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Theoretical energy straggling of 252Cf fission fragments in various absorbers
Nuclear Instruments and Methods 198 (1982) 583-586 North-Holland Publishing Company
THEORETICAL ABSORBERS
ENERGY STRAGGLING
583
O F 2S2Cf F I S S I O N F R A G M E N T S
IN VARIOUS
*
M.B. A L - B E D R I a n d K . A . A . W . A L - H I N D A W I
**
Ph_vsics Department, College of Science, Universi(v of Baghdad, Baghdad, lraq Received 21 July 1981
A new theoretical formula of energy straggling for heavy ions such as 252Cf fission fragments in various stopping media has been calculated in terms of incident heavy ion energy, differential scattering cross-section, electronic and nuclear stopping cross-section (data taken from Lindhard and co-workers [ 1,2]). However, no theory yet exists which is capable of predicting straggling behaviour of fission fragments. Comparison between our theoretical predictions and the previous work of A1-Bedri [3] and Sykes [4] has been made. Good agreement has been achieved at the beginning of the range of fission fragments but towards the end of the range our theoretical results were greater than the experimental results by about 0-20%. This discrepancy could be caused by the stopping cross-section, because the experimental stopping cross-section is greater than the theoretical data by about 0-30% [5-8}.
1. Introduction
Previous work [3-5, 9-13] was concerned with the measurement of energy straggling of light ions (protons and 241Am alpha particles) and heavy ions (252Cf spontaneous fission fragments) in gases, gas mixtures, tissue equivalent gas (TEG), tissue equivalent solid, and solid films. The phenomena of decreasing energy straggling of 24~Am alpha particles with increasing absorber thickness was studied experimentally, particular attention [9,10,12,13] being paid to phenomena occurring near the end of the range of alpha particles. The periodic dependence of energy straggling on the effective atomic numbers of the absorber was investigated in the previous work [16]. It could be caused by the periodic dependence of the stopping cross-section on the atomic n u m b e r of the stopping media [17,18]. The theoretical stopping cross-section by Lindhard et al. [1,2] and the limits of power (s) for the nuclear stopping cross-section [2] in different energy regions were taken into consideration in the theoretical calculation of energy straggling of 252Cf fission fragments in various absorbers.
2. Theoretical calculation
A R these particles will suffer a n u m b e r of collisions with electrons and atoms of the target. In each collision they will transfer a certain amount of energy to the target electron and atom. The statistical nature of collision processes will cause an energy dispersion of the beam of monoenergetic charged particles. The statistical fluctuation in the energy loss of charged particles is defined as follows: ~-~2 = ( A E _ ~ )
2,
(1)
where fl is the standard deviation of a Gaussian distribution, A E is the energy loss of the charged particles passing through a given thickness of the material ( A R ) and AE is the average value of the energy loss of the same identical particle energy passing through the same material thickness ( A R ). The energy straggling parameter 7/can be defined as the full width at half-maximum (fwhm) of a Gaussian distribution as given by: r/---- 2(2 In 2) 1/2 ~ = 2.354 f~.
(2)
The stopping power -dE/dR of a material for charged particles is defined by I C R U [15] as the quotient of d E by d R , where d E is the average energy lost by a charged particle of specified energy in traversing a path length d R as given below:
If a beam of monoenergetic charged particles with initial energy E passes through an absorber of thickness * This work was submitted to the University of Baghdad for the degree of M.Sc. ** Present address: Physics Department, College of Science, University of Mustansiriyah, Baghdad, Iraq. 0167-5087/82/0000-0000/$02.75 © 1982 North-Holland
where N is the n u m b e r of atoms per unit volume, S is the total stopping cross-section (i.e. S = Se + Sn), Se and Sn are the electronic and nuclear stopping cross-sections respectively, T is the energy transfer to an atom in a
M.B.
584
Al-Bedri,
K.A.A.
