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Milne's problem in backscattering Mössbauer spectroscopy
Nuclear Instruments North-Holland
and Methods
MILNE’S PROBLEM
in Physics
Research
B44 (1989) 211-215
IN BACKSCATI’ERING
MijSSBAUER
211
SPECTROSCOPY
G.N. BELOZERSKII Chemical Department
of Leningrad
University, Leningrad,
USSR
A.A. DOZOROV Yaroslaol Institute of Technology, Yaroslaul, USSR Received
6 February
1989 and in revised form 23 June 1989
Transmission of Mossbatter radiation from a certain depth of the scatterer to its surface equations for transmission functions have been formulated for recoil- and recoilless quanta. equations are discussed and some results of the calculations are given.
1. Introduction The classical Milne’s problem considers the radiation transmitted from a certain depth of the sample up to its surface [l-3]. Originally this problem was formulated for star radiation reaching the surface of the Earth and then it found application for a number of other phenomena. A similar problem exists in studies of the escape of neutrons from an object under investigation. In the classical Milne’s problem it is assumed that there is one scattering channel. Hence the transmission function T(x) of radiation from a depth x can be described by a Fredholm integral equation of the second kind T(x)
= T,(x)
+Xjo”K(
Ix-YI)Z-(Y)
dy>
(1)
where E,( 1x -y I) is the kernel of the integral equation, h is the known parameter of the problem. The transmission function T,(x) is the probability of the radiation to escape from the depth x when re-scattering of radiation inside the substance is not accounted for. In scattering spectroscopy with detection of electromagnetic waves at an angle B with respect to the surface of a sample, the transmission function is T,(x) = exp( -px/cos 8), where p is the linear attenuation coefficient of the radiation. In the case of resonant scattering the escaping y-radiation consists of elastically and inelastically scattered quanta. Hence there exist two channels of resonant scattering. There are also 3 channels of nonresonant scattering. The intensity of the scattered radiation has been calculated previously by several authors [4-81 without taking into account rescattering processes. The problem of multiple Mossbauer scattering has been investigated by Mkrtchyan et al. [9,10]. These authors considered the problem of 0168-583X/89/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
has been considered. The integral The approximate solutions of the
Mijssbauer radiation scattering at an angle 0 inside a sample of thickness d. It has been shown that in the case of large values of the reemission probability (the internal conversion coefficient a < 1) or a large effective thickness, the problem cannot be limited only to single scattering. An essential simplification has been made in ref. [lo]. The re-scattering processes have been considered only for the elastic resonant channel of scattering with f’= 1 and pe = 0, where f’ is the Mossbauer factor for the scatterer and pL, is the total linear coefficient of nonresonant absorption in the sample. To calculate the transmission functions, the authors have taken into account that the solution of integral equation (1) can be given with the aid of known tabulated Ambartsumyan’s functions [3]. In this case if one wants to find the transmission functions it is necessary to do a numerical integration. In the one-dimensional model case it becomes possible to find an exact expression for the transmission function. We will try to consider Milne’s problem and take into account all possible channels and multiple scattering for the case of backscattering Mossbauer spectroscopy with detection of the y-quanta.
