Compton profiles and momentum space inequalities

Abstract

Abstract. Rigorous upper and lower bounds to the atomic Compton profile J(q) are obtained for any value of the momentum transferred q in terms of radial expectation values (p~) of the atomic momentum density 7(p). In doing so, a procedure based on moment-theoretic techniques and Chebyshev inequalities has been used. This type of results can be employed to study the compatibility of diverse information obtained by using different models, techniques, numerical calculations or experimental data. The same method allows also to obtain approximations to the Compton profile and to bound other relevant characteristics of J(q). A comparison of the approximations with some previously known Maximum Entropy Approximations is done. In order to test the accuracy of the bounds, a numerical study of the results is carried out in a Hartree-Fock framework for atomic systems. 31.10.+z; 31.15.+q The one-electron densities of a N-electron system in position and momentum spaces, p (r) and 9, (p) respectively, are basic ingredients in the study of many physical properties from a Density Functional Theory perspective, which is nowadays one of the fundamental theories of matter [ I 1. PACS:Much attention has been paid to the relevant role played by the position-space density p (r) in the description of many-electron systems. However, a similar study in terms of the electronic distribution in momentum space has still to be much more worked out. Moreover, in the last decade the electron momentum distribution of atomic and molecular systems has been shown to be experimentally accessible [2][3][4] It is worthy to point out that the height of the peak of the Compton profile may be the most accurately measured quantity in atomic and molecular physics [2]. However, the same accuracy does not occur when Compton experiments require high values of the momentum transferred. In spite of the experimental improvements, the relation between the measured cross sections and Compton profiles is not straightforward, specially at high energies. This is mainly due to the fact that the Compton profile J(q) can only be measured up to some finite value of the momentum transferred qmax, which depends on the type of experiment. So, the computation of quantities such as expectation values of the momentum density 7 (P) from experimental Compton profiles involves extrapolation techniques or the use of analytical models. In Ref.[12], a careful analysis of the expectation values of the momentum density predicted from statistically simulated Compton profiles is carried out. The numerical tests performed in this work indicate that the accuracy in the obtention of the higher order expectation values from J(q) strongly depends on qma×" An asymptotic constraint (calculated in a Hartree-Fock framework) had to be used to reduce drastically the admissible range for these moments and therefore to stabilize the extrapolation.The reciprocal form factor or characteristic function [ 13 -17 ], quantum-mechanical calculations [ 18 -19 ] and information theory with momentum and e...

Compton profile and charge density reconstruction by the maximum entropy method

“…Originally proposed by Jaynes [2,3], the maximization of entropy to construct probability models has been applied to a wide range of physics problems, as well as to estimate electron charge density distribution via inverse problems [4][5][6]. In addition, many authors have used the maximum entropy principle to compare Compton profile experiments and theoretical calculations of atoms and molecules from the knowledge of energy, showing relatively good agreement results [7][8][9].…”

Maximum-entropy analysis of one-particle densities in atoms

“…radioastronomy [45], parameter spectral estimation [46], particle physics [47] or atomic and many-fermion physics [25,[48][49][50]. On the other hand, let us remark that the ME technique provides complementary information to the one obtained from the Stieltjes-Chebyshev method, recently applied to the study of (r) [29], (p) [31] and J(q) [30]. From the latter, rigorous upper and lower bounds on the values of the density are obtained, while the ME method chooses, among all the admissible functions which are compatible with these bounds, the most plausible in the sense described above.…”

“…For atomic systems, some rigorous properties are well known [13][14][15][16]. Moreover, several other structural properties [17][18][19][20][21][22], numerical computations [23][24][25] and inequalities involving the expectation values [21,[26][27][28][29][30][31] L "…”