Laue Geometry Research Articles

Determination of the absolute structure factor for the forbidden (222) reflection in silicon using 0.12-Åγrays

A room-temperature determination of the absolute structure factor for the forbidden (222) reflection in silicon has been conducted at the University of Missouri Research Reactor with 103-keV gamma rays. The measured structure factor of $F(222)=1.456$ is in excellent agreement with five of the earlier intensity measurements, and is significantly different from any value determined using Pendell\"osung techniques. An increase in accuracy over previous intensity measurements by a factor of between 2 and 10 has been achieved and is made possible through the use of monoenergetic, short-wavelength gamma rays, which allow absolute measurements to be made in Laue geometry on relatively thick crystals (\ensuremath{\sim}1 mm) without encountering extinction problems.

Experimental correction for primary and secondary extinction. I. Method

The energy transfer equations are used to solve secondary extinction in a parallel-sided crystal slab in Laue geometry. Practical formulas are given for determination of the refiectivity per unit length, σ(∊), from the measured absolute powers of the diffracted and direct beams at rocking angle ∊ = θ- θB. The calculation assumes non-divergent beams, and the experimental tolerances are considered in detail. The primary extinction factor is defined by yp(∊) = σ(∊)/σkin(∊), where σkin is the kinematical reflectivity. When yp > 0.5 it may be approximated by yp ≃ exp[-(αδ)2], where α is about 0.5 and δ is the average size of the coherent domains when measured in units of the extinction distance Λ, δ = D/Λ. The extinction distance can be varied by changing the X-ray wavelength or polarization, and yp(∊) can be determined from two measurements.

On the polarization of X-rays diffracted in mosaic crystals

The theory of polarized X-ray diffraction in mosaic crystals, which takes into account multiple Bragg scattering, is developed in the framework of transfer equations for the polarization tensors of X-ray beams. Both the depolarization and differences in the phase velocities of polarized X-rays due to Bragg scattering in mosaic crystals are determined. Exact solutions of the equations are obtained for a plane-parallel plate and expressions for the polarization characteristics of reflected and transmitted beams are given for Bragg and Laue geometries. The approach developed may also be applied to neutron or Mössbauer γ-ray diffraction in magnetically ordered mosaic crystals.

Coherence Properties of Dynamically Diffracted X‐Rays

AbstractThe description of dynamical diffraction of X‐rays by means of the mutual coherence functions of primary and diffracted beams is presented. The mutual coherence functions of a beam diffracted from a plane‐parallel crystal in Laue geometry are computed and the influence of the crystal thickness on the coherence width is investigated. The possible method is shown for measurement of a mutual coherence function of an X‐ray.

Core-electron distributions in beryllium

Abstract X-ray diffraction and Compton profile measurements have been extended to larger scattering vectors and electron momenta than used previously in order to check the claims of an expansion of the core electrons in beryllium metal from the free-atom configuration. The absolute reflection powers of the h0.1 reflections with 0.74 < sin θ/λ < 1·40 A−1 were measured from a single-crystal wafer in the symmetrical Laue geometry. Close agreement with the theoretical scattering factors of the free atom is obtained with the thermal parameters B11= 0·460 ± 0·01 A2 and B33 = 0·415 ± 0·01 A2, which are almost identical with the neutron results. The Compton profile measurement was made on a thin polycrystalline foil of Be using an annular 241Am source and a high purity Ge detector. The measurements were corrected for the instrumental factors and for multiple Compton scattering. The results agree completely with the free-atom values for momentum values above p≳2a.u., which substantiates the conclusion of the diffr...

Magnetic field effects on dynamical diffraction of neutrons by perfect crystals

Experiments have been performed on the effect of magnetic fields on dynamical diffraction phenomena of neutrons, using a perfect Si crystal in symmetric Laue geometry. A crystal collimator was used to define the incoming beam. The small change of neutron wavelength when entering a modest magnetic field results in large changes of neutron-propagation direction within the crystal. Energy changes as small as ${10}^{\ensuremath{-}8}$ eV were detectable in the system by observing these directional changes. By combining prism refraction with the magnetic field action, the polarized states could be separated within the crystal and the polarization of these separated beams was studied.

Overcoming of the [formula omitted] reflectivity limit in Laue neutron diffraction

Experimental evidence for the overcoming of the 50% peak reflectivity limit in nuetron diffraction in the Laue geometry is presented. Peak reflectivity up to 75%, in spite of the neutron absorption, has been obtained with a bent germanium crystal. This effect has been explained by a simple model based on the dynamical theory of neutron diffraction and appears as a consequence of the elimination, due to a proper curvature, of the Pendellösung effect.

