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Direct computation of lss parameters for the prediction of fission-fragment ranges and energy-loss in any substance
Nuclear Instruments and Methods in Physics Research B2 (1984) 364-367 North-Holland. Amsterdam
364
DIRECT COMPUTATION OF LSS PARAMETEZRS FOR THE PREDICTION FRAGMENT RANGES AND ENERGY-LOSS IN ANY SUBSTANCE P.A. DICKSTEIN
OF FISSION-
and D. INGMAN
Departmentof N&ear Engineering,Techion - Israel fnsiituteof Technology2 Hoifa 3.2&?0,Israel
Received 9 September 1983 and in revised form 29 February
1984
Analysis of extensive experimental range and energy loss data of 252Cf fission fragments in gases and solids reveals large discrepancies between the ranges calculated according to LSS theory and the experimental data. The same apnlies also to experimental energy loss data. A modified LSS expression which is based on these results enables the prediction of range and energy loss data for 252Cf fragments in any substance. Both the original LSS expressions and the modified expressions used to involve an interpolation procedure and iterative processes. Based on extensive experimental range and energy loss data a direct method for calculating the LSS parameters is introduced, eliminating the need for the interpolation procedure and the iterative process.
1. In~uctiun
2. Fission fragment ranges and energy loss - LSS theory
Range and energy loss predictions of fission fragments in stopping media of any atomic number are of considerable interest for the study of the mechanism of the interaction of energetic heavy ions with matter. Another aspect is the study of the kinetic energy properties of the fragments, which leads to a better understanding of the fission process. In a recent experimental study accurate range data were determined for a large number of individual 252Cf fission-fragments [2,3] as well as accurate energy loss data for mean fission fragments f1,4] in media representing a wide range of atomic numbers throu~out the periodic tabfe. The experimental data revealed large discrepancies between the ranges calculated according to the theory of Lindhard, Sharff and Schiott (LSS) [S] and the experimental data [3,7]. The same applies also to experimental energy loss data [1,4]. Based on these results a modified LSS formulation has been suggested [3,4] which enables one to predict fission fragment ranges and energy loss in any substance with much better accuracy. Both the LSS formulation and the modified formulation involve an interpolation procedure and an iterative process in order to determine the LSS parameters needed for the caiculation of the ranges and energy loss of fission fragments. A new method is presented which enables one to determine directly the values of the LSS parameters. This method eliminates the need for interpolation procedures and iterative processes.
The initial energies associated with fission fragments and their mass range connect them to the medium velocity region of the order of u~Z:/~, where Z, is the nuclear charge of the fragment and uc the Bohr velocity in the hydrogen atom. In the fission fragment energy domain most of the energy will be lost by inelastic ion-electron collisions, resulting in ionization and excitation of the stopping medium, but with decreasing velocity, the contribution of quasi-elastic ion-atom collisions to the stopping process will be more significant. The LSS theory [S] is based on a statistical approach. In this theory, the energy and the integrated path length are transformed into the reduced variables e and p, respectively, where:
0168-583X/84/$03.00 0 Elsevier Science publishers (North-Holland Physics Publishing Division)
B.V.
f=
a4
z,-w(4
p = aa’NyR.
+A*)
E, (2)
mt )
where Rint and E are the integral path length and energy in conventional units, A, and Z, are the mass and atomic number of the stopping medium, N the atomic density, e the electronic charge, and A, and Z, are the mass and atomic number of the projectile-ion. a is the screening parameter given by: a = 0.8853 a,, ( Z:‘3 + Zz/3)-1’2, where a0 = h2/me2 y = 4A,A,/(
is the Bohr radius and y is given by
A, + A2)2.
(41
365
P.A. Dickstein, D. Ingman / LSS parameters for fission -fragment ranges
The reduced
(a+ap),
=
electronic
stopping-power
is
k&*,
(5)
where
k = 5,
0.0793 z:‘22;‘2(
A, + ‘42)3’2 (6)
(zf/3
{ is estimated
+
z;/~)~‘~A;/*&*
’
by LSS to be roughly
{, = zi’6.
(7)
The total reduced stopping-power is the sum of the electronic and nuclear stopping contributions
Fig. 1. LSS (A. k/2) parameter vs LSS k parameter, calculated in the direct method
($),=(E).+(E):
(8)
In the LSS paper eq. (8) was integrated range-energy relation was obtained:
and
,o(r)=;[r1/2-fkA(k,r)],
the
(9)
where A(k, C) is the shortening of the reduced electronic range caused by the nuclear stopping and is given in fig. 5 of the LSS paper. After inserting the numerical constants we obtain: R,,,_~[c’,2_rd(k,r)]
(Zf’3+Zi’3)(A~+A2)2. A,
for c = 250.
