) = 0 (~t(0)
=
½E2(fl)
(19)
We can finally collect these results together to give C(fl,2) in the form
~"(0) = ¼ { E , ( # ) - e -p } C(#,,~)
~b"(0) = } (3E6(fl) + fle -p + e-P}. The calculation of the higher derivatives is equally
=
½n ~_, a.(#)/2 n=0
where
2",
(20)
THE DETECTION EFFICIENCY OF A CYLINDRICAL COUNTER
References
ao=O a,
=
129
!
a2 = ~r~b"(O)/2!
(21)
aa = ?~q~"(0)/3! G r a p h s of the correction function (2/n) C(fl, 2), computed using eqs. (19), (20), (21) and published tables 6) o f the functions E,(z) are shown in fig. 3 for four values of the p a r a m e t e r 2.
1) Curran, Angus and Cockroft, Phil. Mag. 40 (1949) 36. 2) Curran, Cockroft and Insch, Proc. Phys. Sot., A 63 (1950) 845. 3) G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, 1952). 4) Whittaker and Watson, Modern Analysis (Cambridge 1952). 5) A. Erd61yi, Higher Transcendental Functions, Bateman Manuscript Project, Vol. 1 (McGraw Hill, 1953). 6) G. Placzek, Declassified Canadian Report MT-I (NRC-1547) (1947).