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Recoil distributions in isotope separator targets
538
RECOIL DISTRIBUTIONS
Nuclear Instruments and Methods in Physics Research B31(1988) 538-540 North-Holland, Amsterdam
IN ISOTOPE SEPARATOR
TARGETS
K.B. WINTERBON Atomic Energy of Canada Limited Research Company, Chalk River Nuclear Laboratories,
Chalk River, Ontario, Canada KOJ 1JO
Received 4 January 1988
The depth distribution of recoil daughter nuclei following the radioactive decay of implanted parent nuclei has been calculated. Moments of the depth distribution of the daughter nuclei are obtained from tabulated moments of the usual range distributions of the parent and daughter atoms. The daughter nuclei are nearly symmetrically distributed about the mean implantation depth with a variance slightly greater than a third of the square of the mean range of the daughter. The distribution may be represented by a Pearson Type II distribution or by the form exp( - a ) x 1P), with parameter p > 2. An example is given.
In the study of nuclei far from stability, it is often desirable to know the distribution of daughter nuclei originating from a parent that has been implanted into a host material with an on-line isotope separator. Implanted-ion depth distributions are usually calculated
via their moments, and tabulations of calculated moments are available (see, for example, ref. [l]). The moments of the daughter depth distribution can be easily obtained from a convolution of the parent-atom distribution with the depth distribution of the daughters measured from their starting points, as follows (see fig. 1). The parent atom is implanted at normal incidence to a depth xi with a probability density f(q). The daughter recoils at an angle B to the surface normal and goes to a depth x2, relative to xi, with density g(x,, 0) Tabulated moments are for distributions with 0 = 0. Averaging over all recoil angles 19, the daughter depth distribution is h(x)
= /dxf(x,)/dx,/d6’ Xsin Bg(x,,
Taking moments, h,=
0)8(x-x1
-x2).
(I)
i.e.,
M dxx”h(x), / -CC
with f, and g, defined
hn= i
similarly,
(:)fn-rgro,
r=O
where g,, is the 1= 0 Legendre coefficient of the moment g,, from the B integration, and is nonzero only for even r. Thus moments through the fourth are hi =fi, hJ=f3+3fieo,
Fig. 1. The parent nucleus is implanted to a depth xi and the daughter nucleus, recoiling at an angle 8, travels through a further depth x2.
h,=f,+g,o, h,=f,+6fs,o+gm.
(3)
It is convenient to work in units in which g,, the mean range the daughter would have if implanted at normal incidence, is unity. Denoting central moments,
K.B. Winterbon / Recoil distributions i.e.,
moments
about the mean, by F,, G,, and H,,, we
have H2=F2+f+G10,
50 keV ‘~~LLI+ C 137 keV ‘fi:Trn + C
H4=F4+6F2(f+G2,,)
(4) + :G,, + GW).
Typically both parent and daughter atoms will be very much heavier than the target atoms, and both energies will be relatively low, well down in the elasticscattering regime, so to a good approximation the two distributions f and g will differ only in the length scale. Hence in particular, writing the ratio ft/gt = r, F, = r”G,,, and so Hz=:+G,,+r2G2.
(5)
Usually G, is of order 0.1 [l] and often r is rather smaller than 1, in which case the straggling H2 is dominated by the spreading of the daughter, as is to be expected. The skewness of f is typically [l] about 0.3, so F3 = 0.3F,3/=, and the skewness of h is HJH;‘=
= 0.3( F2,‘H2)3’2 -=z0.3
(6)
and may usually be neglected. The kurtosis of F is F,/F: and is typically [l] slightly greater than 3, the value for a Gaussian. The kurtosis of h is H,/Hi = 3 + ( H4 - 3H:)/Hi, and, using G, = G, + G,,, H,-3H,2=(-&++G,,+$G,,+G,-3G;,) +F4-
3F,2.
(7)
In this expression the G,, term is small and may be bounded by G, and 0, while the Ga - 3G& and F4 3F,2 terms are negligible. Tabulated moments [l] include straggling, i.e.,
Mean range Relative straggling Relative transverse straggling Skewness Kurtosis
and relative transverse
straggling,
4.66 pg/cm* 0.0864
9.63 pg/cm* 0.0868
0.443 0.312 3.078
0.428 0.291 3.045
nucleus, ‘ETm, is given a recoil energy of 137 keV. Range moments of the two distributions, interpolated from the’tables of ref. [l], are given in table 1. The scaling factor r = 0.48, and the straggling of h is 0.407, in units of g:, so the relative squared straggling is 1.739. Approximating G,, by G, and 0, respectively, and neglecting the Ga - 3G:o and F4 - 3F: terms, the kurtosis of h is 2.50 or 2.45. The distribution h can be represented within the Pearson system [3,4] as a Type II distribution, of the form (1 - x=/i@)m, with m = (5k - 9)/(6 - 2k) and A2 = 2k/(3 - k), where k is the kurtosis. Here x is measured from the mean of h in units of the straggling. The real distribution would not be expected to be cut off sharply like the Pearson distribution, so it may be better to use a distribution of the form exp( -a 1x 1P) which has variance a- 2’Pr(3/p)r-‘(l/p) and kurtosis k= r(s/p)r(l/p)r-‘(3/p).
