Neutron Scattering

0 z>d
d V(z) = 2nh2b,P, / mz =V, : >_ z > 0 (6 .10)
0 21rh2b2 P2 /m =V2 . > z
if the interfacial rms-roughnesses ai and 62 are neglected. The derivative yields two deltafunctions
: dV(z)/dz-(b2p2-b,pl)S(z)+b,plô(z-d) . Using Eq . (6 .5), the specularily reflected
intensity of a perfectly smooth monolayer system is given by
I(Qz ) - Q4
[ (b2P2 -b,P,) 2 + (b,P, ) 2 +2(b2P2 - b,P, )(b,P, )cos(Qd)] . (6.11)
This means that films cause oscillations in the reflectivity. The period is determined by the fihn
thickness the strength (usually called `contrast') by the difference of the potentials Vl and V2
(see Fig . 6 .6 right) .
Figure 6 .6 : The left graph displays the effect of the surface roughness on the specularily
refected intensity : the rougher the surface the less intensity is refected at large Qz . The right
figure shows refectivities of perfectly smooth monolayer systems . The curves are shifted in
intensity for clarity. The thicker the film the smaller the distance of the so-called Kiessig
fringes . The less the contrast (given by [b2P2-b,p, ]) the less pronounced the oscillations are.
In this way every additional layer appears as an oscillation in the reflectivity curve.
Interface roughnesses can also quite easily be included in the theory and yield a typical
damping of each oscillation. Multilayer systems with different parameters for each layer
generally show very complicate reflectivities . They are usually difficult to analyze especially
because of the so-called 'phase problem' which prevents an unambiguous solution of Eq. (6 .5) .
The ,phase problem" appears when performing the mean square of the complex function
6 .8