Back-Face Bragg Diffraction from a Perfect and Ultralightly Deformed Thick Crystal

Abstract

The intensity of neutrons Bragg diffracted from the back face of a 9 mm thick slab-shaped Si(111) analyzer crystal has been measured experimentally at the ORNL double crystal diffractometer and calculated theoretically. The back-face rocking curve of a strain-free perfect crystal contains two symmetrical peaks which become asymmetrical under an ultrasmall static deformation strain (bending with a radius of tens of km). The asymmetry is shown to be a sensitive measure of both the magnitude and direction of bending. [S0031-9007 (98)06594-6] PACS numbers: 62.20. -x, 61.12.ExThe high transparency of silicon for thermal neutrons creates a unique opportunity to study total Bragg diffraction from a thick crystal (with the thickness T ϳ 1 cm). Such experiments cannot be done with conventional x-ray radiation which cannot penetrate to the back face of a thick crystal. This property of neutron radiation was originally used by C. G. Shull in the early 1970s to measure separately the front-face (FF), back-face (BF), and end-face Bragg reflections from a thick Si crystal by scanning a narrow cadmium slit across the exiting beam [1,2]. However, all of these experiments were carried out in the conditions when the crystal under study was set up at a chosen angle of incidence, u u B , or slightly deviated from the exact Bragg angle, u B , and did not move while the detector was scanned across the diffracted beam.The angular dependence of the intensity of BF reflection, I BF ͑u 2 u B ͒, on u was measured for the first time at the ORNL Bonse-Hart double crystal diffractometer (DCD) [3]. In that experiment the short plate of the Si(111) analyzer channel-cut crystal 3 was covered by Cd, as shown in Fig. 1, to block the FF reflected beam and measure only the BF reflection from the long plate. The incident four-bounce neutron beam (wavelength l 2.59 Å, u B 24.4 ± ) was formed by a sequence of reflections from the single-bounce premonochromator, followed by a triple-bounce channel-cut monochromator crystal (see Fig. 1). The intensity of the experimental BF rocking curve (BFRC), normalized by the peak intensity of the FF reflection, is shown in Fig. 2 (open circles) as a function of the dimensionless angular parameter of dynamical diffraction theory, y ͑u 2 u B ͒͞du B , where du B b c e 2W jF S jNl 2 ͞p sin u B is the half-width of the Darwin plateau, b c is the atomic coherent scattering length, e 2W is the Debye-Waller factor, F S is the structure factor, and N is the number of unit cells per unit volume (in our case u 2 u B 1 arc sec corresponds to y 1.2). The BFRC (Fig. 2) contains two sharp peaks at y 62.2 and a deep minimum at y 0, in contrast to the FF rocking curve (FFRC) from a perfect crystal which has one peak at y 0.The BFRC can be derived from the classical dynamical diffraction theory using the condition (see, e.g., in [4]) that the intensity partially transmitted into a transparent crystal in the vicinity of the Bragg angle can be written as 1 2 R D ͑y͒, where R D ͑y͒ is the Darwin reflectivity function [5]Thus, inside the...

Neutron Back- and Front-Face Bragg Diffraction on a Thin Si Single Crystal Excited by Ultrasound

“…The spatial distribution of the diffracted beam at "no sound" and linear growth of the FFS extinction peak intensity when sound is switched on can be described also in terms of the quasi-classical Kato's approximation, introducing a new extinction length whose value depends on the AW amplitude. This result-up to a numerical factor-coincides with that obtained using expressions (8)(9)(10)(11), taking into account the inelastic scattering of neutrons. However, when US is switched on, the neutron scattering inside a crystal has more complicated behavior and cannot be satisfactorily explained only by the appearance of a new "sonic" extinction length [21].…”

“…τ is the extinction length. Expressions (9,10) are valid if H ⊥ k s , and this corresponds to the conditions of transversal AW in our experiments. The expressions (10) differ from those for the case of X-ray acoustic resonance [18] because the neutrons with a wavelength 0.243 nm have the speed comparable with the speed of ultrasound phonons and between them there is an exchange of energy [14].…”

“…In the both cases Bragg-reflected intensity increases. For example, in [9] [10] it was shown that back-face Bragg diffraction is very sensitive to the ultrasmall static deformation strain.…”

Dynamical diffraction peak splitting in time-of-flight neutron diffraction

“…8,9 This effect is quite well known in steady-state ͑constant wavelength͒ neutron interferometers, where the reflection from the back face must be blocked to obtain good peak profiles. 10,11 It has not been reported before in TOF neutron diffraction.…”

“…͑ii͒ Alternatively, one can calculate the expected beam profile using dynamical theory. 11,15 Finally, ͑iii͒, for a perfect crystal one can rotate the specimen 180°and reacquire the diffraction data. If the weak and strong peaks switch position, the effect is not due to dynamical diffraction.…”