2. Theory Let y-rays from a Mijssbauer source be normally incident on the surface of an uniform bulk sample and the scattered -r-radiation escape at any angles to the surface. Only the resonant radiation will be taken into account whose fraction is f, since the nonresonant fraction contributes to the background only. The scattered y-radiation will be detected with 100% ef-
212
G.N. Belozerskii, A.A. Dororou / Mhe’s
problem in backscattering
ficiency. Assume that the linewidths in the source and in the sample are equal (r/2 = 1). Then taking the intensity of the source as unity MGssbauer radiation will have a Lorentzian distribution
where c = E,, - S - E,,u/c, E, is the position of the centroid of the source line at a zero velocity u, S is the isomer shift, c is the velocity of light. The incident y-quanta reach the depth x with the probability exp( - p( E)x). The total linear coefficient of interaction p(E) is a sum of two contributions: resonant scattering p,(E) = pr/(l + E*) and nonresonant scattering by the electrons of atom pLe, the total coefficient being p(E) = pr( E) + pe. The photoeffect, Rayleigh and Compton scatterings determine the nonresonant part II, = pph + pa + f.~,. The beam of scattered y-quanta is I(e)=/jo”dx/_+:L(E-r)
exp[-p(E)x]
$J-$f’T,(x,
E) + (I
-f’)T,(x)l
+~afaTr(x, E)
+I(1 -fIt)CLa+ PclT,(x)j
dE>
(2)
where (r is the internal conversion coefficient, f’ is the Mossbauer factor for the scatterer, fa is the probability of Rayleigh elastic scattering. The functions T,(x, E) and T,(x) are the escape probabilities of elastically and nonelastically scattered radiation from the depth x to the sample surface. They are dependent on both the probability of radiation to directly escape the depth x to the surface and the probability of re-scattering of the quanta in the vicinity of a certain point at the depth y followed by generation of a resonant or nonresonant quantum and the escape of this quantum to the sample surface. For example, if a resonant quantum is rescattered, the probability of the process is proportional to the following quantity
&P(E)T(Y. E) exd-p(E)(x /Ices
-y)/cos 01 dy
e I.
Integrating over the angle, an integral tained to be used to find the T,(x) functions T,(x)
= 0.5
E2(w)
+
ht
+
equation is oband T,(x, E)
Mijssbauer spectroscopy
T,(x, El
X/%EL@)
Ix -Y I)T,(Y,
0
E) dy
where exp( E,(Y) = J;w
-yt)
dt/t”
is the integral exponent. In (4) T,(y) is the function obtained by means of (3). The first term in braces of (4) gives the escape probability of an elastically scattered resonant y-quantum at a depth x without any intermediate interactions. The second and third terms account for the fact that this quantum may be elastically or inelastically re-scattered at a depth y and then may escape. A conventional detector, due to a poor energy resolution will also detect the Compton-scattered y-quanta. In (3) and (4) the scattering was assumed to be isotropic. This assumption is, of course, not strict, but in the absence of level splitting, it is justified for most of the applications. If scattering at an angle 0 is of interest, eq. (3) and (4) can be simplified. In this case one must replace the integral exponent E*(z) by the ordinary exponent exp( -z/cos 8). If, in addition, in eq. (3) and (4) we take f’ = 1 and pa = pc = 0, we will have the equations of ref. [lo]. Eq. (3) represents the classical Milne’s problem of radiation leaving the solid. Eq. (4) is a generalization of the classical Milne’s problem for the case of Mossbauer radiation. The problem can be formulated as follows. At a certain point in the scatterer a resonant quantum has been produced. What is the probability of the y-quantum to leave the surface of the scatterer with any energy? For samples which are highly enriched in the resonant isotope, i.e. for the cases pr B pe, one can introduce the function T(x,
E)
=f’T,(x,
E) + (1 -f’)T,(x),
which determines the probability for recoil- and recoilless y-quanta to come back to the sample surface. The scattered intensity (2) may be transformed now into
/JCL,)
[
X
/ 0
m-%(~eI~-~I)~n(~)
dy
(4)
1 9
(3) xexp[-I1(E)x]LLr(E)T(x,
E) dE.
G. N. Belozerskii,
The escape probability T(x, fies the equation T(x,
E) =0.5[(1
A.A. Dozorou / Milne’s problem in backscattering
E) for a y-quantum satis-
+f“%(P(E)x)l
-f’)J%(/P)
+K]%(IL@) IX-Yi)T(Y,E) dy. 0 (5)
K = O.Sf’&(
Q/(1
+ Cx).
-f’)&(t)
+0.5ah,
/
+f’E,(at)]
mE,(alt-t’l)T(t’,
dt’,
E)
- l)/[a(l
E) = T,(t,
+ a)].
E) + OSUX,
X /0
==kl(aIt-t’\)Tk(t’,
dt’.
E)
(9)
under a natural assumption that T,(t)
=0.5[(1
-f’)E,(t)
+f’E,(at)].
