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A semiempirical method based on the LSS formalism for estimating electronic energy loss
Nuclear
Instruments
North-Holland,
and Methods
527
in Physics Research B14 (1986) 527-529
Amsterdam
A SEMIEMPIRICAL METHOD BASED ON THE LSS FORMALISM ELECTRONIC ENERGY LOSS P. BANERJEE, &ha
Institute
Received
B. SETH1
of Nuclerrr
30 September
and J.M.
Physrcs, Calcuttu
FOR ESTIMATING
CHATTERJEE
700 064. India
and in revised form 19 November
1985
A modified semiempirical expression for the LSS parameter 5, valid for ions with nuclear applicability for predicting electronic stopping powers is discussed
In the region of low and medium velocities where the LSS formula [1,2] for the electronic energy loss predicts velocity proportional stopping of moving ions in matter, the experimental range and energy loss data differ appreciably from the predicted values. Many authors have attributed this discrepancy to the Z-dependence of the factor .$, appearing in the LSS formula for energy loss, Z, being the charge of the moving ion. In the present work, the Z,-dependence of 5, has been investigated and an empirical expression is obtained which gives closer agreement with the experimental results. The specific energy loss of ions with velocities u less than u~Z~/~ where va is the Bohr velocity in the hydrogen atom, according to LSS, is given by 8mNe2a0Z1Z2 2:‘s
+ 2;‘s)
u 3/2
G
where .$, is defined to be of the order of Zl’“. the subscripts 1 and 2 refer to the moving ion and the stopping medium, respectively, N is the number of atoms per unit volume of the medium, e is the electronic charge and a, is the Bohr radius. However, as pointed out previously the formula is only approximately valid since the factor 5, shows deviations from the value Z,“6. Measurements of range [3,4] and energy loss [5,6] of fission fragments in different media have established large discrepancies between the LSS values and the experimental data. Attempts have been made by many authors to overcome this discrepancy by defining 5, = az; and determining empirical values for a and x that would fit the theory to the experimental data. It is observed however that a and x are not constant for all ion-atom combinations. Aras et al. [7] obtained x = 0168-583X/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
charge
$ 92 is presented
and its
0.211 and a = 1 from the range measurements of 235U fission fragments in aluminium and found the value to fit particularly well for 99Mo and laBa. Birglil et al. [8] found a = 0.655 and x = 0.346 for fourteen 252Cf fission fragments in aluminium. Hontzeas and Blok [9] expressed x as an exponential function of the atomic numbers of the ion and the medium, with a = 1, applicable in the range 74 < Z, < 92. Izmen et al. [lo] observed the parameters a and x to be varying inversely as the mass number (A2) of the medium and obtained expression valid for A, > 22, from the range measurements of two 252Cf fission fragments in four stopping media. Laichter et al. [3] evaluated x for a = 1, from range data of 252Cf fission fragments in a few solid and gaseous media and found an expression for x, linear in A,, which predicts the fission fragment ranges with an accuracy of 20-30% in the limited mass range 90
epzl,
P. Banerjee et d. / Semtempirical
528
(a) CARBON N=
0.725
X = 0.297 p = 3.911.10
tb) NICKEL Q=O.853 X = O.ZL3 p =-3.508r1i4
3-
IC)
i_
SILVER o( I 1.627 x = 0.103
______________--------
p * 5.7a01164
,/' 11
IdI GOLD
fornwla
for electronic
the figure show the LSS value of 5, given by Zi/“. As already established, the LSS values deviate appreciably from the observed .$, values. The present expression for 5, describes its Z,dependence over a wider range of Z,-values (Z, < 92) in contrast to the previously reported expressions, valid in limited ranges. The predicted values of 5, for fission fragments in the mass range 90 < A, < 150, using the present expression agree with the values according to Izmen et al. [lo] to within 10% for the media nickel and silver and within 10-l% for gold. In the same mass region, the predictions based on the empirical expression of Laichter et al. [3] differ by less than 10% with the present method, except at the ends of the mass region, where the disagreement is - lo-20%. The 5, values given by the expression of Hontzeas and Blok 191, applicable in the Z,-range 74 i Z, < 92, however show an overall agreement of < 10% throughout the mass region A, = 90-150, with the present method, for gold ( z, = 79). The parametrisation of 5, in this work in terms of the three parameters a, j3 and x accounts for the variation of .