Compton-profile

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

Beyond the local-density approximation in calculations of Compton profiles... 

Generally, all band theoretical calculations of momentum densities are based on the local-density approximation (LDA) [1] of density functional theory (DFT) [2], The LDA-based band theory can explain qualitatively the characteristics of overall shape and fine structures of the observed Compton profiles (CPs). However, the LDA calculation yields CPs which are higher than the experimental CPs at small momenta and lower at large momenta. Furthermore, the LDA computation always produces more pronounced fine structures which originate in the Fermi surface geometry and higher momentum components than those found in the experiments [3-5]. 

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

Kubo, Y., Sakurai, Y., Tanaka, Y., Nakamura, T., Kawata, H. and Shiotani, N. (1997) Effects of self-interaction correction on Compton profiles of diamond and silicon, J. Phys. Soc. Jpn., 66, Till 2780. 

Sakurai, Y. (1995) High-resolution Compton-profile measurements, Second International Workshop on Compton Scattering and Fermiology, Tokyo, Japan. 

Lundqvist, B.I. and Lyden, C. (1971) Calculated momentum distributions and Compton profiles of interacting conduction electrons in lithium and sodium, Phys. Rev., B4, 3360-3370. 

The i (r (-function was originally introduced as a mathematical intermediate in order to attain high accuracy in calculating EMD or Compton profile J(qz), which is represented under the impulse approximation as [7, 9]... 

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. 

Pattison, P. and Williams, B. (1976) Fermi surface parameters from fourier analysis of Compton profiles, Solid State Commun., 20, 585-588. 

Mueller, F.M. (1977) Anisotropic momentum densities from Compton profiles silicon, Phys. Rev., BIS, 3039-3044. 

Shrilke, W. (1977) The one-dimensional Fourier transform of Compton profiles, Phys. Stat. Sol.(b), 82, 229-235. 

Kramer, B., Krusius, P., Schroder, W. and Schiilke, W. (1977) Fourier-transformed Compton profiles a sensitive probe for the microstructure of semiconductors, Phys. Rev. Lett., 38, 1227-1230. 

Kobayasi, T. (1996) Core-orthogonalization effect on the Compton profiles of valence electrons in Si, Bull. Coll. Med. Sci. Tohoku Univ., 5, 149-164. 

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. 

To provide experimental information about (k) from Compton profile measurements, we performed Compton measurements on Li using an experimental setup described elsewhere [21], The momentum space resolution obtained was APz = 0.12a.u.,... 

A smearing of n(k) at k = kF should influence the Compton profile in the following way at the Fermi-break, the Compton profile should change slope abruptly, changing from the narrow valence electron profile into the much broader profile of the core electrons, if the electrons are modelled as being free without correlation. As... 

We find the second derivative of the Compton profile for q (111), (100) to be broadened beyond the experimental resolution with an additional AE 5eV which is due to the convolution with A(k+q, E + /m), as described in the previous section. For q (110), we find an additional broadening of the second derivative of the order of some eV which we ascribe to lattice effects on the electron correlation, predicted by the GWA-calculation. 

Section 2 describes the experimental determination of Compton profiles of Cu and CU0.953AI o.o47 in some detail. Section 3 describes the data evaluation and Section 4 the method of the reconstruction. Section 5 presents the results and finally Section 6 concludes. 

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. 

Only the valence Compton profiles are needed for the reconstruction of the momentum... 

Compton profile. Furthermore the contribution of the multiple scattered photons to the measured spectra has to be taken into account (for example by a Monte Carlo simulation [6]). Additionally one has to take heed of the fact that the efficiency of the spectrometer is energy dependent, so the data must be corrected for energy dependent effects which are the absorption in the sample and in the air along the beam path, the vertical acceptance of the spectrometer and the reflectivity of the analyzing crystal. 

The relativistic derivation of the relationship between the Compton cross section and the Compton profile leads to a further correction factor [7]. Finally a background subtraction and a normalization of the valence profiles to the number of valence... 

I is the measured intensity, J the Compton profile, M the multiple scattering contribution, K the energy dependent correction factor, B the background, C the normalization constant and Zvai the mean number of valence electrons. Figure 3 shows a valence Compton profile of Cu obtained by this procedure. 

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 main problem in using this method is the statistical error of the Compton profiles which calculates to... 

After calculating the Fourier transform of the Compton profiles one observes that the amplitude of its oscillations becomes smaller than this statistical error when r is greater than 15 a.u. and therefore the B(r) function cannot be used for r > 15 a.u. On the other hand if one wants to get results for Cu with a similar statistical error compared to the results of the Li reconstruction the number of counts needed is given by... 

On the other hand both the momentum densities and the occupation number functions were influenced by the early truncation of the B(r) function due to inadequate statistics. The effect of this influence has to be studied, which could best be done by a reconstruction using calculated Compton profiles. 

Hansen, H., (1980) Reconstruction of the electron momentum distribution from a set of experimental Compton profiles, Hahn Meitner Institute (Berlin), Report HMI B 342. 

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