(10)
If all the T, (k = 1, 2 . . . ) are subsequently substituted into (9) and all the Tk(t, E) values of the resulting equation are replaced with the maximal values, a power series is produced for the parameter p = 0.5ah
0 (7)
where A, =f’(a
In this section we will try to obtain the simplest and most convenient approximate solutions of eq. (7). The integral equation (7) for the three-dimensional problem will be resolved approximately by using two methods. The first one is based on an iterative procedure
(6)
The inhomogeneous second kind Fredholm eqs. (l), (3), (4) and (5) are all of the same type. For simplicity, only eq. (5) will be discussed hereafter. Let us introduce the dimensionless distance t = pex and the parameter a = p( E)//.L,. Then eq. (5) will be
T(t, E) =0.5[(1
4. The approximate solution of the integral equation
Tk+,(t,
The K parameter is
213
MGssbauer spectroscopy
mE,(aIt-t’l)dt’.
rO/
The iterative process is convergent if the relation p -C 1 is satisfied. Hence the estimate follows k
TI(t,
E)
< T,+,(t,
E)
< T,(t,
E) +max
T,.
3. The exact solution of the integral equation
c
A?.
WI=1
(11) The method of solution of the integral equation (5) is known [3]. But the transmission function can be given only through the known tabulated functions and a numerical integration is necessary. The exact solution of the integral equation (5) has not been obtained so far, due to mathematical difficulties. The exact solution of (5) can be obtained only in a hypothetical one-dimensional case, i.e. considering yquanta backscattered along the surface normal. Although this situation is not realised in practice, it is very instructive. In this case the integral exponents E,(z) in (7) should be substituted for the ordinary exponents exp( - z) and we have T(t,
E) = 0.5[(1 -f’) +0.5ah,
/0
exp( -t)
+f’
exp( -at)]
mexp( ---a 1t - t’ I)T(t’,
E)
dt’.
An exact solution of this equation is exp(-t)+Bexp(-/It),
T(t, E)=A
(8)
where
The relation h, c 1 is the condition for the iterative process to converge. Since p,(E) is always less than p(E), the relation above is fulfilled if f’/(l + a) -C 1. The convergence parameter f’/(l + a) is 0.08 for 57Fe. This means that for the 57Fe Miissbauer effect the role of multiple re-scatterings should not be large. If only single re-scattering is taken into account the transmission function is written in the form T2(t.
X
a2 - l)/( p2 - l),
B =/‘a/(/? + 0) - 0.5(1 -f’)( /(P’P=ClJ_.
I>,
a + l)(u
E)
/0
+0.5aX,
mE,(aIt-t’I)T,(t’,
E)
dt’.
(12)
For the general case the analytical integration in (9) or (12) can hardly be made. But it is very easy for t = 0. When the re-scattering of y-quanta is not accounted for we have T,(O) = 0.5. Accounting for single re-scattering, we will obtain a transmission function in the second approximation (12). As /0
?E,(z)E2(bz) =
A = 0.5(1 -f’)(
E) = T,(t,
dz
0.5[1 + ln(1 + b)/b - b ln(1 + l/b)],
we will have - p>
T,(O)
= T,(O){1
-ln(l
+ 0.25X,[(l
+ a)/a)
-f’)(l
+ a ln(1 + l/u)
+f’]}.
In the extreme cases of minimum and maximum enrich-
214
G.N. Belozerskii, A.A. Dororov / M&e’s problem in backscattering
ment of the sample obtain T*(O, a -+ 1) = z-,(O)[l
in the resonant
isotope
Mijssbauer spectroscopy
we will
T 0.5
i OSf’(a
- 1),(1+
1\
(13) 0.4
and T,(O, a z+ 1) = T,(O)[l
+ O.Sf’(l
- O.Sf')/(l +
/
This is equivalent T(r,
E) = jrt(t,
E)/[l
-X,(1
0.2
i \ I \ \
0.1
'\
a=11 \ \
\
\
\
\
\
. -.
