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From nuclear stopping to interatomic potential via the power law formalism
Volume 108A, number 8
PHYSICS LETTERS
22 April 1985
FROM NUCLEAR STOPPING TO INTERATOMIC POTENTIAL VIA T H E P O W E R LAW F O R M A L I S M A.G. W A G H and S.K. G U P T A Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India
Received 31 January 1985; accepted for publication 27 February 1985
A general method for inverting the energy-dependent nuclear stopping cross sections Sn(e ) to derive the interatomic potential V(x) is described. A correspondence between e and x is derived for mapping from Sn(e ) to V(x) and vice versa. The method is illustrated by using the recent range-energy data of keV indium and xenon ions in amorphous silicon to serf-consistently deduce the In-Si and Xe-Si potentials.
The last two decades have witnessed an intense research activity in the field of slowing down of atomic particles in matter at energies in the keV regime. Over this energy range, the particle loses energy primarily in elastic collisions with atoms, i.e. via the so-called nuclear stopping mechanism. The phenomenon As described within the classical framework wherein the scattering and stopping cross sections are derived under the assumption of a specific screened Coulomb potential [ 1 - 4 ] operating between the particle and the atom. The reverse problem of extracting the particleatom potential from the angular distribution of scattered particles measured at several [5] energies and at a single [6] energy, has also been dealt with. There has also been a simplistic attempt [7] to derive the potential from range-energy data for projectile and target atoms o f equal mass. In this letter, we shall formulate from the first principles, a procedure for deducing the interaction potential from the stopping cross sections using the power law [2,4] formalism and demonstrate its efficacy by applying it to the recent range-energy data. The power law formalism considers a repulsive potential of the form V(x)= V c ( x ) u ( x ) ,
u(x)=K/(sxS-1),
(1)
where Vc is the Coulomb potential, u the screening function, s the power of the potential and x the par0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ticle-atom separation expressed in units of the T h o m a s - F e r m i screening radius a. The scattering of particles under the influence of such a potential leads to the reduced nuclear stopping cross section [2,4] Sn(e ) = (de/dP)n = K 2 / S F s e l - 2 / s ,
(2)
where e and p are the reduced energy and range respectively of the particle as defined in ref. [2]. F s is a function of s alone. The estimates of F s obtained o n the basis of impulse approximation [2] are seriously in error even for powers s moderately exceeding 2, as shown by the recent exact [4] computations. The function F s is closely represented [4] by the analytic expression F s = {s2/S[2.04 + (1.55s - 1) -1 ] (1 - l/s)) -1 ,
(3)
for s values lying between 1.5 and 6.0. At keV energies, the scattering of atomic particles is influenced only by the monotonically decreasing repulsive part of the particle-atom potential. For such a potential, each centre-of-mass scattering angle corresponds to a unique impact parameter [5,6,8] and a measured Sn(e ) variation to a unique potential. Thus if the observed energy dependence of S n is described by eq. (2) over a certain e domain, it implies back a potential (1) over a range of separations x. However, there is no a priori knowledge of the x-range where the power s is applicable. We prescribe here an inversion 391
Volume 108A, number 8
PHYSICS LETTERS
procedure wherein the particle-atom potential is represented by the envelope of the family of power potentials deduced from the S n variations in various portions of the e domain, each conforming to a single power s. It will be presently seen that this procedure yields a correspondence between the energy e s and the distance x s where the same s applies. By examining the local logarithmic derivatives of the observed S n variation at several e values, the corresponding powers s are determined and using eqs. (2), (3) and (1), a family of screening functions Us(X ) = [ S n ( e ) l ( F s e t - Z / s ) ] s / 2 / ( s x S - 1
) = Bsxl-s
, (4)
is generated. The screening function of the particleatom potential is represented by the envelope of the family Us(X ) and obtained by applying the condition OUs/OS = 0, i.e. x s = exp [d(ln B s ) / d s ] .
