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Inversion of neutron reflection data
PHYSICA
Physica B 190 (1993) 377-382 North-Holland
SDI: 0921-4526(93)E0064-N
Inversion of neutron reflection data R. Lipperheide and G. Reiss Hahn-Meitner-lnstitut and Freie Universitiit Berlin, Germany
H. Fiedeldey and S.A. Sofianos Department of Physics, University of South Africa, Pretoria, South Africa
H. L e e b Institut f iir Kernphysik, Technische Universitiit Wien, Austria Received 4 December 1992
A method is proposed for determining surface profiles from neutron reflection data by inversion. With the reflectivity and the dwell time reDgarded as measured input, the complex reflection coefficient is determined in the form of a rational function of the neutron momentum, for which the scattering potential is known in analytic form. A numerical example is given.
1. Introduction
The specular reflection of neutrons from plane surfaces is employed to obtain information on the profile of the surface, i.e. its structure in a perpendicular direction [1,2]. The quantity usually measured is the reflectivity r(q)= IR(q)[ 2, where R(q) is the complex reflection coefficient as a function of neutron momentum q. In order to extract the profile from the reflection data unambiguously, one needs to know not only the reflectivity but also the phase of the reflection coefficient, ~bR(q) = I m log R(q), as a function of neutron momentum q. This is the age-old phase problem of diffraction analysis [3-5] which has for a long time resisted a general deductive solution, and has so far prevented the application of any genuine inversion method in this field. An interesting proposal has recently been made to remove the phase ambiguity in Born Correspondence to: R. Lipperheide, Hahn-Meitner-Institut, Postfach 390128, D-1000 Berlin 39, Germany.
approximation by introducing a 'reference edge' [6]. A similar idea has been advanced earlier in X-ray diffraction [7]. In the present note we propose an inversion procedure which makes use of the reflectivity data as well as of the dwell time [8,9], which can be measured via absorption [10,11]. As has been shown in a previous paper [12], the dwell time ~-(q) contains the missing information on the reflection phase ~bR(q) which is needed to determine the profile. For the inversion itself we propose a scheme in terms of complex reflection coefficients represented as essentially rational functions of q, which are associated with a class of potentials (surface profiles) whose form is known analytically. A fit of the rational reflection coefficients to the reflectivity data and the dwell time with an appropriate number of free parameters therefore immediately yields the desired potential. In the next section we present the class of potentials which is associated with the rational reflection (and transmission) coefficients to be
0921-4526/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved
R. Lipperheide et al. / Inversion of neutron reflection data
378
We consider a wave incident from the left with momentum q, and correspondingly seek the solution of eq. (1) in the form of the Jost solution f ( q , x ) with the asymptotic boundary condition
used in this investigation. The inversion method in terms of a 'solvable' class of potentials is taken over, with appropriate modifications, from the Bargmann scheme used in the quantum theory of nonrelativistic particle scattering [13,14]. It is closely related to the rational method of Kay [15] for the inversion in one dimension [16,17]. In section 3 we explain the procedure for fitting the parameters of the rational reflection coefficient to the reflectivity and the dwell time. Finally, in section 4, a schematic example is treated numerically for an illustration of the method.
f(q,x)---~exp[iftx]
for x---~ ~ ,
(2)
where ~ = + ~ q 2 _ U. For x < 0, where U 0 = 0, the solution has the form
f(q,x)-
1
To(q)
[e 'qx + R 0 ( q ) e-iqx],
x-<0,
(3)
Specular reflection is essentially one-dimensional scattering by a potential barrier. We begin with the scattering on a given 'background' potential Uo(x), described by the Schr6dinger equation (' ~- d/dx)
where R 0 and T O are the 'background' reflection and transmission coefficients, respectively. We further introduce a second solution of eq. ( i ) , ~b(q, x), which is linearly independent of the Jost solution (2) and which we call the 'regular solution'. It is defined by the following boundary conditions at x --- 0:
~O"(q, x) + [q2 _ Uo(x)]qJ(q, x) = 0 .
