A method for calculating the photon-atom coherent scattering cross section considering molecular interference effects

Abstract

The present invention discloses a method for calculating a photon-atom coherent scattering cross section that takes into account the molecular interference effect. For a specific molecule, the Thomson differential scattering cross section of each element contained in the molecule is first calculated; secondly, a molecular interference function of the molecule to be calculated is obtained based on experimental measurement or molecular dynamics simulation; then, the molecular interference function is used to correct the atomic shape factor of each element contained in the molecule based on the independent atomic shape factor approximation method to obtain an atomic shape factor that takes into account the molecular interference effect; then, the factor is integrated to obtain an integrated atomic shape factor in a continuous energy ACE format database used by a Monte Carlo program; at the same time, the atomic shape factor that takes into account the molecular interference effect is used to correct the Thomson differential scattering cross section to obtain a differential scattering cross section of a coherent scattering reaction that takes into account the molecular interference effect, and the energy angle integration of the modified Thomson differential scattering cross section can be used to obtain a multi-group coherent scattering cross section used by a deterministic program.

Description

Technical Field

[0001] The present invention relates to the fields of nuclear reactor physics calculation, radiation shielding calculation and medical X-ray imaging, and in particular to a method for calculating a photon-atom coherent scattering cross section taking into account molecular interference effects. Background Art

[0002] The interaction of photons with electrons or the Coulomb field outside an atomic nucleus is called a photon-atom reaction, which primarily includes coherent scattering, incoherent scattering, the photoelectric effect, and the electron pair effect. When the photon energy is lower than the atomic K-shell binding energy, coherent scattering becomes the primary type of photon-atom reaction. The traditional method for calculating the coherent scattering cross section uses the independent atom form factor approximation. This method accurately calculates the coherent scattering cross section for incident photon energies exceeding the atomic K-shell binding energy. However, this method assumes that the atoms interacting with the photons are isolated atoms and that the distribution of the electron density outside the nucleus is spherically symmetric. When the incident photon energy is low and the photon wavelength is of similar order to the bond length in a molecule, using these assumptions results in significant deviations in the calculation of the coherent scattering cross section, failing to accurately describe the angular distribution of the coherently scattered secondary photons. Therefore, for the calculation of the coherent scattering cross section for low-energy photons, a more accurate method is needed to account for the molecular interference effects present in the low-energy photon scattering process. Summary of the Invention

[0003] To solve the problems existing in the above-mentioned prior art, based on the original independent atom form factor approximation method, the present invention proposes a photon-atom coherent scattering cross section calculation method considering the molecular interference effect, which can realize the accurate calculation of the photon-atom coherent scattering cross section for incident photon energy lower than the atomic K-shell electron binding energy.

[0004] To achieve the above object, the present invention adopts the following technical solutions:

[0005] A method for calculating a photon-atom coherent scattering cross section taking into account molecular interference effects comprises the following steps:

[0006] Step 1: For a specific molecule, calculate the Thomson differential scattering cross section of each element according to the type of elements contained in the molecule to be calculated. The Thomson differential scattering cross section is calculated using formula (1):

[0007]

[0008] Where:

[0009] —Thomson differential scattering cross section;

[0010] σ Th —Thomson cross section;

[0011] μ—cosine of the average scattering angle;

[0012] Step 2: Obtain the corresponding molecular interference function according to the type of molecule to be calculated. The molecular interference function is calculated by two methods: experimental measurement and molecular dynamics simulation.

[0013] For the experimental measurement method, the atomic shape factor of the molecule to be calculated is obtained experimentally and divided by the atomic shape factor of the molecule calculated based on the independent atomic shape factor approximation method to obtain the molecular interference function of the corresponding molecule. The molecular interference function is calculated based on the experimental measurement method using formula (2):

[0014]

[0015] Where:

[0016] x—momentum transfer;

[0017] s(x)—molecular interference function;

[0018] F exp (x) — the atomic shape factor of the molecule obtained by experimental measurement;

[0019] F free (x)—the atomic shape factor of the molecule calculated based on the independent atom shape factor approximation method;