W. Al-Hinduwr
single collision, and da is the differential scattering cross section derived from Lindhard et al. [I] which is given by: (4) where C, is a constant which is related to the nuclear stopping cross-section (S,) as written by:
/
Theoretical
Hence
However, /
X
1
2Z,Z,e*
dT,
where MO is the reduced
for TG T,,
/
n m’
oTmT2 da = &+vE,
(12)
T,,, = WE.
The theory of Lindhard et al. [2] for the nuclear ping cross-section S, is written as:
s, = ~(z,z2e2)2~s
’
M,v2
S”
__
_(I-‘/S)ST - (Z-l/S)
s,
The limits of the power (S) for the stopping cross-section of 252Cf fission fragments is greater than 1 and less than 2, (Lindhard [2]), b is the collision diameter and is written as: b=
l/S) Tl-l/s m
m
8S2
(1-i
(I-
oTmT2 do=
cn=;(b2a2~-23S-l)“sXT =
energy srmgghng
mass x
(aul/2)*‘~-m,
i
stop-
‘/S
$5 2
1
El-2/s,
M,M2
M,,=----M,+M*’ M, is the mass of the incident ion with energy E, M, is the mass of the stopping medium, and T, is the maximum energy transfer to the atom: T,, =
4M,M,E
since v = 4M, M2/( M, + M2)2, and a is the distance between the nuclei calculated by the following formula: a = 0.8853 ao( Zf13 + Zi/3)-“2,
0
Our calculation (electronic and integration can heavy ions and 0
do.
0
da,
(
au”*
and thus
(s,),=+.
Now by substituting eq. (14) into eq. (12) one can get the following relationship:
/
(15)
,,7”Tz da = &+2’-2/S(Sn)~vE.
But
x (S,),vEdE, where (dE/dR),= Hence
(16) N(S, + S,)F.
(9)
where R is the mean range of the charged particles of initial energy E moving through a material. This can be calculated from the theoretical stopping power (-d E/d R)E at average energy ,!? by the following relationship:
and so fi2’s-l
__2s - 1 S>l
R(E)=l,“dR=f(g),‘dE.
(14)
(8)
is concerned with the total energy loss nuclear stopping power), so that the be taken over the entire range of the thus eq. (8) becomes:
a2 = NJRd RITmT2
(sJ,=~(Z,Z2e2)2’S ( !pc$ys
(7)
where a, is the radius of the electron orbit (Bohr radius) in the hydrogen atom, Z, and Z, are the atomic number of the incident ion and the absorber, respectively. Bohr [ 191 expressed energy straggling as the standard deviation of a Gaussian distribution as given by: N ARJT”T2
(13)
but S,, at average energy is given by:
(6)
(M,+M2)2=YE’
a2 =AE*=
1,
s>
(10)
v
221s
E2 1 + (SJS”)~
’ (17)
585
M.B. Al-Bedri, K.A.A. W. AI-Hindawi / Theoretical enerD, straggling
When heavy ions interact with matter, they lose energy by ionization, excitation or charge exchange. Theoretical energy straggling parameters ~ (fwhm) for both light and heavy fragments in aluminium have been calculated by using eqs. (17) and (2) as shown in fig. 1, and comparison has been made with the previous experimental results [3]. At the beginning of the range heavy fragments produced an energy straggling greater than light fragments, the two curves were found to intersect at about 0.4 mg c m - 2 mass thickness, while for
an aluminium thickness greater than 0.4 mg c m - 2 light fragments produced an energy straggling greater than heavy fragments. This phenomenon was found experimentally by Sykes and Harris [4] and by A1-Bedri and
0 -
-
THEORETICAL PREDICTION EXPERIMENTAL WORK(AL_BEDR! ?975)
|~(FWHM)252CF LIGHT FRAGMENTS IN AI. 252 THEORETICAL PREDICT)ON ~1 TI(FWHM) CF HEAVY EXPERIMENTAL WORK(AL_BEDRI 7 9 7 5 ) ~ HFRAGMENTS IN AI.
