Bragg harmonics

The channel-cut monochromator is the simplest type employed experimentally. A channel is cut in a perfect crystal (e.g.. Si) to provide two parallel reflecting surfaces that have a particular crystal plane [e.g., the Si (220)] parallel to the surface. The Bragg condition is used to select a particular wavelength and the reflected beam emerges parallel to the incident beam but is vertically displaced by 2D cos d, where D is the distance between the two faces and 0 is the angle between the beam and the Bragg planes. The accuracy of data collected using channel-cut crystal monochromators may be limited due to harmonic con-... 

We propose and demonstrate [Gershnabel 2003] an echo method of reducing the inhomogeneous broadening of Bogoliubov excitations in a harmonically-trapped BEC. Our proposal includes transfer of +q to —q momentum excitations, using a double two-photon Bragg process, in which a substantial... 

Figure 10 Bragg spectra for Bogoliubov (extremely weak Bragg pulses) excitations, (a) Harmonic trap. (b) Flat-bottom trap. Solid line linearized GPE solution, dashed line LDA. y axis ...

In practice the infinite sums can be reduced to a maximum number of harmonics. From Equation (32) it is easily seen that the Bessel functions diminish rapidly with the order of the satellite. The intensity of a Bragg reflection, proportional to the square of the structure factor Equation (35) or Equation (38), is then reduced due to the presence of the Bessel functions. 

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 =

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Another important parameter that may affect the resolution is the higher harmonic contribution from the Bragg reflector. A fused quartz mirror behind the monochromator has been currently used to reject this high harmonic contribution. Therefore, the energy resolution of the spectrometer is just limited by the Darwin width of the rocking curve and the spatial resolution of the position-sensitive detector [8]. 

The harmonics of the fundamental of interest (for the Mn K-edge, this is at 6.54 keV) may also satisfy the Bragg relation. An examination of Fig. 10 shows that for a Wiggler magnet the content of the first harmonic (e.g., 13.08 keV for Mn absorption) can be nearly equal to the flux of the... 

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