[0020] The atomic shape factor of a molecule is calculated based on the independent atom shape factor approximation method using formula (3):

[0021]

[0022] Where:

[0023] n i —the number of atoms of element i in the molecule;

[0024] Z i —the number of protons of element i in the molecule;

[0025] f free (x,Z i )—the atomic shape factor of element i in the molecule calculated based on the independent atom shape factor approximation method;

[0026] For the molecular dynamics simulation method, the radial distribution function between atoms in the molecule is obtained through the molecular dynamics simulation program, and the radial distribution function is further used to calculate the local molecular interference function. The local molecular interference function is calculated based on the molecular dynamics simulation method using formula (4):

[0027]

[0028] Where:

[0029] ρ0—bulk density;

[0030] g ij (r)—radial distribution function of element i and element j;

[0031] s ij (x)—local molecular interference function of element i and element j;

[0032] r—distance from the reference atom;

[0033] For the calculated local molecular interference function, the total molecular interference function of the molecule is obtained by weighted calculation. The calculation of the total molecular interference function adopts formula (5):

[0034]

[0035] Where:

[0036] S(x)—total molecular interference function;

[0037] c i , c j —Number density of element i and element j in the molecule;

[0038] f free,i (x,Z i )—the original value of element i calculated based on the independent atom shape factor approximation method

[0039] Subform factor;

[0040] f free,j (x,Z j )—the atomic shape factor of element j calculated based on the independent atom shape factor approximation method;

[0041] Step 3: Based on the molecular interference function calculated by the experimental measurement method or the molecular dynamics simulation method in step 2, the atomic shape factor of each element in the molecule obtained by the independent atomic shape factor approximation method is corrected to obtain the atomic shape factor of each element in the molecule considering the molecular interference effect. The atomic shape factor of each element considering the molecular interference effect is calculated using formula (6):

[0042]

[0043] Where:

[0044] f mi (x,Z i )—atomic shape factor of element i taking into account molecular interference effects;

[0045] Step 4: Based on the atomic shape factors of each element in the molecule obtained in step 3, taking into account the molecular interference effect, calculate the data required in the continuous energy ACE format database used by the Monte Carlo program and the multi-group cross-section database used by the deterministic program respectively;

[0046] For the continuous energy ACE format database used by the Monte Carlo program, the atomic shape factor and integrated atomic shape factor of each element need to be provided. The atomic shape factor is calculated using formula (6), and the integrated atomic shape factor is calculated using formula (7):

[0047]

[0048] Where:

[0049] A(x 2 ,Z i )—integrated atomic shape factor of element i;

[0050] For the multi-group cross section database used by the deterministic program, the differential scattering cross section of the coherent scattering reaction and the multi-group coherent scattering cross section need to be provided. The differential scattering cross section of the coherent scattering reaction is calculated using formula (8):

[0051]

[0052] Where:

[0053] — differential scattering cross section of the coherent scattering response;

[0054] To calculate the multi-group coherent scattering cross section, it is necessary to first calculate the coherent scattering feed function. The coherent scattering feed function is calculated using formula (9):

[0055]

[0056] Where:

[0057] σ coh (E, E′, μ)—double differential scattering cross section for coherent scattering with incident photon energy E, outgoing photon energy E′, and cosine of the average scattering angle μ;

[0058] P l (μ)—coefficient of the l-th order Legendre polynomial;

[0059] The multi-group coherent scattering cross section is calculated using formula (10):

[0060]

[0061] Where:

[0062] σ coh,l,g→g′—l-th order coherent scattering cross section from the g-th group to the g′-th group;

[0063] Γ coh,l,g′ (E)—the l-th order coherent scattering feeding function of the incident photon with energy E scattered to the g′th group;

[0064] σ coh (E)—continuous energy coherent scattering cross section with incident photon energy E;

[0065] φ l (E)—The l-order weighted energy spectrum of the incident photon energy E.

[0066] Compared with the existing technology, it has the following advantages:

[0067] For photon-atom coherent scattering (PSS) where the incident photon energy is lower than the atomic K-shell electron binding energy, the method presented in this paper can significantly improve the accuracy of the calculation of the PSS cross section. This method improves upon the traditional independent atom form factor approximation method by simultaneously considering the influence of molecular interference effects on the electron density distribution outside the nucleus, thus enabling the calculation of the PSS cross section with molecular interference effects. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 It is a flow chart of the method of the present invention.