0 -
-
-
&
-
&
f
2C
_
%..~
o
~a
i
1
I
Au
3. Results and discussion
o
a
i
i
i
i
2 aluminum thickness ~X {rng. cm~2)
,3
Fig. 1.'Comparison between theoretical energy straggling and experimental work for 252Cf light and heavy fission fragments as a function of aluminum thickness.
THEORETICAL ENERGY STRAGGLING HEAVY FISSION F R A G M E N T S IN
2C
?5
OF25~F
.........
L AI
......
0
i
i 2
\
FRAGMENTS
IN
I
A r - -
-.17
\
i i 4 absorber thickness frng. cm -2)
i 6
i
Fig. 3. Theoretical energy straggling ~H (fwhm) for 252Cf heavy fission fragments in AI, Cu, Ag and Au as a function of
absorber thickness.
Harris [3], and is in a good agreement with the behaviour of the effective charge of 252Cf fission fragments which is rather high at the beginning of the range (22e for heavy fragments and 20e for light fragments) and then decreasing along the path of the range. The effective charge for heavy fragments decreses more rapidly than for light fragments [3] as due to capture and loss of electrons. The theoretical straggling of heavy fragments ~H (fwhm) is plotted as a function of absorber thickness (AR) in figs. 2 and 3 for gases (N, Ne, Ar and Kr and the solids A1, Cu, Ag and Cu respectively, while in figs. 4 and 5 it is plotted for light fragments ~L in gases and solids respectively. The enersy straggling of 2s2 Cf light and heavy fragments is found to be decreasing with increasing absorber thickness. Good agreement was found between our theoretical data and the previous experimental results [3] at the beginning of the range. Towards the end of the range the experimental results
r Xr
THEORETICAL ENERGY STRAGGLING OF 252CF J
Ag----
"~
20
m
THEORETICAL ENERGY STRAGGLING OF
L
252 252CF
LIGHT FISSION F R A G M E N T S
1
Kr Ar " N.........i___
,..~ ~
~ •
~
......... ~ '.,,. ~
".,.
"~ "-~
absorbfr thickne$$ AX(mg.cm -2)
0
4
8bsorbcr thickness~x(mg.cn~ 21
6
Fig. 2. Theoretical energy straggling t/H (fwhm) for 252Cf heavy fission fragments in N, Ne, Ar and Kr as a function of
Fig. 4. Theoretical energy straggling 'qL (fwhm) for 252Cf light fission fragments in N, Ne, Ar and Kr as a function of
absorber thickness.
absorber thickness.
M.B. Al-Bedri, K.A.A.W. Al-Hindawi / Theoretical energy straggling
586 2C
I
Au
THEORETICAL ENERGY STRAGGLING OF
CF LIGHT FISSION
Ag
..............
Cu----
lS
At . . . . . . .
g
"-.....
8
• ..... .... c
"'... "--... i
O0
~
2 absorber
,~ t h i c k n e s s (rng. c m - 2 )
i 6
/
i
o
10
Fig. 5. Theoretical energy straggling 7/L (fwhm) for 252Cf light fission fragments in AI, Cu, Ag and Au as a function of absorber thickness.
were f o u n d to be smaller t h a n the theoretical d a t a by a b o u t 0-20%. This discrepancy between experimental a n d theoretical d a t a is due to the stopping cross-section which is greater t h a n the theoretical one b y a b o u t 0 - 3 0 % [6-8,16]. O u r results are expected to be greater t h a n the experimental results.
4. Conclusion The expected decrease in energy straggling of fission fragments with increasing a b s o r b e r thickness has been observed in all cases. The decrease in stopping power a n d consequently the decrease in energy straggling with decreasing energy is closely related to the effective charge of the fission fragments over the energy range considered. In general, energy straggling of 252 Cf fission fragments has b e e n f o u n d to increase with increasing atomic n u m b e r of the absorber.
