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The nuclear encounter probability
Nuclear Insttuments
and Methods in Physics Research B 94 (1994)
595-596
Beam Interactions with Materials & Atoms
Letter to the Editor ELSEVIER
The nuclear encounter probability P.J.M. Smulders * hkl~~~ Solid .kte Physics,
Materials
Science
Centn?, Universiiy
of Groningen,
Nijenborgh 4, 9747 AG Groningen,
The
Netherlands
Received 25 July 1994
Abstract
This Letter dicussesthe nuclear encounterprobability as usedin ion channeling analysis. A formulation is given, incorporating effects of large beam angles and beam divergence. A critical examination of previous definitions is made.
In a Monte Carlo simulation of channeling phenomena, such as Rutherford backscattering (RBS) under channeling conditions, it is impractical to register only those events where the ion is scattered directly in the direction of the detector. Barrett [ 1 ] realized the advantage of registering instead, at every encounter between an ion and an atom, the probability that such events occur, taking all parameters fixed, except the position of the thermally vibrating atom. From the path of each ion, one thus obtains the nuclear encounter probability for all possible configurations of the atomic positions, for this particular trajectory. The method may be applied to the simulation of processes such as RBS, nuclear reactions, PIXE, or recoil. All these processes can be treated on an equal footing by the use of the normalized nuclear encounter probability (NEP), defined by Barrett as “the ratio of the cumulative probability computed in the course of a series of trajectories to the cumulative probability for an equal number of randomly oriented trajectories through an equal distance within the lattice”. The NEP may be compared to the normalized yield, i.e. the actual yield, divided by the random yield. In most cases the comparison requires approximations however. Apart from problems involved in obtaining the random yield [ 21, the analysis is complicated by the fact that, aher traversing some depth, ions in the channeling experiment have different distributions of energy and angle than they would have in a random system. One might define the NEP as the ratio of the two yields, taking these differences into account. This would require two simulations: one of a channeling experiment, and another of a random experiment. If the NEP is evaluated as a function of depth an additional question arises: should the NEP be normalized by dividing the actual yield at each depth by the random yield at that depth, or by one overall normalization constant? The last method has the advantage that the relative
* Tel.
+31 50 634822, fax +31 50 633471, e-mail [email protected].
0168-583X/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDIO168-583X(94)00325-4
yield versus depth is faithfully represented. The simplest choice for the normalization constant is then the “random” value at zero depth of the nuclear encounter probability. In absence of a detailed specification of the experiment we will adopt this method. For a random system the NEP defined this way has the value 1 at the surface, but may differ from 1 at larger depths, owing to an increase in beam divergence, or to the energy dependence of the cross section. Let us first consider the interaction between a single atom with equilibrium position rj and an ion i. Let Q(r) d3r be the probability that the thermally vibrating atom is present in a volume element dsr, and f(r) the probability that an interaction occurs during the passage of the ion, when the atom is at r. Then the probability for a nuclear encounter is Pji =
s
@(r)f(r)
dv.
For @ we take a Gaussian distribution with one-dimensional standard deviation u: G(r)
= (2rr~~)~~‘~exp(-~r~rj~~/2~~).
(2)
We choose a local coordinate system with its z-axis along the ion’s path before the encounter, such that the equilibrium position of the atom rj = (bji, 0,O). Assuming that f depends on the impact parameter p = (x2 + y*) ‘I* only, we integrate over z, and obtain
For most close encounter processes f is only significant for p << u, and Pji reduces to: (4)
596
WM.
Smulders/Nucl.
Instr. and Meth. in Phys. Res. B 94 (1994) 595-596
where Di) CT= 2~
pf(p)
dp.
