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Multiply peaked energy loss spectra of heavy ions transmitted through polycrystalline foils: An interpretation in terms of channeling
Nuclear Instruments and Methods in Physics Research B2 (1984) 199-201 North-Holland, Amsterdam
199
MULTIPLY PEAKED ENERGY LOSS SPECTRA OF HEAW IONS TRANSMITTED THROUGH POLYCRYSTALLINE FOILS: AN INTERPRETATION IN TERMS OF CHANNELING F. SCHULZ
and W. MICHAEL
Gesellschaft fiir Strahlen
- und Umweltforschung mbH, Physikalisch - Technische A bteilung D - 8042 Neuherberg, Fed. Rep. Germany
In order to understand the multiply peaked structure of the energy loss spectra of low-velocity ions transmitted through thin polycrystalline foils we have carried out a model calculation based upon the idea that the low-loss peaks are due to axially channeled ions. It is assumed that the incident beam, upon striking the target, splits into a channeled and a random component. The emerging ions are described by appropriate angular distributions. The model predicts that the channeled-to-random intensity ratio in the energy observations for 100-600 loss spectra decreases with increasing ion energy as E -‘. This is in reasonable agreement with experimental keV N transmitted through an Au foil with a (111) fibre texture. The calculations suggest a characteristic width of the texture of 5” to 8”.
1. Introduction N ---Au
Energy
loss spectra of (heavy) ions transmitted through polycrystalline foils have been found to exhibit double [l] or even multiply peaked [2] structures which aggravate the determination of the stopping power relevant to a “random” direction [1,2]. Since doubly peaked energy loss spectra have also been observed in measurements involving twinned single crystal foils [3], the low-loss peaks showing up in the spectra for polycrystalline foils have tentatively been attributed to channeling [1,2]. A detailed quantitative analysis has not been presented before. We have recently carried out a thorough investigation of the energy loss of 100-600 keV He, N, Ne and Ar in polycrystalline Al and Au (as well as in amorphous C) [2,4]. The energy loss spectra revealed a systematic variation of the relative intensity of the different loss components as a function of the impact energy. Two examples of spectra recorded in the forward direction (I$, = 0”) with an angular resolution of 0.1” are depicted in fig. 1. Separation into the random and (apparently two) channeled components is assisted by measurements at non-zero angles of observation (&, = 1.3’, dashed curves in fig. 1) in which case the low-loss (channeled) components are suppressed relative to the high-loss (random) component [2,4]. Knowing the random-to-channeled intensity ratio quantitatively as a function of the ion energy, a comparison with theoretical results appears to be desirable. In this paper we described a model which allows a calculation of the random-to-channeled intensity ratio of swift ions after transmission through polycrystalline foils. The targets are characterized by a certain distribution of crystallite orientations. X-ray diffraction analysis of the Au foils 0168-583X/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
wp,, = 130 pg /cm2
* ia Y
500 I
zr’ l.oz
0 z
520
I
I
lb1 E,,=150keV
/
0.6 -
0.L-
: I’
0.2 JX -
I I I I
‘! 7
’
t :,, ‘a.*“J i 7
: : :
/‘\
i
:
,/’ ---
I .;;.
:
I 8 I I
100
I
:-!
0.8 -
0
540 I
:,,\
.’ I
110 ENERGY
I
‘\ ..
I
120
130
IkeV)
Fig. 1. Typical energy spectra of nitrogen ions after transmission through a thin polycrystalline gold foil. (a) Beam energy 600 keV, (b) 150 keV. The full circles and the dashed curves denote spectra recorded in the direction of propagation of the incident beam (&a = 0”) and at @a, = 1.3’, respectively 141.
employed in refs. [2] and [4] showed that the (111) pole intensity was enhanced by a factor of - 10 compared to a random powder sample. In this paper we will thereIII. ENERGY
LOSS
F. Schulz, W. Michael / Multiply peaked energv loss spectra
200
fore restrict ing.
ourselves
to a discussion
of axial channel-
where 0, is the unit vector in the direction tion. The total contribution of channeled intensity recorded in the direction a, is
of observaions to the
2. Model hkl
We consider a target foil composed of microcrystals with a certain distribution of low directions relative to the axis of the incident beam. It is assumed that an individual crystallite extends through the whole thickness of the foil, as illustrated in fig. 2a. Let the probability for finding an axial channel (hkl) within. the solid angle da, around the unit vector 9, be g,,,(@i,a,)dQr
“g&arc
cos(Qi
*Qa,))df&.