References [I] J. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128. [2] J. Lindhard, M. Scharff and H.E. Schie~tt, Kgl. Dan. Vid. Selsk. Mat.-Fys. Medd. 33 (1963) no. 14. [3] M.B. Al-Bedri and S.J. Harris, Nucl. Instr. and Meth. 124 (1975) 125.
[4] D.A. Sykes and S.J. Harris, Nucl. Instr. and Meth. 97 (1971) 203. [5] M.B. A1-Bedri, in press. [6] R. Muller and F. Gonnenwein, Nucl. Instr. and Meth. 91 (1971) 357. [7] C.D. Moak and M.D. Brown, Phys. Rev. 149 (1966) 244. [8] R. Bimbot, S. Della Negra, D. Gardes, H. Gauvin, F. Fleury and F. Hubert, Nucl. Instr. and Meth. 153 (1978) 161. [9] M.B. A1-Bedri and S.J. Harris, Health Phys. 28 (1975) 816. [10] M.B. A1-Bedri, S.J. Harris and D.A. Sykes, Nucl. Instr. and Meth. 106 (1973) 241. [I I] M.B. AI-Bedri, S.J. Harris and H.G.F.S. Parish,Rad. Eff. 27 (1976) 183. [12] D.A. Sykes and S.J. Harris, Nucl. Instr. and Meth 94 (1971) 39. [13] D.A. Sykes and S.J. Harris, Nucl. Instr. and Meth. 101 (1972) 423. [14] N. Bohr, Vgl. Dan. Vid. Selsk Mat.-Fys. Medd. 18 (1948) no. 8. [15] ICRU, Recommendation of the ICRU, Natl. Bur. Handbk 85 (1964). [16] M.B. Al-Bedri, Ph.D. Thesis, University of Surrey (1974). [17] C.C. Rousseau, W.K. Chu and D. Powers, Phys. Rev. A4 (1971) t060. [18] W.K. Chu and D. Powers, Phys. Lett. A38 (1972) 267. [19] N. Bohr, Phil. Mag. 30 (1915) 581.
THEORETICAL ABSORBERS
ENERGY STRAGGLING
583
O F 2S2Cf F I S S I O N F R A G M E N T S
IN VARIOUS
*
M.B. A L - B E D R I a n d K . A . A . W . A L - H I N D A W I
**
Ph_vsics Department, College of Science, Universi(v of Baghdad, Baghdad, lraq Received 21 July 1981
A new theoretical formula of energy straggling for heavy ions such as 252Cf fission fragments in various stopping media has been calculated in terms of incident heavy ion energy, differential scattering cross-section, electronic and nuclear stopping cross-section (data taken from Lindhard and co-workers [ 1,2]). However, no theory yet exists which is capable of predicting straggling behaviour of fission fragments. Comparison between our theoretical predictions and the previous work of A1-Bedri [3] and Sykes [4] has been made. Good agreement has been achieved at the beginning of the range of fission fragments but towards the end of the range our theoretical results were greater than the experimental results by about 0-20%. This discrepancy could be caused by the stopping cross-section, because the experimental stopping cross-section is greater than the theoretical data by about 0-30% [5-8}.
1. Introduction
Previous work [3-5, 9-13] was concerned with the measurement of energy straggling of light ions (protons and 241Am alpha particles) and heavy ions (252Cf spontaneous fission fragments) in gases, gas mixtures, tissue equivalent gas (TEG), tissue equivalent solid, and solid films. The phenomena of decreasing energy straggling of 24~Am alpha particles with increasing absorber thickness was studied experimentally, particular attention [9,10,12,13] being paid to phenomena occurring near the end of the range of alpha particles. The periodic dependence of energy straggling on the effective atomic numbers of the absorber was investigated in the previous work [16]. It could be caused by the periodic dependence of the stopping cross-section on the atomic n u m b e r of the stopping media [17,18]. The theoretical stopping cross-section by Lindhard et al. [1,2] and the limits of power (s) for the nuclear stopping cross-section [2] in different energy regions were taken into consideration in the theoretical calculation of energy straggling of 252Cf fission fragments in various absorbers.