th(q,0) = 0 ,
2. The inversion scheme
(1)
q2l
(4)
The background solutions f ( q , x ) and ~b(q,x) taken at the complex momenta /3, and a m, respectively (n, m = 1, 2 . . . . . N), are now used to define, for x > 0, a function
The potential Uo(x ) is generally complex, vanishes to the left of the origin, Uo(x ) = 0 for x < 0, and becomes constant in the substrate to the right, Uo(x ) = 0 for x > £ (cf. fig. 1).
eiqX
~b'(q,0) = - 1 .
uo
T(q)e iqx
R(q)e-iq x
x, : 0
R
xa
x
Fig. 1. One-dimensional neutron scattering by a barrier (specular reflection from a plane surface). The curves labelled U and Uo represent the total and background potentials, respectively. The homogeneous substrate starts at ~ and is represented by the constant potential 0. The points xl, x2 delimit the interval across which the dwell time is calculated.
379
R. Lipperheide et al. / Inversion of neutron reflection data
u(4 + arO -~-_
F(q,x)=f(q,x) X
Ilam.(~) - amq(~)f(~., x) lf(q, x) ll
(5)
II~m.(x)lf
where d m = ~ - U , fln=W'fl 2 - 0 , with the sign of the square root chosen such that sgn Im d m = sgn Im am, and similarly for /3,. Here we have introduced
W[~)(am, X), f(q, X)] a2m _ q2
amq(X) =
F'( q) - iqF( q) F'(q) + iqF(q) '
R(q) =
,
(6)
r(q)-
1 + R(q) F(q) '
(12)
where, using continuity at x = 0, the functions F(q) =--F(q, 0) and F'(q) = F'(q, 0) are obtained from eq. (5) in the following way. Equation (7) yields, with the boundary conditions (2) for
¢(am, X), amq(O) = f( q)/(a2m _ q2), t
(7)
amn(X ) = amo,(X ) ,
where W[a, b] = ab' - a'b is the Wronskian, and II'"ll denotes the determinant. In constructing the function (5) we make use of the fact that ¢ ( q , x ) is an even function of q, and freely choose Im a m < 0, so that ~b(am, x) - exp[idmX] for x--~ ~. Following refs. [13,14] it can be verified that the function F(q, x) is the Jost solution to the Schr6dinger equation (1) with the potential Uo(x ) replaced by
U(x)=O,
x
d2 U(x) = Uo(x ) - 2 ~ logllamn(X)[[ , x > 0 .
(8)
The transformation (5-8) can in principle also be performed for non-cut-off potentials [16]. The Jost solution F(q, x) for x-< 0 is given as in eq. (3) with the coefficients R 0 and TO replaced by the coefficients R(q) and T(q) associated with the potential U(x):
F(q, x) = ~
1
[e 'qx + R(q) e-'qx],
X <--O.
