The isotropic Compton profile in C-like ions: individual shells analysis

Ahmed, Riam Amer; Ameen, Wissam Ahmed

doi:10.37652/juaps.2025.163980.1613

[1] Compton AH.A Quantum Theory of the Scattering of X-rays by Light Elements. Physical Review. 1923;21(5):483-502.10.1103/PhysRev.21.483

[2] Cooper MJ. Compton Scattering and Electron Momentum Determination. Reports on Progress in Physics. 1985;48(4):415-81. 10.1088/0034-4885/48/4/001

[3] Qiao CK, Wei JW, Chen L. An Overview of the Compton Scattering Calculation. Crystals. 2021;11(5):525.10.3390/cryst11050525

[4] Hubbell JH, Veigele WJ, Briggs EA, Brown RT, Cromer DT, Howerton RJ. Atomic Form Factors, Incoherent Scattering Functions, and Photon Scattering Cross Sections. Journal of Physical and Chemical Reference Data. 1975;4(3):471-538. 10.1063/1.555523

[5] Klein O, Nishina Y. Über die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac. Zeitschrift für Physik. 1929;52(11–12):853-68. 10.1007/BF01366453

[6] Currat R, DeCicco PD, Weiss RJ. Impulse Approximation in Compton Scattering.Physical Review B. 1971;4(12):4256-61.10.1103/PhysRevB.4.4256

[7] Bergstrom PM, Pratt RH. An Overview of the Theories Used in Compton Scattering Calculations. Radiation Physics and Chemistry. 1997;50(1):3-29. 10.1016/S0969-806X(97)00022-4

[8] Vijayakumar R, Shivaramu, Ramamurthy N, Ford MJ. Compton Scattering Study of Electron Momentum Distribution in Lithium Fluoride Using 662 keV Gamma Radiations. Physica B: Condensed Matter. 2008;403(23–24):4309-13. 10.1016/j.physb.2008.09.022

[9] Eisenberger P.Compton-Profile Measurements of N2, O2, and Ne Using Silver and Molybdenum X-Rays. Physical Review A. 1972;5(2):628-35. 10.1103/PhysRevA.5.628

[10] Biggs F, Mendelsohn LB, Mann JB.Hartree–Fock Compton Profiles for the Elements. Atomic Data and Nuclear Data Tables. 1975;16(3):201-309. 10.1016/0092-640X(75)90030-3

[11] Brown RE, Smith VH. Compton Profiles of the First-Row Atoms: First-Order Profiles and Correlation Effects.Molecular Physics. 1977;34(3):713-29. 10.1080/00268977700102051

[12] Gadre SR, Gejji SP, Chakravorty SJ. Hartree–Fock Momentum Expectation Values for Atoms and Ions. Atomic Data and Nuclear Data Tables. 1983;28(3):477-91.10.1016/0092-640X(83)90003-7

[13] Esquivel RO, Tripathi AN, Sagar RP, Smith VH. Accurate One-Electron Momentum-Space Properties for the Lithium Isoelectronic Sequence. Journal of Physics B: Atomic, Molecular and Optical Physics. 1992;25(13):2925-41. 10.1088/0953-4075/25/13/003

[14] Talukdar B, Sarkar A, Roy SN, Sarkar P. Effect of Electron Correlation on Momentum Properties of Helium Atom. Chemical Physics Letters. 2003;381(1–2):67-73.10.1016/j.cplett.2003.09.074

[15] Al-Sharif AI. Atomic Compton Profile of Neon Calculated from an Accurate Kohn–Sham Potential. Physical Review A. 2005;72(1):012703. 10.1103/PhysRevA.72.012703

[16] Khera S, Mathur S, Ahuja BL. Electronic Structure of Hafnium: A Compton Profile Study. Pramana – Journal of Physics. 2007;68(1):91-8. 10.1007/s12043-007-0010-z

[17] Alzubadi AA, Albayati KH.Calculation of Theoretical Isotropic Compton Profile for Many Particle Systems.Modern Physics Letters B. 2010;24(14):1601-14.10.1142/S0217984910023396

[18] Özdogan T, Eraslan M. Analytic Calculation of Momentum Distribution and Compton Profiles of Atoms Using Hartree–Fock–Roothaan Method: Applications to Atoms 2 ≤ Z ≤ 10. In: Advances in Quantum Chemistry. vol. 73. Elsevier; 2016. p. 173-82.10.1016/bs.aiq.2015.09.002

[19] Mukherjee N, Roy AK. Analysis of Compton Profile through Information Theory in H-like Atoms inside Impenetrable Sphere. Journal of Physics B: Atomic, Molecular and Optical Physics. 2020;53(23):235002.10.1088/1361-6455/abbe28