(11)
Some values of k = k(p) have been listed in table 2. The indefinite integrals of these distributions are required to determine the number of daughter nuclei emitted from the foil. These integrals are R(y)
dxh(x)/jm
=/’ -m
For the Pearson
F,=%,+F,,,
T = ( F,o -
Table 1 Moments of range distributions, interpolated from ref. [l]
H,=F3,
+ (+ + :G, + fG,
539
in isotope separator targets
dxh(x). -cc
distribution,
R(y)=B,(m+1,1/2)/2B(m+1,1/2),
T, say
(12)
y = 1 - y2/A2 F,2/2)
F,
(8)
t
so
F20/F2 = (1 + 2T)/3
and
Table 2 Kurtosis k vs p [eq. (9)]
F22=F2-F20.
If more accuracy is desired in the higher moments, from moments G,, and G& may be estimated figures in ref. [2] as follows.
the the
G,, = ;( G, + 2M,,TG;‘=),
(9)
where Ml2 is the mixed third moment and Go=
+(G,+
G,~(~M==T+ $M~~T=)),
plotted
in ref. [2],
(10)
where M2= and MO4 are plotted in ref. [2]. Consider as an example ‘ZzLu implanted into a carbon target at 50 keV. Upon a: decay, the daughter
k
P
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
6.00 4.69 3.93 3.42 3.06 2.78 2.56 2.38 2.23 2.11 2.00
540
K.B. Winterbon / Recoil distributions in isotope separator targets
References
and for the exponential,
(13) where the B’s and r’s are complete and incomplete beta and gamma functions, respectively. In an infinite medium, it would be easy, albeit tedious, to extend the above analysis to more intermediate decays, but if any substantial numbers of daughters escape, it is necessary instead after each decay to construct an intermediate distribution, such as the Pearson or exponential, truncate it at the foil surface, and repeat the convolution. I thank Dieter Schardt for proposing this problem, and Vem Koslowsky and Erik Hagberg for comments on the manuscript.
[l] K.B. Winterbon, Ion Implantation Range and Energy Deposition Distributions, vol. 2 (Plenum, New York, 1975). [2] K.B. Winterbon, Radiat. Eff. 30 (1976) 85. [3] ES. Pearson and H.O. Hartley (eds.) Biometrika Tables for Statisticians, vol. II (Cambridge University Press, 1972). [4] W.P. Elderton and N.L. Johnson, Systems of Frequency Curves (Cambridge University Press, 1969).
RECOIL DISTRIBUTIONS
Nuclear Instruments and Methods in Physics Research B31(1988) 538-540 North-Holland, Amsterdam
IN ISOTOPE SEPARATOR
TARGETS
K.B. WINTERBON Atomic Energy of Canada Limited Research Company, Chalk River Nuclear Laboratories,
Chalk River, Ontario, Canada KOJ 1JO
Received 4 January 1988
The depth distribution of recoil daughter nuclei following the radioactive decay of implanted parent nuclei has been calculated. Moments of the depth distribution of the daughter nuclei are obtained from tabulated moments of the usual range distributions of the parent and daughter atoms. The daughter nuclei are nearly symmetrically distributed about the mean implantation depth with a variance slightly greater than a third of the square of the mean range of the daughter. The distribution may be represented by a Pearson Type II distribution or by the form exp( - a ) x 1P), with parameter p > 2. An example is given.
In the study of nuclei far from stability, it is often desirable to know the distribution of daughter nuclei originating from a parent that has been implanted into a host material with an on-line isotope separator. Implanted-ion depth distributions are usually calculated
via their moments, and tabulations of calculated moments are available (see, for example, ref. [l]). The moments of the daughter depth distribution can be easily obtained from a convolution of the parent-atom distribution with the depth distribution of the daughters measured from their starting points, as follows (see fig. 1). The parent atom is implanted at normal incidence to a depth xi with a probability density f(q). The daughter recoils at an angle B to the surface normal and goes to a depth x2, relative to xi, with density g(x,, 0) Tabulated moments are for distributions with 0 = 0. Averaging over all recoil angles 19, the daughter depth distribution is h(x)
= /dxf(x,)/dx,/d6’ Xsin Bg(x,,
Taking moments, h,=
0)8(x-x1
-x2).
(I)
i.e.,
M dxx”h(x), / -CC
with f, and g, defined
hn= i
similarly,
(:)fn-rgro,
r=O
where g,, is the 1= 0 Legendre coefficient of the moment g,, from the B integration, and is nonzero only for even r. Thus moments through the fourth are hi =fi, hJ=f3+3fieo,
Fig. 1. The parent nucleus is implanted to a depth xi and the daughter nucleus, recoiling at an angle 8, travels through a further depth x2.
h,=f,+g,o, h,=f,+6fs,o+gm.
