scholarly journals The four-loop slope of the Dirac form factor

2019 ◽  
Author(s):  
Stefano Laporta
Keyword(s):  
Form Factor ◽  
Dirac Form Factor

scholarly journals Three-Loop Slope of the Dirac Form Factor and the1SLamb Shift in Hydrogen

2000 ◽  
Vol 84 (8) ◽  
pp. 1673-1676 ◽  
Author(s):  
Kirill Melnikov ◽  
Timo van Ritbergen
Keyword(s):  
Form Factor ◽  
Dirac Form Factor

scholarly journals The 4-loop slope of the Dirac form factor

2020 ◽  
Vol 234 ◽  
pp. 01009
Author(s):  
Stefano Laporta
Keyword(s):  
Form Factor ◽  
Hydrogen Atom ◽  
Energy Levels ◽  
Elliptic Integrals ◽  
Loop Contribution ◽  
One Dimensional ◽  
Dirac Form Factor ◽  
Complete Elliptic Integrals

The 4-loop contribution to the slope of the Dirac form factor in QED has been evaluated with 1100 digits of precision. The value is $ {m^2}F_1^{(4)'}(0) = {\rm{0}}{\rm{.886545673946443145836821730610315359390424032660064745}} \cdots {\left( {{\alpha \over \pi }} \right)^4} $. We have also obtained a semi-analytical fit to the numerical value. The expression contains harmonic polylogiπ2iπ iπ arithms of argument $ {e^{{{i\pi } \over 3}}},{e^{{{2i\pi } \over 3}}},{e^{{{i\pi } \over 2}}} $, one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits. We show the correction to the energy levels of the hydrogen atom due to the slope.

scholarly journals NEUTRON CHARGE RADIUS: RELATIVISTIC EFFECTS AND THE FOLDY TERM

2003 ◽  
Vol 18 (31) ◽  
pp. 5767-5779 ◽  
Author(s):  
W. R. B. DE ARAÚJO ◽  
T. FREDERICO ◽  
M. BEYER ◽  
H. J. WEBER
Keyword(s):  
Wave Function ◽  
Form Factor ◽  
Charge Radius ◽  
Relativistic Effects ◽  
Spin Coupling ◽  
Coupling Scheme ◽  
Neutron Radius ◽  
Quark Spin ◽  
Dirac Form Factor ◽  
Front Model

The neutron charge radius is studied within a light-front model with different spin coupling schemes and wave functions. The cancellation of the contributions from the Foldy term and Dirac form factor to the neutron charge form factor is verified for large nucleon sizes and it is independent of the detailed form of quark spin coupling and wave function. For the physical nucleon our results for the contribution of the Dirac form factor to the neutron radius are insensitive to the form of the wave function while they strongly depend on the quark spin coupling scheme.

The effect of temperature on high-resolution TEM image contrast

1995 ◽  
Vol 53 ◽  
pp. 640-641
Author(s):  
T. Geipel ◽  
W. Mader ◽  
P. Pirouz
Keyword(s):  
Form Factor ◽  
Inelastic Scattering ◽  
Accurate Method ◽  
Waller Factor ◽  
Effect Of Temperature ◽  
Specimen Thickness ◽  
Influence Of Temperature ◽  
Debye Waller Factor ◽  
Influence Of Temperature On ◽  
Atomic Form Factor

Temperature affects both elastic and inelastic scattering of electrons in a crystal. The Debye-Waller factor, B, describes the influence of temperature on the elastic scattering of electrons, whereas the imaginary part of the (complex) atomic form factor, fc = fr + ifi, describes the influence of temperature on the inelastic scattering of electrons (i.e. absorption). In HRTEM simulations, two possible ways to include absorption are: (i) an approximate method in which absorption is described by a phenomenological constant, μ, i.e. fi; - μfr, with the real part of the atomic form factor, fr, obtained from Hartree-Fock calculations, (ii) a more accurate method in which the absorptive components, fi of the atomic form factor are explicitly calculated. In this contribution, the inclusion of both the Debye-Waller factor and absorption on HRTEM images of a (Oll)-oriented GaAs crystal are presented (using the EMS software.Fig. 1 shows the the amplitudes and phases of the dominant 111 beams as a function of the specimen thickness, t, for the cases when μ = 0 (i.e. no absorption, solid line) and μ = 0.1 (with absorption, dashed line).

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