&, is the unit vector in the direction of propagation of the incident beam. If an ion strikes a crystallite, the probability for becoming channeled in the low-index direction [ hkl] along 0, is 3;rkl(%e)
= .C&arc
The total fraction
co%%
s&1).
of channeled
ions is 0)
The fraction of non-channeled nent) is simply r,(Q,)
=
1-
ions (“random”
K(Qe)-
compo-
(2)
We assume that ions, once fed into a channel on entrance, remain channeled until they leave the target. The angular distribution of ions emerging from a (hkl) channel is denoted by
kl)
misalianed ysthite
random
Similarly the angular distribution of the random component is denoted n,(Q,,) so that the “random” intensity in 0, becomes Ytf%e)
= [I -
Y,mJ1%v4,)~
(4
The two different distributions are sketched schematically in fig. 2a. The target normal n may be tilted with respect to &I,, by an angle (Y..r~and 0, define the plane of incidence in which the energy Ioss spectra of refs. [2) and [4] were measured, cf. fig. 2b. The actual calculations were carried out for Au foils with two different crystallite orientations. Firstly, we assumed an isotropic distribution of fee directions, i.e. ghk, = chkJ2?r, with the multiplicities 3, 4, 6 and 12 for (lOO), (ill), (110) and (211), respectively. The second choice was a (Ill) textured structure [.5] characterized by a parameter p, giii(&,e)
= Q/2n)(l+
E”‘) exp(
-f4,0).
(5)
where the (111) pole axis is assumed to be parallel to 9,. The channeling probability is taken to be of the form ihklfel,a)
= (1 - xikf)
exp(-e~,~/2u~,~~~)~
(6)
where &, is the minimum yield (random fraction) observed in an experiment involving a single crystal. We make use of the empirical formula suggested by Barrett
where N is the number density of target atoms, dhk, the atomic spacing along (hkf), u1 the one-dimensional rms vibrational amplitude and $J,,~, = #1,2(hkf) is the half width at half minimum of the channeling dip. In the energy region of interest we employ the relation [7] qhkl = (1.16Z,ZzeZa/d,2,,E)1’3,
(8)
where Z, and Z, are the atomic numbers of the ion and the target atoms, respectively, e is the elementary charge, E is the ion energy and a = 0.885 ao/( 2:/3 -t Z~/3)1/2, u. is the Bohr radius, u,,.~&, in eq. (6) is derived from eq. (8) through the simple relation *in,hkl
=
(21n 2)-i’*$~~~,.
(9)
The angular distribution of channeled ions on exit is assumed to be characterized by the same width as the channeling probability, i.e. Fig. 2. Schematic illustration of ion trans~ssion through a poiycrystalline foil, (a) microscopic, (b) macroscopic picture. Ei, and E,,, are the energies of the incident and outgoing ions, respectively.
a&.hkl=
aan,hkf=
ahkl’
Thus (10)
201
F. Schulz, W. Michael / Mulrip1y peaked energy loss spectra
-
ENERGY
(111) flbre
lkeVl
Fig. 3. Channeled-to-random intensity ratio of nitrogen ions transmitted through a polycrystalline gold foil as a function of the mean energy ( Ei, + E0,,)/2. E,,, is the mean energy of the emerging random component [2]. The solid lines represent the results of the present calculations ( y = In 2,‘)~) for tilting angle
(Y= 0”. The corresponding experimental data [4] are denoted by full circles. The open symbols represent ratios [4] measured with the target normal tilted by an angle a (see inset).
Similarly, the angular dom component is ~~(@,,a) = (2ac,Z)-t
distribution
of the emerging
exp( -8&/2c,Z).