(9)
amq(O) = O,
(13) (14)
where f(q) =-f(q, 0). Using
f( q, x)Ilam,,(x) - a.,q(X)f(B., x)/f( q, x)[[ = Ilam.(x)ll[f(q,x) - E f(B., x)a.m(X)amq(X)] -' tim
(15) and (a-l) ' = - a - l a ' a -1, we arrive at
n ( q + [~.) F(q) = f ( q ) H-~---~m--m) '
(16)
F'(q) = l(q)S(q)F(q),
(17)
where l(q) = i f ( q , O)/f(q, 0) and
l ( q ) ( a 2 - - q2)_- l(fl.)(a~-- fiE) S(q) = II 2 2 - q 2) l(q)(am2 - [3.)([3. I[
×
Ila~- ~lI-' 1
(18)
The reflection coefficient thus becomes
R(q) =
l(q)S(q) - iq l(q)S(q) +iq
(19)
and the transmission coefficient is obtained from eq. (12) as
We have
F(q,O)-
(11)
1 + R(q) T(q) '
F'(q,O) =iq
1 - R(q) T(q) '
2iq 1 H(4-ffm) T(q) = l(q)S(q) + iq f(q) 1I(4 + fin) "
(20)
(10) and therefore
These are the reflection and transmission coefficients which are associated with the 'solvable'
R. Lipperheide et al. / Inversion o f neutron reflection data
380
class of potentials (6). They are rational functions of q if the background potential vanishes (Uo(x) -- 0, so that go(q) -- 0 and l(q) = iq). For a real potential we must have R ( q ) = R * ( - q ) and T( q) -- T * ( - q ) . In this case the parameters am, /3n must either be purely imaginary, or they must come in pairs (a,,, 0/,.),
d rl/2(q) - sin(~bR(q) -- 2qxa) q
(25)
(/3., -/3"). For a simple example, we take an imaginary pair 0/= - i a , /3 = ib (b > a > 0) and find (b 2 --
a2)/2
R( q) - q2 + ibq - (b z - a2)/2 '
(21)
where t(q)= [T(q)l z and ~ ( q ) =
Im log T(q). In the present case, however, where the reflection and transmission coefficients are represented by the analytic expressions (19) and (20), the simplest way to use the sum rule (25) is to rewrite it as
q( q + ia) T(q) = q2 + ibq - (b z - a2)/2 '
(22)
vr(q) = Im[R*(q)-~q R(q)] - 2Xar(l )
+ Im[R(q) e -i2°xl] ~ Im
2(b 2 - a2)a 2
U(x) = (b sinh ax + a cosh ax) 2 "
q (23)
3. Determination of the reflection coefficient from the experimental data The absolute value of the refection coefficient, r(q) = IR(q)[ 2, is directly measurable. The reflection phase dpR(q), on the other hand, can only be determined via other quantities. As such we propose to use the dwell time [8] x2
if
r(q)= o
Iq,(q, x)l~ dx
(24)
[ T*(q) -~qa r(q) ]
+ t(q)(x 2 - xl)
(26)
and to substitute here the expressions (19) and (20), with r(q), t(q) = 1 - r(q) and ~(q) considered known. The parameters 0/,. and /3~ are adjusted to fit the reflectivity in conjunction with relation (26).
4. Numerical example We present a simple schematic example in order to demonstrate the features of the rational representation. We consider the case of four pairs of complex parameters,
X1
where v = h q / m is the incident velocity. This quantity can be regarded as the time spent by the neutron in the interval (x~, x2) during the scattering [8]. The dwell time can be determined experimentally by measuring the absorption rate in the surface layer doped with a weak absorber of known absorption rate [10,11]. When the limits of this interval lie outside the scattering potential, and if the absorption is negligible (real potentials), the dwell time satisfies the sum rule [9,12]
a 1 = 0.118 - i0.170,
/31 = 0.127 + i0.172,,
0/2 ~
/32 = - / 3 ~ ,
- - 0 / 1*,
a 3 = 0.185 - i0.141,
/33 = 0.184 + i0.146,
0/4 -~- - - O / 3
/3, = - / 3 1 ,
,
(27)
and vanishing background potential, U ( x ) = O. The corresponding potential U(x) is shown in fig. 2, and the reflectivity r(q) and the dwell time r(q), or rather the 'dwell length' w'(q), are shown in fig. 3. The solution of the inverse problem then takes the following route. The
R. Lipperheide et al. / Inversion of neutron reflection data
5o
'°V
381
U(x)
t
i
i
x~ -- 0
~
I0
J
i
x [nm]
i
l
,
20
Fig. 2. Neutron potential U(x) (cf. formula (8)) for the set of parameters (27).
parameters are known, the potential is calculated using eq. (6), and reproduces the potential of fig. 3. This example is of course only schematic and much too simple. For actual applications an appropriate background potential and a fairly large number of parameters would clearly be needed to fit the data.
I0
0.1
0.01 0
0.1 q [nm"]
[nm]
5. Conclusion vr(q)
15
5 0
0.1
q m rn-"] Fig. 3. Reflectivity r(q) and dwell length or(q) for the potential of fig. 2.