i--; +A,(1
a=l.l 1 \ \ \ \ \,
-OSE,(at)].
to E)[l
:
0.3 ' 1 I I
Both (13) and (14) show that the contribution of rescattering processes for “Fe is about 2.5%. The second method of approximate solution of (7) is based on the fact that the integral exponent E, (a 1t’ f I) increases quickly in the vicinity of I’ = t and the probability value T(t) changes smoothly enough. Then in (7) as a first approximation one can take T( t’) = T(t). In this case an approximate solution can be obtained E) = T,(t,
‘,
\
a)]. 04)
T(t,
\ II \
CY)]
-OSE,(at))f
1
\
\
-c ?--'
Fig. 1.
(15)
because A, -C 1. Let us suppose that scattering takes place on the surface. If the enrichment is low (a --) 1) then the approximation (15) will coincide with the exact expression (13). If the enrichment is high (a Z+ 1) then T(0,
E) = Y&(O)[l + 0.5f’/(l
+
of numerical calculations are presented in fig. 1 and table 1. The figure presents one-dimensional functions [eq. (8)] (dashed curves) and three-dimensional functions (12) (solid curves), the parameter u being 1.1 and 11. The table gives some results of the transmission functions. T is the exact solution (8) for a one-dimensional case. Tl is the approximate solution (10) for a one- or three-Dimensions problem calculated without taking into account the re-scattering processes. T’ is the approximate solution (12) for a one- or three-dimensional problem if the single re-scattering processes is accounted for. All numerical calculations were made with f’ = 0.7 and a = 8.2, i.e. with respect to the isotope 57Fe. It should be noted that if the parameter a decreases, the problem parameter X, increases and the contribution of the re-scattering processes is more significant.
a)].
The result of these calculations differs from that obtained with the exact expression (14) but not drastically. Expression (14) also satisfies the one-dimensional problem. The exact solution (8) makes it possible to compare the exact and approximate solutions and allows the error in using the approximate solutions for the one-dimensional problem to be estimated. Numerical calculations performed using (S), (10) and (12) have shown that the contribution of the re-scattering increases with the depth of the layer where the scattering occurs and with increase of the p(E) value. In the essential range where x changes, i.e. where T(x, of double re-scattering E) > 0.1, the cont~bution processes should be accounted for only when the accuracy required for the solution is better than 1%. Results
Table 1 The exact and approximate transmission functions
ID
3D
a =11
a =I.1
Dimension of problem
t =O.l
0.5
1
0.1
0.5
1
r, 7-l
0.4502 0.4502 0.4493
0.2940 0.2940 0.2929
0.1726 0.1726 0.1717
0.2670 0.2662 0.2522
0.0995 0.0990 0.0924
0.0593 0.0590 0.0552
T2 G
0.3561 0.3550
0.1549 0.1540
0.0617 0.0672
0.1636 0.1533
0.0526 0.0492
0.0237 0.0223
T
G. N. Belozerskii, A.A. Dozorov / Milne’s problem in backscattering Miissbauer spectroscopy
References [l] P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, New York, 1953). [2] B. Noble, Methods based on the Wiener-Hopf Technique (Pergamon, London, New York, 1958). [3] V.V. Sobolev, Kurs teoretitcheskoi astrofiziki (Nauka, Moscow, 1985). [4] P. Debrunner and R.J. Morrison, Rev. Sci. Instr. 36 (1965) 145.
215
[S] B. Balko and G.R. Hoy, Phys. Rev. BlO (1974) 36. [6] B. Balko and G.R. Hoy, Phys. Rev. BlO (1974) 4523. [7] J.J. Bara and B.F. Bogacz, Nucl. Instr. and Meth. 186 (1981) 561. (81 D.C. Champeney. Rep. Progr. Phys. 42 (1979) 1017. [9] A.R. Mkrtchyan and R.G. Gabrielyan, Astrofizika 20 (1984) 607. [lo] A.R. Mkrtchyan, R.G. Gabrielyan, A.H. Martirossyan and A.Sh. Grigoryan, Phys. Status Solidi (b) 139 (1987) 583.