[0069] Figure 2 is the Thomson differential scattering cross section of hydrogen and oxygen in water molecules.

[0070] Figure 3 It is the radial distribution function of HH bonds, OH bonds and OO bonds in water molecules.

[0071] Figure 4 It is the local molecular interference function of the HH bond, OH bond and OO bond in the water molecule.

[0072] Figure 5 is the total molecular interference function of water molecules.

[0073] Figure 6 It is the atomic shape factor of hydrogen and oxygen elements in water molecules taking into account the molecular interference effect. DETAILED DESCRIPTION

[0074] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0075] The present invention is a method for calculating the photon-atom coherent scattering cross section taking into account the molecular interference effect. When the incident photon energy is lower than the atomic K-shell electron binding energy, the influence of the molecular interference effect on the photon-atom coherent scattering cross section needs to be considered. Therefore, the present invention improves the traditional independent atom shape factor approximation method to achieve accurate calculation of the photon-atom coherent scattering cross section when the incident photon energy is lower than the atomic K-shell electron binding energy, such as Figure 1 As shown, the present invention includes the following steps:

[0076] Step 1: For a specific molecule (such as water molecule), according to the type of elements contained in the molecule to be calculated (such as water molecule contains hydrogen and oxygen elements, the Thomson differential scattering cross sections of hydrogen and oxygen elements are as follows Figure 2 As shown), calculate the Thomson differential scattering cross section of each element. The Thomson differential scattering cross section is calculated using formula (1):

[0077]

[0078] Where:

[0079] —Thomson differential scattering cross section;

[0080] σ Th —Thomson cross section;

[0081] μ—cosine of the average scattering angle;

[0082] Step 2: According to the type of molecule to be calculated (such as water molecules), the corresponding molecular interference function is obtained, where the molecular interference function is calculated through two methods: experimental measurement and molecular dynamics simulation;

[0083] For the experimental measurement method, the atomic shape factor of the molecule to be calculated is obtained experimentally and divided by the atomic shape factor of the molecule calculated based on the independent atomic shape factor approximation method to obtain the molecular interference function of the corresponding molecule. The molecular interference function is calculated based on the experimental measurement method using formula (2):

[0084]

[0085] Where:

[0086] x—momentum transfer;

[0087] s(x)—molecular interference function;

[0088] F exp (x) — the atomic shape factor of the molecule obtained by experimental measurement;

[0089] F free(x)—the atomic shape factor of the molecule calculated based on the independent atom shape factor approximation method;

[0090] The atomic shape factor of a molecule is calculated based on the independent atom shape factor approximation method using formula (3):

[0091]

[0092] Where:

[0093] n i —the number of atoms of element i in the molecule;

[0094] Z i —the number of protons of element i in the molecule;

[0095] f free (x,Z i )—the atomic shape factor of element i in the molecule calculated based on the independent atom shape factor approximation method;

[0096] For the molecular dynamics simulation method, the radial distribution function between atoms in the molecule is obtained by a molecular dynamics simulation program (such as the GROMACS program) (such as the radial distribution function of the HH bond, OH bond and OO bond in the water molecule, the radial distribution function of the HH bond, OH bond and OO bond in the water molecule is as follows Figure 3 As shown), the radial distribution function is further used to calculate the local molecular interference function (such as the local molecular interference function of the HH bond, OH bond and OO bond in the water molecule, the local molecular interference function of the HH bond, OH bond and OO bond in the water molecule is shown as Figure 4 As shown), the local molecular interference function is calculated based on the molecular dynamics simulation method using formula (4):

[0097]

[0098] Where:

[0099] ρ0—bulk density;

[0100] g ij (r)—radial distribution function of element i and element j;

[0101] s ij (x)—local molecular interference function of element i and element j;

[0102] r—distance from the reference atom;