The characteristic width scattering theory [8].
ur was derived
ran-
(11) from multiple
The predicted intensity ratios yc/yr are seen to be in good agreement with the experimental results, provided we assume a fibre texture with 5” I y I 8” or an isotopic distribution of crystallites. Measurements performed with a tilted foil (cy > 0’) support the idea that the foil was characterized by a broad texture rather than by an isotropic distribution of crystallites (in the latter case u,/u, should be nearly independent of cy). Some additional comments are in order at this point. The assumption of Gaussian distributions constitutes only a simple approximation. Only in the case of a fibre texture, however, the results were found to be sensitive to the actual values of uhk,. The Gaussian tails of the channeled distributions may be responsible for the fact that the calculated intensity ratios yF/y, were found to drop off more slowly with increasing angle of observation than observed (not shown in fig. 3). The slight difference in the energy dependence of the experimental and theoretical y&-ratios may be due to the fact that dechanneling was neglected in the calculations. This effect should become more important as the impact energy is reduced.
4. Conclusion In spite of the simplicity of our model, the reasonable agreement between experiment and calculations is encouraging. Several improvements of the model are conceivable. On the experimental side it would be desirable to perform measurements on foils with a wellcharacterized orientation of crystallites.
We would like to thank our colleague K. Wittmaack for constant encouragement during this work and for assistance in writing this paper.
References 3. Results [l] P. Mertens, Nucl. Instr. and Meth. 149 (1978) 149; Thin Fig. 3 shows the channeled-to-random intensity ratio of nitrogen ions transmitted through a polycrystalline gold foil as a function of the (mean) ion energy. The straight lines relate to calculations based upon the assumption of a target having a (111) fibre texture of width y = In 2/).~. The dashed curve represents the results obtained for an isotropic distribution of crystallites. The calculated data relate to an angle of observation 0,, = 0”. The corresponding experimental data are denoted by full circles.
Solid Films 60 (1979) 313. [2] [3] [4] [5]
F. Schulz and W. Brandt, Phys. Rev. B26 (1982) 4864. J. Berttiger and F. Bason, Rad. Eff. 2 (1969) 105. F. Schulz, unpublished.
H.H. Andersen, K.N. Tu and J.F. Ziegler, Nucl. Instr. and Meth. 149 (1978) 247. [6] J.H. Barrett, Phys. Rev. B3 (1971) 1527. [7] A. van Wijngaarden, E. Reuther and J.N. Bradford, Canad. J. Phys. 47 (1969) 411. (81 P. Sigmund and K.B. Winterbon, Nucl. Instr. and Meth. 119 (1974) 541.
III. ENERGY
LOSS
199
MULTIPLY PEAKED ENERGY LOSS SPECTRA OF HEAW IONS TRANSMITTED THROUGH POLYCRYSTALLINE FOILS: AN INTERPRETATION IN TERMS OF CHANNELING F. SCHULZ
and W. MICHAEL
Gesellschaft fiir Strahlen
- und Umweltforschung mbH, Physikalisch - Technische A bteilung D - 8042 Neuherberg, Fed. Rep. Germany
In order to understand the multiply peaked structure of the energy loss spectra of low-velocity ions transmitted through thin polycrystalline foils we have carried out a model calculation based upon the idea that the low-loss peaks are due to axially channeled ions. It is assumed that the incident beam, upon striking the target, splits into a channeled and a random component. The emerging ions are described by appropriate angular distributions. The model predicts that the channeled-to-random intensity ratio in the energy observations for 100-600 loss spectra decreases with increasing ion energy as E -‘. This is in reasonable agreement with experimental keV N transmitted through an Au foil with a (111) fibre texture. The calculations suggest a characteristic width of the texture of 5” to 8”.
1. Introduction N ---Au
Energy
loss spectra of (heavy) ions transmitted through polycrystalline foils have been found to exhibit double [l] or even multiply peaked [2] structures which aggravate the determination of the stopping power relevant to a “random” direction [1,2]. Since doubly peaked energy loss spectra have also been observed in measurements involving twinned single crystal foils [3], the low-loss peaks showing up in the spectra for polycrystalline foils have tentatively been attributed to channeling [1,2]. A detailed quantitative analysis has not been presented before. We have recently carried out a thorough investigation of the energy loss of 100-600 keV He, N, Ne and Ar in polycrystalline Al and Au (as well as in amorphous C) [2,4]. The energy loss spectra revealed a systematic variation of the relative intensity of the different loss components as a function of the impact energy. Two examples of spectra recorded in the forward direction (I$, = 0”) with an angular resolution of 0.1” are depicted in fig. 1. Separation into the random and (apparently two) channeled components is assisted by measurements at non-zero angles of observation (&, = 1.3’, dashed curves in fig. 1) in which case the low-loss (channeled) components are suppressed relative to the high-loss (random) component [2,4]. Knowing the random-to-channeled intensity ratio quantitatively as a function of the ion energy, a comparison with theoretical results appears to be desirable. In this paper we described a model which allows a calculation of the random-to-channeled intensity ratio of swift ions after transmission through polycrystalline foils. The targets are characterized by a certain distribution of crystallite orientations. X-ray diffraction analysis of the Au foils 0168-583X/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
wp,, = 130 pg /cm2
* ia Y
500 I
zr’ l.oz
0 z
520
I
I
lb1 E,,=150keV
/
0.6 -
0.L-
: I’
0.2 JX -
I I I I
‘! 7
’
t :,, ‘a.*“J i 7
: : :
/‘\
i
:
,/’ ---
I .;;.