(5)
On the other hand, if the potential is known and Sn(e ) is to be constructed therefrom as the envelope of piecemeal power curves of the type (2) generated using the known potential, a similar condition O(Sn)s/ Os = 0 leads to
22 April 1985
for estimating Sn(es). It has been shown previously [9] that simple potentials so matched to the true potentials at appropriate x values do reproduce the corresponding scattering cross sections. It is nevertheless important to check the self-consistency of the inversion procedure by computing Sn(e ) integrals using the inferred potential and tallying them with the original data. The knowledge of particle --atom interactions at separations of 0.I to 1 A provides an insight into the atomic collisions and radiation damage phenomena of keV atomic particles and vice versa. Over the last few years, with the emergence of high precision techniques for measuring particle ranges of the order of 100 A, a unique probe for exploring the interaction between two atoms with overlapping electron clouds has become available. As a sequel to such measurements, a strong oscillatory Z 1-dependence of nuclear stopping [10,11] of ions in amorphous silicon has been discovered lately. For two specific ion species, viz. indium and xenon which lie close to the maximum and minimum respectively in the Z l-oscillations of reduced range, the projected range data [10,11 ] is available
In es = - ~ s2{ d In [Fs(sBs) 2 / s ] / d s } , 0.50V ~-
e s = B s x s s exp{-½ s [d ln(FsS2/S)/d in s] },
:
~
'
I
,"
0.2t0
i.e. xs/b s
T
~
or, using eq. (5),
"~
Xs(es/Bs)l/s = e x p { - ~1 [d ln(Fss2/S)/d Ins]} ,
(6) b s being the distance of closest approach during a head-on collision in the s-potential, expressed in units ofa. Eq. (6) represents the required correspondence between es and x s. Th e ratio x s / b s is a function ors alone regardless of the potential used. A simple analytic fit, viz. Xs/b s = 1 + (2.94s - 3.57) -1 ,
I
.~
_~
t
O./,-
( de,
~'-~'J,, o.2
/"/""~
(7)
reproduces the relation (6) within 0.1% for s values lying between 1.75 and 8.0. The particle-atom potential is thus deduced over the x-range given by eq. (6), using the Sn(e ) data. The envelope approach for inferring the potential is equivalent to replacing the actual potential with power potentials matched in value and slope at distances x s 392
005
I
o.1 od5 o.o, o.'o2 o.6s o'.,o o.~ Fig. 1. The range-energy data [10,11 ] in reduced units for indium (circles) and xenon (squares) ions in amorphous silicon. The smooth fits (top) to the data and the nuclear stopping cross sections (bottom) derived therefrom are depicted by the solid and dashed curves for indium and xenon respectively.
Volume 108A, number 8
PHYSICS LETTERS
over a wide e domain. We shall now derive the I n - S i and X e - S i potentials from this data using the inversion formalism outlined above. The measured projected ranges of indium and xenon have been converted into total ranges p in ref. [11 ] us'ing standard projection factors and plotted against e (cf. fig. 2 in ref. [11]). The p - e data is reproduced here in fig. 1 and shown by circles and squares for indium and xenon respectively. The Z 1 -dependence of 19 represented by the splitting between the two sets of data is seen to become more conspicuous at lower e values. A smooth variation [ 12] of the type p = 1.6(2/v - 1)(2.558v - 1)e v ,
I
v=l+Clne, is fitted to each set of the p - e data. The best fits obtained with C values o f 0.0824 for indium and 0.0606 for xenon are depicted in fig. 1 by the solid and dashed curves respectively. The reciprocal of dp/de calculated by differentiating eq. (8) represents the total stopping cross section from which (de/dP)n is derived by subtracting the electronic [2] contribution
(de/dp) e = [O.0793Z2/3(Z2/A2) 1/2 (9)
Here Z denotes the atomic number, A the atomic mass and the subscripts 1 and 2 stand for the particle and atom respectively. The electronic contribution to the total stopping cross section rises from about 5% at e = 0.004 to around 10% at e = 0.1 for both the projectile species. Thus the corrections for projection factor and electronic stopping are both small and variations therein do not give rise to significant errors in the conversion of projected range data to Sn(e ). The nuclear stopping cross sections thus computed are plotted in fig. 1 for indium (solid curve) and xenon (dashed curve). The corresponding interaction potentials inferred using the envelope formalism and expressed in units o f the Coulomb potential at the screening radius are depicted in fig. 2. Using analytic fits to the two potentials, the (de/dP)n integrals were numerically computed and found identical to those in fig. 1, thus confirming the validity of the present prescription for de-
~ I
'
I r I
'
I
'
I '
I
0
~- 0.05 %
';%"'. ,%,,
N 0.02
> II
0.01
X 0.OO5
\\
N 3
\
0.002
(8)
where the parameter v itself depends weakly on e as
X (1 +A2/A1)3/2(Z 2/3 +Z2/3) -3/4 ] e 1/2 .
