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On a method for sampling the energy loss in electron transport
Volume S1A, number 7
PHYSICS LETTERS
21 April 1975
ON A METHOD FOR SAMPLING THE ENERGY LOSS IN ELECTRON TRANSPORT D.W. VANDE PUTFE and DJ. CUSENS Isotoperand Radiation Dn’ision, Atomic Energy Board, Private Bag 1256, Pretoria, South Africa Received 27 January 1975 A biased sampling method is proposed, allowing the use of the Blunck-Westphal energy loss distribution in Monte Carlo electron transport.
Two expressions for the probability f(Q) dQ of an electron of initial energy E0 losing an amount of energy between Q and Q + dQ by ionization, when travelling a path of lengths, are available, viz, that of Landau [1] and that of Blunck and Westphal [2]. However, the corresponding values of the average loss are systematically lower than the values predicted by the eXperimentally verified Bethe formula for continuous slowing down of charged partIcles by ionization [3]. Therefore, when performing Monte Carlo simulations destined to compute e.g. detector response [4,5] metallic foil electron reflection and transmission coefficients [4,71or energy deposition in matter [6,7], one is confronted with the fact that pathlengths obtamed when using these distributions, are too long. Some palliatives have been proposed, at different levels of the calculation. Wittig ([4] and presumably in [5] as well), manipulates the “X.,” constants in the Blunck-Westphal formulas [2], thus actually modifying the distributions prior to their use; Patau [6] on the other hand, multiplies the Monte Carlo-obtained path lengths by a suitable corrective factor, thus effecting an “a posteriori” correction. We propose a biased sampling of the unmodified distribution, thereby introducing the correction at the sampling level. If the sampled Q is greater than or equal to the Bethe average ~B’ it is accepted, whereas, if Q
Table 1 Average energy loss (MeV) for 1.4 14 MeV electrons normally incident on Al, Cu and Pb foils. -
s (cm) Al 0.0026 Al 0.0212 Cu 0.00071
-
QB 0.00906 0.07213 0.00872
~
-
Q (Landau)
Q (Bl. and West.)
0.01055 0.08420 0.01022
0.00832 0.07132 0.00802
0.06252
0.0501
~
Pb 0.00394
0.05269
,
sampled Q is accepted regardless. This method merely equalizes the number of hits to the left and right of QB• Table 1 shows a few results of this sampling technique when applied to single distributions. It appears that the values of the sampled Q (Bl. and West.) are in __________________________________ is
-
~ ~
~5LAHOAU
~,
Eo
y=
/QB
J f(Q) dQ / J QB
BLUNCK WESTPHAL
-
I
1(Q) dQ.
0
If ,~~ y, a new Q and a new t~are chosen and the tests repeated a maximum of ten times, after which the
_____
0
__________________________ 402
40’
0 I 54eV)
405
405
0~tO
0,12
Fig. 1. Blunck-Westphal and Landau distributions for 1 MeV electrons incident on 0.0097 cm of aluminium.
393
Volume 51A, number 7
PHYSICS LETI’ERS
Table 2 Comparison of average total path lengths (cm) obtained by the Bethe formula (upper value) and the Monte Carlo method (lower value). E
0 (MeV)
Beryllium
Aluminium
Lead
0.65230 0.65781
0.44857 0.45388
0.14373 0.14701
1.0
0.29113 0.29410
0.20446 0.20768
0.07030 0.07192
0.1
0.00918 0.00869
0.00683 0.00627
0.00275 0.00200
2.0
—
good agreement with QB; those of the sampled Q (Landau) are too high. This is due to the particular shape of the Landau distribution inasmuch as it extends towards high Q values (fig. 1). To further confirm the validity of this type of biased sampling from
394
21 April 1975
the Blunck-Westphal distribution, it was successfully used in Monte Carlo calculations to determine average total path lengths, as can be seen in table 2. The efficiency of the process, i.e. the ratio of successes to trials, in ~ in graphite, 43% in aluminium, 48% in copper, ~
in silver and 60% in lead.
References (1] L.D. Landau, J. Phys. USSR 8(1944)201. [2] [3] [4) [5]
0. Blunck and K. Westphal, Zeit. Phys. 130 (1951) 641. D.W. Vande Putte, report PEL-235 (1974). S. Wittig, report IKF-20 (1968). M. Waldschmidt and S. Wittig, Nucl. Instr. Meths. 64 (1968) 189. [6) J.-P. Patau, thesis, University of Toulouse (1972). [7) M.J. Berger, Methods in computational physics, (Vol. 1), ed. B. Alder (Academic Press, New York, 1963).