and Methods
MILNE’S PROBLEM
in Physics
Research
B44 (1989) 211-215
IN BACKSCATI’ERING
MijSSBAUER
211
SPECTROSCOPY
G.N. BELOZERSKII Chemical Department
of Leningrad
University, Leningrad,
USSR
A.A. DOZOROV Yaroslaol Institute of Technology, Yaroslaul, USSR Received
6 February
1989 and in revised form 23 June 1989
Transmission of Mossbatter radiation from a certain depth of the scatterer to its surface equations for transmission functions have been formulated for recoil- and recoilless quanta. equations are discussed and some results of the calculations are given.
1. Introduction The classical Milne’s problem considers the radiation transmitted from a certain depth of the sample up to its surface [l-3]. Originally this problem was formulated for star radiation reaching the surface of the Earth and then it found application for a number of other phenomena. A similar problem exists in studies of the escape of neutrons from an object under investigation. In the classical Milne’s problem it is assumed that there is one scattering channel. Hence the transmission function T(x) of radiation from a depth x can be described by a Fredholm integral equation of the second kind T(x)
= T,(x)
+Xjo”K(
Ix-YI)Z-(Y)
dy>
(1)
where E,( 1x -y I) is the kernel of the integral equation, h is the known parameter of the problem. The transmission function T,(x) is the probability of the radiation to escape from the depth x when re-scattering of radiation inside the substance is not accounted for. In scattering spectroscopy with detection of electromagnetic waves at an angle B with respect to the surface of a sample, the transmission function is T,(x) = exp( -px/cos 8), where p is the linear attenuation coefficient of the radiation. In the case of resonant scattering the escaping y-radiation consists of elastically and inelastically scattered quanta. Hence there exist two channels of resonant scattering. There are also 3 channels of nonresonant scattering. The intensity of the scattered radiation has been calculated previously by several authors [4-81 without taking into account rescattering processes. The problem of multiple Mossbauer scattering has been investigated by Mkrtchyan et al. [9,10]. These authors considered the problem of 0168-583X/89/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
has been considered. The integral The approximate solutions of the
Mijssbauer radiation scattering at an angle 0 inside a sample of thickness d. It has been shown that in the case of large values of the reemission probability (the internal conversion coefficient a < 1) or a large effective thickness, the problem cannot be limited only to single scattering. An essential simplification has been made in ref. [lo]. The re-scattering processes have been considered only for the elastic resonant channel of scattering with f’= 1 and pe = 0, where f’ is the Mossbauer factor for the scatterer and pL, is the total linear coefficient of nonresonant absorption in the sample. To calculate the transmission functions, the authors have taken into account that the solution of integral equation (1) can be given with the aid of known tabulated Ambartsumyan’s functions [3]. In this case if one wants to find the transmission functions it is necessary to do a numerical integration. In the one-dimensional model case it becomes possible to find an exact expression for the transmission function. We will try to consider Milne’s problem and take into account all possible channels and multiple scattering for the case of backscattering Mossbauer spectroscopy with detection of the y-quanta.