22 April 1985
I , I J I , I , I , h., I 3.0 5.0 7.0 9.0 3C
Fig. 2. The In-Si (solid curve) and Xe-Si (dashed curve) interaction potentials deduced from the data in fig. 1. The dotted curve represents the rms average of 500 ab-initio [13 ] potentials.
ducing the interaction potential. The success o f the inversion procedure stems from the exactness of the function F s [4] used. Potentials deduced similarly using the LNS approximation [2] for F s do not reproduce the S n curves on forward computation. In an earlier work [13], the interatomic potentials were calculated ab-initio for 500 ion-target combinations. The rms average of these potentials is represented by the dotted curve in fig. 2, having taken into account the different definition of the screening radius used by the authors. The average potential is evidently discrepant with the I n - S i and X e - S i potentials, underestimating their average by 10 to 40%, thus highlighting the scope for improvement in the prevalent theoretical understanding of interatomic potentials.
References [ 1 ] O.B. Firsov, Soy. Phys. JETP 5 (1957) 1192; 7 (1958) 308; 9 (1959) 1076. [2] J. Lindhard, M. Scharff and H.E. Schi~tt, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (1963) No. 14; J. Lindhard, V. Nielsen and M. Seharff, Mat. Fys. Medd. Dan. Vid. Selsk. 36 (1968) No. 10. [3] W.D. Wilson, L.G. Haggmark and J.P. Biersack, Phys. Rev. B15 (1977) 2458. [4] A.G. Wagh and S.K. Gupta, Phys. Lett. 101A (1984) 482.
[5] F.C. Hoyt, Phys. Rev. 55 (1939) 664.
393
Volume 108A, number 8
PHYSICS LETTERS
[6] O.B. Firsov, Zh. Eksp. Teor. Fiz. 24 (1953) 279; G.H. Lane and E. Everhart, Phys. Rev. 120 (1960) 2064. [ 7 ] D.P. CorkhiU and G. Carter, Phil. Mag. 11 (1965) l 31. [8] I.M. Torrens, Interatomic potentials (Academic Press, New York, 1972). [9] G. Leibfried and O.S. Oen, J. Appl. Phys. 33 (1962) 2257; C. Lehmann and M.T. Robinson, Phys. Rev. 134 (1964) A37.
394
22 April 1985
[10] F. Besenbacher, J. B~bttiger, T. Laursen, P. Loftager and W. M611er, Nucl. Instrum. Methods 170 (1980) 183. [ 11 ] J. Berthold and S. Kalbitzer, Nucl. lnstrum. Methods 209/210 (1983) 13. [12] S.K. Gupta and P.K. Bhattacharya, Phys. Rev. B29 (1984) 2449. [13] J.P. Biersack and J.F. Ziegler, in: Springer series in electrophysics, Vol. 10, eds. H. Ryssel and H. Glawischnig (Springer, Berlin, 1982) p. 122.
