Momentum transfer vector

Fig. 3 shows the difference diffraction pattern obtained by subtracting the pattern of the molecules in the So state from that of the molecules in the Si electronic state. For this calculation we assume that the wave vector of the incoming electron beam is ko = 100 A 1. The ordinate is the projection of the momentum transfer vector, s, onto the plane of the detector, i.e.,... 

As co) increases, the experimental scattering lobe is no longer symmetric to Zcol but turns its symmetry axis essentially into the direction of the momentum-transfer vector K=kout—kjn as indicated by 9 in Fig. 32. In Born s approximation the lobe would be exactly symmetric to K. This reflects the fact that electronic angular momentum of the atom is transferred into linear momentum of the scattered electron. 

Small-angle x-ray scattering requires a special detector arrangement to measure at small and very small angles (0.0002° to 6°). In this technique, the intensity of the scattered photons, I q), is monitored as a function of the absolute value of the momentum transfer vector, q, for an elastic scattering event ... 

Here PMV denotes density-matrix elements, K = k0 - k, is the momentum transfer vector and Za is the nuclear charge of nucleous a. 

An incident monochromatic beam scattered by a sample is analyzed with a detector at a general position in space (see Fig. 2). Incident and scattered neutrons are regarded as plane waves whose wavevectors are ki and kf, respectively. ( jfco = 27t/Ao and kf = 2tt/ /, where Ao and A/ are the incident and scattered wavelengths, respectively.) The momentum transfer vector is Q — ki kf. 

The full scattering function (6) should be represented in five dimensions. Partial landscape views in three dimensions are shown in Fig. 6 [Ikeda 2002], Energy transfer values correspond to proton modes and the momentum transfer vector spans various planes in reciprocal space, (Qi, Qj), such as Qk - 0, with i, j or k = x, y or z. The maps of intensity are graphic views of the orientation of the displacement vector for each mode. Visual examination confirms that vibrational coordinates remain unchanged in the various excited states and Hr- a,. nz are relevant quantum numbers. 

The previous derivation of the reflection and transmission coefficients correctly describes the intensity of reflected neutrons at any value of momentum transfer vector. However, there is a useful alternative derivation, which gives a highly analytical function describing the reflectivity. This derivation is based on the Born approximation and is often referred to as reflectivity in the kinematic limit. Suppose there are two arbitrary but different SLD profiles pi(z) and p2 z) and one wishes to determine the separate reflectivities Ri(Q,z) and -R2(Gz) for the two scattering potentials. The solution to the problem is described by combining Eqs. (3.15) and (3.17)... 

Figure 3.14 X-ray reflectivity as a function of momentum transfer vector for a gold-chromium-covered quartz substrate. The curve corresponds to the measurement made before electrochemistry. The curve for the post-electrochemical measurement is... 

The reader will appreciate that the scattering law of a particular vibrational transition, given by Eq. (2.32), is written in terms of the scalar (or inner, or dot) product of the neutron momentum transfer vector, Q, and the atomic displacement vector, . This product is a function of the angle between the vectors and allows some interesting single crystal effects to be examined. 

Because the three vibrations are degenerate, if the q components along one direction are reduced they are compensated by the larger components along other directions. Thus, for example, if qy = q =0 then qx = Q and substituting this into Eq. (2.32) gives the same result. The direction of the momentum transfer vector is irrelevant and it may be represented by its magnitude alone, Q. 

Whilst retaining the conventional symbol for a normal coordinate, Q, we have distinguished it firom the momentum transfer vector, Q, by its font.) The normal coordinate system is important since it represents the displacement vectors in Cartesian coordinates and simplifies the expressions for the kinetic and potential energy. This allows us to rewrite Eqs. (4.6) and (4.11) as ... 

FIGURE 5.1.5 X-ray scattering experimental setups (a and b) and some results (c and d) for pentacene thin films on SiOj. (Ruiz, R. et al., Appl. Phys. Lett, 85, 4926, 2004.) Specular orientation reveals information about the ordering parallel to the substrate where pentacene films have high crystalline ordering. In (a) and (c), the momentum transfer vector is normal to the surface in (b) and (d), the momentum transfer is mostly in the plane of the surface. 

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