$, with Z, for 2, < 92 in carbon, nickel, silver and gold. It is expected that the present method can be extended to other media as well and used for predicting specific energy loss values in the linear regions of (dE/dx),, vs. velocity curves. The effect of Z,-oscillations, dominant at ion velocities c’/uO 7 2, is a major source of uncertainty in the tabulations of Northcliffe and Schilling [19]. Although the Z,-oscillations were introduced in the Ziegler [20] tables, they still underestimate the stopping power values at the maxima of the Z,-oscillations [16]. In the present method the Z,-oscillations do not cause any uncertainty in the predicted stopping powers since the parameters in the expression for 5, are characteristic of the medium. The effect of Z,-oscillations is however not accounted for; the present method furnishes a mean value of .$,, in the regions of Z,-oscillations which are significant at rather low energies (4 0.05 MeV/nucleon) and decrease rapidly with increasing velocity. Computed electronic energy loss values using the modified 6, can nevertheless be useful in the analysis of Doppler shift attenuation data, wherein the uncertainty in the derivation of short lifetimes of excited nuclear states arises mainly from the insufficient knowledge of the electronic stopping powers of the recoiling ions in the stopping medium. The experimental stopping powers are, however, not strictly velocity proportional in the region of velocities < u0 Z,?/‘. The velocity range in which the deviations from linearity are negligible or comparable to the experimental errors varies from about 2u, to about 50, as Z, varies from 2 to 92. It is interesting to note that, using the mean velocities in the above mentioned velocity range, even with the limited set of experimental data used for determining the parameters in 5,. the
2/fy-_---il:::--i:
I
a=
1.224
x = 0.201
p = -3.791rl6'
I
1'
01 *
’
20
I
’
LO
r
’
60
5
’
80
’
100
2,
Fig. 1. Variation of 5, with 2, in the media (a) carbon, (b) nickel, (c) silver and (d) gold. Continuous curves are the least squares fits of the experimental data to the semiempirical expression for g, proposed in this work. The broken curves show the LSS values given by Z:/‘; A ref. [ill, 0 ref. 1121, 0 ref. 1131, x ref. 114). D ref. [15], l ref. 1161, v ref. 1171 and v ref. 1181.
where the parameters Q, p and x are characteristic of the medium. The least squares fits of this function to the &,-values based on the available experimental data are shown by continuous curves in figs. l(a), (b), (c) and. (d) for the media carbon, nickel, silver and gold. respectively. The corresponding values of the parameters (Y, ,8 and x are also given in the figure. The broken lines in
energv losses
P. Baner~ee et al. / Semiempirrcal formula for electronrc energy losses
(dE/dx),, values are usually reproduced by the present method to within 5% and in a few cases within lo-20% of the experimental results in the specified velocity regions. At higher ion velocities approaching u0 Z2/3 the experimental (dE/dx),, - velocity curves deviite, often appreciably, from linearity contrary to the predictions of the LSS theory, and hence the predicted (dE/dx),, method
values also
tend
using to
the modified deviate
from
5,
in the present
the
experimental
values.
The possibility of extrapolation of the present expression for 5, to other media where experimental data are scarce, depends on the exact form of the Z,-dependence of the parameters (Y, x and j3, which are characteristic of the medium. The variation of these parameters with Z, is not expected to be smooth due to the effects of Z, oscillations. With the availability of more data on stopping power measurements it may be possible to interpolate between close lying Z, values away from the regions of maxima or minima in the Z, oscillations.
References [l] J. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128. [2] J. Lindhard, M. Scharff and H.E. Schiott, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 33 (1963) no. 14. [3] Y. Laichter and N.H. Shafrir, Nucl. Phys. A371 (1981) 45. [4] M. Pickering and J.M. Alexander, Phys. Rev. C6 (1972) 332.
