Relativistic Form Factors Research Articles

A study of integrable form factors in massless relativistic AdS2

In this paper we initiate the study of form factors for the massless scattering of integrable AdS2 superstrings, where the difference-form of the S-matrix can be exploited to implement the relativistic form factor bootstrap. The non-standard nature of the S-matrix implies that traditional methods do not apply. We use the fact that the massless AdS2S-matrix is a limit of a better-behaved S-matrix found by Fendley. We show that the previously conjectured massless AdS2 dressing factor coincides with the limit of the De Martino-Moriconi improved dressing factor for the Fendley S-matrix. We then solve the form factor constraints in the two-particle case. Along the way we find a method to construct integral representations of relativistic dressing factors satisfying specific assumptions, and use it to obtain analytic proofs of crossing and unitarity relations.

Preliminary Empirical study of the Rayleigh to Compton scattering ratio for elements at gamma energy 59.5 keV using Si(Li) detector

The characteristic parameters of a material relation to photon interactions such as: mass attenuation coefficient, effective atomic number, effective electron density are required to provide essential data in diverse works such as nuclear diagnostic and cancer radiotherapy, industrial irradiation, radiation dosimetry, radiation protection and shielding, analyzing of the concentration of elements and radioactive isotopes. In this paper, the theoretical models such as non-relativistic form factor (NRFF), relativistic form factor (RFF), and modified form factor (MFF) were used to calculate the ratio Rayleigh-Compton for elements with at gamma energy 59.5 keV. The results showed that there was a discrepancy between the theoretical modes at a high atomic number. The mean value of the Rayleigh-Compton ratio depends on the atomic number, which shows the quadratic function of the correlation coefficient R2 = 0.996 as well. Besides, the experimental system was set-up and measured for some targets such as aluminum, copper, and lead at a scattering angle 150o using 241Am source by Si(Li) detector to confirm the theoretical values. The preliminary result showed that there was a good agreement with experimental and theoretical results is lower than 20%. Further investigation will be measured by the samples for more detailed evaluation.

The coherent to Compton scattering differential cross section ratios of some fifth period elements in the external magnetic field

Some fifth-period elements Y, Zr, Nb, Mo, Ru, Rh, Pd, Ag and Cd have been obtained in powder form and converted into pellets by applying pressure (~ 1 GPa). They have annealed at 1000 °C (except Cd ~ 300 °C). In the external magnetic fields (0.4 and 0.8 T) and without magnetic field, the coherent to Compton differential cross-sectional ratios has been investigated by using energy-dispersive X-ray fluorescence system. The data were analyzed via Origin 9.0 computer program. The results were calculated experimentally and theoretically. The theoretical results were obtained with two different methods, and these results were compared with the experimental results. These methods are relativistic form factor theory and non-relativistic form factor theory at without magnetic field. It was seen that the coherent to Compton differential cross-sectional ratios of the elements used in the present study increases with the increasing of the magnetic field and atomic number.

Measurement of coherent to Compton scattering differential cross-section ratios of cadmium at different momentum transfer factors

The coherent to Compton scattering ratio is a useful technique because it is independent of the density of the material and the material is defined only by the atomic number. Also, this ratio is used where changes in the atomic number are important and changes in the attenuation coefficient are too small to detect. In this study, coherent to Compton scattering ratios of cadmium have been calculated at different scattering angles (90°, 95°, 100°, 105°, 110°, 115°, 120°, 125°, and 130°). A high purity germanium detector was used. The coherent to Compton scattering differential cross-section ratios were plotted as a function of the scattering angles and constituted a best fit curve. It was observed that the coherent to Compton differential cross-section ratios decrease with an increasing scattering angle. The experimental results agree well with the results of the non-relativistic and relativistic form factor method.

The photon interaction cross sections of blood

The photon interaction cross sections which contain the photoelectric absorption, coherent (Rayleigh) and incoherent (Compton) scattering are calculated for blood in the energy range of 1–150 keV. The molecular photoelectric absorption cross section is achieved from atomic cross section data. The molecular coherent (Rayleigh) and incoherent (Compton) scattering cross sections are estimated using relativistic modified form factor and incoherent scattering function approximation, respectively. The molecular form factors updated including molecular interference effects and incoherent scattering functions were also calculated. With the inclusion of the molecular interference effects in the coherent (Rayleigh) scattering, the more accurate data of radiation scattered from blood will be provided. The results are compared with references values in literature. It is found that the theoretical attenuation coefficients are in good agreement with experimental values.