PHYSICS LETTERS
22 April 1985
FROM NUCLEAR STOPPING TO INTERATOMIC POTENTIAL VIA T H E P O W E R LAW F O R M A L I S M A.G. W A G H and S.K. G U P T A Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India
Received 31 January 1985; accepted for publication 27 February 1985
A general method for inverting the energy-dependent nuclear stopping cross sections Sn(e ) to derive the interatomic potential V(x) is described. A correspondence between e and x is derived for mapping from Sn(e ) to V(x) and vice versa. The method is illustrated by using the recent range-energy data of keV indium and xenon ions in amorphous silicon to serf-consistently deduce the In-Si and Xe-Si potentials.
The last two decades have witnessed an intense research activity in the field of slowing down of atomic particles in matter at energies in the keV regime. Over this energy range, the particle loses energy primarily in elastic collisions with atoms, i.e. via the so-called nuclear stopping mechanism. The phenomenon As described within the classical framework wherein the scattering and stopping cross sections are derived under the assumption of a specific screened Coulomb potential [ 1 - 4 ] operating between the particle and the atom. The reverse problem of extracting the particleatom potential from the angular distribution of scattered particles measured at several [5] energies and at a single [6] energy, has also been dealt with. There has also been a simplistic attempt [7] to derive the potential from range-energy data for projectile and target atoms o f equal mass. In this letter, we shall formulate from the first principles, a procedure for deducing the interaction potential from the stopping cross sections using the power law [2,4] formalism and demonstrate its efficacy by applying it to the recent range-energy data. The power law formalism considers a repulsive potential of the form V(x)= V c ( x ) u ( x ) ,
u(x)=K/(sxS-1),
(1)
where Vc is the Coulomb potential, u the screening function, s the power of the potential and x the par0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ticle-atom separation expressed in units of the T h o m a s - F e r m i screening radius a. The scattering of particles under the influence of such a potential leads to the reduced nuclear stopping cross section [2,4] Sn(e ) = (de/dP)n = K 2 / S F s e l - 2 / s ,
(2)
where e and p are the reduced energy and range respectively of the particle as defined in ref. [2]. F s is a function of s alone. The estimates of F s obtained o n the basis of impulse approximation [2] are seriously in error even for powers s moderately exceeding 2, as shown by the recent exact [4] computations. The function F s is closely represented [4] by the analytic expression F s = {s2/S[2.04 + (1.55s - 1) -1 ] (1 - l/s)) -1 ,
(3)
for s values lying between 1.5 and 6.0. At keV energies, the scattering of atomic particles is influenced only by the monotonically decreasing repulsive part of the particle-atom potential. For such a potential, each centre-of-mass scattering angle corresponds to a unique impact parameter [5,6,8] and a measured Sn(e ) variation to a unique potential. Thus if the observed energy dependence of S n is described by eq. (2) over a certain e domain, it implies back a potential (1) over a range of separations x. However, there is no a priori knowledge of the x-range where the power s is applicable. We prescribe here an inversion 391
Volume 108A, number 8
PHYSICS LETTERS
procedure wherein the particle-atom potential is represented by the envelope of the family of power potentials deduced from the S n variations in various portions of the e domain, each conforming to a single power s. It will be presently seen that this procedure yields a correspondence between the energy e s and the distance x s where the same s applies. By examining the local logarithmic derivatives of the observed S n variation at several e values, the corresponding powers s are determined and using eqs. (2), (3) and (1), a family of screening functions Us(X ) = [ S n ( e ) l ( F s e t - Z / s ) ] s / 2 / ( s x S - 1
) = Bsxl-s
, (4)
is generated. The screening function of the particleatom potential is represented by the envelope of the family Us(X ) and obtained by applying the condition OUs/OS = 0, i.e. x s = exp [d(ln B s ) / d s ] .