[20] Al-Jibbouri H.Compton Profile of 1s²-State for 2 ≤ Z ≥ 10. Indian Journal of Pure & Applied Physics. 2021;59(11):752-5.10.56042/ijpap.v59i11.45354

[21] Wan JJ, Gu J, Wu ZY, Wu F, Li J, Qiao HX.High-Accuracy Calculation of Nonrelativistic Compton Profile for H-like Ions. Results in Physics. 2023;50:106562.10.1016/j.rinp.2023.106562

[22] Mohammed MN. Compton Profile Investigation for Aluminum (Al).Tikrit Journal of Pure Science. 2024;29(1):81-8.10.25130/tjps.v29i1.1441

[23] Wan JJ, Wu F, Gu J. High-Accuracy Calculation on Relativistic Compton Profile of H-like Ions.The European Physical Journal Plus. 2025;140(5):434. 10.1140/epjp/s13360-025-06378-x

[24] Clementi E, Roetti C. Roothaan–Hartree–Fock Atomic Wavefunctions: Basis Functions and Their Coefficients for Ground and Certain Excited States of Neutral and Ionized Atoms, Z ≤ 54.Atomic Data and Nuclear Data Tables. 1974;14(3–4):177-478. 10.1016/S0092-640X(74)80016-1

[25] Wolfram Research, Inc . Mathematica, Version 12.1. Champaign, Illinois: Wolfram Research, Inc.; 2020. Computer software. Available from: https://www.wolfram.com/mathematica/

[26] Hinde RJ. Quantum Chemistry, 5th Edition (by Ira N. Levine). Journal of Chemical Education. 2000;77(12):1564. Book review. 10.1021/ed077p1564

[27] Sherrill CD. An Introduction to Hartree–Fock Molecular Orbital Theory; 2000. Lecture notes, School of Chemistry and Biochemistry, Georgia Institute of Technology. Available from: http://vergil.chemistry.gatech.edu/notes/hf-intro/

[28] Hartree DR, Hartree W. Self-Consistent Field, with Exchange, for Beryllium. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences. 1935;150(869):9-33. 10.1098/rspa.1935.0085

[29] Echenique P, Alonso JL. A Mathematical and Computational Review of Hartree–Fock SCF Methods in Quantum Chemistry. Molecular Physics. 2007;105(23–24):3057-98. 10.1080/00268970701757875

[30] Goeppert-Mayer M, Sklar AL. Calculations of the Lower Excited Levels of Benzene. The Journal of Chemical Physics. 1938;6(10):645-52. 10.1063/1.1750138

[31] Komarov FF, Temkin MM. On the Momentum Representation of Slater-Type Orbitals. Journal of Physics B: Atomic and Molecular Physics. 1976;9(10):L255-8.10.1088/0022-3700/9/10/001

[32] Roothaan CCJ. New Developments in Molecular Orbital Theory. Reviews of Modern Physics. 1951;23(2):69-89.10.1103/RevModPhys.23.69

[33] Kaijser P, Smith, Jr VH.Evaluation of Momentum Distributions and Compton Profiles for Atomic and Molecular Systems. In: Advances in Quantum Chemistry. vol. 10. Academic Press; 1977. p. 37-76. 10.1016/S0065-3276(08)60578-X

[34] Thakkar AJ. The Momentum Density Perspective of the Electronic Structure of Atoms and Molecules. In: Rice SA, editor. Advances in Chemical Physics. vol. 128. Wiley; 2003. p. 303-52. 10.1002/0471484237.ch5

[35] Koga T.Interelectronic Angle Densities of Atoms in Momentum Space.Theoretical Chemistry Accounts. 2003;110(2):79-84.10.1007/s00214-003-0450-6

[36] Löwdin PO. Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction.Physical Review. 1955;97(6):1474-89. 10.1103/PhysRev.97.1474

[37] Mallick B.A Quantum Theory of the Scattering of Soft X-ray from Compton Effect Caused Electron Standing Wave in the Materials.Turkish Journal of Physics. 2011;35(2):75-114.10.3906/fiz-1101-81

[38] Jaiswal P, Shukla A. Kinetically Balanced Gaussian Basis-Set Approach to Relativistic Compton Profiles of Atoms. Physical Review A. 2007;75(2):022504.10.1103/PhysRevA.75.022504