.1.66
x 10s
(10) Range measurements usually give the projected range of the ions in the beam direction and not the Ri,, values. The transformation from R, to R int as a function of A,, A,, and c is given in fig. 8 of the LSS paper. Dealing with fission fragment, the most probable charge Z, [6] was taken as the representative value for
The values of ikA( k, c) calculated according to eq. (11) are in very good agreement with those obtained from fig. 5 in the LSS paper. This holds for values of k between 0.1 and 0.4, and values of c between 20 and 1000. These values limit the region of interest when dealing with fission fragments. A series of figures was prepared, plotting ikA( k, c) versus k for several values of t. Such a figure is shown in fig. 1. This series of figures enables one to determine 4 kA (k, C) for every k and every e of interest. Combining eqs. (9) and (1) we get: Rinr=P(~)f(Z1,Z2,‘,,‘2),
R,
Z,.
3. A semi-analytical parameter
derivation
of the LSS
A(k, E)
(12)
where
(zf’3+ f(Z,,
z2,
‘4,,
A2)
=
zy3)(‘4, +A2)2 1.66 x 105A,
.
Based on the semi-analytical expression for ikA(k, c) given in eq. (ll), and the expression for p(r) given in eq. (9) another series of figures was prepared, plotting p(c) versus k for given values of E. A representative figure of this series is given in fig. 2. This series of figures enables one to determine p(c) for every k and every c of interest and then immediately calculate Rint according to eq. (12).
Rq. (10) used to be solved for k by an iterative process because A (k, C) was not known analytically. Furthermore, fig. 5 in the LSS paper gives values of fkA (k, C) for single values of k only, namely k = 0.1, 0.2,0.4 and 1.6. This gives rise to the need for interpolation processes as the fission fragments have a large variety of k values. We developed a semi-analytical expression for tkA( k, c) of the form jkA(k,c)=(ak2+bk+c)
(
l-
+
1
,
(11)
where a = + 15.166, b = - 12.15, c = + 3.813, and n = 0.35 + 0.07 In c.
Fig. 2. LSS dimensionless lated in the direct method
range p vs LSS k parameter, for c = 250.
calcu
P.A. Dicksiein, D. Ingman / LSS parameters for fission -fragmentranges
366 4. Direct
calculation
of LSS
k parameter
2
It is known from the literature and has been shown by accurate range measurements of individual 252Cf fission fragments in a large variety of stopping media that there are large discrepancies between the ranges calculated according to eq. (9) and the experimental data [3,7]. The same applies also to experimental energy loss data [1,4]. In order to overcome this discrepancy Laichter and Shafrir have replaced eq. (7) by [3,4]: 3, = z:.
(13)
e
-._LI~-.II~I-IIL.Ll
1.33
%a0
Fig. 3. The X parameter vs Z,. x [9], solid line - LSS recommended
where
k= + D,k + D, = 0,
M = 0.0793( Z,Z,)“=(A,
be substituted
( 1 ‘-7
D,=
/
b+2f
,\
3
D, =
f was defined
in eq. (12), a, b, c, 1) were defined
in eq.
(II). The positive solution for k as derived from eq. (14) is the LSS k parameter. Se, as given in eq. (13), may now
Table 1 LSS modified X values of the ‘43Ce fission fragment (Z, = 55.90) from the spontaneous fission of 252Cf calculated for several stopping media. X, - values of X calculated in the direct method. X, - values of X calculated with the iterative process. z2
Xl
XD
1 7 10 18 22 28 36 41 54 79
0.508 0.236 0.153 0.239 0.210 0.115 0.208 0.196 0.241 0.116
0.42 0.236 0.156 0.241 0.251 0.160 0.201 0.198 0.239 0.185
q/3
533
hbb
799
This work, 0 - Aras et al. value, X = 0.166
for the X.
eq. (6) yields:
X = ln( k/M)/ln
(
R int
-
-1
in eq. (6) to be solved
Rearranging
where: 1
x10
z2
The X values were determined to satisfy eqs. (8) and (9) respectively, using experimental energy loss and range data. Because of the discrepancies stated above k could not be derived from eq. (6) but by solving eq. (10) in an iterative process. We now substitute fkA (k, c) as given in eq. (11) into eq. (10). Rearranging, we get a second order equation for k of the form: (14)
399
2.66
Z,,
+
Z;/3)3’4Ay2@
(15)
+ /Q3’= .