:
I 8 I I
100
I
:-!
0.8 -
0
540 I
:,,\
.’ I
110 ENERGY
I
‘\ ..
I
120
130
IkeV)
Fig. 1. Typical energy spectra of nitrogen ions after transmission through a thin polycrystalline gold foil. (a) Beam energy 600 keV, (b) 150 keV. The full circles and the dashed curves denote spectra recorded in the direction of propagation of the incident beam (&a = 0”) and at @a, = 1.3’, respectively 141.
employed in refs. [2] and [4] showed that the (111) pole intensity was enhanced by a factor of - 10 compared to a random powder sample. In this paper we will thereIII. ENERGY
LOSS
F. Schulz, W. Michael / Multiply peaked energv loss spectra
200
fore restrict ing.
ourselves
to a discussion
of axial channel-
where 0, is the unit vector in the direction tion. The total contribution of channeled intensity recorded in the direction a, is
of observaions to the
2. Model hkl
We consider a target foil composed of microcrystals with a certain distribution of low directions relative to the axis of the incident beam. It is assumed that an individual crystallite extends through the whole thickness of the foil, as illustrated in fig. 2a. Let the probability for finding an axial channel (hkl) within. the solid angle da, around the unit vector 9, be g,,,(@i,a,)dQr
“g&arc
cos(Qi
*Qa,))df&.
&, is the unit vector in the direction of propagation of the incident beam. If an ion strikes a crystallite, the probability for becoming channeled in the low-index direction [ hkl] along 0, is 3;rkl(%e)
= .C&arc
The total fraction
co%%
s&1).
of channeled
ions is 0)
The fraction of non-channeled nent) is simply r,(Q,)
=
1-
ions (“random”
K(Qe)-
compo-
(2)
We assume that ions, once fed into a channel on entrance, remain channeled until they leave the target. The angular distribution of ions emerging from a (hkl) channel is denoted by
kl)
misalianed ysthite
random
Similarly the angular distribution of the random component is denoted n,(Q,,) so that the “random” intensity in 0, becomes Ytf%e)
= [I -
Y,mJ1%v4,)~
(4
The two different distributions are sketched schematically in fig. 2a. The target normal n may be tilted with respect to &I,, by an angle (Y..r~and 0, define the plane of incidence in which the energy Ioss spectra of refs. [2) and [4] were measured, cf. fig. 2b. The actual calculations were carried out for Au foils with two different crystallite orientations. Firstly, we assumed an isotropic distribution of fee directions, i.e. ghk, = chkJ2?r, with the multiplicities 3, 4, 6 and 12 for (lOO), (ill), (110) and (211), respectively. The second choice was a (Ill) textured structure [.5] characterized by a parameter p, giii(&,e)
= Q/2n)(l+
E”‘) exp(
-f4,0).
(5)
where the (111) pole axis is assumed to be parallel to 9,. The channeling probability is taken to be of the form ihklfel,a)
= (1 - xikf)
exp(-e~,~/2u~,~~~)~
(6)
where &, is the minimum yield (random fraction) observed in an experiment involving a single crystal. We make use of the empirical formula suggested by Barrett
where N is the number density of target atoms, dhk, the atomic spacing along (hkf), u1 the one-dimensional rms vibrational amplitude and $J,,~, = #1,2(hkf) is the half width at half minimum of the channeling dip. In the energy region of interest we employ the relation [7] qhkl = (1.16Z,ZzeZa/d,2,,E)1’3,
(8)
where Z, and Z, are the atomic numbers of the ion and the target atoms, respectively, e is the elementary charge, E is the ion energy and a = 0.885 ao/( 2:/3 -t Z~/3)1/2, u. is the Bohr radius, u,,.~&, in eq. (6) is derived from eq. (8) through the simple relation *in,hkl
=
(21n 2)-i’*$~~~,.
