Equations Hall coefficient

Equation (A40) demonstrates the well-known vanishing of the Hall coefficient when rppnl = rnnpl. This effect has often been observed as temperature is raised, usually in a sample which is p-type at low temperature (Look, 1975). At this point we should mention the various definitions of p-and n-type. 

The variation of the Hall coefficient and the conductivity of zinc oxide as a function of oxygen pressure has been studied by many workers (27,32,40-42). The conclusions reached are that the number of carriers varies with oxygen pressure approximately according to the equation... 

Consider the phenomenological equations under the conditions JY - - 0. (a) Express VXT in terms of J. (b) Find VyT in terms of J. (c) Express the resistivity and Hall coefficient in terms of appropriate LLj and compare these... 

Impose the conditions Jy = Jf =0 on the phenomenological equations. Express Vy T and V T in terms of Jx. Express the resistivity and Hall coefficient in terms of appropriate and compare your results to those in the text. 

The thermoelectric properties were measured at 300 K for the FGM and its component layers separated from the FGM. The electrical conductivity ((T) and Hall coefficient (Rjj) were measured by the 6-probe method for the FGM and by the van der Pauw configuration for the components cut from the FGM using Pt-wire electrodes. The carrier concentration (n) and Hall mobility (/ h) calculated using the equation n=lleR (e electric charge) and / H= h respectively. The thermoelectric power (a) at 300 K was estimated from the linear relationship between thermoelectromotive force (EMF) and temperature difference within 5 K. 

The application of a magnetic field on a MIEC placed in a galvanic cell affects the current density equation which now contains also the effect of the Lorentz force.The use of a magnetic field should enable the determination of the Hall coefficient and magneto resistance. The interpretation of the results is straightforward and can be done in terms of the two-band semiconductor modeF when the composition of the MIEC is uniform and there is one dominant mobile ionic species and one dominant electronic species. Uniformity holds when the two electrodes have the same composition, are reversible, and a steady state has been reached. 

The Hall coefficient is defined as Ru=Ey/jxBz. Putting jx = o-Ex into Equation 18.24, we find... 

We shall see in later chapters that it is possible in semiconductors (and in some metals) for holes in the electrons band structure to act as positively charged carriers. Had the carriers been holes, the signs in the equation of motion would have been reversed resulting in a positive Hall coefficient. (Note This simplified derivation of the Hall effect is only... 

The HaU voltage may be determined using Equation 18.18. However, it is first necessary to compute the Hall coefficient (1 h) from Equation 18.20b as... 

Ordinaiy differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractious are uouhuear because the coefficients of Xi j change with time. Therefore, numerical methods of integration with respect to time must be enmloyed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear Numerical Initial Value Problems in Ordinaiy Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

The variational principle leads to the following equations describing the molecular orbital expansion coefficients, c. , derived by Roothaan and by Hall ... 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

A paper published by Hall and Selinger [3] points out that an empirical formula relating the concentration (c) to the coefficient of variation (CV) is also known as the precision (cr). They derive the origin of the trumpet curve using a binomial distribution explanation. Their final derived relationship becomes equation 72-2 ... 

Second virial coefficients represent the first approximation to the system equation of state. Yethiraj and Hall [148] obtained the compressibility factor, i.e., pV/kgTn, for small stars. They found no significant differences with respect to the linear chains in the pressure vs volume behavior. Escobedo and de Pablo [149] performed simulations in the NPT ensemble (constant pressure) with an extended continuum configurational bias algorithm to determine volumetric properties of small branched chains with a squared-well attractive potential... 

Z is the nuclear charge, R-r is the distance between the nucleus and the electron, P is the density matrix (equation 16) and (qv Zo) are two-electron integrals (equation 17). f is an exchange/correlation functional, which depends on the electron density and perhaps as well the gradient of the density. Minimizing E with respect to the unknown orbital coefficients yields a set of matrix equations, the Kohn-Sham equations , analogous to the Roothaan-Hall equations (equation 11). 

We have m x m equations because each of the m spatial MO s i// we used (recall that there is one HF equation for each ip, Eqs. 5.47) is expanded with m basis functions. The Roothaan-Hall equations connect the basis functions (p (contained in the integrals F and S, Eqs. 5.55, above), the coefficients c, and the MO energy levels . Given a basis set energy levels e. The overall electron distribution in the molecule can be calculated from the total wavefunction P, which... 

To use the Roothaan-Hall equations we want them in standard eigenvalue-like form so that we can diagonalize the Fock matrix F of Eq. 5.57 to get the coefficients c and the energy levels e, just as we did in connection with the extended Hiickel method (Section 4.4.1). The procedure for diagonalizing F and extracting the c s and e s and is exactly the same as that explained for the extended Hiickel method (although here the cycle is iterative, i.e. repetitive, see below) ... 

The overlap matrix. SCF-type semiempirical methods take the overlap matrix as a unit matrix, S = 1, so S vanishes from the Roothaan-Hall equations FC = SCe without the necessity of using an orthogonalizing matrix to transform these equations into standard eigenvalue form FC = Ce so that the Fock matrix can be diagonalized to give the MO coefficients and energy levels (Sections 4.4.3 and 4.4.1 Section 5.2.3.6.2). 

To assign values to the molecular orbital coefficients, c, many computational methods apply Hartree-Fock theory (which is based on the variational method).44 This uses the result that the calculated energy of a system with an approximate, normalized, antisymmetric wavefunction will be higher than the exact energy, so to obtain the optimal wavefunction (of the single determinant type), the coefficients c should be chosen such that they minimize the energy E, i.e., dEldc = 0. This leads to a set of equations to be solved for cMi known as the Roothaan-Hall equations. For the closed shell case, the equations are... 

The Roothaan Hall equations are nonlinear because the Fock matrix F/tv depends upon the orbital coefficients through the density matrix expression (6.111). Solution... 

---