INDEX Fourier series

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

In this demonstration of a Fourier series we will use only cosine waves to reproduce the shadow image of the black squares. The procedure itself is rather straightforward, we just need to know the proper values for the amplitude A and the index h for each wave. The index h determines the frequency, i.e. the number of full waves trains per unit cell along the a-axis, and the amplitude determines the intensity of the areas with high (black) potential. As outlined in Figure 4, the Fourier synthesis for the present case is the sum of the following terms ... 

Index giving type of polar plot or number of samples or number of terms in Fourier series Net number of encirclements of the point (-1,0) on the complex plane... 

Equations (4.60) and (4.61) are also able to yield the vector recurrence relations for the case of a skew bias field, that is, when vectors h and n are not parallel. In this case one should ascribe to each bi as many as 21+1 components, corresponding to different values of the azimuthal index m. Another problem, involving vector recurrence relations, is a steady-state nonlinear oscillations of bi in a high-AC field. To study the harmonic content of the nonlinear response, one has to expand all the moments b t)l in the Fourier series. Then the Fourier coefficients may be treated as components of a... 

Immediately after the introduction of a constant refractive index Bethe developed a dispersion theory of electron diffraction which is very closely related to the Darwin-Ewald theory. In this theory the propagation of de Broglie waves through a crystal is investigated, the potential being expanded in a triple Fourier series in terms of the contributions of the individual lattice planes hkl. Thus Vq in Schrodinger s equation is replaced by a triple Fourier series with the coefficients In accordance with this assumption, the solution... 

Figure 6.13. Evolution of the logarithm of the Fourier series cosine coefficient according to the square of the Miller index I...

We can, however, only set up as many conditions of this kind as there are incommensurable frequencies in the motion. This may be seen as follows. By way of example we consider the case of two degrees of freedom the index in the Fourier series then contains the sum v-yTi H" VgTg, where T , t2 are integers. If, e.g., v- = where k is an integer, piTi + + 2)- N"ow /ct + can take any... 

We have incorporated the terms with negative exponents into the same sum with the other terms by allowing the summation index to take on negative as well as positive values. The function being represented by a Fourier series does not have to be a real function. If it is a real function, the coefficients and bn will be real and the coefficients c will be complex. 

Common graphical representations of these data include plots of Ci (or A/t) and 9k > which are referred to as the magnitude and. phase spectra, respectively. As is implied by the indexing of the spectral data by the integer k, the Fourier series of a continous-time waveform is itself discrete in character, that is, it takes on nonzero values at only discrete, isolated points of the frequency axis, namely, the frequencies Q.k = 2itk/T. 

Variations of Fourier series were developed over time to solve certain practical problems. For example, the sine and cosine building-block functions are not concentrated near any particular x value. No position information can be gleaned from just the coefficients at a single frequency index n. On the other hand, an individual sample /(jc ) of a smooth function / indicates roughly how big the function is near x, but confers no frequency information. What was sought was a collection of nice functions w = w k x), with two indices n, k, such that an arbitrary function / = fix) could be written as a superposition ... 

An integral transform is similar to a functional series, except that it contains an integration instead of a summation, which corresponds to an integration variable instead of a summation index. The integrand contains two factors, as does a term of a functional series. The first factor is the transform, which plays the same role as the coefficients of a power series. The second factor is the basis function, which plays the same role as the set of basis functions in a functional series. We discuss two types of transforms, Fourier transforms and Laplace transforms. 

A(u = complex pulse area D = number of distinguishable permutations of the pairs (o, a) (er) = expectation value of the molecular dipole operator E = energy of level i E t) = electric field vector E t) = component along ju of the electric field vector E cd) = amplitude of the Fourier component of E at frequency (o E = amplitude of the monochromatic wave at frequency a> E = component along ju of E E t) = envelope of the quasi-monochromatic field at central frequency a> H(t) = time-dependent total Hamiltonian Hq = unperturbed Hamiltonian in the absence of electromagnetic fields = intensity of the monochromatic wave at frequency w N = number density of molecules = refractive index at frequency macroscopic polarization vector P< ) = w-th order term in the series expansion of P in powers of... 

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