Coefficients Even harmonics

Thus, the perhaps unfamiliar constitutive relations (2.23)-(2.25) yield familiar results when the fields are time harmonic moreover, because of (2.26) and (2.27), physical meaning can now be attached to the phenomenological coefficients even for arbitrarily time-dependent fields. 

The inherent feature of the quadratic SR under study is that the bias field is the sole cause of even harmonics in the spectrum. Due to the symmetry considerations, they must vanish at H —> 0. This means that S2 (E, —> 0) = 0. In Eq. (4.290) this limit is ensured by the proportionality of S2 to B. According to the second of Eqs. (4.253), the coefficients consists only of the odd equilibrium moments of the distribution (4.226). Since the function Wo is even in x at E, = 0, the odd moments vanish. 

The compounds K5Nb3OFi8 and Rb5Nb3OFi8 display promising properties for their application in electronics and optics. The compounds can be used as piezoelectric and pyroelectric elements due to sufficient piezo- and pyroelectric coefficients coupled with very low dielectric permittivity. In addition, the materials can successfully be applied in optic and optoelectronic systems due to their wide transparency range. High transparency in the ultraviolet region enables use of the materials as multipliers of laser radiation frequencies up to the second, and even fourth optical harmonic generation. 

A similar convergence is found for the third harmonic generation process at the lower of the two frequencies, 671.5 nm. At the higher frequency, 476.5 nm, the Taylor approximations for the third harmonic generation hyperpolarizability converge only very slowly, even with a tenth-order Taylor approximation a one-percent accuracy is not obtained. This accuracy, however, is still achieved with a [1,2] Fade approximant calculated from the dispersion coefficients up to sixth order. 

Only even orders are taken into account in Eqs. 2.83 and 2.84 due to the presence of the inversion center in the diffraction pattern. The number of harmonic coefficients C and terms k(h) varies depending on lattice symmetry and desired harmonic order L. The low symmetry results in multiple terms (triclinic has 5 terms for L = 2) and therefore, low orders 2 or 4 are usually sufficient. High symmetry requires fewer terms (e.g. cubic has only 1 term for L = 4), so higher orders may be required to adequately describe preferred orientation. The spherical harmonics approach is realized in GSAS. ... 

In a diffraction experiment the crystal reflection coordinates (cj), p) are determined by the reflection index (h) while the sample coordinates (ij/, y) are determined by the orientation of the sample on the diffractometer. This formulation assumes that the probability surface is smooth and can be described by a sum of spherical harmonic terms, kfl and kjf, that depend on h and sample orientation, respectively, to some maximum harmonic order, L (ref. 25). The coefficients Cf" then determine the strength and details of the texture. Notably, only the even order, L = 2n, terms in these harmonic sums affect the intensity of Bragg reflections the odd order terms in the ODF are invisible to diffraction. 

If the quantitative texture analysis is not of interest the sample is not rotated on a goniometer and only one or a small number of patterns are recorded. Because the number of points in the space (T, y) is not sufficient, one expects that the refined harmonic coefficients give only a rough description of the texture, even if the texture correction is very good. An extreme case is the Bragg-Brentano geometry. In this case in Equations (41-43) we must take (0 = 9, x =

[Pg.348]

However, just as in the case of the linear harmonic oscillator, the infinite series so obtained is not a satisfactory wave function for general values of X, because its value increases so rapidly with increasing as to cause the total wave function to become infinite as increases without limit. In order to secure an acceptable wave function it is necessary to cause the scries to break off after a finite number of terms. The condition that the series break off at the term an " + ml, where n is an even integer, is obtained from 17-22c by putting n + 2 in place of v and equating the coefficient of a , to zero. This yields the result... 

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

To each value of q, a countable infinite set of characteristic values of a is associated for which x t) is an odd or even function that is njr-periodic in time, n being an integer. Series approximations for the characteristic values are obtained by expressing the integral-order Mathieu function as a series of harmonic oscillations, plugging the resultant expression into Eq. (20.8), and equating coefficients of each (orthogonal) frequency component to zero. These laborious calculations yield infinite series in q where each coefficient of q can be expressed as a continued Iraction [4]. 

The term is a constant offset, or the average of the waveform. The b and c coefficients are the weights of the wth harmonic cosine and sine terms. If the function is purely even about t = 0 (this is a boundary condition like that discussed in Chapter 4), only cosines are required to represent it, and only the b terms would be nonzero. Similarly, if the function is odd, only the terms would be required. A general function Fper(0 will require sinusoidal harmonics of arbitrary amplitudes and phases. The magnitude and phase of the mth harmonic in the Fourier series can be found by ... 

---