In table 1 are listed some values of modified LSS X-values calculated in both the direct method and the iterative method. The X values in this table were calculated for Z, = 55.90 which is the most probable charge of the ‘43Ce fission fragment from the spontaneous fission of 252Cf. The values of X are given for several stopping media. The standard deviation of the X-values iteratively calculated, from the X-values directly calculated for Z, = 55.90 is 0.03. In fig. 3 are plotted the experimentally based X values, as calculated in this work, versus Z,, the atomic number of the absorbing medium. Also are plotted X values calculated with the formula introduced by Aras et al. [9], and the recommended LSS value, X= 0.166.
5. Conclusions There are large discrepancies between the X-values calculated on the base of experimental results, and the recommended LSS value, X = 0.166. The mean value for X, calculated in this work for 20 different fission fragments and 11 stopping media, is found to be X= 0.222. The expression for X, introduced by Aras [9], is valid, according to the author, only for 74 < Z, < 92 and it gives X-values which are very close to the LSS X-value. An oscillatory behaviour of the X-value is revealed when plotting experimentally based X-values versus Z,,
PA. Dickstein, D. Wingman/ LSS parameters for Jssion -fragment ranges
in literature as Z,-oscillations [4,8]. Further investigation and analysis of this oscillatory behaviour might enable the development of an improved expression for the fitted LSS X-values, an expression which will take into account the oscillatory behaviour, and will enable the determination of X-values for every combination of fission fragment and absorbing medium. known
References [l] M. Hakim and N.H. Shafrir, Can. J. Phy. 49 (1971) 3024. 121 Y. Laichter and N.H. Shafrir, Nucl. Instr. and Meth. 177 (1980) 459.
367
[3] Y. Laichter and N.H. Shafrir, Nucl. Phys. A371 (1981) 45. [4] Y. Laichter and N.H. Shafrir, Nucl. Phys. A394 (1983) 77. [5] J. Lindhard, M. Scharff and H.E. Schiott, Kgl. Dan. Vid. Seisk Mat. Fys. Medd. 33 no. 14 (1963). [6] D.R. Nethaway, Lawrence Livermore Laboratory, Report UCRL-51640 (1974). [7] M. Pickering and J.M. Alexander, Phys. Rev. C6 (1972) 332. [S] P. Dickstein, Y. Laichter and N.H. Shafrir, Proc. Conf. on Nuclear data for science and technology, Antwerp, Be&urn (1982) p. 57. [9] N.K. Aras, M.P. Menon and GE. Gordon, Nucl. Phys. 69 (1965) 337.
364
DIRECT COMPUTATION OF LSS PARAMETEZRS FOR THE PREDICTION FRAGMENT RANGES AND ENERGY-LOSS IN ANY SUBSTANCE P.A. DICKSTEIN
OF FISSION-
and D. INGMAN
Departmentof N&ear Engineering,Techion - Israel fnsiituteof Technology2 Hoifa 3.2&?0,Israel
Received 9 September 1983 and in revised form 29 February
1984
Analysis of extensive experimental range and energy loss data of 252Cf fission fragments in gases and solids reveals large discrepancies between the ranges calculated according to LSS theory and the experimental data. The same apnlies also to experimental energy loss data. A modified LSS expression which is based on these results enables the prediction of range and energy loss data for 252Cf fragments in any substance. Both the original LSS expressions and the modified expressions used to involve an interpolation procedure and iterative processes. Based on extensive experimental range and energy loss data a direct method for calculating the LSS parameters is introduced, eliminating the need for the interpolation procedure and the iterative process.
1. In~uctiun
2. Fission fragment ranges and energy loss - LSS theory
Range and energy loss predictions of fission fragments in stopping media of any atomic number are of considerable interest for the study of the mechanism of the interaction of energetic heavy ions with matter. Another aspect is the study of the kinetic energy properties of the fragments, which leads to a better understanding of the fission process. In a recent experimental study accurate range data were determined for a large number of individual 252Cf fission-fragments [2,3] as well as accurate energy loss data for mean fission fragments f1,4] in media representing a wide range of atomic numbers throu~out the periodic tabfe. The experimental data revealed large discrepancies between the ranges calculated according to the theory of Lindhard, Sharff and Schiott (LSS) [S] and the experimental data [3,7]. The same applies also to experimental energy loss data [1,4]. Based on these results a modified LSS formulation has been suggested [3,4] which enables one to predict fission fragment ranges and energy loss in any substance with much better accuracy. Both the LSS formulation and the modified formulation involve an interpolation procedure and an iterative process in order to determine the LSS parameters needed for the caiculation of the ranges and energy loss of fission fragments. A new method is presented which enables one to determine directly the values of the LSS parameters. This method eliminates the need for interpolation procedures and iterative processes.
