J . Phys. Chem. 1993, 97, 1722
1722
COMMENTS y;(z) = -ik(z), and upon performing the first iteration one has
Approximate Solutions to Neutron Reflectivity Problems in One Dimension M. Krech
(
Fachbereich Physik. Bergische UniversitBt Wuppertal, Postfach 10 01 27, 0-5600 Wuppertal 1 , Federal Republic of Germany Received: March 24, 1992 I comment on the approximate solution of the neutron reflectivity problem given by Zhou, Chen, and Felcher in ref 1 on the basis of the time-independent Schradinger equation in one dimension. A criterion for the validity of this approximation is given. It is argued that a substantial improvement over the WKB approximation in general cannot be achieved. A fundamental problem in reflectivity theory and experiments is the calculation of the interaction potential from, e.g., neutron reflectivity data. In the direct approach one guesses the shape of the potential and then calculates the transmission and reflection coefficients from the solution of the Schradinger equation for neutron reflection. This has been done in ref 1. The inverse problem is in general still unresolved and provides a wide field of present research. An overview of recent developments concerning the inverse reflection problem is given in ref 2. In the following I will comment on the direct approach to the neutron reflection problem presented in ref 1. In the notation of ref 1 the one-dimensional Schradinger equation for the wave function U(z)reads
+
V”(z) kZ(Z)U(Z) = 0 (1) (see eq 1.1 in ref l), where k2(z) = (2m/h2)(E - V(z)). V ( z ) is the scattering potential and E the energy eigenvalue. Now a differential equation for the function y ( z ) defined by
= W)/W
(2) can be derived by taking the first derivative of eq 2. One finds Y(Z)
y’(2)
+ y2(2) + k2(2) = 0
::;;y2
y;(z) = -ik(z) 1 - i-
(3)
where eq 1 has been used. If a solution of eq 3 is known, then eq 2 can be integrated directly, giving (4) Consequently, eq 1 can be solved by solving the two first-order differential equations (3) and (2) in sequence. Equation 3 is a special case of the so-called Riccati differential equation. If one assumes that the potential, Le., the function k(z),is slowly varying in terms of the position z, then eq 3 suggests an iterative scheme for the calculation of the function y(z). It reads Y ’I, ( 2 ) + Y , 2 W + k 2 ( z )= 0 (5) wherk n is an index indicating the number of iterations and y - , ( z ) = 0. For n = 0 one finds the two solutions $ ( z ) = ik(z) and
0022-365419312097-1722$04.00/0
Equation 6 together with eq 4 gives two linearly independent approximate solutions of eq 1, which can be combined to give
where UII and U Iare ~ constants. Equation 7 is equivalent to eq 2.14a in ref 1. The calculation of the reflection and transmission coefficients presented in ref 1 is based on the approximate solution Ul( z ) ,which has been achieved by iterating eq 5 up ton = 1. This is only justified under the aforementioned assumption that k ( z ) and therefore y ( z ) are slowly varying functions. This implies b’,,-l(z)l