An experimental study on the angular dependence of coherent to Compton scattering differential cross-section ratios of calcium

The coherent to Compton scattering cross-section ratio of calcium was measured experimentally for several polar scattering angles (90°, 100°, 110°, 120°, and 130°) and azimuthal scattering angles (30°, 20°, 10°, 0°, -10°, and -20°). A high purity germanium semiconductor detector with a resolution of 199.6 eV at 5.9 keV was used. The theoretical values of the coherent to Compton scattering differential cross-section of calcium were calculated on the basis of nonrelativistic form factors, relativistic form factors, and S-matrix theories. It was observed that the experimental results were most compatible with the nonrelativistic form factor theory.

Polar and azimuthal angular dependence of coherent to incoherent scattering differential cross-section ratios of Au at 59.54 keV

Abstract Coherent to incoherent differential cross-section ratios of Au have been measured for several polar scattering angles (90°, 100°, 110°, 120° and 130°) and azimuthal angles (30°, 20°, 10°, 0°, −10°, and −20°) at 59.54 keV photon energy by using high purity germanium (HPGe) detector, which has a resolution of 199.6 eV at the 5.9 keV. The samples were excited with 59.54 keV gamma rays emitted from 3.7×10 9 Bq (100 mCi) Am 241 point source. The intensity ratios were corrected due to the photopeak efficiency of gamma detector and absorption of photons in the target and air. The experimental values obtained in this study were compared with those estimated on the basis of the non-relativistic form factors and relativistic form factors.

Form factor for a two-particle system within a relativistic quasipotential approach: Case of arbitrary masses and of a vector current

A new relativistic form factor for a bound two-particle system was obtained for the case of a vector current. The present consideration was performed within the relativistic quasipotential approach based on the covariant Hamiltonian formulation of quantum field theory by going over to the three-dimensional relativistic configuration representation for the case of interaction between two relativistic spinless particles of arbitrary mass.

Measurement of effective atomic number and Rayleigh-to-Compton cross-section ratio for 145 keV gamma photons

The effective atomic numbers of composites and Rayleigh-to-Compton cross-section ratio of elements, in the range 6 ≤ Z ≤ 82 for 145 keV gamma photons, are determined experimentally. An HPGe (high purity germanium) semiconductor detector is placed, at scattering angle of 90°, to record the spectra originating from interactions of incident gamma photons with the target under investigations. The intensity ratio of Rayleigh-to-Compton scattered peaks, corrected for photo-peak efficiency of gamma detector and absorption of photons in the target and air column, is plotted as a function of atomic number and a best fit-curve is constituted. From this fit-curve, the respective effective atomic numbers of the scientific samples are determined. The measured values of cross-section ratio increases non-linearly with increase in atomic number and are found to agree with theoretical predictions based upon non-relativistic form factor, relativistic form factor, modified form factor and S-matrix theory.

Form factor of a relativistic two-particle system within the relativistic quasipotential approach: Case of an arbitrary masses and a scalar current

A new relativistic form factor for a bound two-particle system was obtained for the case of a scalar current. The respective analysis was performed within the relativistic quasipotential approach based on a covariant Hamiltonian formulation of quantum field theory by going over to a three-dimensional relativistic configuration representation for the case of the interaction of two relativistic spinless particles that have arbitrary masses.

An experimental study on cross-section ratio of coherent to incoherent scattering for 145 keV incident gamma photons

The coherent (Rayleigh) to incoherent (Compton) scattering cross-section ratio of elements, in the range 6 ≤ Z ≤ 82, are determined experimentally for 145 keV incident gamma photons. An HPGe (High purity germanium) semiconductor detector is employed, at scattering angle of 50°, 70° and 90°, to record the spectra originating from interactions of incident gamma photons with the target under investigation. The intensity ratio of Rayleigh to Compton scattered peaks observed in the recorded spectra, and corrected for photo-peak efficiency of gamma detector and absorption of photons in the target and air, along with the other required parameters provides the differential cross-section ratio. The measured values of cross-section ratio are found to agree with theoretical predictions (corresponding to 4.939, 6.704 and 8.264 Å−1 photon momentum transfer) based upon non-relativistic form factor, relativistic form factor, modified form factor and S-matrix theory.