[0103] For the calculated local molecular interference function, the total molecular interference function of the molecule is obtained by weighted calculation (such as the total molecular interference function of water molecules, the total molecular interference function of water molecules is as follows Figure 5The total molecular interference function is calculated using formula (5):

[0104]

[0105] Where:

[0106] S(x)—total molecular interference function;

[0107] c i , c j —Number density of element i and element j in the molecule;

[0108] f free,i (x,Z i )—the original value of element i calculated based on the independent atom shape factor approximation method

[0109] Subform factor;

[0110] f free,j (x,Z j )—the atomic shape factor of element j calculated based on the independent atom shape factor approximation method;

[0111] Step 3: Based on the molecular interference function calculated by the experimental measurement method or the molecular dynamics simulation method in step 2, the atomic shape factor of each element in the molecule obtained by the independent atomic shape factor approximation method is corrected to obtain the atomic shape factor of each element in the molecule considering the molecular interference effect (such as the atomic shape factor of hydrogen and oxygen elements in water molecules considering the molecular interference effect, the atomic shape factor of hydrogen and oxygen elements in water molecules considering the molecular interference effect such as Figure 6 The atomic shape factor of each element considering the molecular interference effect is calculated using formula (6):

[0112]

[0113] Where:

[0114] f mi (x,Z i )—atomic shape factor of element i taking into account molecular interference effects;

[0115] Step 4: Based on the atomic shape factors of each element in the molecule obtained in step 3, taking into account the molecular interference effect, calculate the data required in the continuous energy ACE format database used by the Monte Carlo program and the multi-group cross-section database used by the deterministic program respectively;

[0116] For the continuous energy ACE format database used by the Monte Carlo program, the atomic shape factor and integrated atomic shape factor of each element need to be provided. The atomic shape factor is calculated using formula (6), and the integrated atomic shape factor is calculated using formula (7):

[0117]

[0118] Where:

[0119] A(x 2 ,Z i )—integrated atomic shape factor of element i;

[0120] For the multi-group cross section database used by the deterministic program, the differential scattering cross section of the coherent scattering reaction and the multi-group coherent scattering cross section need to be provided. The differential scattering cross section of the coherent scattering reaction is calculated using formula (8):

[0121]

[0122] Where:

[0123] — differential scattering cross section of the coherent scattering response;

[0124] To calculate the multi-group coherent scattering cross section, it is necessary to first calculate the coherent scattering feed function. The coherent scattering feed function is calculated using formula (9):

[0125]

[0126] Where:

[0127] σ coh (E, E′, μ)—double differential scattering cross section for coherent scattering with incident photon energy E, outgoing photon energy E′, and cosine of the average scattering angle μ;

[0128] P l (μ)—coefficient of the lth-order Legendre polynomial.

[0129] The multi-group coherent scattering cross section is calculated using formula (10):

[0130]

[0131] Where:

[0132] σ coh,l,g→g′ —l-th order coherent scattering cross section from the g-th group to the g′-th group;

[0133] Γ coh,l,g′ (E)—the l-th order coherent scattering feeding function of the incident photon with energy E scattered to the g′th group;

[0134] σcoh (E)—continuous energy coherent scattering cross section with incident photon energy E;

[0135] φ l (E)—The l-order weighted energy spectrum of the incident photon energy E.

[0136] In step 2, it is mentioned that the molecular interference function can be obtained by using experimental measurement methods and molecular dynamics simulation methods. The present invention has no limitation on the experimental measurement process and the software used in the molecular dynamics simulation.

[0137] Step 3 mentions correcting the atomic shape factor obtained based on the independent atomic shape factor approximation method. The sources of the atomic shape factor data given based on the independent atomic shape factor approximation method are diverse. The present invention has no restrictions on the method for obtaining the atomic shape factor based on the independent atomic shape factor approximation method.