529
[51 M. Hakim and N.H. Shafrir, Can. J. Phys. 49 (1971) 3024. [61 Y. Laichter and N.H. Shafrir, Nucl. Phys. A394 (1983) 77. [71 N.K. Aras, M.P. Menon and G.E. Gordon, Nucl. Phys. 69 (1965) 337. Acta 18 PI 0. Birgiil, I. Glmez and N.K. Aras, Radio&mica (1972) 198. 191 S. Hontzeas and H. Blok, Phys. Scripta 4 (1971) 229. DOI A. Izmen, 0. Birgiil and N.K. Aras, J. Inorg. Nucl. Chem. 36 (1974) 25. Proc. Phys. Sot. 77 [ill D.I. Porat and K. Ramavataram, (1961) 97. Proc. Phys. Sot. 78 WI D.I. Porat and K. Ramavataram, (1961) 1135. G.C. Ball, G.J. v31 J.S. Forster, D. Ward, H.R. Andrews, Costa, W.G. Davies and I.V. Mitchell, Nucl. Instr. and Meth. 136 (1976) 349. [I41 H. Pape, H.G. Clerc and K.H. Schmidt, Z. Phys. A286 (1978) 159. P51 P.A. Dickstein and D. Ingman, Nucl. Instr. and Meth. B2 (1984) 364. Y. Laichter, W.F.W. Schneider and P. [I61 H. Geissel, Armbruster, Nucl. Instr. and Meth. 194 (1982) 21. Nucl. Instr. and Meth. 91 [I71 R. Miller and F. Gonnenwein, (1971) 357. U81 S. Kahn and V. Forgue, Phys. Rev. 163 (1967) 290. 1191 L.C. Northcliffe and R.F. Schilling, Nucl. Data A7 (1970) 233. of Stopping Cross-sections for PO1 J.F. Ziegler, Handbook Energetic Ions in all Elements (Pergmon, New York, 1980).
Instruments
North-Holland,
and Methods
527
in Physics Research B14 (1986) 527-529
Amsterdam
A SEMIEMPIRICAL METHOD BASED ON THE LSS FORMALISM ELECTRONIC ENERGY LOSS P. BANERJEE, &ha
Institute
Received
B. SETH1
of Nuclerrr
30 September
and J.M.
Physrcs, Calcuttu
FOR ESTIMATING
CHATTERJEE
700 064. India
and in revised form 19 November
1985
A modified semiempirical expression for the LSS parameter 5, valid for ions with nuclear applicability for predicting electronic stopping powers is discussed
In the region of low and medium velocities where the LSS formula [1,2] for the electronic energy loss predicts velocity proportional stopping of moving ions in matter, the experimental range and energy loss data differ appreciably from the predicted values. Many authors have attributed this discrepancy to the Z-dependence of the factor .$, appearing in the LSS formula for energy loss, Z, being the charge of the moving ion. In the present work, the Z,-dependence of 5, has been investigated and an empirical expression is obtained which gives closer agreement with the experimental results. The specific energy loss of ions with velocities u less than u~Z~/~ where va is the Bohr velocity in the hydrogen atom, according to LSS, is given by 8mNe2a0Z1Z2 2:‘s
+ 2;‘s)
u 3/2
G
where .$, is defined to be of the order of Zl’“. the subscripts 1 and 2 refer to the moving ion and the stopping medium, respectively, N is the number of atoms per unit volume of the medium, e is the electronic charge and a, is the Bohr radius. However, as pointed out previously the formula is only approximately valid since the factor 5, shows deviations from the value Z,“6. Measurements of range [3,4] and energy loss [5,6] of fission fragments in different media have established large discrepancies between the LSS values and the experimental data. Attempts have been made by many authors to overcome this discrepancy by defining 5, = az; and determining empirical values for a and x that would fit the theory to the experimental data. It is observed however that a and x are not constant for all ion-atom combinations. Aras et al. [7] obtained x = 0168-583X/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
charge
$ 92 is presented
and its
0.211 and a = 1 from the range measurements of 235U fission fragments in aluminium and found the value to fit particularly well for 99Mo and laBa. Birglil et al. [8] found a = 0.655 and x = 0.346 for fourteen 252Cf fission fragments in aluminium. Hontzeas and Blok [9] expressed x as an exponential function of the atomic numbers of the ion and the medium, with a = 1, applicable in the range 74 < Z, < 92. Izmen et al. [lo] observed the parameters a and x to be varying inversely as the mass number (A2) of the medium and obtained expression valid for A, > 22, from the range measurements of two 252Cf fission fragments in four stopping media. Laichter et al. [3] evaluated x for a = 1, from range data of 252Cf fission fragments in a few solid and gaseous media and found an expression for x, linear in A,, which predicts the fission fragment ranges with an accuracy of 20-30% in the limited mass range 90
