Magnetic form factor

Brown P J 1999 Magnetic form factors International Tables for Crystallography 2n6 edn, vol C, ed A J C Wilson and E Prince (Dordrecht Kluwer) section 4.4.5... 

Stewart, R.F. (1980) Partitioning of Hartree-Fockatomic form factors into core and valence shells, In Electron and Magnetization Densities in Molecules and Crystals, Becker, P. (Ed.), Plenum Press, New York, pp. 427 131. 

Concerning induced orbital moments of U-based intermetallic compounds, many PND experiments have been performed and have shown that the ratio iL/ -is can be used as a measure of the hybridisation [42-44] (in the light actinides, orbital and spin moments are oppositely directed and the neutron magnetic form factors are highly sensitive to the ratio uL/us). Indeed, this ratio is reduced as compared to the free ion expectations (Figure 4). 

Kennedy, S.J., Brown P.J. and Coles, B.R. (1993) A polarised neutron study ofthe magnetic form factors in CeFe2, J. Phys. Cond. Mat., 5(29), 5169-5178. 

Electron spin resonance, nuclear magnetic resonance, and neutron diffraction methods allow a quantitative determination of the degree of covalence. The reasonance methods utilize the hyperfine interaction between the spin of the transferred electrons and the nuclear spin of the ligands (Stevens, 1953), whereas the neutron diffraction methods use the reduction of spin of the metallic ion as well as the expansion of the form factor [Hubbard and Marshall, 1965). The Mossbauer isomer shift which depends on the total electron density of the nucleus (Walker et al., 1961 Danon, 1966) can be used in the case of Fe. It will be particularly influenced by transfer to the empty 4 s orbitals, but transfer to 3 d orbitals will indirectly influence the 1 s, 2 s, and 3 s electron density at the nucleus. 

Since the magnetic interaction vector q is known, it is possible to deduce the magnetic form factor f( ). Although possible when using unpolarized neutrons, its measurement is much more precise using polarized neutron diffraction by a (single domain) ferromagne-... 

Since the application of pressure may modify strongly the charge density distribution in a solid, and therefore affects the orbital more than the spin moment, magnetic form factors and magnetic anisotropy may become much more pressure-dependent than usually assumed. 

We have seen above that calculation of the corrections of order a"(Za) m (n > 1) reduces to calculation of higher order corrections to the properties of a free electron and to the photon propagator, namely to calculation of the slope of the electron Dirac form factor and anomalous magnetic moment, and to calculation of the leading term in the low-frequency expansion of the polarization operator. Hence, these contributions to the Lamb shift are independent of any features of the bound state. A nontrivial interplay between radiative corrections and binding effects arises first in calculation of contributions of order a Za) m, and in calculations of higher order terms in the combined expansion over a and Za. 

The Sachs electric and magnetic form factors are defined as (see, e.g. [3])... 

Respective corrections are written via the slope of the Dirac form factor and the anomalous magnetic moment exactly as in Subsect. 7.3.4. The only difference is that the contributions to the form factors are produced by the hadronic vacuum polarization. 

Fig. 9.1. Electron anomalons magnetic moment contribntion to HFS. Bold dot corresponds to the Panh form factor...

We will first consider the contributions generated only by the elastic intermediate nuclear states. This means that calculating this correction we will treat the nucleus as a particle which interacts with the photons via its nontrivial Sachs electric and magnetic form factors in (6.8). 

The recoil part of the proton size correction of order Za)Ep was first considered in [9, 10]. In these works existence of the nontrivial nuclear form factors was ignored and the proton was considered as a heavy particle without nontrivial momentum dependent form factors but with an anomalous magnetic moment. The result of such a calculation is most conveniently written in terms of the elementary proton Fermi energy Ep which does not include the contribution of the proton anomalous magnetic moment (compare (10.2) in the muonium case). Calculation of this correction coincides almost exactly... 

The contribution due to the three-loop slope of the Dirac form factor was the last unknown contribution to the hydrogen energy levels at order a3(Za)4. The two other contributions come from the three-loop electron anomalous magnetic moment and the three-loop vacuum polarization correction to the Coulomb propagator. These contributions can be extracted from the literature [10,13]. 

We have displayed the contributions due to the three-loop slope of the Dirac form factor, the three-loop anomalous magnetic moment of the electron and the three-loop photon vacuum polarization separately. Thanks to the cancellation between these contributions, the correction turns out to be quite small numerically. 

The proposed experiment on Is hfs will test the accuracy and verify the validity of the PSI experiment on the excited states. With improved accuracy it may yield a result which can be used as a sum rule for the magnetic form factor of the proton. The uncertainty of this is on the same level as the proton polarizability contribution. 

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