2. Theory Let y-rays from a Mijssbauer source be normally incident on the surface of an uniform bulk sample and the scattered -r-radiation escape at any angles to the surface. Only the resonant radiation will be taken into account whose fraction is f, since the nonresonant fraction contributes to the background only. The scattered y-radiation will be detected with 100% ef-
212
G.N. Belozerskii, A.A. Dororou / Mhe’s
problem in backscattering
ficiency. Assume that the linewidths in the source and in the sample are equal (r/2 = 1). Then taking the intensity of the source as unity MGssbauer radiation will have a Lorentzian distribution
where c = E,, - S - E,,u/c, E, is the position of the centroid of the source line at a zero velocity u, S is the isomer shift, c is the velocity of light. The incident y-quanta reach the depth x with the probability exp( - p( E)x). The total linear coefficient of interaction p(E) is a sum of two contributions: resonant scattering p,(E) = pr/(l + E*) and nonresonant scattering by the electrons of atom pLe, the total coefficient being p(E) = pr( E) + pe. The photoeffect, Rayleigh and Compton scatterings determine the nonresonant part II, = pph + pa + f.~,. The beam of scattered y-quanta is I(e)=/jo”dx/_+:L(E-r)
exp[-p(E)x]
$J-$f’T,(x,
E) + (I
-f’)T,(x)l
+~afaTr(x, E)
+I(1 -fIt)CLa+ PclT,(x)j
dE>
(2)
where (r is the internal conversion coefficient, f’ is the Mossbauer factor for the scatterer, fa is the probability of Rayleigh elastic scattering. The functions T,(x, E) and T,(x) are the escape probabilities of elastically and nonelastically scattered radiation from the depth x to the sample surface. They are dependent on both the probability of radiation to directly escape the depth x to the surface and the probability of re-scattering of the quanta in the vicinity of a certain point at the depth y followed by generation of a resonant or nonresonant quantum and the escape of this quantum to the sample surface. For example, if a resonant quantum is rescattered, the probability of the process is proportional to the following quantity
&P(E)T(Y. E) exd-p(E)(x /Ices
-y)/cos 01 dy
e I.
Integrating over the angle, an integral tained to be used to find the T,(x) functions T,(x)
= 0.5
E2(w)
+
ht
+
equation is oband T,(x, E)
Mijssbauer spectroscopy
T,(x, El
X/%EL@)
Ix -Y I)T,(Y,
0
E) dy
where exp( E,(Y) = J;w
-yt)
dt/t”
is the integral exponent. In (4) T,(y) is the function obtained by means of (3). The first term in braces of (4) gives the escape probability of an elastically scattered resonant y-quantum at a depth x without any intermediate interactions. The second and third terms account for the fact that this quantum may be elastically or inelastically re-scattered at a depth y and then may escape. A conventional detector, due to a poor energy resolution will also detect the Compton-scattered y-quanta. In (3) and (4) the scattering was assumed to be isotropic. This assumption is, of course, not strict, but in the absence of level splitting, it is justified for most of the applications. If scattering at an angle 0 is of interest, eq. (3) and (4) can be simplified. In this case one must replace the integral exponent E*(z) by the ordinary exponent exp( -z/cos 8). If, in addition, in eq. (3) and (4) we take f’ = 1 and pa = pc = 0, we will have the equations of ref. [lo]. Eq. (3) represents the classical Milne’s problem of radiation leaving the solid. Eq. (4) is a generalization of the classical Milne’s problem for the case of Mossbauer radiation. The problem can be formulated as follows. At a certain point in the scatterer a resonant quantum has been produced. What is the probability of the y-quantum to leave the surface of the scatterer with any energy? For samples which are highly enriched in the resonant isotope, i.e. for the cases pr B pe, one can introduce the function T(x,
E)
=f’T,(x,
E) + (1 -f’)T,(x),
which determines the probability for recoil- and recoilless y-quanta to come back to the sample surface. The scattered intensity (2) may be transformed now into
/JCL,)
[
X
/ 0
m-%(~eI~-~I)~n(~)
dy
(4)
1 9
(3) xexp[-I1(E)x]LLr(E)T(x,
E) dE.
G. N. Belozerskii,
The escape probability T(x, fies the equation T(x,
E) =0.5[(1
A.A. Dozorou / Milne’s problem in backscattering
E) for a y-quantum satis-
+f“%(P(E)x)l
-f’)J%(/P)
+K]%(IL@) IX-Yi)T(Y,E) dy. 0 (5)
K = O.Sf’&(
Q/(1
+ Cx).
-f’)&(t)
+0.5ah,
/
+f’E,(at)]
mE,(alt-t’l)T(t’,
dt’,
E)
- l)/[a(l
E) = T,(t,
+ a)].
E) + OSUX,
X /0
==kl(aIt-t’\)Tk(t’,
dt’.
E)
(9)
under a natural assumption that T,(t)
=0.5[(1
-f’)E,(t)
+f’E,(at)].