(5)
J 0
An exception is the analysis of PIXE measurements where relevant impact parameters are comparable to the thermal vibration amplitude. If the inner shell ionization probability f(p) is app~ximated by a Gaussian (with one-dimensional standard deviation bo), Eq. (4) still holds with u* replaced byu’+b; [3]. In a Monte Carlo simulation for a set of M trajectories Pi; is evaluated for each binary collision separately, and the estimated total number of nuclear encounters is:
(6) i=l
i
Since the interaction takes place in a very small region around rj, from a macroscopic point of view the encounter takes place at rj. To obtain the nuclear encounter pmbabili~ as a function of depth the encounters are divided into depth bins k of equal width AZ. From here on z is the axis along which the depth is measured. The sum over j is over all atoms in such a depth interval. The dependence of LTon the ion energy and angle may be incorporated in Eq. (6). In the following we assume that such effects are negligible, and treat c as a constant. In a random system the probability to find an atom in dsr is N dsr, with N the atomic density. Let si be the path length travelled by ion i in interval AZ, which presumably is small enough to approximate the path by a straight line, at angle @i with the z axis: si = AZ,/ COS&.
(7)
The number of nuclear encounters per depth interval in a random system is then:
Thus, we finally obtain for the normalized NEP: (9)
channeling simulations in such a way that the random value is 1 at all depths. In a slightly different method of no~a~zation in~oduced by Dygo et al. 143, this fictive process is used, but, in addition, the contribution of each ion-atom encounter is normalized separately, and a trajectory- and depth-dependent factor cos (ilk appears in the summation of Eq. (9). As a consequence encounters with identical impact parameters that occur at the same position, and only differ in the ion’s direction to the z-axis, have different cont~butions to the NEI? Contributions from trajectories that make larger angles with the axis are systematically underestimated, relative to trajectories with small angles. Although in most practical cases these differences will cancel out, this method is inferior to the above. The close encounter probabili~ is sometimes defined as the convolution of the flux distribution of the ions with the probability distribution of the vibrating atom [ 51. This definition closely corresponds to Eq. (6), when the sum over i is replaced by a surface integral, provided the convolution is carried out in a plane perpendicular to the ion velocity. Only then the distance between ion and atom in the plane may be equated to the impact parameter. If instead the plane is tilted by an angle 9 then this distance is a factor between 1 and l/ cos + larger than in the transverse plane. For a uniform flux this leads to a reduction of the result by a factor cos#. This effect may be largely compensated by omitting the (1,’ cos @j-t in the equivalent of Eq. (9). Neve~eless this method is only useful as long as the beam.angle as well as the beam divergence are small. Like in Eq. (6) an energy dependent cross section may be folded into the flux distribution. In Barrett’s original formulation of the NEP [I ,3] the distances bji are calculated in a plane perpendicular to a row of atoms, rather than pe~endicular to the ion’s trajectory. Thus the NEP is underestimated by a factor in the order of cos (/I. Some further differences between Eq. (9) and Barrett’s formula am, that the contributions in Eq. (9) are not restricted to the atoms of one particular row, and the distance d between atoms along a row is replaced by the more general notion of a depth step AZ, along an arbitrary z axis. The present form is suitable for atomic structures other than single crystals, such as quasi crystals and multilayers. References [l]
As discussed above, we take here for $ the initial direction. For a parallel beam of ions (l/ cos (cl)-’ reduces to cos*. An alternative is to use a depth dependent (l/ cos @)k in Eq. (9) as obtained from the trajectories of the channeling simulation. This would correspond to no~alizing by a@rive random process, where the random system is identicai to the crystalline one except for a thin “random” layer at depth z. This method may be of use to represent results of
J.H. Barrett, Phys. Rev. B 3 f 1971) 1527. [ 21 A. Dygo, W.N. Lennacd and 1.V. Mitcfiell, Nucl. Ins% and Meth. B 74 (1994)581;and A. Dygo, W.N. Lennard, LV. Mitchell and P.J.M. Smolders, Nucl. Instr. and Meth. B 90 (1994) 161. [3] D. Comedi, R. Kalish and J.H. Barrett, Nucl. Instr. and Me&. B 63 (1992) 451. [4] A. Dygo, W.N. Lennard and LV. Mitchell, Nucl. Instr. and Meth. B 90 ( 1994) 142. [ 51 L.C. Feldman, J.W. Mayer and S.T. Picraux. Materials Analysis by Ion Channeling (Academic Press, New York, 1982).
