Magnetization and magnetic susceptibility

Hamiltonian Eq. (1.24) is formally equivalent to that describing the dipolar interaction between two spins s and S2 whose sum is S. Actually, in organic radicals where spin-orbit interactions are negligibly small, the dipolar interaction between the two electron spins in an S = 1 system causes ZFS. 

Let us now take an S manifold, unsplit at zero magnetic field, with no orbital angular momentum. The sum of the energies in a magnetic field would be zero 

If we consider that gefxBBoMs C kT, the exponential is well approximated by (1 — ge/xBBoMs/kT). It follows from Eq. (1.28) that the denominator is just 25+1, and the energy is given by (see Appendix VI) 

From the classical treatment, the energy of the system is also given by the product of the induced magnetic moment along the field, iijnd (/+nd = A0i(/x), see Eqs. (1.26) and (1.27)) and the external magnetic field B0 (cf. Eq. (1.3))  

In other words, the induced magnetic moment per particle is just proportional to the expectation value (5Z). Note that the value of (5Z) is referred to a single spin 5 and to its fractional occupancy of the energy level ladder. In the case of a spin ensemble, the value of (5Z) provides the average value of Sz of the 

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Magnetization and magnetic susceptibility 204 Nmr spectra of singlet species 208 Other measurements 208 Theoretical approaches 209... 

Figure 17. Magnetization and magnetic susceptibility at the transition from the superconducting to the normal state in a magnetic field a) type 1 superconductor (cr > 0), b) type 2 superconductor (cr < 0) [9,10].

Flood (1978) has measured the magnetization and magnetic susceptibility of ErHj, finding Curie-Weiss behavior from 4.2 to 2K and evidence for an antiferromagnetic transition. The low-field effective moment is 6.77 + 0.27/Zb-... 

Gouy balance A balance for the determination of magnetic susceptibility. The sample is weighed in and out of a magnetic field and the susceptibility is calculated from the difference in weights. 

NmI point The temperature at which magnetic susceptibility becomes normal. See anti-ferromagnetism and ferromagnetism. 

Figure A2.3.29 Calculation of the critical temperature and the critical exponent y for the magnetic susceptibility of Ising lattices in different dimensions from high-temperature expansions.

Equation (A2.5.20) is the Curie-Weiss law, and the temperature at which the magnetic susceptibility becomes infinite, is the Curie temperature. Below this temperature the substance shows spontaneous magnetization and is ferromagnetic. Nonnally the Curie temperature lies between 1 and 10 K. However, typical ferromagnetic materials like iron have very much larger values for quantum-mechanical reasons that will not be pursued here. 

A third exponent y, usually called the susceptibility exponent from its application to the magnetic susceptibility x in magnetic systems, governs what m pure-fluid systems is the isothennal compressibility k, and what in mixtures is the osmotic compressibility, and detennines how fast these quantities diverge as the critical point is approached (i.e. as > 1). 

Other magnetic measurements of catalysts include electron paramagnetic resonance and magnetic susceptibility. Although those are not as common as NMR, they can be used to look at the properties of paramagnetic and ferromagnetic samples. Examples of these applications can be found in the literature [87. 

Co(II), Ni(n), Cu(n), and Zn(II) complexes of Schiff bases derived from 4-aryl-2-aminothiazoles and salicylaldehyde have been prepared, and structure 276 (Scheme 170) was established by magnetic susceptibility measurements and by infrared, electronic, and mass spectra (512). 

Although the size separation/classification methods are adequate in some cases to produce a final saleable mineral product, in a vast majority of cases these produce Httle separation of valuable minerals from gangue. Minerals can be separated from one another based on both physical and chemical properties (Fig. 8). Physical properties utilized in concentration include specific gravity, magnetic susceptibility, electrical conductivity, color, surface reflectance, and radioactivity level. Among the chemical properties, those of particle surfaces have been exploited in physico-chemical concentration methods such as flotation and flocculation. The main objective of concentration is to separate the valuable minerals into a small, concentrated mass which can be treated further to produce final mineral products. In some cases, these methods also produce a saleable product, especially in the case of industrial minerals. 

Fig. 3. Magnetic susceptibility of plutonium metal as a function of temperature where represent initially high density plutonium and , cast plutonium...

Uranium metal is weaMy paramagnetic, with a magnetic susceptibility of 1.740 X 10 A/g at 20°C, and 1.804 x 10 A/g (A = 10 emu) at 350°C (51). Uranium is a relatively poor electrical conductor. Superconductivity has been observed in a-uranium, with the value of the superconducting temperature, being pressure-dependent. This was shown to be a result of the fact that there are actually three transformations within a-uranium (37,52). 

The ESR spectrum of the pyridazine radical anion, generated by the action of sodium or potassium, has been reported, and oxidation of 6-hydroxypyridazin-3(2//)-one with cerium(IV) sulfate in sulfuric acid results in an intense ESR spectrum (79TL2821). The self-diffusion coefficient and activation energy, the half-wave potential (-2.16 eV) magnetic susceptibility and room temperature fluorescence in-solution (Amax = 23 800cm life time 2.6 X 10 s) are reported. 

A dramatic decrease in the magnitude of the magnetic susceptibility anisotropy is observed on going from thiirane to the open-chain analog, dimethyl sulfide, and has been attributed to non-local or ring-current effects (70JCP(52)5291). The decrease also is observed to a somewhat lesser extent in oxirane relative to dimethyl ether. 

Solids separation based on density loses its effectiveness as the particle size decreases. For particles below 100 microns, separation methods make use of differences in the magnetic susceptibility (magnetic separation), elec trical conductivity (electrostatic separation), and in the surface wettability (flotation and selec tive flocculation). Treatment of ultrafine solids, say smaller than 10 microns can also be achieved by utilizing differences in dielectric and electrophoretic properties of the particles. 

The magnetic susceptibility of a material (%, volume susceptibility) is dimensionless and is defined as the ratio of induced magnetization to magnetic field intensity. It is expressed as... 

NOTE Extensive listing of magnetic susceptibilities of elements and organic and inorganic compounds can be found in G, Foex, Tables de constantes et donnes numeiiques, Massou et Cie, Paris, 1957,... 

In addition, the magnetic characteristics of a material can change as a function of stress (e.g., unannealed series 316 stainless steel can be magnetic after machining), temperature, pressure, and physical and chemical treatment. Therefore, when two paramagnetic materials with similar magnetic susceptibilities are to be separated, the possibility that pretreatment will facilitate subsequent separation should be studied. 

The Civo.strearn operate.s with a peak held oF 4 te.sla and a magnetic Force oF25() T /rn allow.s. separation oF minerals with magnetic susceptibilities in the order oF l(h ernii/g. In essence the Crvostrearn is an industrial-scale version oF the well laiown laboratory Frantz Iso-dvnarnic Separator,... 

The measured magnetic susceptibility of multilayer buckytubes for xb S approximately half xg of graphite. This can be interpreted as follows. We recall that crystalline graphite is a semimetal with a small band overlap and a low density of carriers (10 /cm )[33, 34] the in-plane effective mass is small (rn ... 

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