Directional Compton profiles

No directional information is obtained from gas-phase experiments because the molecules are freely rotating. In the solid state, the rotational motion can be frozen, and the directional Compton profile can be written as... 

In the special case that the scattering vector is parallel to one of the coordinate axes, these expressions look much simpler. For example, if q is parallel to the z axis, the directional Compton profile, expressed in Cartesian coordinates, is simply the marginal momentum density along the axis ... 

There are two main methods [145] for the reconstruction of n( p) from the directional Compton profile. In the Fourier-Hankel method [145,162], the directional Compton profile is expanded as... 

A vast number of directional Compton profiles have been measured for ionic and metallic solids, but none for free molecules. Nevertheless, several calculations of directional Compton profiles for molecules have been performed as another means of analyzing the momentum density. 

Subsequent work on graphical analysis of anisotropic momentum densities, directional Compton profiles, and their differences in diatomic molecules was reported by several groups, including Kaijser and Smith [195,316], Ramirez [317-319], Matcha, Pettit, Ramirez, and Mclntire [320-328], Leung and Brion [329,330], Simas et al. [331], Rozendaal and Baerends [332,333], Cooperand Allan [334], Anchell and Harriman [138], and Rerat et al. [335,336]. 

Experimentally, the momentum density is closely connected with the Compton profile, the spectrum of scattered radiation. Within the impulse approximation (Kilby, 1965 Eisenberger and Platzman, 1970), the directional Compton profile is given by... 

Fig. 20. Differences in the directional Compton profiles for (a) the lscr, state and (b) the 2po-u state of the H2+ system. All values in atomic units. (Reproduced from Koga and Morita, 1981a.)...

There are two main methods for the reconstruction of 7T(p) from the directional Compton profile. In the Fourier-Hankel method [33,51], a spherical harmonic expansion of the directional Compton profile is inverted term-by-term to obtain the corresponding expansion of /T(p). In the Fourier reconstraction method [33,34], the reciprocal form factor B0) is constructed a ray at a time by Fourier transformation of the measured J(q) along that same direction. Then the electron momentum density is obtained from B( ) by using the inverse of Eq. (22). A vast number of directional Compton profiles have been measured for ionic and metallic solids, but none for free molecules. Nevertheless, several calculations of directional Compton profiles for molecules have been performed as another means of analyzing the momentum density. 

The Fourier transforms of Eq. (51) can be performed in closed form for most commonly used basis sets. Moreover, formulas and techniques for the computation of the spherically averaged momentum density, isotropic and directional Compton profiles, and momentum moments have been worked out for both Gaussian- and Slater-type basis sets. Older work on the methods and formulas has been summarized in a review article by Kaijser and Smith [79]. A bibliography of more recent methodological work can be found in another review article [11]. Advantages and disadvantages of various types of basis sets, including many unconventional ones, have been analyzed from a momentum-space perspective [80-82]. Section 19.7 describes several illustrative computations chosen primarily from my own work for convenience. 

Taking the photon scattering vector q in z-direction, the dynamical structure factor is related to the Compton profile J(pz) by... 

Experimentally, the EMD function p(q) can be reconstructed from a set of Compton profiles J qz ) s, and B( r) from the EMD. However, A Air) is not a direct experimental product. By combining the experimental B(r) with theoretical B aik (r), we need to derive a semiexperimental AB(r). Since the atomic image is very weak, many problems must be cleared in experimental resolution, in reconstruction (for example, selection of a set of directions and range of qzs), in various deconvolution procedures and so on. First of all, high resolution experiments are desirable. 

Nara, H., Kobayasi, T., Takegahara, K., Cooper, M.J. and Timms, D.N. (1994) Optimal number of directions in reconstructing 3D momentum densities from Compton profiles of semiconductors, Computational Materials Sci., 2, 366-374. 

The Compton profile measurements on Cu and Cu 953AI0047 were performed at ID 15b of the ESRF. Figure 1 shows the setup of the scanning-type Compton spectrometer used. It consists of a Si (311) monochromator (M), a Ge (440) analyzer (A) and a Nal detector (D). The signal of an additional Ge solid state detector (SSD) was used for normalization. ES, CS and DS denote the entrance slit, the collimator slit and the detector slit, respectively. For each sample 10 different directions were measured with approximately 1.5-2 x 103 7 total counts per direction. The incident energy was 57.68 keV for the Cu and 55.95 keV for the Cuo.953Alo.047 measurement. 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

Other moments of momentum can also be obtained directly from the Compton profile without first going through the numerical differentiation of Eq. (5.64), which is prone to roundoff and truncation errors. In particular, several groups [9,188-191] independently reported one or more of the sum rules... 

The momentum distribution of H is derived from the measured NCS TOF-spectra by standard procedures [Mayers 1994 Mayers 2004], The distribution J(y) (often called "Compton profile" [Sears 1984 Watson 1996]) is proportional to the density of protons with momentum component hy along the direction of the neutron-proton momentum transfer hq. J(y) at scattering angle 0 = 66° is shown in Fig. 13 (full line). Here hy is the H-momentum component (before collision) along the direction of momentum transfer hq. 

To provide experimental information about w(l from Compton profile measurements, we performed Compton measurements on Li using an experimental setup described elsewhere [21]. Tbe momentum space resolution obtained was = O.lZa.u., 11 different directions of q were measured. From the obtained profiles we calculated their Fourier transforms and took from these the ii(R) values, given by triangles in Figure 12. 

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