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An analytical formula of K-shell photoeffect for intermediate and high Z atoms
NUCLEAR
INSTRUMENTS
AND
METHODS
IO 4
(I972)
73-75; ©
NORTH-HOLLAND
PUBLISHING
CO.
AN ANALYTICAL F O R M U L A OF K-SHELL P H O T O E F F E C T FOR INTERMEDIATE AND HIGH Z ATOMS YIN YIN HLA, CHIT KHAING
and M A Y SU
Nuclear Physics Laboratory, Arts and Science University, Rangoon, Burma and A. M. G H O S E *
I.A.E.A. Visiting Professor to the Revolutionary Govt. of the Union of Burma Received 8 M a y 1972 Theoretical calculation o f photoeffect is n o t available in analytic form a n d extraction o f photoelectric cross-sections for specific element a n d p h o t o n energy require a formidable a m o u n t o f c o m p u t a t i o n time. To remove this difficulty an analytical f o r m u l a for K-shell photoeffect was developed semi-empirically which is valid for intermediate a n d heavy elements (Z~> 50) a n d for
p h o t o n energy above 150 keV. T h e f o r m u l a agrees a l m o s t exactly with the high energy limit calculated exactly by Pratt while at intermediate energies it agrees with screened potential calculation o f Schmickley a n d Pratt as well as with the experimental results available in this region.
Theoretical calculations of the K-shell photoelectric effect have been made by several workers t-7) using high speed computers. Of all the existing calculations the most accurate are those of Schmickly and Pratt 8) who used the relativistic Coulomb wave functions in a screened central potential and a multi-term partial wave expansion for the wave function of the ejected electron. Unfortunately no simple analytical formula of these results exists in the literature and the evaluation of K-shell photoelectric cross-sections for a particular element and gamma energy requires a formidable amount of computation time. It is the object of the present investigation to develop a simple semi-empirical formula which can yield the cross-section within a reasonable error limit over a wide energy range. Since the high-energy limit of the K-shell photoeffect has been accurately calculated by Pratt9), the form of the empirical formula is so chosen that it reproduces the high-energy limit due to Pratt and for an other energy range reproduces the values calculated by Schmickley and PrattS). Starting from the following expression due to Sauter 1°) for K-shell photoelectric cross-sections using relativistic Born approximations:
where
[
/mc2\5-]
o
{~
4,0 = 3
\.,cV
'
hv + m c 2 mc 2
1
137
'
Using the notation X = hv/mc z
and
a = aZ,
we get the following expression in the limit of low energy
~=~o
1+
~
~/
where a5 1
~o = ~ 0 o - - - - . cg
x
In the high energy limit Sauter's equation reduced to aK= a o. Calculations of Nagasaka 11) for the high energy photoeffect have indicated that the structure of eq. (1) can be retained even at higher energies. Pratt 9) has shown that more accurately the limiting K-shell photoeffect is given by
7(7-2)
+ - - ×
y+l
aK = ao F ( a ) .
2~(~2_1)~
~_,f(~2_
(2)
, (1)
(3)
Pratt has also given a semi-empirical formula for F ( a ) which can be justified on theoretical grounds when a is not too large. We have expressed F ( a ) with a
* Present address: Bose Institute, Calcutta-9, India.
73
74
YIN
YIN
HLA
et al.
1000
\ 10C
i
•
10
iO 2
--
PRESENT WORK SCHNIKLEY & PRATT
o t_
d
1
Z=92
• Z=82 2=78 2=74
,¢
01 Z=60
Z=50
0.01 100
i 1000
Z=84 Z=79
i 10 0 0 0
io
L 1000
100
E (keV)
Fig. 1. C o m p a r i s o n
i 10000 E (keV)
of present
formula
with theoretical
values.
reasonable degree of accuracy by
F(a) = e x p ( - 3 . 6 0 a + 1.79a 2).
(4)
(6) Basing on these above facts we have chosen the form of the proposed semi-empirical relation as
= O+ )°F8 j To make eq. (5) agree with eq. (3) we must have A(a) = 2~x/2 F(a).
It is clear from the form of eq. (6) that for intermediate and heavy atoms it was not possible to express B(a) as a function of atomic number only, but it showed a slight variation with x. This was taken into account by changing B(a) in eq. (6) to B(a,x) and expressing this by
C(a)
B(a,x) = B(a) -- - -
X
Eq. (5) is too simple and to fit it with Schmickley and Pratt and with experimental cross-sections well a new term B(a) is introduced in the equation. Therefore the equation takes the form
The values of B(a) and C(a) were evaluated and can be expressed in the form B(a) = 7.25 e x p ( - 2 . 7 5 a ) - 8 . 6 5 e x p ( - 1.61 a ) - 0 . 4 9 - 1.37a,
AN ANALYTICAL FORMULA OF K-SHELL PHOTOEFFECT and C (a) = 0.09575 - 0.00383 (90 - 137 a). Therefore, the final form of the empirical formula stands as, ~K=CrO
(
+x~ 1 2/
1~[-8x/23 2.7,, 865e-1.61 {.x-~/ +7.25e- . "
- 0 . 4 9 - 1 . 3 7 a ] + 2 . / 2 e x p ( - 3 . 6 O a + 1.79a 2) _ J x ~_ 0.09575 -- 0.00383(90--x 2-" 137 a)} .