(5)
On the other hand, if the potential is known and Sn(e ) is to be constructed therefrom as the envelope of piecemeal power curves of the type (2) generated using the known potential, a similar condition O(Sn)s/ Os = 0 leads to
22 April 1985
for estimating Sn(es). It has been shown previously [9] that simple potentials so matched to the true potentials at appropriate x values do reproduce the corresponding scattering cross sections. It is nevertheless important to check the self-consistency of the inversion procedure by computing Sn(e ) integrals using the inferred potential and tallying them with the original data. The knowledge of particle --atom interactions at separations of 0.I to 1 A provides an insight into the atomic collisions and radiation damage phenomena of keV atomic particles and vice versa. Over the last few years, with the emergence of high precision techniques for measuring particle ranges of the order of 100 A, a unique probe for exploring the interaction between two atoms with overlapping electron clouds has become available. As a sequel to such measurements, a strong oscillatory Z 1-dependence of nuclear stopping [10,11] of ions in amorphous silicon has been discovered lately. For two specific ion species, viz. indium and xenon which lie close to the maximum and minimum respectively in the Z l-oscillations of reduced range, the projected range data [10,11 ] is available
In es = - ~ s2{ d In [Fs(sBs) 2 / s ] / d s } , 0.50V ~-
e s = B s x s s exp{-½ s [d ln(FsS2/S)/d in s] },
:
~
'
I
,"
0.2t0
i.e. xs/b s
T
~
or, using eq. (5),
"~
Xs(es/Bs)l/s = e x p { - ~1 [d ln(Fss2/S)/d Ins]} ,
(6) b s being the distance of closest approach during a head-on collision in the s-potential, expressed in units ofa. Eq. (6) represents the required correspondence between es and x s. Th e ratio x s / b s is a function ors alone regardless of the potential used. A simple analytic fit, viz. Xs/b s = 1 + (2.94s - 3.57) -1 ,
I
.~
_~
t
O./,-
( de,
~'-~'J,, o.2
/"/""~
(7)
reproduces the relation (6) within 0.1% for s values lying between 1.75 and 8.0. The particle-atom potential is thus deduced over the x-range given by eq. (6), using the Sn(e ) data. The envelope approach for inferring the potential is equivalent to replacing the actual potential with power potentials matched in value and slope at distances x s 392
005
I
o.1 od5 o.o, o.'o2 o.6s o'.,o o.~ Fig. 1. The range-energy data [10,11 ] in reduced units for indium (circles) and xenon (squares) ions in amorphous silicon. The smooth fits (top) to the data and the nuclear stopping cross sections (bottom) derived therefrom are depicted by the solid and dashed curves for indium and xenon respectively.
Volume 108A, number 8
PHYSICS LETTERS
over a wide e domain. We shall now derive the I n - S i and X e - S i potentials from this data using the inversion formalism outlined above. The measured projected ranges of indium and xenon have been converted into total ranges p in ref. [11 ] us'ing standard projection factors and plotted against e (cf. fig. 2 in ref. [11]). The p - e data is reproduced here in fig. 1 and shown by circles and squares for indium and xenon respectively. The Z 1 -dependence of 19 represented by the splitting between the two sets of data is seen to become more conspicuous at lower e values. A smooth variation [ 12] of the type p = 1.6(2/v - 1)(2.558v - 1)e v ,
I
v=l+Clne, is fitted to each set of the p - e data. The best fits obtained with C values o f 0.0824 for indium and 0.0606 for xenon are depicted in fig. 1 by the solid and dashed curves respectively. The reciprocal of dp/de calculated by differentiating eq. (8) represents the total stopping cross section from which (de/dP)n is derived by subtracting the electronic [2] contribution
(de/dp) e = [O.0793Z2/3(Z2/A2) 1/2 (9)
Here Z denotes the atomic number, A the atomic mass and the subscripts 1 and 2 stand for the particle and atom respectively. The electronic contribution to the total stopping cross section rises from about 5% at e = 0.004 to around 10% at e = 0.1 for both the projectile species. Thus the corrections for projection factor and electronic stopping are both small and variations therein do not give rise to significant errors in the conversion of projected range data to Sn(e ). The nuclear stopping cross sections thus computed are plotted in fig. 1 for indium (solid curve) and xenon (dashed curve). The corresponding interaction potentials inferred using the envelope formalism and expressed in units o f the Coulomb potential at the screening radius are depicted in fig. 2. Using analytic fits to the two potentials, the (de/dP)n integrals were numerically computed and found identical to those in fig. 1, thus confirming the validity of the present prescription for de-
~ I
'
I r I
'
I
'
I '
I
0
~- 0.05 %
';%"'. ,%,,
N 0.02
> II
0.01
X 0.OO5
\\
N 3
\
0.002
(8)
where the parameter v itself depends weakly on e as
X (1 +A2/A1)3/2(Z 2/3 +Z2/3) -3/4 ] e 1/2 .