(9)
The angular distribution of channeled ions on exit is assumed to be characterized by the same width as the channeling probability, i.e. Fig. 2. Schematic illustration of ion trans~ssion through a poiycrystalline foil, (a) microscopic, (b) macroscopic picture. Ei, and E,,, are the energies of the incident and outgoing ions, respectively.
a&.hkl=
aan,hkf=
ahkl’
Thus (10)
201
F. Schulz, W. Michael / Mulrip1y peaked energy loss spectra
-
ENERGY
(111) flbre
lkeVl
Fig. 3. Channeled-to-random intensity ratio of nitrogen ions transmitted through a polycrystalline gold foil as a function of the mean energy ( Ei, + E0,,)/2. E,,, is the mean energy of the emerging random component [2]. The solid lines represent the results of the present calculations ( y = In 2,‘)~) for tilting angle
(Y= 0”. The corresponding experimental data [4] are denoted by full circles. The open symbols represent ratios [4] measured with the target normal tilted by an angle a (see inset).
Similarly, the angular dom component is ~~(@,,a) = (2ac,Z)-t
distribution
of the emerging
exp( -8&/2c,Z).
The characteristic width scattering theory [8].
ur was derived
ran-
(11) from multiple
The predicted intensity ratios yc/yr are seen to be in good agreement with the experimental results, provided we assume a fibre texture with 5” I y I 8” or an isotopic distribution of crystallites. Measurements performed with a tilted foil (cy > 0’) support the idea that the foil was characterized by a broad texture rather than by an isotropic distribution of crystallites (in the latter case u,/u, should be nearly independent of cy). Some additional comments are in order at this point. The assumption of Gaussian distributions constitutes only a simple approximation. Only in the case of a fibre texture, however, the results were found to be sensitive to the actual values of uhk,. The Gaussian tails of the channeled distributions may be responsible for the fact that the calculated intensity ratios yF/y, were found to drop off more slowly with increasing angle of observation than observed (not shown in fig. 3). The slight difference in the energy dependence of the experimental and theoretical y&-ratios may be due to the fact that dechanneling was neglected in the calculations. This effect should become more important as the impact energy is reduced.
4. Conclusion In spite of the simplicity of our model, the reasonable agreement between experiment and calculations is encouraging. Several improvements of the model are conceivable. On the experimental side it would be desirable to perform measurements on foils with a wellcharacterized orientation of crystallites.
We would like to thank our colleague K. Wittmaack for constant encouragement during this work and for assistance in writing this paper.
References 3. Results [l] P. Mertens, Nucl. Instr. and Meth. 149 (1978) 149; Thin Fig. 3 shows the channeled-to-random intensity ratio of nitrogen ions transmitted through a polycrystalline gold foil as a function of the (mean) ion energy. The straight lines relate to calculations based upon the assumption of a target having a (111) fibre texture of width y = In 2/).~. The dashed curve represents the results obtained for an isotropic distribution of crystallites. The calculated data relate to an angle of observation 0,, = 0”. The corresponding experimental data are denoted by full circles.
Solid Films 60 (1979) 313. [2] [3] [4] [5]
F. Schulz and W. Brandt, Phys. Rev. B26 (1982) 4864. J. Berttiger and F. Bason, Rad. Eff. 2 (1969) 105. F. Schulz, unpublished.
H.H. Andersen, K.N. Tu and J.F. Ziegler, Nucl. Instr. and Meth. 149 (1978) 247. [6] J.H. Barrett, Phys. Rev. B3 (1971) 1527. [7] A. van Wijngaarden, E. Reuther and J.N. Bradford, Canad. J. Phys. 47 (1969) 411. (81 P. Sigmund and K.B. Winterbon, Nucl. Instr. and Meth. 119 (1974) 541.
III. ENERGY
LOSS