	The isotropic Compton profile in C-like ions: individual shells analysis
Journal of University of Anbar for Pure Science
Volume 20, Issue 2, August 2026, Pages 274-283 PDF (579.87 K)
Document Type: Research Paper
DOI: 10.37652/juaps.2025.163980.1613
Authors
Riam Amer Ahmed; Wissam Ahmed Ameen^*
Physics Department, College of Science, University of Anbar, Anbar, Iraq
Abstract
The Compton profiles of intra- and inter-shell interactions in the carbon atom were calculated and analyzed using the Roothaan-Hartree-Fock method. To examine the effect of increasing nuclear charge, the study was extended to include selected carbon-like ions: N⁺, O²⁺, and F³⁺. All calculations were performed for the ground state and reported in atomic units. A progressive narrowing of the orbitals with increasing nuclear charge appears to reduce the height of the central peak and extend the momentum distribution toward higher momentum values. The crossover points between the profiles were also identified, revealing regions in which the momentum density expands in response to ionization. The Compton profile of intra-shell interactions appears to contain two main regions: one near q = 0 and another at intermediate q values. For inter-shell interactions, more than one region was observed, reflecting the contribution of exchange processes. Overall, the findings are in good agreement with the published literature.
Keywords
Compton profile; Momentum space; Roothaan-Hartree-Fock method

