Neutron Scattering

6.5 The Regime of the Total External Reflection : Exact Solution of the Wave Equation
It is obvious that the Born approximation [Eqs . (6 .5) and (6.13)] fails for Q,--->O because
the intensity would become infinite at Q=O. The reason is that multiple scattering processes are
neglected within the Born approximation . For surface sensitive experiments multiple scattering
processes become essential at very small angles [7] . An exact description of the scatternd
intensity can be deduced for a perfectly smooth surface from quantum theory .
Starting point is the Schrödinger equation
b z
-2m
n
4+V(r) I`V(r) = EW(r) (6.15)
for the wave function of the neutrons `F(r) . The energy of the neutrons is given by
E = bzk z 1(2m,) with the mean value k=2ir1,~ of the wave vector k (the incident and the
outgoing beam have identical k because elastic scattering is assumed) . For a homogeneous
sample the potential is determined by Eq. (6 .6), thus
r ll r z l1
LA+Ckz -4nY,bj p~ I J
'F(r)=l A+kz C1--Eb i
p i
IJ
'F(~=~4+kr ]`PL)=0 (6.16)
with the wave vector k, inside the medium (see Fig. 6 .4) . From Eq . (6.16) it is justified to
introduce the refraction index n,=k,/k ofthe material. In very good approximation one yields
z
n, =1-
2 ~,b;p ; =1-8 1 (6.17)
for the refraction index which is a number close to 1 for neutrons of approximate 1Â
wavelength (the correction 8, is called dispersion and is on the order of 10-5 . . .10-6)
By introducing the refraction index the basic principles of optics can be applied for all
further considerations . First of all it is remarkable that for many materials n, is smaller than 1
(because b; is usually and pi always positive, thus 8, is usually positive) . This means that the
transmitted beam is refracted towards the sample surface (0,