PHYSICS LETTERS
21 April 1975
ON A METHOD FOR SAMPLING THE ENERGY LOSS IN ELECTRON TRANSPORT D.W. VANDE PUTFE and DJ. CUSENS Isotoperand Radiation Dn’ision, Atomic Energy Board, Private Bag 1256, Pretoria, South Africa Received 27 January 1975 A biased sampling method is proposed, allowing the use of the Blunck-Westphal energy loss distribution in Monte Carlo electron transport.
Two expressions for the probability f(Q) dQ of an electron of initial energy E0 losing an amount of energy between Q and Q + dQ by ionization, when travelling a path of lengths, are available, viz, that of Landau [1] and that of Blunck and Westphal [2]. However, the corresponding values of the average loss are systematically lower than the values predicted by the eXperimentally verified Bethe formula for continuous slowing down of charged partIcles by ionization [3]. Therefore, when performing Monte Carlo simulations destined to compute e.g. detector response [4,5] metallic foil electron reflection and transmission coefficients [4,71or energy deposition in matter [6,7], one is confronted with the fact that pathlengths obtamed when using these distributions, are too long. Some palliatives have been proposed, at different levels of the calculation. Wittig ([4] and presumably in [5] as well), manipulates the “X.,” constants in the Blunck-Westphal formulas [2], thus actually modifying the distributions prior to their use; Patau [6] on the other hand, multiplies the Monte Carlo-obtained path lengths by a suitable corrective factor, thus effecting an “a posteriori” correction. We propose a biased sampling of the unmodified distribution, thereby introducing the correction at the sampling level. If the sampled Q is greater than or equal to the Bethe average ~B’ it is accepted, whereas, if Q
Table 1 Average energy loss (MeV) for 1.4 14 MeV electrons normally incident on Al, Cu and Pb foils. -
s (cm) Al 0.0026 Al 0.0212 Cu 0.00071
-
QB 0.00906 0.07213 0.00872
~
-
Q (Landau)
Q (Bl. and West.)
0.01055 0.08420 0.01022
0.00832 0.07132 0.00802
0.06252
0.0501
~
Pb 0.00394
0.05269
,
sampled Q is accepted regardless. This method merely equalizes the number of hits to the left and right of QB• Table 1 shows a few results of this sampling technique when applied to single distributions. It appears that the values of the sampled Q (Bl. and West.) are in __________________________________ is
-
~ ~
~5LAHOAU
~,
Eo
y=
/QB
J f(Q) dQ / J QB
BLUNCK WESTPHAL
-
I
1(Q) dQ.
0
If ,~~ y, a new Q and a new t~are chosen and the tests repeated a maximum of ten times, after which the
_____
0
__________________________ 402
40’
0 I 54eV)
405
405
0~tO
0,12
Fig. 1. Blunck-Westphal and Landau distributions for 1 MeV electrons incident on 0.0097 cm of aluminium.
393
Volume 51A, number 7
PHYSICS LETI’ERS
Table 2 Comparison of average total path lengths (cm) obtained by the Bethe formula (upper value) and the Monte Carlo method (lower value). E
0 (MeV)
Beryllium
Aluminium
Lead
0.65230 0.65781
0.44857 0.45388
0.14373 0.14701
1.0
0.29113 0.29410
0.20446 0.20768
0.07030 0.07192
0.1
0.00918 0.00869
0.00683 0.00627
0.00275 0.00200
2.0
—
good agreement with QB; those of the sampled Q (Landau) are too high. This is due to the particular shape of the Landau distribution inasmuch as it extends towards high Q values (fig. 1). To further confirm the validity of this type of biased sampling from
394
21 April 1975
the Blunck-Westphal distribution, it was successfully used in Monte Carlo calculations to determine average total path lengths, as can be seen in table 2. The efficiency of the process, i.e. the ratio of successes to trials, in ~ in graphite, 43% in aluminium, 48% in copper, ~
in silver and 60% in lead.
References (1] L.D. Landau, J. Phys. USSR 8(1944)201. [2] [3] [4) [5]
0. Blunck and K. Westphal, Zeit. Phys. 130 (1951) 641. D.W. Vande Putte, report PEL-235 (1974). S. Wittig, report IKF-20 (1968). M. Waldschmidt and S. Wittig, Nucl. Instr. Meths. 64 (1968) 189. [6) J.-P. Patau, thesis, University of Toulouse (1972). [7) M.J. Berger, Methods in computational physics, (Vol. 1), ed. B. Alder (Academic Press, New York, 1963).