References
[1] Compton AH.A Quantum Theory of the Scattering of X-rays by Light Elements. Physical Review. 1923;21(5):483-502.10.1103/PhysRev.21.483 [2] Cooper MJ. Compton Scattering and Electron Momentum Determination. Reports on Progress in Physics. 1985;48(4):415-81. 10.1088/0034-4885/48/4/001 [3] Qiao CK, Wei JW, Chen L. An Overview of the Compton Scattering Calculation. Crystals. 2021;11(5):525.10.3390/cryst11050525 [4] Hubbell JH, Veigele WJ, Briggs EA, Brown RT, Cromer DT, Howerton RJ. Atomic Form Factors, Incoherent Scattering Functions, and Photon Scattering Cross Sections. Journal of Physical and Chemical Reference Data. 1975;4(3):471-538. 10.1063/1.555523 [5] Klein O, Nishina Y. Über die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac. Zeitschrift für Physik. 1929;52(11–12):853-68. 10.1007/BF01366453 [6] Currat R, DeCicco PD, Weiss RJ. Impulse Approximation in Compton Scattering.Physical Review B. 1971;4(12):4256-61.10.1103/PhysRevB.4.4256 [7] Bergstrom PM, Pratt RH. An Overview of the Theories Used in Compton Scattering Calculations. Radiation Physics and Chemistry. 1997;50(1):3-29. 10.1016/S0969-806X(97)00022-4 [8] Vijayakumar R, Shivaramu, Ramamurthy N, Ford MJ. Compton Scattering Study of Electron Momentum Distribution in Lithium Fluoride Using 662 keV Gamma Radiations. Physica B: Condensed Matter. 2008;403(23–24):4309-13. 10.1016/j.physb.2008.09.022 [9] Eisenberger P.Compton-Profile Measurements of N2, O2, and Ne Using Silver and Molybdenum X-Rays. Physical Review A. 1972;5(2):628-35. 10.1103/PhysRevA.5.628 [10] Biggs F, Mendelsohn LB, Mann JB.Hartree–Fock Compton Profiles for the Elements. Atomic Data and Nuclear Data Tables. 1975;16(3):201-309. 10.1016/0092-640X(75)90030-3 [11] Brown RE, Smith VH. Compton Profiles of the First-Row Atoms: First-Order Profiles and Correlation Effects.Molecular Physics. 1977;34(3):713-29. 10.1080/00268977700102051 [12] Gadre SR, Gejji SP, Chakravorty SJ. Hartree–Fock Momentum Expectation Values for Atoms and Ions. Atomic Data and Nuclear Data Tables. 1983;28(3):477-91.10.1016/0092-640X(83)90003-7 [13] Esquivel RO, Tripathi AN, Sagar RP, Smith VH. Accurate One-Electron Momentum-Space Properties for the Lithium Isoelectronic Sequence. Journal of Physics B: Atomic, Molecular and Optical Physics. 1992;25(13):2925-41. 10.1088/0953-4075/25/13/003 [14] Talukdar B, Sarkar A, Roy SN, Sarkar P. Effect of Electron Correlation on Momentum Properties of Helium Atom. Chemical Physics Letters. 2003;381(1–2):67-73.10.1016/j.cplett.2003.09.074 [15] Al-Sharif AI. Atomic Compton Profile of Neon Calculated from an Accurate Kohn–Sham Potential. Physical Review A. 2005;72(1):012703. 10.1103/PhysRevA.72.012703 [16] Khera S, Mathur S, Ahuja BL. Electronic Structure of Hafnium: A Compton Profile Study. Pramana – Journal of Physics. 2007;68(1):91-8. 10.1007/s12043-007-0010-z [17] Alzubadi AA, Albayati KH.Calculation of Theoretical Isotropic Compton Profile for Many Particle Systems.Modern Physics Letters B. 2010;24(14):1601-14.10.1142/S0217984910023396 [18] Özdogan T, Eraslan M. Analytic Calculation of Momentum Distribution and Compton Profiles of Atoms Using Hartree–Fock–Roothaan Method: Applications to Atoms 2 ≤ Z ≤ 10. In: Advances in Quantum Chemistry. vol. 73. Elsevier; 2016. p. 173-82.10.1016/bs.aiq.2015.09.002 [19] Mukherjee N, Roy AK. Analysis of Compton Profile through Information Theory in H-like Atoms inside Impenetrable Sphere. Journal of Physics B: Atomic, Molecular and Optical Physics. 2020;53(23):235002.10.1088/1361-6455/abbe28 [20] Al-Jibbouri H.Compton Profile of 1s²-State for 2 ≤ Z ≥ 10. Indian Journal of Pure & Applied Physics. 2021;59(11):752-5.10.56042/ijpap.v59i11.45354 [21] Wan JJ, Gu J, Wu ZY, Wu F, Li J, Qiao HX.High-Accuracy Calculation of Nonrelativistic Compton Profile for H-like Ions. Results in Physics. 2023;50:106562.10.1016/j.rinp.2023.106562 [22] Mohammed MN. Compton Profile Investigation for Aluminum (Al).Tikrit Journal of Pure Science. 2024;29(1):81-8.10.25130/tjps.v29i1.1441 [23] Wan JJ, Wu F, Gu J. High-Accuracy Calculation on Relativistic Compton Profile of H-like Ions.The European Physical Journal Plus. 2025;140(5):434. 10.1140/epjp/s13360-025-06378-x [24] Clementi E, Roetti C. Roothaan–Hartree–Fock Atomic Wavefunctions: Basis Functions and Their Coefficients for Ground and Certain Excited States of Neutral and Ionized Atoms, Z ≤ 54.Atomic Data and Nuclear Data Tables. 1974;14(3–4):177-478. 10.1016/S0092-640X(74)80016-1 [25] Wolfram Research, Inc . Mathematica, Version 12.1. Champaign, Illinois: Wolfram Research, Inc.; 2020. Computer software. Available from: https://www.wolfram.com/mathematica/ [26] Hinde RJ. Quantum Chemistry, 5th Edition (by Ira N. Levine). Journal of Chemical Education. 2000;77(12):1564. Book review. 10.1021/ed077p1564 [27] Sherrill CD. An Introduction to Hartree–Fock Molecular Orbital Theory; 2000. Lecture notes, School of Chemistry and Biochemistry, Georgia Institute of Technology. Available from: http://vergil.chemistry.gatech.edu/notes/hf-intro/ [28] Hartree DR, Hartree W. Self-Consistent Field, with Exchange, for Beryllium. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences. 1935;150(869):9-33. 10.1098/rspa.1935.0085 [29] Echenique P, Alonso JL. A Mathematical and Computational Review of Hartree–Fock SCF Methods in Quantum Chemistry. Molecular Physics. 2007;105(23–24):3057-98. 10.1080/00268970701757875 [30] Goeppert-Mayer M, Sklar AL. Calculations of the Lower Excited Levels of Benzene. The Journal of Chemical Physics. 1938;6(10):645-52. 10.1063/1.1750138 [31] Komarov FF, Temkin MM. On the Momentum Representation of Slater-Type Orbitals. Journal of Physics B: Atomic and Molecular Physics. 1976;9(10):L255-8.10.1088/0022-3700/9/10/001 [32] Roothaan CCJ. New Developments in Molecular Orbital Theory. Reviews of Modern Physics. 1951;23(2):69-89.10.1103/RevModPhys.23.69 [33] Kaijser P, Smith, Jr VH.Evaluation of Momentum Distributions and Compton Profiles for Atomic and Molecular Systems. In: Advances in Quantum Chemistry. vol. 10. Academic Press; 1977. p. 37-76. 10.1016/S0065-3276(08)60578-X [34] Thakkar AJ. The Momentum Density Perspective of the Electronic Structure of Atoms and Molecules. In: Rice SA, editor. Advances in Chemical Physics. vol. 128. Wiley; 2003. p. 303-52. 10.1002/0471484237.ch5 [35] Koga T.Interelectronic Angle Densities of Atoms in Momentum Space.Theoretical Chemistry Accounts. 2003;110(2):79-84.10.1007/s00214-003-0450-6 [36] Löwdin PO. Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction.Physical Review. 1955;97(6):1474-89. 10.1103/PhysRev.97.1474 [37] Mallick B.A Quantum Theory of the Scattering of Soft X-ray from Compton Effect Caused Electron Standing Wave in the Materials.Turkish Journal of Physics. 2011;35(2):75-114.10.3906/fiz-1101-81 [38] Jaiswal P, Shukla A. Kinetically Balanced Gaussian Basis-Set Approach to Relativistic Compton Profiles of Atoms. Physical Review A. 2007;75(2):022504.10.1103/PhysRevA.75.022504
Statistics Article View: 124 PDF Download: 107