2. Theoretical calculation
A R these particles will suffer a n u m b e r of collisions with electrons and atoms of the target. In each collision they will transfer a certain amount of energy to the target electron and atom. The statistical nature of collision processes will cause an energy dispersion of the beam of monoenergetic charged particles. The statistical fluctuation in the energy loss of charged particles is defined as follows: ~-~2 = ( A E _ ~ )
2,
(1)
where fl is the standard deviation of a Gaussian distribution, A E is the energy loss of the charged particles passing through a given thickness of the material ( A R ) and AE is the average value of the energy loss of the same identical particle energy passing through the same material thickness ( A R ). The energy straggling parameter 7/can be defined as the full width at half-maximum (fwhm) of a Gaussian distribution as given by: r/---- 2(2 In 2) 1/2 ~ = 2.354 f~.
(2)
The stopping power -dE/dR of a material for charged particles is defined by I C R U [15] as the quotient of d E by d R , where d E is the average energy lost by a charged particle of specified energy in traversing a path length d R as given below:
If a beam of monoenergetic charged particles with initial energy E passes through an absorber of thickness * This work was submitted to the University of Baghdad for the degree of M.Sc. ** Present address: Physics Department, College of Science, University of Mustansiriyah, Baghdad, Iraq. 0167-5087/82/0000-0000/$02.75 © 1982 North-Holland
where N is the n u m b e r of atoms per unit volume, S is the total stopping cross-section (i.e. S = Se + Sn), Se and Sn are the electronic and nuclear stopping cross-sections respectively, T is the energy transfer to an atom in a
M.B.
584
Al-Bedri,
K.A.A.
W. Al-Hinduwr
single collision, and da is the differential scattering cross section derived from Lindhard et al. [I] which is given by: (4) where C, is a constant which is related to the nuclear stopping cross-section (S,) as written by:
/
Theoretical
Hence
However, /
X
1
2Z,Z,e*
dT,
where MO is the reduced
for TG T,,
/
n m’
oTmT2 da = &+vE,
(12)
T,,, = WE.
The theory of Lindhard et al. [2] for the nuclear ping cross-section S, is written as:
s, = ~(z,z2e2)2~s
’
M,v2
S”
__
_(I-‘/S)ST - (Z-l/S)
s,
The limits of the power (S) for the stopping cross-section of 252Cf fission fragments is greater than 1 and less than 2, (Lindhard [2]), b is the collision diameter and is written as: b=
l/S) Tl-l/s m
m
8S2
(1-i
(I-
oTmT2 do=
cn=;(b2a2~-23S-l)“sXT =
energy srmgghng
mass x
(aul/2)*‘~-m,
i
stop-
‘/S
$5 2
1
El-2/s,
M,M2
M,,=----M,+M*’ M, is the mass of the incident ion with energy E, M, is the mass of the stopping medium, and T, is the maximum energy transfer to the atom: T,, =
4M,M,E
since v = 4M, M2/( M, + M2)2, and a is the distance between the nuclei calculated by the following formula: a = 0.8853 ao( Zf13 + Zi/3)-“2,
0
Our calculation (electronic and integration can heavy ions and 0
do.
0
da,
(
au”*
and thus
(s,),=+.
Now by substituting eq. (14) into eq. (12) one can get the following relationship:
/
(15)
,,7”Tz da = &+2’-2/S(Sn)~vE.
But
x (S,),vEdE, where (dE/dR),= Hence
(16) N(S, + S,)F.
(9)
where R is the mean range of the charged particles of initial energy E moving through a material. This can be calculated from the theoretical stopping power (-d E/d R)E at average energy ,!? by the following relationship:
and so fi2’s-l
__2s - 1 S>l
R(E)=l,“dR=f(g),‘dE.