(10)
If all the T, (k = 1, 2 . . . ) are subsequently substituted into (9) and all the Tk(t, E) values of the resulting equation are replaced with the maximal values, a power series is produced for the parameter p = 0.5ah
0 (7)
where A, =f’(a
In this section we will try to obtain the simplest and most convenient approximate solutions of eq. (7). The integral equation (7) for the three-dimensional problem will be resolved approximately by using two methods. The first one is based on an iterative procedure
(6)
The inhomogeneous second kind Fredholm eqs. (l), (3), (4) and (5) are all of the same type. For simplicity, only eq. (5) will be discussed hereafter. Let us introduce the dimensionless distance t = pex and the parameter a = p( E)//.L,. Then eq. (5) will be
T(t, E) =0.5[(1
4. The approximate solution of the integral equation
Tk+,(t,
The K parameter is
213
MGssbauer spectroscopy
mE,(aIt-t’l)dt’.
rO/
The iterative process is convergent if the relation p -C 1 is satisfied. Hence the estimate follows k
TI(t,
E)
< T,+,(t,
E)
< T,(t,
E) +max
T,.
3. The exact solution of the integral equation
c
A?.
WI=1
(11) The method of solution of the integral equation (5) is known [3]. But the transmission function can be given only through the known tabulated functions and a numerical integration is necessary. The exact solution of the integral equation (5) has not been obtained so far, due to mathematical difficulties. The exact solution of (5) can be obtained only in a hypothetical one-dimensional case, i.e. considering yquanta backscattered along the surface normal. Although this situation is not realised in practice, it is very instructive. In this case the integral exponents E,(z) in (7) should be substituted for the ordinary exponents exp( - z) and we have T(t,
E) = 0.5[(1 -f’) +0.5ah,
/0
exp( -t)
+f’
exp( -at)]
mexp( ---a 1t - t’ I)T(t’,
E)
dt’.
An exact solution of this equation is exp(-t)+Bexp(-/It),
T(t, E)=A
(8)
where
The relation h, c 1 is the condition for the iterative process to converge. Since p,(E) is always less than p(E), the relation above is fulfilled if f’/(l + a) -C 1. The convergence parameter f’/(l + a) is 0.08 for 57Fe. This means that for the 57Fe Miissbauer effect the role of multiple re-scatterings should not be large. If only single re-scattering is taken into account the transmission function is written in the form T2(t.
X
a2 - l)/( p2 - l),
B =/‘a/(/? + 0) - 0.5(1 -f’)( /(P’P=ClJ_.
I>,
a + l)(u
E)
/0
+0.5aX,
mE,(aIt-t’I)T,(t’,
E)
dt’.
(12)
For the general case the analytical integration in (9) or (12) can hardly be made. But it is very easy for t = 0. When the re-scattering of y-quanta is not accounted for we have T,(O) = 0.5. Accounting for single re-scattering, we will obtain a transmission function in the second approximation (12). As /0
?E,(z)E2(bz) =
A = 0.5(1 -f’)(
E) = T,(t,
dz
0.5[1 + ln(1 + b)/b - b ln(1 + l/b)],
we will have - p>
T,(O)
= T,(O){1
-ln(l
+ 0.25X,[(l
+ a)/a)
-f’)(l
+ a ln(1 + l/u)
+f’]}.
In the extreme cases of minimum and maximum enrich-
214
G.N. Belozerskii, A.A. Dororov / M&e’s problem in backscattering
ment of the sample obtain T*(O, a -+ 1) = z-,(O)[l
in the resonant
isotope
Mijssbauer spectroscopy
we will
T 0.5
i OSf’(a
- 1),(1+
1\
(13) 0.4
and T,(O, a z+ 1) = T,(O)[l
+ O.Sf’(l
- O.Sf')/(l +
/
This is equivalent T(r,
E) = jrt(t,
E)/[l
-X,(1
0.2
i \ I \ \
0.1
'\
a=11 \ \
\
\
\
\
\
. -.
i--; +A,(1
a=l.l 1 \ \ \ \ \,
-OSE,(at)].