22 April 1985
I , I J I , I , I , h., I 3.0 5.0 7.0 9.0 3C
Fig. 2. The In-Si (solid curve) and Xe-Si (dashed curve) interaction potentials deduced from the data in fig. 1. The dotted curve represents the rms average of 500 ab-initio [13 ] potentials.
ducing the interaction potential. The success o f the inversion procedure stems from the exactness of the function F s [4] used. Potentials deduced similarly using the LNS approximation [2] for F s do not reproduce the S n curves on forward computation. In an earlier work [13], the interatomic potentials were calculated ab-initio for 500 ion-target combinations. The rms average of these potentials is represented by the dotted curve in fig. 2, having taken into account the different definition of the screening radius used by the authors. The average potential is evidently discrepant with the I n - S i and X e - S i potentials, underestimating their average by 10 to 40%, thus highlighting the scope for improvement in the prevalent theoretical understanding of interatomic potentials.
References [ 1 ] O.B. Firsov, Soy. Phys. JETP 5 (1957) 1192; 7 (1958) 308; 9 (1959) 1076. [2] J. Lindhard, M. Scharff and H.E. Schi~tt, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (1963) No. 14; J. Lindhard, V. Nielsen and M. Seharff, Mat. Fys. Medd. Dan. Vid. Selsk. 36 (1968) No. 10. [3] W.D. Wilson, L.G. Haggmark and J.P. Biersack, Phys. Rev. B15 (1977) 2458. [4] A.G. Wagh and S.K. Gupta, Phys. Lett. 101A (1984) 482.
[5] F.C. Hoyt, Phys. Rev. 55 (1939) 664.
393
Volume 108A, number 8
PHYSICS LETTERS
[6] O.B. Firsov, Zh. Eksp. Teor. Fiz. 24 (1953) 279; G.H. Lane and E. Everhart, Phys. Rev. 120 (1960) 2064. [ 7 ] D.P. CorkhiU and G. Carter, Phil. Mag. 11 (1965) l 31. [8] I.M. Torrens, Interatomic potentials (Academic Press, New York, 1972). [9] G. Leibfried and O.S. Oen, J. Appl. Phys. 33 (1962) 2257; C. Lehmann and M.T. Robinson, Phys. Rev. 134 (1964) A37.
394
22 April 1985
[10] F. Besenbacher, J. B~bttiger, T. Laursen, P. Loftager and W. M611er, Nucl. Instrum. Methods 170 (1980) 183. [ 11 ] J. Berthold and S. Kalbitzer, Nucl. lnstrum. Methods 209/210 (1983) 13. [12] S.K. Gupta and P.K. Bhattacharya, Phys. Rev. B29 (1984) 2449. [13] J.P. Biersack and J.F. Ziegler, in: Springer series in electrophysics, Vol. 10, eds. H. Ryssel and H. Glawischnig (Springer, Berlin, 1982) p. 122.