Numerical analysis of the orientational motion of neutral particles in crystals

226

In the respective normalization of columns and rows of the matrices Se and Sg-l along X‘=A-lA, (B=QiX,, with the diagonalization of B, they reduce simultaneously all the matrices For this we must take C-EXiX1) to symmetric form. 6o=

cf+UT

~,

Z,+c~D'WT

es=

UT -cf 2pVDWT

, f=s’V-qu.

It is easy to verify that for U*>(SC)~ the norms of Sg and Sg-' are uniformly bounded. If system (1) is written in curvilinear coordinates it becomes non-homogeneous and metric coefficients appear as factors in the matrices B and C. The transformations presented in the paper are applied to include these factors in a. p, 7 The use of these transformations for a conservative system of Euler equations is analogous to /2/. Application of the results conpresented here to the difference method in /3/ for writing the equations in strict servative form in an arbitrary system of coordinates enables us to perform calculations with Courant numbers of order 100, eliminating the difficulty with the approximation of a free Calculations were made for a non-viscous three-dimensional flow with condensation of term. points on the surface of the body around which the flow occurs (up to ten inside any boundary layer with numbers Pie-10')for investigating the flow parameters in entropy layers. The author thanks Yu.D. Shevelev

and V.A. Aleksin

for useful

discussions.

REFERENCES 1. GODUNOV S.K., ZABRODIN A.V. and IVANOV M.YA., Numerical Solution of n-dimensional Problems of Gas-Dynamics, Nauka, Moscow, 1976. and simulataneous symmetrization 2. WARMING R.F., BEAM M.J. and HYETI B.J., Diagonalization of the gas dynamic matrices, Math. Comput., 29, 132, 1037-1045, 1975. method for calculating 3. PCGORELOV N.V. and SHEVBLBV YU.D., A marching explicit-implicit supersonic flow round a body, Zh. vychisl. Mat. mat. Fir., 25, 9, 1391-1400, 1985. Translated

U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain

Vo1.28,No.3,pp.226-227,1988

by S.R.

0041-5553/88 $lO.OO+O.OO 20.1989 Pergamon Press plc

NUMERICAL ANALYSIS OF THE ORIENTATIONALMOTION OF NEUTRAL PARTICLES IN CRYSTALS* ** V.I. VYSOTSKII,

R.N. KUZ'MIN

and N.P. SAVENKOVA

The intensive development and progress of aspects of the physics of the channelling of charged particles in crystals is stimulating a search for an analogous effect for neutral particles (in particular, neutrons) with a magnetic moment p. In /l/ the physical mechanism of the phenomenon of an ordered force was discovered, that enables stable orientational motion in a non-magnetic lattice to occur and is related to an intense of particles with momentum p magnetic field with amplitude Hm =~sZ.v&c (2,'is the charge on the atoms and So is the area of an elementary cell), and that arises in a crystal in a system at rest moving with velocity u particles with mass M. The corresponding potential has the form

R=h*/m.c~Z'~~ is the screening radius, u is the amplitude of thermal vibrations, o(a) Here is the probability integral, and a is the distance between planes (the width of the channel). In this paper two mathematical models are introduced that correspond to the physical 2-r/a. and a problem described above. Equations are given in dimensionless form taking potential V(ri=ZMaWLxlih2. For the first mathematical model we consider the Sturm-Liouville equation with periodicity conditions at the ends of an interval Y"(s)-[V(z)--hlY(z)=O, O
1401-V87,1987

(ia)

227

s (L

Y

(0)= Y (a)*

Y’ (0) = I/’ (a)9

1y(z)

I’dr

=

(lb)

1.

0

It is required to calculate all values of the parameter h that correspond to non-trivial solutions of problem (1). The second mathematical procedure reduces to investigating the zonal structure of a spectrum and the calculation of separate wave functions of the Schrddinger problem with periodic potential:

(2b) The calculation of the magnification factor of the reaction k in an energy band using squares of the wave functions found in (1) and (2) is of separate physical interest. Note that in the case of real potentials it is not possible to solve problems (1) and (2) analytically. Problem (1) is solved by means of a difference method, i.e. on a uniform mesh ehN introduced in the interval [O,a] a difference scheme is constructed that approximatesproblem (1) up to second order (see /2/), in this case to investigate the zonal structure of the spectrum the eigenfunction has to be represented in the form K(r)-Y(+)&, where k is the wave number. Calculation of problem (1) without the special representation of u(z) corresponds to a calculation with wave number k=O. In this case the solution of (1) reduces to solving the complete eigenvalue problem for a tridiagonal Hermitian matrix with elements a>-, and a,.,, that are non-zero. To solve (2) a pseudospectral method was used on the fact that periodic functions that satisfy Dirichlet conditions can be expanded in Fourier series. This enables the solution of (2) to be reduced to solving the complete eigenvalue problem for a completely filled Hermitian matrix. The accuracy of the method was determined experimentally on a model problem of "step" type that can be solved analytically. As a result of the numerical calculations it is clear that in the physical process under investigation the channelling of particles is possible with a zonal structure with a possible increase (!Ll>l) and decrease (&~,
Translated

by

S.R.