Pure Hydroxyapatite Phantoms for the Calibration of in Vivo X-ray Fluorescence Systems of Bone Lead and Strontium Quantification

Plaster of Paris [poP, CaSO4·(1)/(2) H2O] is the standard phantom material used for the calibration of in vivo X-ray fluorescence (IVXRF)-based systems of bone metal quantification (i.e bone strontium and lead). Calibration of IVXRF systems of bone metal quantification employs the use of a coherent normalization procedure which requires the application of a coherent correction factor (CCF) to the data, calculated as the ratio of the relativistic form factors of the phantom material and bone mineral. Various issues have been raised as to the suitability of poP for the calibration of IVXRF systems of bone metal quantification which include its chemical purity and its chemical difference from bone mineral (a calcium phosphate). This work describes the preparation of a chemically pure hydroxyapatite phantom material, of known composition and stoichiometry, proposed for the purpose of calibrating IVXRF systems of bone strontium and lead quantification as a replacement for poP. The issue with contamination by the analyte was resolved by preparing pure Ca(OH)2 by hydroxide precipitation, which was found to bring strontium and lead levels to <0.7 and <0.3 μg/g Ca, respectively. HAp phantoms were prepared from known quantities of chemically pure Ca(OH)2, CaHPO4·2H2O prepared from pure Ca(OH)2, the analyte, and a HPO4(2-) containing setting solution. The final crystal structure of the material was found to be similar to that of the bone mineral component of NIST SRM 1486 (bone meal), as determined by powder X-ray diffraction spectrometry.

Spontaneous electromagnetic fluctuations in unmagnetized plasmas. V. Relativistic form factors of weakly damped/amplified thermal modes

Closed analytical expressions for the spontaneously emitted fluctuation spectra of weakly damped/amplified fluctuations in unmagnetized plasmas are derived using fully relativistic dispersion functions and form factors for the important class of isotropic thermal plasmas. Especially, an electron-proton plasma is investigated in the limits of nonrelativistic and ultrarelativistic plasma temperatures. The results confirm the earlier nonrelativistic approach for positive values of the imaginary frequency γ and complete the nonrelativistic treatment for negative values of γ&amp;lt;0. The well known electrostatic, collective Langmuir mode also occurs within the relativistic theory of spontaneously emitted fluctuations and is analytically and graphically identified. For the first time, the ultrarelativistic temperature limit is analyzed for subluminal phase speeds, which leads to the identification of an additional, so far unknown, longitudinal subluminal collective mode.

Spontaneous electromagnetic fluctuations in unmagnetized plasmas. IV. Relativistic form factors of aperiodic Lorentzian modes

Closed analytical expressions for the electromagnetic fluctuation spectra in unmagnetized plasmas are derived using fully relativistic dispersion functions and form factors for the important class of isotropic form-invariant Lorentzian plasma particle distribution functions. Such distribution functions occur frequently in cosmic plasmas due to the presence of suprathermal charged particles and energetic cosmic ray particles. The results are illustrated for the important special case of aperiodic fluctuations. The collective, transverse, damped aperiodic mode, discovered before in nonrelativistic Maxwellian particle distributions, also exists in Lorentzian electron-proton particle distributions, now with the damping rate γ∝−k3 for all wavenumber values, resulting from the presence of relativistic particles in the tail of the Lorentzian distribution. For longitudinal electric field, fluctuations no damped or growing aperiodic collective mode exists in Lorentzian plasmas. The existence of a damped, collective, transverse, aperiodic mode is not in conflict with earlier general instability studies excluding the existence of growing aperiodic collective modes in isotropic plasmas.

Spontaneous electromagnetic fluctuations in unmagnetized plasmas. II. Relativistic form factors of aperiodic thermal modes

General expressions for the electromagnetic fluctuation spectra in unmagnetized plasmas are derived using fully relativistic dispersion functions and form factors for the important class of isotropic plasma particle distribution functions including in particular relativistic Maxwellian distributions. In order to obtain fluctuation spectra valid in the entire complex frequency plane, the proper analytical continuations of the unmagnetized form factors and dispersion functions are presented. The results are illustrated for the important special case of isotropic Maxwellian particle distribution functions providing in particular the thermal fluctuations of aperiodic modes. No restriction to the plasma temperature value is made, and the electromagnetic fluctuation spectra of ultrarelativistic thermal plasmas are calculated. The fully relativistic calculations also provide more general results in the limit of nonrelativistic plasma temperatures being valid in the entire complex frequency plane. They complement our earlier results in paper I and III of this series for negative values of the imaginary part of the frequency. A new collective, transverse, damped aperiodic mode with the damping rate γ∝−k−5/3 is discovered in an isotropic thermal electron-proton plasma with nonrelativistic temperatures.