epzl,
P. Banerjee et d. / Semtempirical
528
(a) CARBON N=
0.725
X = 0.297 p = 3.911.10
tb) NICKEL Q=O.853 X = O.ZL3 p =-3.508r1i4
3-
IC)
i_
SILVER o( I 1.627 x = 0.103
______________--------
p * 5.7a01164
,/' 11
IdI GOLD
fornwla
for electronic
the figure show the LSS value of 5, given by Zi/“. As already established, the LSS values deviate appreciably from the observed .$, values. The present expression for 5, describes its Z,dependence over a wider range of Z,-values (Z, < 92) in contrast to the previously reported expressions, valid in limited ranges. The predicted values of 5, for fission fragments in the mass range 90 < A, < 150, using the present expression agree with the values according to Izmen et al. [lo] to within 10% for the media nickel and silver and within 10-l% for gold. In the same mass region, the predictions based on the empirical expression of Laichter et al. [3] differ by less than 10% with the present method, except at the ends of the mass region, where the disagreement is - lo-20%. The 5, values given by the expression of Hontzeas and Blok 191, applicable in the Z,-range 74 i Z, < 92, however show an overall agreement of < 10% throughout the mass region A, = 90-150, with the present method, for gold ( z, = 79). The parametrisation of 5, in this work in terms of the three parameters a, j3 and x accounts for the variation of .$, with Z, for 2, < 92 in carbon, nickel, silver and gold. It is expected that the present method can be extended to other media as well and used for predicting specific energy loss values in the linear regions of (dE/dx),, vs. velocity curves. The effect of Z,-oscillations, dominant at ion velocities c’/uO 7 2, is a major source of uncertainty in the tabulations of Northcliffe and Schilling [19]. Although the Z,-oscillations were introduced in the Ziegler [20] tables, they still underestimate the stopping power values at the maxima of the Z,-oscillations [16]. In the present method the Z,-oscillations do not cause any uncertainty in the predicted stopping powers since the parameters in the expression for 5, are characteristic of the medium. The effect of Z,-oscillations is however not accounted for; the present method furnishes a mean value of .$,, in the regions of Z,-oscillations which are significant at rather low energies (4 0.05 MeV/nucleon) and decrease rapidly with increasing velocity. Computed electronic energy loss values using the modified 6, can nevertheless be useful in the analysis of Doppler shift attenuation data, wherein the uncertainty in the derivation of short lifetimes of excited nuclear states arises mainly from the insufficient knowledge of the electronic stopping powers of the recoiling ions in the stopping medium. The experimental stopping powers are, however, not strictly velocity proportional in the region of velocities < u0 Z,?/‘. The velocity range in which the deviations from linearity are negligible or comparable to the experimental errors varies from about 2u, to about 50, as Z, varies from 2 to 92. It is interesting to note that, using the mean velocities in the above mentioned velocity range, even with the limited set of experimental data used for determining the parameters in 5,. the
2/fy-_---il:::--i:
I
a=
1.224
x = 0.201
p = -3.791rl6'
I
1'
01 *
’
20
I
’
LO
r
’
60
5
’
80
’
100
2,
Fig. 1. Variation of 5, with 2, in the media (a) carbon, (b) nickel, (c) silver and (d) gold. Continuous curves are the least squares fits of the experimental data to the semiempirical expression for g, proposed in this work. The broken curves show the LSS values given by Z:/‘; A ref. [ill, 0 ref. 1121, 0 ref. 1131, x ref. 114). D ref. [15], l ref. 1161, v ref. 1171 and v ref. 1181.