'measured' functions r(q) and oz(q) of fig. 3 are inserted in relation (26) with t ( q ) = 1 - r ( q ) . Then the four pairs of parameters (27) are retrieved by fitting the functions R(q) and T(q) in the form (18-20) to relation (26). Once these
We have proposed an inversion scheme for neutron reflection employing a rational representation of the reflection coefficient. Although we consider a certain class of reflection functions (which, however, we believe to be comprehensive enough to include realistic cases) the method does not make use of any approximation regarding the one-dimensional scattering process. Because of the restriction of the potential to the half-axis x > 0 , the reflection coefficients are different from those obtained on the full axis [16]. The present inversion scheme is based exclusively on physical observables. It employs the dwell time sum rule (25) just as our previous work [12] does, and provides an alternative to solving the associated differential equation for the reflection phase [12].
382
R. Lipperheide et al. / Inversion o f neutron reflection data
References [1] T.P. Russel, Mater. Sci. Rep. 5 (1990) 171. [2] J. Penfold and R.K. Thomas, J. Phys.: Condens. Matter 2 (1990) 1369. [3] J.M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975). [4] R.E. Burge, M.A. Fiddy, A.H. Greenaway and G. Ross, Proc. Roy. Soc. London A 350 (1976) 191; M.V. Klibanov and P.E. Sacks, J. Math. Phys. 33 (1992) 3813. [5] D. Moras, A.D. Podjarny and J.C. Thierry (eds.), Crystallographic Computing 5 (Oxford University Press, 1991). [6] D.S. Sivia, W.A. Hamilton, G.S. Smith, T.P. Rieker and R. Pynn, J. Apph Phys. 70(2) (1991) 732; D.S. Sivia, W.A. Hamilton and G.S. Smith, Physica B 173 (1991) 121. [7] W. Lesslauer and J.K. Blaisie, Acta Cryst. A 27 (1971) 456. [8] E.H. Hauge, J.P. Falck and T.A. Fjeldly, Phys. Rev. B 36 (1987) 4203.
[9] E.H. Hauge and J.A. St0vneng, Rev. Mod. Phys. 61 (1989) 917. [10] M. B/ittiker, in: Electronic Properties of Multilayers and Low-Dimensional Superconductors, eds. J.M. Chamberlain, L. Eaves and J.C. Portal (Plenum Press, New York, 1990) p. 297. [11] R. Golub, S. Felber, R. G~ihler and E. Gutsmiedl, Phys. Lett, A 148 (1990) 27. [12] H. Fiedeldey, R. Lipperheide, H. Leeb and S.A. Sofianos, Phys. Lett. A 170 (1992) 347. [13] W.R. Theis, Z. Naturf. l l A (1956) 889. [14] R. Lipperheide and H. Fiedeldey, Z. Phys. A 286 (1978) 45; 301 (1981) 81. [15] I. Kay, Comm. Pure Apph Math. 13 (1960) 371. [16] K. Chadan and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, Berlin, 1989; 2nd ed.). [17] F. Calogero and A. Degasperis, Spectral Transform and Solitons, Vol. 1 (North-Holland, Amsterdam, 1982).
Physica B 190 (1993) 377-382 North-Holland
SDI: 0921-4526(93)E0064-N
Inversion of neutron reflection data R. Lipperheide and G. Reiss Hahn-Meitner-lnstitut and Freie Universitiit Berlin, Germany
H. Fiedeldey and S.A. Sofianos Department of Physics, University of South Africa, Pretoria, South Africa
H. L e e b Institut f iir Kernphysik, Technische Universitiit Wien, Austria Received 4 December 1992
A method is proposed for determining surface profiles from neutron reflection data by inversion. With the reflectivity and the dwell time reDgarded as measured input, the complex reflection coefficient is determined in the form of a rational function of the neutron momentum, for which the scattering potential is known in analytic form. A numerical example is given.