The initial energies associated with fission fragments and their mass range connect them to the medium velocity region of the order of u~Z:/~, where Z, is the nuclear charge of the fragment and uc the Bohr velocity in the hydrogen atom. In the fission fragment energy domain most of the energy will be lost by inelastic ion-electron collisions, resulting in ionization and excitation of the stopping medium, but with decreasing velocity, the contribution of quasi-elastic ion-atom collisions to the stopping process will be more significant. The LSS theory [S] is based on a statistical approach. In this theory, the energy and the integrated path length are transformed into the reduced variables e and p, respectively, where:
0168-583X/84/$03.00 0 Elsevier Science publishers (North-Holland Physics Publishing Division)
B.V.
f=
a4
z,-w(4
p = aa’NyR.
+A*)
E, (2)
mt )
where Rint and E are the integral path length and energy in conventional units, A, and Z, are the mass and atomic number of the stopping medium, N the atomic density, e the electronic charge, and A, and Z, are the mass and atomic number of the projectile-ion. a is the screening parameter given by: a = 0.8853 a,, ( Z:‘3 + Zz/3)-1’2, where a0 = h2/me2 y = 4A,A,/(
is the Bohr radius and y is given by
A, + A2)2.
(41
365
P.A. Dickstein, D. Ingman / LSS parameters for fission -fragment ranges
The reduced
(a+ap),
=
electronic
stopping-power
is
k&*,
(5)
where
k = 5,
0.0793 z:‘22;‘2(
A, + ‘42)3’2 (6)
(zf/3
{ is estimated
+
z;/~)~‘~A;/*&*
’
by LSS to be roughly
{, = zi’6.
(7)
The total reduced stopping-power is the sum of the electronic and nuclear stopping contributions
Fig. 1. LSS (A. k/2) parameter vs LSS k parameter, calculated in the direct method
($),=(E).+(E):
(8)
In the LSS paper eq. (8) was integrated range-energy relation was obtained:
and
,o(r)=;[r1/2-fkA(k,r)],
the
(9)
where A(k, C) is the shortening of the reduced electronic range caused by the nuclear stopping and is given in fig. 5 of the LSS paper. After inserting the numerical constants we obtain: R,,,_~[c’,2_rd(k,r)]
(Zf’3+Zi’3)(A~+A2)2. A,
for c = 250.
.1.66
x 10s
(10) Range measurements usually give the projected range of the ions in the beam direction and not the Ri,, values. The transformation from R, to R int as a function of A,, A,, and c is given in fig. 8 of the LSS paper. Dealing with fission fragment, the most probable charge Z, [6] was taken as the representative value for
The values of ikA( k, c) calculated according to eq. (11) are in very good agreement with those obtained from fig. 5 in the LSS paper. This holds for values of k between 0.1 and 0.4, and values of c between 20 and 1000. These values limit the region of interest when dealing with fission fragments. A series of figures was prepared, plotting ikA( k, c) versus k for several values of t. Such a figure is shown in fig. 1. This series of figures enables one to determine 4 kA (k, C) for every k and every e of interest. Combining eqs. (9) and (1) we get: Rinr=P(~)f(Z1,Z2,‘,,‘2),
R,
Z,.
3. A semi-analytical parameter
derivation
of the LSS
A(k, E)
(12)
where
(zf’3+ f(Z,,
z2,
‘4,,
A2)
=
zy3)(‘4, +A2)2 1.66 x 105A,
.
Based on the semi-analytical expression for ikA(k, c) given in eq. (ll), and the expression for p(r) given in eq. (9) another series of figures was prepared, plotting p(c) versus k for given values of E. A representative figure of this series is given in fig. 2. This series of figures enables one to determine p(c) for every k and every c of interest and then immediately calculate Rint according to eq. (12).
Rq. (10) used to be solved for k by an iterative process because A (k, C) was not known analytically. Furthermore, fig. 5 in the LSS paper gives values of fkA (k, C) for single values of k only, namely k = 0.1, 0.2,0.4 and 1.6. This gives rise to the need for interpolation processes as the fission fragments have a large variety of k values. We developed a semi-analytical expression for tkA( k, c) of the form jkA(k,c)=(ak2+bk+c)
(
l-
+
1
,
(11)
where a = + 15.166, b = - 12.15, c = + 3.813, and n = 0.35 + 0.07 In c.
