Momentum density directional Compton profiles

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 ... 

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... 

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. 

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. 

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. 

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. 

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