1. Introduction
The specular reflection of neutrons from plane surfaces is employed to obtain information on the profile of the surface, i.e. its structure in a perpendicular direction [1,2]. The quantity usually measured is the reflectivity r(q)= IR(q)[ 2, where R(q) is the complex reflection coefficient as a function of neutron momentum q. In order to extract the profile from the reflection data unambiguously, one needs to know not only the reflectivity but also the phase of the reflection coefficient, ~bR(q) = I m log R(q), as a function of neutron momentum q. This is the age-old phase problem of diffraction analysis [3-5] which has for a long time resisted a general deductive solution, and has so far prevented the application of any genuine inversion method in this field. An interesting proposal has recently been made to remove the phase ambiguity in Born Correspondence to: R. Lipperheide, Hahn-Meitner-Institut, Postfach 390128, D-1000 Berlin 39, Germany.
approximation by introducing a 'reference edge' [6]. A similar idea has been advanced earlier in X-ray diffraction [7]. In the present note we propose an inversion procedure which makes use of the reflectivity data as well as of the dwell time [8,9], which can be measured via absorption [10,11]. As has been shown in a previous paper [12], the dwell time ~-(q) contains the missing information on the reflection phase ~bR(q) which is needed to determine the profile. For the inversion itself we propose a scheme in terms of complex reflection coefficients represented as essentially rational functions of q, which are associated with a class of potentials (surface profiles) whose form is known analytically. A fit of the rational reflection coefficients to the reflectivity data and the dwell time with an appropriate number of free parameters therefore immediately yields the desired potential. In the next section we present the class of potentials which is associated with the rational reflection (and transmission) coefficients to be
0921-4526/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved
R. Lipperheide et al. / Inversion of neutron reflection data
378
We consider a wave incident from the left with momentum q, and correspondingly seek the solution of eq. (1) in the form of the Jost solution f ( q , x ) with the asymptotic boundary condition
used in this investigation. The inversion method in terms of a 'solvable' class of potentials is taken over, with appropriate modifications, from the Bargmann scheme used in the quantum theory of nonrelativistic particle scattering [13,14]. It is closely related to the rational method of Kay [15] for the inversion in one dimension [16,17]. In section 3 we explain the procedure for fitting the parameters of the rational reflection coefficient to the reflectivity and the dwell time. Finally, in section 4, a schematic example is treated numerically for an illustration of the method.
f(q,x)---~exp[iftx]
for x---~ ~ ,
(2)
where ~ = + ~ q 2 _ U. For x < 0, where U 0 = 0, the solution has the form
f(q,x)-
1
To(q)
[e 'qx + R 0 ( q ) e-iqx],
x-<0,
(3)
Specular reflection is essentially one-dimensional scattering by a potential barrier. We begin with the scattering on a given 'background' potential Uo(x), described by the Schr6dinger equation (' ~- d/dx)
where R 0 and T O are the 'background' reflection and transmission coefficients, respectively. We further introduce a second solution of eq. ( i ) , ~b(q, x), which is linearly independent of the Jost solution (2) and which we call the 'regular solution'. It is defined by the following boundary conditions at x --- 0:
~O"(q, x) + [q2 _ Uo(x)]qJ(q, x) = 0 .
th(q,0) = 0 ,
2. The inversion scheme
(1)
q2l
(4)
The background solutions f ( q , x ) and ~b(q,x) taken at the complex momenta /3, and a m, respectively (n, m = 1, 2 . . . . . N), are now used to define, for x > 0, a function
The potential Uo(x ) is generally complex, vanishes to the left of the origin, Uo(x ) = 0 for x < 0, and becomes constant in the substrate to the right, Uo(x ) = 0 for x > £ (cf. fig. 1).
eiqX
~b'(q,0) = - 1 .
uo
T(q)e iqx
R(q)e-iq x
x, : 0
R
xa
x
Fig. 1. One-dimensional neutron scattering by a barrier (specular reflection from a plane surface). The curves labelled U and Uo represent the total and background potentials, respectively. The homogeneous substrate starts at ~ and is represented by the constant potential 0. The points xl, x2 delimit the interval across which the dwell time is calculated.
