Nuclear Bragg scattering

S x — t) where the I vectors denote the positions of the unit cells. 

871 u% = Bn is the angular independent temperature factor generally quoted. 

The techniques of single-crystal diffractometry have been discussed by Arndt and WUlis (51). We should note that extinction is a very serious problem in the determination of accurate magnetic intensity data from single crystals. Although extinction must always be accounted for in conventional crystallographic studies, it is particularly important to make proper correction in polarized neutron experiments where the ratio of magnetic to nuclear structure factors is determined. 

If not, incorrect conclusions about magnetic moment reductions and the shapes of form factors may be made. Discussion of the extinction corrections (52) in form factor determinations has recently been given in the case of Tb(OH)s (55) where the intensity of some reflections was reduced by as much as 90%, and K2NaCrFe (22). Because of the small crystal size, extinction has not been observed to be significant in any work with polycrystalline samples, which is one of the principal advantages of the latter technique. Preferred orientation can be a nuisance in powder work (especially with X-rays) but does not appear to have been significant in the experiments discussed below. 

Scattering from a polycrystaliine sample takes place into Debye-Scherrer cones with the k direction as axis and semiangles 2d defined by sm6=rl2k. The total cross section associated with each cone is [from (3.15)] 

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Fig. 1.10 Coherent excitation and coherent de-excitation of the nucleus for the nuclear forward scattering and the nuclear Bragg scattering. Coherent de-excitation of the resonant scattering creates the quantum beat with a frequency of Q...

Fig. 16. Constant-6 scans near zone centers, with [6 = (002)] and without [g = (001)1 allowed nuclear Bragg spots. Solid lines correspond to eq. (2.1) with parameters chosen for best fit to data. The inset shows how the real o> = 0 susceptibility, as deduced from scattering data, varies with temperature for the two momentum transfers probed the dashed line represents bulk (g = 0) susceptibility. (From Aeppli et al. 1987.)...

Indeed, neutron scattering performed on TmSeo.45Teo.55 at 1.5 K and 15kbar at this temperature revealed only nuclear Bragg peaks and no AF peaks could be discerned. 

From this evaluation we derive the Curie temperature 70 = 5.07 K and the critical exponent P = 0.32 0.03 Tc — T)/Tc = 0.002-i-0.3). A last important result of the neutron scattering at this pressure is that the lattice constant at 2 K is 5.90 A which is significantly smaller than at ambient conditions (6.137 A) indicating that the sample is indeed in the collapsed intermediate-valent phase. The variation of the temperature through Tc has no effect on the position of the nuclear Bragg reflections which again proves the coexistence of ferromagnetism and intermediate valenee but with the absence of a hybridization gap. 

In special cases, e.g., when weak nuclear peaks may arise from magnetoelastic distortions, the separation of magnetic and nuclear Bragg peaks may be difficult, but in the vast majority of cases it does not constitute a problem. This is different from the case of inelastic scattering in that case, as we will see, the separation is often difficult. 

Water on Smectites. Compared to vermiculites, smectites present a more difficult experimental system because of the lack of stacking order of the layers. For these materials, the traditional technique of X-ray diffraction, either using the Bragg or non-Bragg intensities, is of little use. Spectroscopic techniques, especially nuclear magnetic resonance and infrared, as well as neutron and X-ray scattering have provided detailed information about the position of the water molecules, the dynamics of the water molecule motions, and the coordination about the interlayer cations. 

Magnetically Ordered Systems. For magnetically ordered systems the discussion is analogous to that for nuclear scattering. By comparison with (3.15) the magnetic Bragg cross section is... 

Figure 3. Experimental geometry for dephasing measurements by three-pulse scattering technique. Pump pulses (1 and 2) produce a grating response that diffracts probe pulse (3) into two orders. In grating experiments described in Section III (nuclear phase coherence), t = 0 and probe pulse is incident at the phase-matching angle for Bragg diffraction into only one order.

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