(14)
(8)
is concerned with the total energy loss nuclear stopping power), so that the be taken over the entire range of the thus eq. (8) becomes:
a2 = NJRd RITmT2
(sJ,=~(Z,Z2e2)2’S ( !pc$ys
(7)
where a, is the radius of the electron orbit (Bohr radius) in the hydrogen atom, Z, and Z, are the atomic number of the incident ion and the absorber, respectively. Bohr [ 191 expressed energy straggling as the standard deviation of a Gaussian distribution as given by: N ARJT”T2
(13)
but S,, at average energy is given by:
(6)
(M,+M2)2=YE’
a2 =AE*=
1,
s>
(10)
v
221s
E2 1 + (SJS”)~
’ (17)
585
M.B. Al-Bedri, K.A.A. W. AI-Hindawi / Theoretical enerD, straggling
When heavy ions interact with matter, they lose energy by ionization, excitation or charge exchange. Theoretical energy straggling parameters ~ (fwhm) for both light and heavy fragments in aluminium have been calculated by using eqs. (17) and (2) as shown in fig. 1, and comparison has been made with the previous experimental results [3]. At the beginning of the range heavy fragments produced an energy straggling greater than light fragments, the two curves were found to intersect at about 0.4 mg c m - 2 mass thickness, while for
an aluminium thickness greater than 0.4 mg c m - 2 light fragments produced an energy straggling greater than heavy fragments. This phenomenon was found experimentally by Sykes and Harris [4] and by A1-Bedri and
0 -
-
THEORETICAL PREDICTION EXPERIMENTAL WORK(AL_BEDR! ?975)
|~(FWHM)252CF LIGHT FRAGMENTS IN AI. 252 THEORETICAL PREDICT)ON ~1 TI(FWHM) CF HEAVY EXPERIMENTAL WORK(AL_BEDRI 7 9 7 5 ) ~ HFRAGMENTS IN AI.
0 -
-
-
&
-
&
f
2C
_
%..~
o
~a
i
1
I
Au
3. Results and discussion
o
a
i
i
i
i
2 aluminum thickness ~X {rng. cm~2)
,3
Fig. 1.'Comparison between theoretical energy straggling and experimental work for 252Cf light and heavy fission fragments as a function of aluminum thickness.
THEORETICAL ENERGY STRAGGLING HEAVY FISSION F R A G M E N T S IN
2C
?5
OF25~F
.........
L AI
......
0
i
i 2
\
FRAGMENTS
IN
I
A r - -
-.17
\
i i 4 absorber thickness frng. cm -2)
i 6
i
Fig. 3. Theoretical energy straggling ~H (fwhm) for 252Cf heavy fission fragments in AI, Cu, Ag and Au as a function of
absorber thickness.
Harris [3], and is in a good agreement with the behaviour of the effective charge of 252Cf fission fragments which is rather high at the beginning of the range (22e for heavy fragments and 20e for light fragments) and then decreasing along the path of the range. The effective charge for heavy fragments decreses more rapidly than for light fragments [3] as due to capture and loss of electrons. The theoretical straggling of heavy fragments ~H (fwhm) is plotted as a function of absorber thickness (AR) in figs. 2 and 3 for gases (N, Ne, Ar and Kr and the solids A1, Cu, Ag and Cu respectively, while in figs. 4 and 5 it is plotted for light fragments ~L in gases and solids respectively. The enersy straggling of 2s2 Cf light and heavy fragments is found to be decreasing with increasing absorber thickness. Good agreement was found between our theoretical data and the previous experimental results [3] at the beginning of the range. Towards the end of the range the experimental results
r Xr
THEORETICAL ENERGY STRAGGLING OF 252CF J
Ag----
"~
20
m
THEORETICAL ENERGY STRAGGLING OF
L
252 252CF
LIGHT FISSION F R A G M E N T S
1
Kr Ar " N.........i___
,..~ ~
~ •
~
......... ~ '.,,. ~
".,.