Fig. 2. LSS dimensionless lated in the direct method
range p vs LSS k parameter, for c = 250.
calcu
P.A. Dicksiein, D. Ingman / LSS parameters for fission -fragmentranges
366 4. Direct
calculation
of LSS
k parameter
2
It is known from the literature and has been shown by accurate range measurements of individual 252Cf fission fragments in a large variety of stopping media that there are large discrepancies between the ranges calculated according to eq. (9) and the experimental data [3,7]. The same applies also to experimental energy loss data [1,4]. In order to overcome this discrepancy Laichter and Shafrir have replaced eq. (7) by [3,4]: 3, = z:.
(13)
e
-._LI~-.II~I-IIL.Ll
1.33
%a0
Fig. 3. The X parameter vs Z,. x [9], solid line - LSS recommended
where
k= + D,k + D, = 0,
M = 0.0793( Z,Z,)“=(A,
be substituted
( 1 ‘-7
D,=
/
b+2f
,\
3
D, =
f was defined
in eq. (12), a, b, c, 1) were defined
in eq.
(II). The positive solution for k as derived from eq. (14) is the LSS k parameter. Se, as given in eq. (13), may now
Table 1 LSS modified X values of the ‘43Ce fission fragment (Z, = 55.90) from the spontaneous fission of 252Cf calculated for several stopping media. X, - values of X calculated in the direct method. X, - values of X calculated with the iterative process. z2
Xl
XD
1 7 10 18 22 28 36 41 54 79
0.508 0.236 0.153 0.239 0.210 0.115 0.208 0.196 0.241 0.116
0.42 0.236 0.156 0.241 0.251 0.160 0.201 0.198 0.239 0.185
q/3
533
hbb
799
This work, 0 - Aras et al. value, X = 0.166
for the X.
eq. (6) yields:
X = ln( k/M)/ln
(
R int
-
-1
in eq. (6) to be solved
Rearranging
where: 1
x10
z2
The X values were determined to satisfy eqs. (8) and (9) respectively, using experimental energy loss and range data. Because of the discrepancies stated above k could not be derived from eq. (6) but by solving eq. (10) in an iterative process. We now substitute fkA (k, c) as given in eq. (11) into eq. (10). Rearranging, we get a second order equation for k of the form: (14)
399
2.66
Z,,
+
Z;/3)3’4Ay2@
(15)
+ /Q3’= .
In table 1 are listed some values of modified LSS X-values calculated in both the direct method and the iterative method. The X values in this table were calculated for Z, = 55.90 which is the most probable charge of the ‘43Ce fission fragment from the spontaneous fission of 252Cf. The values of X are given for several stopping media. The standard deviation of the X-values iteratively calculated, from the X-values directly calculated for Z, = 55.90 is 0.03. In fig. 3 are plotted the experimentally based X values, as calculated in this work, versus Z,, the atomic number of the absorbing medium. Also are plotted X values calculated with the formula introduced by Aras et al. [9], and the recommended LSS value, X= 0.166.
5. Conclusions There are large discrepancies between the X-values calculated on the base of experimental results, and the recommended LSS value, X = 0.166. The mean value for X, calculated in this work for 20 different fission fragments and 11 stopping media, is found to be X= 0.222. The expression for X, introduced by Aras [9], is valid, according to the author, only for 74 < Z, < 92 and it gives X-values which are very close to the LSS X-value. An oscillatory behaviour of the X-value is revealed when plotting experimentally based X-values versus Z,,
PA. Dickstein, D. Wingman/ LSS parameters for Jssion -fragment ranges
in literature as Z,-oscillations [4,8]. Further investigation and analysis of this oscillatory behaviour might enable the development of an improved expression for the fitted LSS X-values, an expression which will take into account the oscillatory behaviour, and will enable the determination of X-values for every combination of fission fragment and absorbing medium. known
References [l] M. Hakim and N.H. Shafrir, Can. J. Phy. 49 (1971) 3024. 121 Y. Laichter and N.H. Shafrir, Nucl. Instr. and Meth. 177 (1980) 459.
367
[3] Y. Laichter and N.H. Shafrir, Nucl. Phys. A371 (1981) 45. [4] Y. Laichter and N.H. Shafrir, Nucl. Phys. A394 (1983) 77. [5] J. Lindhard, M. Scharff and H.E. Schiott, Kgl. Dan. Vid. Seisk Mat. Fys. Medd. 33 no. 14 (1963). [6] D.R. Nethaway, Lawrence Livermore Laboratory, Report UCRL-51640 (1974). [7] M. Pickering and J.M. Alexander, Phys. Rev. C6 (1972) 332. [S] P. Dickstein, Y. Laichter and N.H. Shafrir, Proc. Conf. on Nuclear data for science and technology, Antwerp, Be&urn (1982) p. 57. [9] N.K. Aras, M.P. Menon and GE. Gordon, Nucl. Phys. 69 (1965) 337.