where the parameters Q, p and x are characteristic of the medium. The least squares fits of this function to the &,-values based on the available experimental data are shown by continuous curves in figs. l(a), (b), (c) and. (d) for the media carbon, nickel, silver and gold. respectively. The corresponding values of the parameters (Y, ,8 and x are also given in the figure. The broken lines in
energv losses
P. Baner~ee et al. / Semiempirrcal formula for electronrc energy losses
(dE/dx),, values are usually reproduced by the present method to within 5% and in a few cases within lo-20% of the experimental results in the specified velocity regions. At higher ion velocities approaching u0 Z2/3 the experimental (dE/dx),, - velocity curves deviite, often appreciably, from linearity contrary to the predictions of the LSS theory, and hence the predicted (dE/dx),, method
values also
tend
using to
the modified deviate
from
5,
in the present
the
experimental
values.
The possibility of extrapolation of the present expression for 5, to other media where experimental data are scarce, depends on the exact form of the Z,-dependence of the parameters (Y, x and j3, which are characteristic of the medium. The variation of these parameters with Z, is not expected to be smooth due to the effects of Z, oscillations. With the availability of more data on stopping power measurements it may be possible to interpolate between close lying Z, values away from the regions of maxima or minima in the Z, oscillations.
References [l] J. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128. [2] J. Lindhard, M. Scharff and H.E. Schiott, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 33 (1963) no. 14. [3] Y. Laichter and N.H. Shafrir, Nucl. Phys. A371 (1981) 45. [4] M. Pickering and J.M. Alexander, Phys. Rev. C6 (1972) 332.
529
[51 M. Hakim and N.H. Shafrir, Can. J. Phys. 49 (1971) 3024. [61 Y. Laichter and N.H. Shafrir, Nucl. Phys. A394 (1983) 77. [71 N.K. Aras, M.P. Menon and G.E. Gordon, Nucl. Phys. 69 (1965) 337. Acta 18 PI 0. Birgiil, I. Glmez and N.K. Aras, Radio&mica (1972) 198. 191 S. Hontzeas and H. Blok, Phys. Scripta 4 (1971) 229. DOI A. Izmen, 0. Birgiil and N.K. Aras, J. Inorg. Nucl. Chem. 36 (1974) 25. Proc. Phys. Sot. 77 [ill D.I. Porat and K. Ramavataram, (1961) 97. Proc. Phys. Sot. 78 WI D.I. Porat and K. Ramavataram, (1961) 1135. G.C. Ball, G.J. v31 J.S. Forster, D. Ward, H.R. Andrews, Costa, W.G. Davies and I.V. Mitchell, Nucl. Instr. and Meth. 136 (1976) 349. [I41 H. Pape, H.G. Clerc and K.H. Schmidt, Z. Phys. A286 (1978) 159. P51 P.A. Dickstein and D. Ingman, Nucl. Instr. and Meth. B2 (1984) 364. Y. Laichter, W.F.W. Schneider and P. [I61 H. Geissel, Armbruster, Nucl. Instr. and Meth. 194 (1982) 21. Nucl. Instr. and Meth. 91 [I71 R. Miller and F. Gonnenwein, (1971) 357. U81 S. Kahn and V. Forgue, Phys. Rev. 163 (1967) 290. 1191 L.C. Northcliffe and R.F. Schilling, Nucl. Data A7 (1970) 233. of Stopping Cross-sections for PO1 J.F. Ziegler, Handbook Energetic Ions in all Elements (Pergmon, New York, 1980).