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This geometry is defined so that the inward normal to the crystal surface lies between the incident and reflected beams. The incident plane wave in figure 2.6 produces two waves as characterised by the wavevectors W j A and W2A. which then define the diffracted wave. As in the kinemadcal approximation it is important to appreciate exactly what quantities are being physically measured when performing diffraction experiments in the dynamical limit. O nce again it is the integrated reflectivity (ie. the area under the rocking curve) which is measured in these investigations. The "intrinsic" rocking curve (for an incident plane w ave), produced by samples in the Laue orientation can be seen in figure 2.7.

In the Laue geometry the reflected beam passes through the sample m ass and thus the angular reflectivity will be dependent on the crystal thickness. This dependence is illustrated for an incident spherical wave in figure 2.8 where the total reflectivity is plotted by integrating over the crystal rotation angle as a function o f the crystal thickness. The expression which describes the total reflectivity takes the following form.

(44).

J0(u) is a Bessel function o f rank 0, and A is the reduced crystal thickness as defined by ( x to / A ) . The limiting v alue o f W(A) for large A is given a s * /2. and as such the oscillating W(A) function will converge to a limiting reflectivity value. This limiting value which is defined fo r a non absorbing crystal can be described by the following expression.

In te g ra te d R ef le ct iv it y ( f t )

Figure 2.7. The rocking curve for a perfectly flat crystal

in t h r I ■ i r n l i i kn n M it a lin n f n r a in o u lM t n l o n . m uu n c n u u o n r o r a in ciae n i p la n e wave.

Reflectivity 3 t

Figure 2JL The reflectivity plotted as a function o f a reduced crystal thickness for an incident spherical wave.

2 Vc sin26 ^45)

2.6b T h e Bragg o r reflec tio n geom etry.

The Bragg geometry, which is defined such that the outward normal to the crystal surface lies between the incident and diffracted beams is illustrated in the reciprocal space representation o f figure 2.9. An interesting situation arises when the incident beam is located at W on the spherical wavevector surface, as the normal to the crystal surface does not intercept the tw o branches o f the dispersion surface. In this case the incident wave will be totally reflected. The angular range over which a sample will exhibit total reflection is simply described by the range o f incident wavevectors which lie on the sphere from Rj to R2 and is given by the expression,

K Vc sin20 (46)

where e is termed the Darwin width. A typical rocking curve (for an incident plane wave), produced from a sample which is in symmetrical Bragg geometry is shown in figure 2.10. The flat top o f this rocking curve represents the angular region in which the sample undergoes total reflection. T h e Darwin width corresponds to o f the FWHM o f this

The reflectivity decreases steadily with increasing deviation from the Bragg angle on either side o f the Darwin width. Clearly the width at the base o f the rocking curve is greater than the Darwin w idth and hence in the limit o f an infinitely thick, non absorbing crystal in symmetrical geometry the reflectivity has a limiting value which is described by,

Figure 2.9. An enlargem ent of the dispersion surface fo r the specific case o f a sam ple in Bragg sym m etrical geometry

Figure 2.10. T he rocking curve for a crystal in the Bragg geometry.

Reflectivity 31

which is exactly twice the value o f the reflectivity predicted by the Laue geometry. When dealing with real crystals o f finite thickness, the reflectivity can be expressed as a function o f thickness as described below and illustrated in Fig. 2.8.

where A =

A (48)

where A is the extinction length which is again defined from the diameter (S1S2) o f the dispersion surface, but because o f the change in geometry o f the incident wave it takes the form of,

A t vety low values o f A (when t < A), the dynamical expressions for the Laue and Bragg integrated reflectivities in the dynamical theory have approximately the same value. Moreover, as this value approaches zero the tangent o f these two curves describes the kinemadcal reflectivity as a function of the reduced crystal thickness (A).

It should be noted that in the dynamical limit the reflectivity ( i t ) is proportional to the magnitude o f the structure factor (F(H)) and not to its square as in the case o f the Kinemadcal theory. Also fo r neutron path lengths in the sample (t), greater than the extinction length (A) the reflectivity will be nearly constant which is not the case in the idnematical theory (figure 2.8)