Claims

1. A method for calculating the photon-atom coherent scattering cross section taking into account the molecular interference effect, characterized by: The following steps are involved: Step 1: For a specific molecule, calculate the Thomson differential scattering cross section of each element according to the type of elements contained in the molecule to be calculated. The Thomson differential scattering cross section is calculated using formula (1): Where: —Thomson differential scattering cross section; σ Th —Thomson cross section; μ—cosine of the average scattering angle; Step 2: Obtain the corresponding molecular interference function according to the type of molecule to be calculated. The molecular interference function is calculated by two methods: experimental measurement and molecular dynamics simulation. For the experimental measurement method, the atomic shape factor of the molecule to be calculated is obtained experimentally and divided by the atomic shape factor of the molecule calculated based on the independent atomic shape factor approximation method to obtain the molecular interference function of the corresponding molecule. The molecular interference function is calculated based on the experimental measurement method using formula (2): Where: x—momentum transfer; s(x)—molecular interference function; F exp (x) — the atomic shape factor of the molecule obtained by experimental measurement; F free (x)—the atomic shape factor of the molecule calculated based on the independent atom shape factor approximation method; The atomic shape factor of a molecule is calculated based on the independent atom shape factor approximation method using formula (3): Where: n i —the number of atoms of element i in the molecule; Z i —the number of protons of element i in the molecule; f free (x,Z i )—the atomic shape factor of element i in the molecule calculated based on the independent atom shape factor approximation method; For the molecular dynamics simulation method, the radial distribution function between atoms in the molecule is obtained through the molecular dynamics simulation program, and the radial distribution function is further used to calculate the local molecular interference function. The local molecular interference function is calculated based on the molecular dynamics simulation method using formula (4): Where: ρ0—bulk density; g ij (r)—radial distribution function of element i and element j; s ij (x)—local molecular interference function of element i and element j; r—distance from the reference atom; For the calculated local molecular interference function, the total molecular interference function of the molecule is obtained by weighted calculation. The calculation of the total molecular interference function adopts formula (5): Where: S(x)—total molecular interference function; c i , c j —Number density of element i and element j in the molecule; f free,i (x,Z i )—the original value of element i calculated based on the independent atom shape factor approximation method Subform factor; f free,j (x,Z j )—the atomic shape factor of element j calculated based on the independent atom shape factor approximation method; Step 3: Based on the molecular interference function calculated by the experimental measurement method or the molecular dynamics simulation method in step 2, the atomic shape factor of each element in the molecule obtained by the independent atomic shape factor approximation method is corrected to obtain the atomic shape factor of each element in the molecule considering the molecular interference effect. The atomic shape factor of each element considering the molecular interference effect is calculated using formula (6): Where: f mi (x,Z i )—atomic shape factor of element i taking into account molecular interference effects; Step 4: Based on the atomic shape factors of each element in the molecule obtained in step 3, taking into account the molecular interference effect, calculate the data required in the continuous energy ACE format database used by the Monte Carlo program and the multi-group cross-section database used by the deterministic program respectively; For the continuous energy ACE format database used by the Monte Carlo program, the atomic shape factor and integrated atomic shape factor of each element need to be provided. The atomic shape factor is calculated using formula (6), and the integrated atomic shape factor is calculated using formula (7): Where: A(x 2 ,Z i )—integrated atomic shape factor of element i; For the multi-group cross section database used by the deterministic program, the differential scattering cross section of the coherent scattering reaction and the multi-group coherent scattering cross section need to be provided. The differential scattering cross section of the coherent scattering reaction is calculated using formula (8): Where: — differential scattering cross section of the coherent scattering response; To calculate the multi-group coherent scattering cross section, it is necessary to first calculate the coherent scattering feed function. The coherent scattering feed function is calculated using formula (9): Where: σ coh (E, E′, μ)—double differential scattering cross section for coherent scattering with incident photon energy E, outgoing photon energy E′, and cosine of the average scattering angle μ; P l (μ)—coefficient of the l-th order Legendre polynomial; The multi-group coherent scattering cross section is calculated using formula (10): Where: σ coh,l,g→g′ —l-th order coherent scattering cross section of the g-th to g′-th groups; Γ coh,l,g′ (E)—the l-th order coherent scattering feeding function of the incident photon with energy E scattered to the g′th group; σ coh (E)—continuous energy coherent scattering cross section with incident photon energy E; φ l (E)—The l-order weighted energy spectrum of the incident photon energy E.

Images