379
R. Lipperheide et al. / Inversion of neutron reflection data
u(4 + arO -~-_
F(q,x)=f(q,x) X
Ilam.(~) - amq(~)f(~., x) lf(q, x) ll
(5)
II~m.(x)lf
where d m = ~ - U , fln=W'fl 2 - 0 , with the sign of the square root chosen such that sgn Im d m = sgn Im am, and similarly for /3,. Here we have introduced
W[~)(am, X), f(q, X)] a2m _ q2
amq(X) =
F'( q) - iqF( q) F'(q) + iqF(q) '
R(q) =
,
(6)
r(q)-
1 + R(q) F(q) '
(12)
where, using continuity at x = 0, the functions F(q) =--F(q, 0) and F'(q) = F'(q, 0) are obtained from eq. (5) in the following way. Equation (7) yields, with the boundary conditions (2) for
¢(am, X), amq(O) = f( q)/(a2m _ q2), t
(7)
amn(X ) = amo,(X ) ,
where W[a, b] = ab' - a'b is the Wronskian, and II'"ll denotes the determinant. In constructing the function (5) we make use of the fact that ¢ ( q , x ) is an even function of q, and freely choose Im a m < 0, so that ~b(am, x) - exp[idmX] for x--~ ~. Following refs. [13,14] it can be verified that the function F(q, x) is the Jost solution to the Schr6dinger equation (1) with the potential Uo(x ) replaced by
U(x)=O,
x
d2 U(x) = Uo(x ) - 2 ~ logllamn(X)[[ , x > 0 .
(8)
The transformation (5-8) can in principle also be performed for non-cut-off potentials [16]. The Jost solution F(q, x) for x-< 0 is given as in eq. (3) with the coefficients R 0 and TO replaced by the coefficients R(q) and T(q) associated with the potential U(x):
F(q, x) = ~
1
[e 'qx + R(q) e-'qx],
X <--O.
(9)
amq(O) = O,
(13) (14)
where f(q) =-f(q, 0). Using
f( q, x)Ilam,,(x) - a.,q(X)f(B., x)/f( q, x)[[ = Ilam.(x)ll[f(q,x) - E f(B., x)a.m(X)amq(X)] -' tim
(15) and (a-l) ' = - a - l a ' a -1, we arrive at
n ( q + [~.) F(q) = f ( q ) H-~---~m--m) '
(16)
F'(q) = l(q)S(q)F(q),
(17)
where l(q) = i f ( q , O)/f(q, 0) and
l ( q ) ( a 2 - - q2)_- l(fl.)(a~-- fiE) S(q) = II 2 2 - q 2) l(q)(am2 - [3.)([3. I[
×
Ila~- ~lI-' 1
(18)
The reflection coefficient thus becomes
R(q) =
l(q)S(q) - iq l(q)S(q) +iq
(19)
and the transmission coefficient is obtained from eq. (12) as
We have
F(q,O)-
(11)
1 + R(q) T(q) '
F'(q,O) =iq
1 - R(q) T(q) '
2iq 1 H(4-ffm) T(q) = l(q)S(q) + iq f(q) 1I(4 + fin) "
(20)
(10) and therefore
These are the reflection and transmission coefficients which are associated with the 'solvable'
R. Lipperheide et al. / Inversion o f neutron reflection data
380
class of potentials (6). They are rational functions of q if the background potential vanishes (Uo(x) -- 0, so that go(q) -- 0 and l(q) = iq). For a real potential we must have R ( q ) = R * ( - q ) and T( q) -- T * ( - q ) . In this case the parameters am, /3n must either be purely imaginary, or they must come in pairs (a,,, 0/,.),
d rl/2(q) - sin(~bR(q) -- 2qxa) q
(25)
(/3., -/3"). For a simple example, we take an imaginary pair 0/= - i a , /3 = ib (b > a > 0) and find (b 2 --
a2)/2
R( q) - q2 + ibq - (b z - a2)/2 '
(21)
where t(q)= [T(q)l z and ~ ( q ) =
Im log T(q). In the present case, however, where the reflection and transmission coefficients are represented by the analytic expressions (19) and (20), the simplest way to use the sum rule (25) is to rewrite it as
q( q + ia) T(q) = q2 + ibq - (b z - a2)/2 '
(22)
vr(q) = Im[R*(q)-~q R(q)] - 2Xar(l )
+ Im[R(q) e -i2°xl] ~ Im
2(b 2 - a2)a 2
U(x) = (b sinh ax + a cosh ax) 2 "
q (23)