Fig. 1 compares the present results with both the theoretical calculations o f Schmickley and Pratt and experimental results where available. It will be seen that the agreement of the equation is quite satisfactory. We conclude, therefore, that this semi-empirical equation can be used with a reasonable degree of accuracy for rapid estimation of the K-shell photoelectric cross-sections of all elements of the periodic table above Z -- 50 and for all energies above 150 keV. Since the high energy limit of eq. (7) agrees with the
75
exact calculation of Pratt, we feel that the formula can be safely used above 5 MeV, where exact c o m p u t a t i o n of photoelectric cross-sections has not yet been carried out. The authors wish to thank Dr. M a u n g M a u n g Kha, Rector and Prof. San Tha Aung, Head of the Department of Physics, Arts and Science University, R a n g o o n for their kind interest in this work.
References 1) H. R. Hulme, J. McDougall, R. A. Buckingham and R. H. Fowler, Proc. Roy. Soc. (London) A149 (1935) 131. 2) S. Hultherg, B. C. H. Nagerl and P. O. M. Olsson, Arkiv Fysik 20 (1961) 555. a) B. C. H. Nagel, Arkiv Fysik 18 (1960) 1. 4) R.H. Pratt, R.D. Levee, R.L. Pexton and W. Aron, Phys. Rev. 134 (1964) A 898. 5) H. Hall and E. Sullivan, Phys. Rev. 152 (1967) 4. 6) j. j. Matese and W. R. Johnson, Phys. Rev. 140 (1969) A I. 7) G. Rakavy and A. Ron, Phys. Letters 19 (1965)207; Phys. Rev. 159 (1967) 50. 8) R. D. Schmickley and R. H. Pratt, Phys. Rev. 164 (1967) 105. 9) R. H. Pratt, Phys. Rev. 117 (1960) 1017. lo) F. Santer, Ann. Physik 11 (1931) 454. 11) F. G. Nagasaka, Thesis (Notre Dame University, Indiana, Aug. 1955).
INSTRUMENTS
AND
METHODS
IO 4
(I972)
73-75; ©
NORTH-HOLLAND
PUBLISHING
CO.
AN ANALYTICAL F O R M U L A OF K-SHELL P H O T O E F F E C T FOR INTERMEDIATE AND HIGH Z ATOMS YIN YIN HLA, CHIT KHAING
and M A Y SU
Nuclear Physics Laboratory, Arts and Science University, Rangoon, Burma and A. M. G H O S E *
I.A.E.A. Visiting Professor to the Revolutionary Govt. of the Union of Burma Received 8 M a y 1972 Theoretical calculation o f photoeffect is n o t available in analytic form a n d extraction o f photoelectric cross-sections for specific element a n d p h o t o n energy require a formidable a m o u n t o f c o m p u t a t i o n time. To remove this difficulty an analytical f o r m u l a for K-shell photoeffect was developed semi-empirically which is valid for intermediate a n d heavy elements (Z~> 50) a n d for
p h o t o n energy above 150 keV. T h e f o r m u l a agrees a l m o s t exactly with the high energy limit calculated exactly by Pratt while at intermediate energies it agrees with screened potential calculation o f Schmickley a n d Pratt as well as with the experimental results available in this region.
Theoretical calculations of the K-shell photoelectric effect have been made by several workers t-7) using high speed computers. Of all the existing calculations the most accurate are those of Schmickly and Pratt 8) who used the relativistic Coulomb wave functions in a screened central potential and a multi-term partial wave expansion for the wave function of the ejected electron. Unfortunately no simple analytical formula of these results exists in the literature and the evaluation of K-shell photoelectric cross-sections for a particular element and gamma energy requires a formidable amount of computation time. It is the object of the present investigation to develop a simple semi-empirical formula which can yield the cross-section within a reasonable error limit over a wide energy range. Since the high-energy limit of the K-shell photoeffect has been accurately calculated by Pratt9), the form of the empirical formula is so chosen that it reproduces the high-energy limit due to Pratt and for an other energy range reproduces the values calculated by Schmickley and PrattS). Starting from the following expression due to Sauter 1°) for K-shell photoelectric cross-sections using relativistic Born approximations:
where
[
/mc2\5-]
o
{~
4,0 = 3
\.,cV
'
hv + m c 2 mc 2
1
137
'
Using the notation X = hv/mc z
and
a = aZ,
we get the following expression in the limit of low energy
~=~o
1+
~
~/
where a5 1
~o = ~ 0 o - - - - . cg
x
In the high energy limit Sauter's equation reduced to aK= a o. Calculations of Nagasaka 11) for the high energy photoeffect have indicated that the structure of eq. (1) can be retained even at higher energies. Pratt 9) has shown that more accurately the limiting K-shell photoeffect is given by
7(7-2)
+ - - ×
y+l
aK = ao F ( a ) .
