Magnetic fields external

The interaction with a magnetic field may similarly be written in term of magnetic dipole, quadrupole etc. moments (there is no magnetic monopole, corresponding to electric charge). Since the magnetic interaction is substantially smaller in magnitude than the electric, only the dipole term is normally considered. 

T he dipole moment m depends on the total angular momentum, which may be written in terms of the orbital angular moment operator L and the total electron spin S. 

The second derivative is the magnetizahility (the corresponding macroscopic quantity is called the magnetic susceptibility x). 

The second derivative is the magnetizability (the corresponding maeroseopic quantity is called the magnetie susceptibility /). 

As for the electric field, this leads to the definition of the dipole and magnetizability as first and second derivatives of the total energy with respect to the magnetic field. 

---

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

In addition, there could be a mechanical or electromagnetic interaction of a system with an external entity which may do work on an otherwise isolated system. Such a contact with a work source can be represented by the Hamiltonian U p, q, x) where x is the coordinate (for example, the position of a piston in a box containing a gas, or the magnetic moment if an external magnetic field is present, or the electric dipole moment in the presence of an external electric field) describing the interaction between the system and the external work source. Then the force, canonically conjugate to x, which the system exerts on the outside world is... 

The little atomic magnets are of course quantum mechanical, but Weiss s original theory of paramagnetism and ferromagnetism (1907) [7] predated even the Boln atom. He assumed that in addition to the external magnetic field Bq, there was an additional internal molecular field B. proportional to the overall magnetization M of the sample. 

Figure A2.5.19. Isothemis showing the reduced external magnetic field B. = P Bq/ZcTj versus the order parameter s = for various reduced temperatures J = TIT. ...

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. 

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

Figure Bl.15.6. The EPR spectrum of tire perinaphthenyl radical in mineral oil taken at room temperature. (A) First derivative of the EPR absorption x with respect to the external magnetic field, Bq. (B) Integrated EPR spectrum.

In electron-spin-echo-detected EPR spectroscopy, spectral infomiation may, in principle, be obtained from a Fourier transfomiation of the second half of the echo shape, since it represents the FID of the refocused magnetizations, however, now recorded with much reduced deadtime problems. For the inhomogeneously broadened EPR lines considered here, however, the FID and therefore also the spin echo, show little structure. For this reason, the amplitude of tire echo is used as the main source of infomiation in ESE experiments. Recording the intensity of the two-pulse or tliree-pulse echo amplitude as a function of the external magnetic field defines electron-spm-echo- (ESE-)... 

The important outcome from this transformation is that now the non-adiabatic coupling term t is incorporated in the Schrodinger equation in the same way as a vector potential due to an external magnetic field. In other words, X behaves like a vector potential and therefore is expected to fulfill an equation of the kind [111a]... 

FIGURE 13 3 (a) In the absence of an external magnetic field the nuclear spins of the protons are randomly oriented (b) In the presence of an external magnetic field Xq the nuclear spins are oriented so that the resulting nuclear magnetic moments are aligned either parallel or antiparallel to Xq The lower energy orientation is the one parallel to Xq and more nuclei have this orientation... 

FIGURE 13 4 An external magnetic field causes the two nuclear spin states to have different energies The difference in energy AE is proportional to the strength of the applied field... 

FIGURE 13 6 The induced magnetic field of the elec trons in the carbon-hydrogen bond opposes the external magnetic field The resulting magnetic field ex perienced by the proton and the carbon is slightly less than Xr,... 

Section 13 3 In the presence of an external magnetic field the +j and —5 nuclear spin states of a proton have slightly different energies... 

Section 13 4 The energy required to flip the spin of a proton from the lower energy spin state to the higher state depends on the extent to which a nucleus is shielded from the external magnetic field by the molecule s electrons... 

Shielding (Section 13 4) Effect of a molecule s electrons that decreases the strength of an external magnetic field felt by a proton or another nucleus... 

The application of external magnetic fields to the sample during analysis presents considerable problems and constraints for electron techniques, whereas external fields have no influence on MOKE. 

---