3. Determination of the reflection coefficient from the experimental data The absolute value of the refection coefficient, r(q) = IR(q)[ 2, is directly measurable. The reflection phase dpR(q), on the other hand, can only be determined via other quantities. As such we propose to use the dwell time [8] x2
if
r(q)= o
Iq,(q, x)l~ dx
(24)
[ T*(q) -~qa r(q) ]
+ t(q)(x 2 - xl)
(26)
and to substitute here the expressions (19) and (20), with r(q), t(q) = 1 - r(q) and ~(q) considered known. The parameters 0/,. and /3~ are adjusted to fit the reflectivity in conjunction with relation (26).
4. Numerical example We present a simple schematic example in order to demonstrate the features of the rational representation. We consider the case of four pairs of complex parameters,
X1
where v = h q / m is the incident velocity. This quantity can be regarded as the time spent by the neutron in the interval (x~, x2) during the scattering [8]. The dwell time can be determined experimentally by measuring the absorption rate in the surface layer doped with a weak absorber of known absorption rate [10,11]. When the limits of this interval lie outside the scattering potential, and if the absorption is negligible (real potentials), the dwell time satisfies the sum rule [9,12]
a 1 = 0.118 - i0.170,
/31 = 0.127 + i0.172,,
0/2 ~
/32 = - / 3 ~ ,
- - 0 / 1*,
a 3 = 0.185 - i0.141,
/33 = 0.184 + i0.146,
0/4 -~- - - O / 3
/3, = - / 3 1 ,
,
(27)
and vanishing background potential, U ( x ) = O. The corresponding potential U(x) is shown in fig. 2, and the reflectivity r(q) and the dwell time r(q), or rather the 'dwell length' w'(q), are shown in fig. 3. The solution of the inverse problem then takes the following route. The
R. Lipperheide et al. / Inversion of neutron reflection data
5o
'°V
381
U(x)
t
i
i
x~ -- 0
~
I0
J
i
x [nm]
i
l
,
20
Fig. 2. Neutron potential U(x) (cf. formula (8)) for the set of parameters (27).
parameters are known, the potential is calculated using eq. (6), and reproduces the potential of fig. 3. This example is of course only schematic and much too simple. For actual applications an appropriate background potential and a fairly large number of parameters would clearly be needed to fit the data.
I0
0.1
0.01 0
0.1 q [nm"]
[nm]
5. Conclusion vr(q)
15
5 0
0.1
q m rn-"] Fig. 3. Reflectivity r(q) and dwell length or(q) for the potential of fig. 2.
'measured' functions r(q) and oz(q) of fig. 3 are inserted in relation (26) with t ( q ) = 1 - r ( q ) . Then the four pairs of parameters (27) are retrieved by fitting the functions R(q) and T(q) in the form (18-20) to relation (26). Once these
We have proposed an inversion scheme for neutron reflection employing a rational representation of the reflection coefficient. Although we consider a certain class of reflection functions (which, however, we believe to be comprehensive enough to include realistic cases) the method does not make use of any approximation regarding the one-dimensional scattering process. Because of the restriction of the potential to the half-axis x > 0 , the reflection coefficients are different from those obtained on the full axis [16]. The present inversion scheme is based exclusively on physical observables. It employs the dwell time sum rule (25) just as our previous work [12] does, and provides an alternative to solving the associated differential equation for the reflection phase [12].
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R. Lipperheide et al. / Inversion o f neutron reflection data
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