and Methods in Physics Research B 94 (1994)
595-596
Beam Interactions with Materials & Atoms
Letter to the Editor ELSEVIER
The nuclear encounter probability P.J.M. Smulders * hkl~~~ Solid .kte Physics,
Materials
Science
Centn?, Universiiy
of Groningen,
Nijenborgh 4, 9747 AG Groningen,
The
Netherlands
Received 25 July 1994
Abstract
This Letter dicussesthe nuclear encounterprobability as usedin ion channeling analysis. A formulation is given, incorporating effects of large beam angles and beam divergence. A critical examination of previous definitions is made.
In a Monte Carlo simulation of channeling phenomena, such as Rutherford backscattering (RBS) under channeling conditions, it is impractical to register only those events where the ion is scattered directly in the direction of the detector. Barrett [ 1 ] realized the advantage of registering instead, at every encounter between an ion and an atom, the probability that such events occur, taking all parameters fixed, except the position of the thermally vibrating atom. From the path of each ion, one thus obtains the nuclear encounter probability for all possible configurations of the atomic positions, for this particular trajectory. The method may be applied to the simulation of processes such as RBS, nuclear reactions, PIXE, or recoil. All these processes can be treated on an equal footing by the use of the normalized nuclear encounter probability (NEP), defined by Barrett as “the ratio of the cumulative probability computed in the course of a series of trajectories to the cumulative probability for an equal number of randomly oriented trajectories through an equal distance within the lattice”. The NEP may be compared to the normalized yield, i.e. the actual yield, divided by the random yield. In most cases the comparison requires approximations however. Apart from problems involved in obtaining the random yield [ 21, the analysis is complicated by the fact that, aher traversing some depth, ions in the channeling experiment have different distributions of energy and angle than they would have in a random system. One might define the NEP as the ratio of the two yields, taking these differences into account. This would require two simulations: one of a channeling experiment, and another of a random experiment. If the NEP is evaluated as a function of depth an additional question arises: should the NEP be normalized by dividing the actual yield at each depth by the random yield at that depth, or by one overall normalization constant? The last method has the advantage that the relative
* Tel.
+31 50 634822, fax +31 50 633471, e-mail [email protected].
0168-583X/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDIO168-583X(94)00325-4
yield versus depth is faithfully represented. The simplest choice for the normalization constant is then the “random” value at zero depth of the nuclear encounter probability. In absence of a detailed specification of the experiment we will adopt this method. For a random system the NEP defined this way has the value 1 at the surface, but may differ from 1 at larger depths, owing to an increase in beam divergence, or to the energy dependence of the cross section. Let us first consider the interaction between a single atom with equilibrium position rj and an ion i. Let Q(r) d3r be the probability that the thermally vibrating atom is present in a volume element dsr, and f(r) the probability that an interaction occurs during the passage of the ion, when the atom is at r. Then the probability for a nuclear encounter is Pji =
s
@(r)f(r)
dv.
For @ we take a Gaussian distribution with one-dimensional standard deviation u: G(r)
= (2rr~~)~~‘~exp(-~r~rj~~/2~~).
(2)
We choose a local coordinate system with its z-axis along the ion’s path before the encounter, such that the equilibrium position of the atom rj = (bji, 0,O). Assuming that f depends on the impact parameter p = (x2 + y*) ‘I* only, we integrate over z, and obtain
For most close encounter processes f is only significant for p << u, and Pji reduces to: (4)
596
WM.
Smulders/Nucl.
Instr. and Meth. in Phys. Res. B 94 (1994) 595-596
where Di) CT= 2~
pf(p)
dp.