"~ "-~
absorbfr thickne$$ AX(mg.cm -2)
0
4
8bsorbcr thickness~x(mg.cn~ 21
6
Fig. 2. Theoretical energy straggling t/H (fwhm) for 252Cf heavy fission fragments in N, Ne, Ar and Kr as a function of
Fig. 4. Theoretical energy straggling 'qL (fwhm) for 252Cf light fission fragments in N, Ne, Ar and Kr as a function of
absorber thickness.
absorber thickness.
M.B. Al-Bedri, K.A.A.W. Al-Hindawi / Theoretical energy straggling
586 2C
I
Au
THEORETICAL ENERGY STRAGGLING OF
CF LIGHT FISSION
Ag
..............
Cu----
lS
At . . . . . . .
g
"-.....
8
• ..... .... c
"'... "--... i
O0
~
2 absorber
,~ t h i c k n e s s (rng. c m - 2 )
i 6
/
i
o
10
Fig. 5. Theoretical energy straggling 7/L (fwhm) for 252Cf light fission fragments in AI, Cu, Ag and Au as a function of absorber thickness.
were f o u n d to be smaller t h a n the theoretical d a t a by a b o u t 0-20%. This discrepancy between experimental a n d theoretical d a t a is due to the stopping cross-section which is greater t h a n the theoretical one b y a b o u t 0 - 3 0 % [6-8,16]. O u r results are expected to be greater t h a n the experimental results.
4. Conclusion The expected decrease in energy straggling of fission fragments with increasing a b s o r b e r thickness has been observed in all cases. The decrease in stopping power a n d consequently the decrease in energy straggling with decreasing energy is closely related to the effective charge of the fission fragments over the energy range considered. In general, energy straggling of 252 Cf fission fragments has b e e n f o u n d to increase with increasing atomic n u m b e r of the absorber.
References [I] J. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128. [2] J. Lindhard, M. Scharff and H.E. Schie~tt, Kgl. Dan. Vid. Selsk. Mat.-Fys. Medd. 33 (1963) no. 14. [3] M.B. Al-Bedri and S.J. Harris, Nucl. Instr. and Meth. 124 (1975) 125.
[4] D.A. Sykes and S.J. Harris, Nucl. Instr. and Meth. 97 (1971) 203. [5] M.B. A1-Bedri, in press. [6] R. Muller and F. Gonnenwein, Nucl. Instr. and Meth. 91 (1971) 357. [7] C.D. Moak and M.D. Brown, Phys. Rev. 149 (1966) 244. [8] R. Bimbot, S. Della Negra, D. Gardes, H. Gauvin, F. Fleury and F. Hubert, Nucl. Instr. and Meth. 153 (1978) 161. [9] M.B. A1-Bedri and S.J. Harris, Health Phys. 28 (1975) 816. [10] M.B. A1-Bedri, S.J. Harris and D.A. Sykes, Nucl. Instr. and Meth. 106 (1973) 241. [I I] M.B. AI-Bedri, S.J. Harris and H.G.F.S. Parish,Rad. Eff. 27 (1976) 183. [12] D.A. Sykes and S.J. Harris, Nucl. Instr. and Meth 94 (1971) 39. [13] D.A. Sykes and S.J. Harris, Nucl. Instr. and Meth. 101 (1972) 423. [14] N. Bohr, Vgl. Dan. Vid. Selsk Mat.-Fys. Medd. 18 (1948) no. 8. [15] ICRU, Recommendation of the ICRU, Natl. Bur. Handbk 85 (1964). [16] M.B. Al-Bedri, Ph.D. Thesis, University of Surrey (1974). [17] C.C. Rousseau, W.K. Chu and D. Powers, Phys. Rev. A4 (1971) t060. [18] W.K. Chu and D. Powers, Phys. Lett. A38 (1972) 267. [19] N. Bohr, Phil. Mag. 30 (1915) 581.