Source of statistical noises in the Monte Carlo sampling techniques for coherently scattered photons

Detailed comparisons of the predictions of the Relativistic Form Factors (RFFs) and Modified Form Factors (MFFs) and their advantages and shortcomings in calculating elastic scattering cross sections can be found in the literature. However, the issues related to their implementation in the Monte Carlo (MC) sampling for coherently scattered photons is still under discussion. Secondly, the linear interpolation technique (LIT) is a popular method to draw the integrated values of squared RFFs/MFFs (i.e. ) over squared momentum transfer (). In the current study, the role/issues of RFFs/MFFs and LIT in the MC sampling for the coherent scattering were analyzed. The results showed that the relative probability density curves sampled on the basis of MFFs are unable to reveal any extra scientific information as both the RFFs and MFFs produced the same MC sampled curves. Furthermore, no relationship was established between the multiple small peaks and irregular step shapes (i.e. statistical noise) in the PDFs and either RFFs or MFFs. In fact, the noise in the PDFs appeared due to the use of LIT. The density of the noise depends upon the interval length between two consecutive points in the input data table of and has no scientific background. The probability density function curves became smoother as the interval lengths were decreased. In conclusion, these statistical noises can be efficiently removed by introducing more data points in the data tables.

Erratum: Fully relativistic form factor for Thomson scattering [Phys. Rev. E81, 036411 (2010)

Erratum: Fully relativistic form factor for Thomson scattering [Phys. Rev. E<b>81</b>, 036411 (2010)

Electron density distribution in vanadium and niobium fromγ-ray diffraction

The electron density distribution in paramagnetic vanadium and niobium is derived via multipole refinement of high-quality single-crystal diffraction data, complete up to sin\ensuremath{\theta}/\ensuremath{\lambda} $=$ 1.9 \AA{}${}^{\ensuremath{-}1}$, collected at room temperature with 316.5 keV $\ensuremath{\gamma}$ radiation. Four $\ensuremath{\gamma}$ lines in the energy range 200--600 keV have been used to extrapolate extinction-free low-order structure factors. Both metals display charge asphericity due to preferential occupancy of the ${t}_{2g}$ orbital. The asphericity in V is in quantitative agreement with ab initio calculations, but not with earlier experimental results, laying at rest a long-standing debate. In both metals, the $d$-electron density is strongly contracted relative to the free atoms. The high-precision thermal parameters are in good agreement with some of the older results. No support is found in the data for anharmonic contributions to the thermal parameters, contrary to claims in previous reports on V. Attention is paid to the 3$d$-4$s$ occupation problem in V, indicating no change in electronic configuration between atom and metal. Relativistic form factor issues are also discussed. It is pointed out that for the analysis of crystallographic data nonrelativistic form factors are to be preferred over their relativistic counterparts.

Calculation of the total Rayleigh scattering cross sections of photons in the energy range of 30–50 keV for Nb and Mo elements

The total Rayleigh scattering cross sections are calculated in the energy range of 30–50 keV using the modified relativistic form factors (MRFF), non-relativistic form factors (NFF), and relativistic form factors (RFF). Numerical calculations are made for energies above the K-threshold comparisons, for medium- Z elements (Nb and Mo), primarily at small momentum transfers. The integration range is divided into intervals using a modified formula to take into account the smaller scattering angles. The calculated cross sections based on MRFF, NFF, and RFF are presented and compared with the tabulated theoretical values. It is found that the values calculated using the NFF and RFF are smaller than the tabulated values with a ratio of 3–7%. It is also observed that the smallest total Rayleigh scattering cross section values are obtained using the MRFF.

Fully relativistic form factor for Thomson scattering

We derive a fully relativistic form factor for Thomson scattering in unmagnetized plasmas valid to all orders in the normalized electron velocity, beta[over ]=v[over ]/c. The form factor is compared to a previously derived expression where the lowest order electron velocity, beta[over], corrections are included [J. Sheffield, (Academic Press, New York, 1975)]. The beta[over ] expansion approach is sufficient for electrostatic waves with small phase velocities such as ion-acoustic waves, but for electron-plasma waves the phase velocities can be near luminal. At high phase velocities, the electron motion acquires relativistic corrections including effective electron mass, relative motion of the electrons and electromagnetic wave, and polarization rotation. These relativistic corrections alter the scattered emission of thermal plasma waves, which manifest as changes in both the peak power and width of the observed Thomson-scattered spectra.