2~(~2_1)~
~_,f(~2_
(2)
, (1)
(3)
Pratt has also given a semi-empirical formula for F ( a ) which can be justified on theoretical grounds when a is not too large. We have expressed F ( a ) with a
* Present address: Bose Institute, Calcutta-9, India.
73
74
YIN
YIN
HLA
et al.
1000
\ 10C
i
•
10
iO 2
--
PRESENT WORK SCHNIKLEY & PRATT
o t_
d
1
Z=92
• Z=82 2=78 2=74
,¢
01 Z=60
Z=50
0.01 100
i 1000
Z=84 Z=79
i 10 0 0 0
io
L 1000
100
E (keV)
Fig. 1. C o m p a r i s o n
i 10000 E (keV)
of present
formula
with theoretical
values.
reasonable degree of accuracy by
F(a) = e x p ( - 3 . 6 0 a + 1.79a 2).
(4)
(6) Basing on these above facts we have chosen the form of the proposed semi-empirical relation as
= O+ )°F8 j To make eq. (5) agree with eq. (3) we must have A(a) = 2~x/2 F(a).
It is clear from the form of eq. (6) that for intermediate and heavy atoms it was not possible to express B(a) as a function of atomic number only, but it showed a slight variation with x. This was taken into account by changing B(a) in eq. (6) to B(a,x) and expressing this by
C(a)
B(a,x) = B(a) -- - -
X
Eq. (5) is too simple and to fit it with Schmickley and Pratt and with experimental cross-sections well a new term B(a) is introduced in the equation. Therefore the equation takes the form
The values of B(a) and C(a) were evaluated and can be expressed in the form B(a) = 7.25 e x p ( - 2 . 7 5 a ) - 8 . 6 5 e x p ( - 1.61 a ) - 0 . 4 9 - 1.37a,
AN ANALYTICAL FORMULA OF K-SHELL PHOTOEFFECT and C (a) = 0.09575 - 0.00383 (90 - 137 a). Therefore, the final form of the empirical formula stands as, ~K=CrO
(
+x~ 1 2/
1~[-8x/23 2.7,, 865e-1.61 {.x-~/ +7.25e- . "
- 0 . 4 9 - 1 . 3 7 a ] + 2 . / 2 e x p ( - 3 . 6 O a + 1.79a 2) _ J x ~_ 0.09575 -- 0.00383(90--x 2-" 137 a)} .
Fig. 1 compares the present results with both the theoretical calculations o f Schmickley and Pratt and experimental results where available. It will be seen that the agreement of the equation is quite satisfactory. We conclude, therefore, that this semi-empirical equation can be used with a reasonable degree of accuracy for rapid estimation of the K-shell photoelectric cross-sections of all elements of the periodic table above Z -- 50 and for all energies above 150 keV. Since the high energy limit of eq. (7) agrees with the
75
exact calculation of Pratt, we feel that the formula can be safely used above 5 MeV, where exact c o m p u t a t i o n of photoelectric cross-sections has not yet been carried out. The authors wish to thank Dr. M a u n g M a u n g Kha, Rector and Prof. San Tha Aung, Head of the Department of Physics, Arts and Science University, R a n g o o n for their kind interest in this work.
References 1) H. R. Hulme, J. McDougall, R. A. Buckingham and R. H. Fowler, Proc. Roy. Soc. (London) A149 (1935) 131. 2) S. Hultherg, B. C. H. Nagerl and P. O. M. Olsson, Arkiv Fysik 20 (1961) 555. a) B. C. H. Nagel, Arkiv Fysik 18 (1960) 1. 4) R.H. Pratt, R.D. Levee, R.L. Pexton and W. Aron, Phys. Rev. 134 (1964) A 898. 5) H. Hall and E. Sullivan, Phys. Rev. 152 (1967) 4. 6) j. j. Matese and W. R. Johnson, Phys. Rev. 140 (1969) A I. 7) G. Rakavy and A. Ron, Phys. Letters 19 (1965)207; Phys. Rev. 159 (1967) 50. 8) R. D. Schmickley and R. H. Pratt, Phys. Rev. 164 (1967) 105. 9) R. H. Pratt, Phys. Rev. 117 (1960) 1017. lo) F. Santer, Ann. Physik 11 (1931) 454. 11) F. G. Nagasaka, Thesis (Notre Dame University, Indiana, Aug. 1955).