to E)[l
:
0.3 ' 1 I I
Both (13) and (14) show that the contribution of rescattering processes for “Fe is about 2.5%. The second method of approximate solution of (7) is based on the fact that the integral exponent E, (a 1t’ f I) increases quickly in the vicinity of I’ = t and the probability value T(t) changes smoothly enough. Then in (7) as a first approximation one can take T( t’) = T(t). In this case an approximate solution can be obtained E) = T,(t,
‘,
\
a)]. 04)
T(t,
\ II \
CY)]
-OSE,(at))f
1
\
\
-c ?--'
Fig. 1.
(15)
because A, -C 1. Let us suppose that scattering takes place on the surface. If the enrichment is low (a --) 1) then the approximation (15) will coincide with the exact expression (13). If the enrichment is high (a Z+ 1) then T(0,
E) = Y&(O)[l + 0.5f’/(l
+
of numerical calculations are presented in fig. 1 and table 1. The figure presents one-dimensional functions [eq. (8)] (dashed curves) and three-dimensional functions (12) (solid curves), the parameter u being 1.1 and 11. The table gives some results of the transmission functions. T is the exact solution (8) for a one-dimensional case. Tl is the approximate solution (10) for a one- or three-Dimensions problem calculated without taking into account the re-scattering processes. T’ is the approximate solution (12) for a one- or three-dimensional problem if the single re-scattering processes is accounted for. All numerical calculations were made with f’ = 0.7 and a = 8.2, i.e. with respect to the isotope 57Fe. It should be noted that if the parameter a decreases, the problem parameter X, increases and the contribution of the re-scattering processes is more significant.
a)].
The result of these calculations differs from that obtained with the exact expression (14) but not drastically. Expression (14) also satisfies the one-dimensional problem. The exact solution (8) makes it possible to compare the exact and approximate solutions and allows the error in using the approximate solutions for the one-dimensional problem to be estimated. Numerical calculations performed using (S), (10) and (12) have shown that the contribution of the re-scattering increases with the depth of the layer where the scattering occurs and with increase of the p(E) value. In the essential range where x changes, i.e. where T(x, of double re-scattering E) > 0.1, the cont~bution processes should be accounted for only when the accuracy required for the solution is better than 1%. Results
Table 1 The exact and approximate transmission functions
ID
3D
a =11
a =I.1
Dimension of problem
t =O.l
0.5
1
0.1
0.5
1
r, 7-l
0.4502 0.4502 0.4493
0.2940 0.2940 0.2929
0.1726 0.1726 0.1717
0.2670 0.2662 0.2522
0.0995 0.0990 0.0924
0.0593 0.0590 0.0552
T2 G
0.3561 0.3550
0.1549 0.1540
0.0617 0.0672
0.1636 0.1533
0.0526 0.0492
0.0237 0.0223
T
G. N. Belozerskii, A.A. Dozorov / Milne’s problem in backscattering Miissbauer spectroscopy
References [l] P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, New York, 1953). [2] B. Noble, Methods based on the Wiener-Hopf Technique (Pergamon, London, New York, 1958). [3] V.V. Sobolev, Kurs teoretitcheskoi astrofiziki (Nauka, Moscow, 1985). [4] P. Debrunner and R.J. Morrison, Rev. Sci. Instr. 36 (1965) 145.
215
[S] B. Balko and G.R. Hoy, Phys. Rev. BlO (1974) 36. [6] B. Balko and G.R. Hoy, Phys. Rev. BlO (1974) 4523. [7] J.J. Bara and B.F. Bogacz, Nucl. Instr. and Meth. 186 (1981) 561. (81 D.C. Champeney. Rep. Progr. Phys. 42 (1979) 1017. [9] A.R. Mkrtchyan and R.G. Gabrielyan, Astrofizika 20 (1984) 607. [lo] A.R. Mkrtchyan, R.G. Gabrielyan, A.H. Martirossyan and A.Sh. Grigoryan, Phys. Status Solidi (b) 139 (1987) 583.