(3)
It is convenient to work in units in which g,, the mean range the daughter would have if implanted at normal incidence, is unity. Denoting central moments,
K.B. Winterbon / Recoil distributions i.e.,
moments
about the mean, by F,, G,, and H,,, we
have H2=F2+f+G10,
50 keV ‘~~LLI+ C 137 keV ‘fi:Trn + C
H4=F4+6F2(f+G2,,)
(4) + :G,, + GW).
Typically both parent and daughter atoms will be very much heavier than the target atoms, and both energies will be relatively low, well down in the elasticscattering regime, so to a good approximation the two distributions f and g will differ only in the length scale. Hence in particular, writing the ratio ft/gt = r, F, = r”G,,, and so Hz=:+G,,+r2G2.
(5)
Usually G, is of order 0.1 [l] and often r is rather smaller than 1, in which case the straggling H2 is dominated by the spreading of the daughter, as is to be expected. The skewness of f is typically [l] about 0.3, so F3 = 0.3F,3/=, and the skewness of h is HJH;‘=
= 0.3( F2,‘H2)3’2 -=z0.3
(6)
and may usually be neglected. The kurtosis of F is F,/F: and is typically [l] slightly greater than 3, the value for a Gaussian. The kurtosis of h is H,/Hi = 3 + ( H4 - 3H:)/Hi, and, using G, = G, + G,,, H,-3H,2=(-&++G,,+$G,,+G,-3G;,) +F4-
3F,2.
(7)
In this expression the G,, term is small and may be bounded by G, and 0, while the Ga - 3G& and F4 3F,2 terms are negligible. Tabulated moments [l] include straggling, i.e.,
Mean range Relative straggling Relative transverse straggling Skewness Kurtosis
and relative transverse
straggling,
4.66 pg/cm* 0.0864
9.63 pg/cm* 0.0868
0.443 0.312 3.078
0.428 0.291 3.045
nucleus, ‘ETm, is given a recoil energy of 137 keV. Range moments of the two distributions, interpolated from the’tables of ref. [l], are given in table 1. The scaling factor r = 0.48, and the straggling of h is 0.407, in units of g:, so the relative squared straggling is 1.739. Approximating G,, by G, and 0, respectively, and neglecting the Ga - 3G:o and F4 - 3F: terms, the kurtosis of h is 2.50 or 2.45. The distribution h can be represented within the Pearson system [3,4] as a Type II distribution, of the form (1 - x=/i@)m, with m = (5k - 9)/(6 - 2k) and A2 = 2k/(3 - k), where k is the kurtosis. Here x is measured from the mean of h in units of the straggling. The real distribution would not be expected to be cut off sharply like the Pearson distribution, so it may be better to use a distribution of the form exp( -a 1x 1P) which has variance a- 2’Pr(3/p)r-‘(l/p) and kurtosis k= r(s/p)r(l/p)r-‘(3/p).
(11)
Some values of k = k(p) have been listed in table 2. The indefinite integrals of these distributions are required to determine the number of daughter nuclei emitted from the foil. These integrals are R(y)
dxh(x)/jm
=/’ -m
For the Pearson
F,=%,+F,,,
T = ( F,o -
Table 1 Moments of range distributions, interpolated from ref. [l]
H,=F3,
+ (+ + :G, + fG,
539
in isotope separator targets
dxh(x). -cc
distribution,
R(y)=B,(m+1,1/2)/2B(m+1,1/2),
T, say
(12)
y = 1 - y2/A2 F,2/2)
F,
(8)
t
so
F20/F2 = (1 + 2T)/3
and
Table 2 Kurtosis k vs p [eq. (9)]
F22=F2-F20.
If more accuracy is desired in the higher moments, from moments G,, and G& may be estimated figures in ref. [2] as follows.
the the
G,, = ;( G, + 2M,,TG;‘=),
(9)
where Ml2 is the mixed third moment and Go=
+(G,+
G,~(~M==T+ $M~~T=)),
plotted
in ref. [2],
(10)
where M2= and MO4 are plotted in ref. [2]. Consider as an example ‘ZzLu implanted into a carbon target at 50 keV. Upon a: decay, the daughter
k
P
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
6.00 4.69 3.93 3.42 3.06 2.78 2.56 2.38 2.23 2.11 2.00
540
K.B. Winterbon / Recoil distributions in isotope separator targets
References
and for the exponential,
(13) where the B’s and r’s are complete and incomplete beta and gamma functions, respectively. In an infinite medium, it would be easy, albeit tedious, to extend the above analysis to more intermediate decays, but if any substantial numbers of daughters escape, it is necessary instead after each decay to construct an intermediate distribution, such as the Pearson or exponential, truncate it at the foil surface, and repeat the convolution. I thank Dieter Schardt for proposing this problem, and Vem Koslowsky and Erik Hagberg for comments on the manuscript.
[l] K.B. Winterbon, Ion Implantation Range and Energy Deposition Distributions, vol. 2 (Plenum, New York, 1975). [2] K.B. Winterbon, Radiat. Eff. 30 (1976) 85. [3] ES. Pearson and H.O. Hartley (eds.) Biometrika Tables for Statisticians, vol. II (Cambridge University Press, 1972). [4] W.P. Elderton and N.L. Johnson, Systems of Frequency Curves (Cambridge University Press, 1969).