Curie’s equation

In an external magnetic field, Bq, the magnetization, Mq, of an ensemble of spins I is given by Curie s equation (10-13) ... 

Curie s equation applies to gases, solutions, and some crystals. For other crystals a more general equation, the Weiss equation, may be used (derived by P. Weiss in 1907). Weiss assumed that the local magnetic field orienting the dipoles is equal to the applied field plus an added field proportional to the magnetic volume polarization M ... 

Where C is a Curie s constant and the equation is called Curie s equation. Ferromagnetism and antiferromagnetism are deviations from the Curie law. From Figure 23, it is seen that in both cases, the magnetic susceptibihty changes suddenly at a temperature Tc - this temperature is called Curie temperature. For antiferromagnetism, the Curie temperature is also called Neel temperature. 

This is clearly more complicated than having to deal with the function U. An analogous relation holds for the dependence of H on Ho, this requires the replacement of q and ao respectively by Hq and xo. The above equation is greatly simplified by introducing Curie s Law we obtain... 

Here T represents the absolute temperature, and C is a constant that is characteristic of the substance and known as its Curie constant. Equation 19-8 expresses what is known as Curie s law ... 

Let us now derive phenomenological equations of the kind (5.193) corresponding to the expression (5.205). As has been mentioned before, each flux is a linear function of all thermodynamic forces. However the fluxes and thermodynamic forces that are included in the expression (5.205) for the dissipative function, have different tensor properties. Some fluxes are scalars, others are vectors, and the third one represents a second rank tensor. This means that their components transform in different ways under the coordinate transformations. As a result, it can be proven that if a given material possesses some symmetry, the flux components cannot depend on all components of thermodynamic forces. This fact is known as Curie s symmetry principle. The most widespread and simple medium is isotropic medium, that is, a medium, whose properties in the equilibrium conditions are identical for all directions. For such a medium the fluxes and thermodynamic forces represented by tensors of different ranks, cannot be linearly related to each other. Rather, a vector flux should be linearly expressed only through vectors of thermodynamic forces, a tensor flux can be a liner function only of tensor forces, and a scalar flux - only a scalar function of thermodynamic forces. The said allows us to write phenomenological equations in general form... 

The values of the paramagnetic part of the molal susceptibility (per heme) at 24° of the substances studied in this investigation are collected in Table VI. If it be assumed that these values result from the independent orientation of the magnetic moments of the hemes and that Curie s law is applicable, they correspond to the values of the magnetic moment /jl, in Bohr magnetons, shown in the last column of the table, calculated with the equation... 

Taking into account certain restrictions originating from the symmetry properties of isotropic liquids at equilibrium (Curie s theorem), after some tensor algebra we obtain the Navier-Stokes equations for single-component atomic fluids. 

These coefficients must be multiplied by the density of air and tissue, respectively. Figure 15.7.2-1 depicts a radiation fallout field. Let C be the curie activity/m. The radiation into a unit area receptor at z = 1 m above the ground. The area is emitting C r dr d0 gammas/s. These are attenuated in the air as exp(-p, R) and geometrically as l/(4 7t R). The radiation received by the receptor is given by equation 15.7.2-1 which becomes 15.7.2-2 by a change of... 

Table I reports the observed NMR linewidths for the H/3 protons of the coordinating cysteines in a series of iron-sulfur proteins with increasing nuclearity of the cluster, and in different oxidation states. We have attempted to rationalize the linewidths on the basis of the equations describing the Solomon and Curie contributions to the nuclear transverse relaxation rate [Eqs. (1) and (2)]. When dealing with polymetallic systems, the S value of the ground state has been used in the equations. When the ground state had S = 0, reference was made to the S of the first excited state and the results were scaled for the partial population of the state. In addition, in polymetallic systems it is also important to account for the fact that the orbitals of each iron atom contribute differently to the populated levels. For each level, the enhancement of nuclear relaxation induced by each iron is proportional to the square of the contribution of its orbitals (54). In practice, one has to calculate the following coefficient for each iron atom ...

This equation is called the Curie law and relates the equilibrium magnetization M0 to the strength of the magnetic field B0. The constants have the following meaning I is the nuclear spin quantum number (see below), y is the gyromagnetic ratio specific for a given isotope, h is Planck s constant, kB is Boltzmann s constant, N is the number of nuclei and T is the temperature. 

S.4.3.2 Model of Hillert and Jarl. In his original treatment, Inden (1976) used a complicated but closed expression for the enthalpy, but had to use a series expansion in order to calculate the entropy. Hillert and Jarl (1978) therefore decided to convert the Cp expression directly through a series expansion which substantially simplifies the overall calculation and leads to a maximum error of only 1-2 J/mol at the Curie temperature of Fe. The equivalent equations to those used by Inden (1976) are given by... 

Once the Curie contribution to R2M is estimated and subtracted, the contribution of contact and dipolar interactions can be estimated by examining the correlation time dependence of the paramagnetic relaxation depicted in Figs. 3.9 and 3.11. It appears that the maximum for R m occurs at dipolar term and at contact term. Taking for simplicity xf Ip = r °", this means that in the intermediate situation where ft>s T p > 1 > relative importance of the contact term is even smaller than that estimated in the fast motion limit. The equation for R2M has non-dispersive terms in both the dipolar and contact contributions (accounting for one-fifth and one-half of the total effect measured in the fast motion limit respectively), and therefore the conclusions drawn in the fast motion limit are still qualitatively correct. 

E2S(S + 1) is a generic notation that holds for dipolar, contact or Curie relaxation. In the equation for Curie relaxation there is a S2(S + l)2 term because a further S(S + 1) term is contained in the E2 term. 

In general, the Curie constant C from the slope of l/x versus Tcan be used with the above equation to find the spin quantum number S of the spin carriers, assuming g 2 for organic spin units. The // T criterion for the Curie law is readily achieved in an external field of 1000 Oe at 1 K for calibration, 10,000 Oe (1 T) — 1 cm-1 — 1.33 cal/mol 5.6 J/mol — 0.7 K. If there is a weak, generalized interaction between spin units, one can apply a mean-field, generalized correction, 9 (the Weiss constant), to the Curie law to get the so-called Curie-Weiss law as follows in Equation (4) ... 

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