(5)
J 0
An exception is the analysis of PIXE measurements where relevant impact parameters are comparable to the thermal vibration amplitude. If the inner shell ionization probability f(p) is app~ximated by a Gaussian (with one-dimensional standard deviation bo), Eq. (4) still holds with u* replaced byu’+b; [3]. In a Monte Carlo simulation for a set of M trajectories Pi; is evaluated for each binary collision separately, and the estimated total number of nuclear encounters is:
(6) i=l
i
Since the interaction takes place in a very small region around rj, from a macroscopic point of view the encounter takes place at rj. To obtain the nuclear encounter pmbabili~ as a function of depth the encounters are divided into depth bins k of equal width AZ. From here on z is the axis along which the depth is measured. The sum over j is over all atoms in such a depth interval. The dependence of LTon the ion energy and angle may be incorporated in Eq. (6). In the following we assume that such effects are negligible, and treat c as a constant. In a random system the probability to find an atom in dsr is N dsr, with N the atomic density. Let si be the path length travelled by ion i in interval AZ, which presumably is small enough to approximate the path by a straight line, at angle @i with the z axis: si = AZ,/ COS&.
(7)
The number of nuclear encounters per depth interval in a random system is then:
Thus, we finally obtain for the normalized NEP: (9)
channeling simulations in such a way that the random value is 1 at all depths. In a slightly different method of no~a~zation in~oduced by Dygo et al. 143, this fictive process is used, but, in addition, the contribution of each ion-atom encounter is normalized separately, and a trajectory- and depth-dependent factor cos (ilk appears in the summation of Eq. (9). As a consequence encounters with identical impact parameters that occur at the same position, and only differ in the ion’s direction to the z-axis, have different cont~butions to the NEI? Contributions from trajectories that make larger angles with the axis are systematically underestimated, relative to trajectories with small angles. Although in most practical cases these differences will cancel out, this method is inferior to the above. The close encounter probabili~ is sometimes defined as the convolution of the flux distribution of the ions with the probability distribution of the vibrating atom [ 51. This definition closely corresponds to Eq. (6), when the sum over i is replaced by a surface integral, provided the convolution is carried out in a plane perpendicular to the ion velocity. Only then the distance between ion and atom in the plane may be equated to the impact parameter. If instead the plane is tilted by an angle 9 then this distance is a factor between 1 and l/ cos + larger than in the transverse plane. For a uniform flux this leads to a reduction of the result by a factor cos#. This effect may be largely compensated by omitting the (1,’ cos @j-t in the equivalent of Eq. (9). Neve~eless this method is only useful as long as the beam.angle as well as the beam divergence are small. Like in Eq. (6) an energy dependent cross section may be folded into the flux distribution. In Barrett’s original formulation of the NEP [I ,3] the distances bji are calculated in a plane perpendicular to a row of atoms, rather than pe~endicular to the ion’s trajectory. Thus the NEP is underestimated by a factor in the order of cos (/I. Some further differences between Eq. (9) and Barrett’s formula am, that the contributions in Eq. (9) are not restricted to the atoms of one particular row, and the distance d between atoms along a row is replaced by the more general notion of a depth step AZ, along an arbitrary z axis. The present form is suitable for atomic structures other than single crystals, such as quasi crystals and multilayers. References [l]
As discussed above, we take here for $ the initial direction. For a parallel beam of ions (l/ cos (cl)-’ reduces to cos*. An alternative is to use a depth dependent (l/ cos @)k in Eq. (9) as obtained from the trajectories of the channeling simulation. This would correspond to no~alizing by a@rive random process, where the random system is identicai to the crystalline one except for a thin “random” layer at depth z. This method may be of use to represent results of
J.H. Barrett, Phys. Rev. B 3 f 1971) 1527. [ 21 A. Dygo, W.N. Lennacd and 1.V. Mitcfiell, Nucl. Ins% and Meth. B 74 (1994)581;and A. Dygo, W.N. Lennard, LV. Mitchell and P.J.M. Smolders, Nucl. Instr. and Meth. B 90 (1994) 161. [3] D. Comedi, R. Kalish and J.H. Barrett, Nucl. Instr. and Me&. B 63 (1992) 451. [4] A. Dygo, W.N. Lennard and LV. Mitchell, Nucl. Instr. and Meth. B 90 ( 1994) 142. [ 51 L.C. Feldman, J.W. Mayer and S.T. Picraux. Materials Analysis by Ion Channeling (Academic Press, New York, 1982).