Wavelength-dependent reflectivities of a few neutron monochromators

Wavelength-dependent reflectivity measurements of neutron monochromators, Pb(200), Ge(111), Zn(10\bar 10) and Be(10\bar 10) were made with a triple-axis spectrometer. The first two axes, in antiparallel arrangement, were employed to yield a monochromatic neutron beam. Pb(200) reflectivity was measured in reflection geometry. The other monochromators were studied in Laue (transmission) geometry. Experimental relative reflectivity values for Ge(111) for 0.83<λ<2.39 Å compare favorably with the corresponding calculated values. For all the other monochromators there is fair agreement for λ > 1.27 Å between the relative, measured and calculated, reflectivity values. The general behaviour of the experimental (reflectivity) values, both in the absolute and relative sense, is not in good agreement with the calculations. The measured integrated reflectivities of Ge(111), Pb(200), Zn(10\bar 10) and Be(10\bar 10) reflections are ≃0.1, 0.16, 0.5 and 0.9 of the respective calculated values.

Dynamical Theory of Neutron Diffraction by Magnetic Crystals for Laue Geometry

AbstractThe dynamical diffraction of thermal neutrons by magnetic crystals is being considered for the case of the symmetrical Laue‐geometry. It is shown that the results depend on the angle between BA, the average value of the magnetic field in the crystal, and BH, the Fourier‐transform of the magnetic field. The neutron wave function in the crystal consists of four wave‐fields, which have the form |Ψ&gt; =[Ψ0|χ0&gt; +ΨH|χH&gt; · exp (i Hr)] · exp (ik′r), where |χ0&gt; and |χH&gt; are spin states. In the general case |X0&gt; and |X)H&gt; differ from each other. The ratio of the reflected to the incident intensity is derived for the incidence of neutrons polarized parallel or antiparallel to BA and for the incidence of unpolarized neutrons, for the case of pure magnetic diffraction, and for the case that BA is normal to BH. In both cases, for each of the two spin‐directions of the incident beam, there are up to two Braggpeaks. For unpolarized neutrons there are up to three peaks.

Diffraction properties of a copper mosaic crystal: model calculations based on γ-ray diffractometry

The mosaic structure of a large copper single-crystal was determined experimentally with 0.03 Å, radiation and found to be inhomogeneous. The influence of inhomogeneities on the diffraction process was studied computationally for the symmetrical Laue geometry as a function of the wavelength of the scattered radiation. The calculations were based on Darwin's theory of extinction. Zachariasen's extinction factor y was found to be a product of two factors, y′ and y′′: y′ ≤ 1 is related to the secondary extinction occurring in the crystal if the mosaic structure is assumed to be homogeneous, y′′ ≤ 1 represents the increase of secondary extinction due to certain inhomogeneities in the mosaic structure. In the investigated copper crystal y′′ is equal to 0.89 for y′ = 0.85, thus the relative error Δy/y in the extinction factor would be of the order of 12% if the inhomogeneities in the mosaic structure are not taken into account. The value of y′′ depends on the degree of inhomogeneity in the mosaic structure of the sample which has to be determined experimentally. The wavelength dependence of y′′ cannot be described by a simple law allowing for extrapolation to λ→ 0. Inhomogeneities cause a relative decrease of the peak reflectivity of about 20% for peak values smaller than 0.3. If the peak reflectivity approaches its saturation value of 0.5, the half width of the diffraction pattern decreases by approximately 30% with respect to the result of calculations where the inhomogeneities in the mosaic structure are not taken into account.

The dynamical scattering amplitude of an imperfect crystal. III. A dynamical diffraction equation for topography in the spatial coordinate representation

Based on a general dynamical theory of diffraction, an integral equation for dynamical diffraction in imperfect crystals is obtained in the spatial coordinate representation. This equation is derived for diffraction topography in the symmetrical Laue geometry from the basic dynamical equation of diffraction previously derived in the momentum representation. Discussion is presented on an interpretation of this integral equation beyond the given diffraction geometry. This equation can be considered as a basic equation for diffraction topography.

X-ray diffraction in crystals of intermediate perfection. I. Calculation of the integral diffracted power for flat and curved crystals in symmetrical Bragg geometry

A method for the treatment of x-ray extinction and absorption is developed for thick flat crystals in symmetrical Bragg geometry and for curved crystals in symmetrical Bragg geometry where the diffraction region may be approximated as a thin crystal in Laue geometry. This method permits the calculation of diffracting power vs wavelength. The crystals may have the full range of properties between the ideally imperfect and perfect cases. The parameters describing the mosaic blocks (mosaic block size and mosaic block misorientation) are related to the dislocation arrangement for crystals of Zachariasen's type I (where extinction is primarily controlled by the mosaic block misorientation). It is indicated that for crystals of type II (where extinction is primarily controlled by the mosaic block size) the mosaic blocks are usually due to defects other than dislocations.

The preparation and characterization of single crystal silver films

Single crystal films ≈0.5 μm thick with dislocations densities ≈10 −7 lines/cm 2 have been prepared by vacuum evaporation and epitaxial growth on cleavage surfaces of NaCl. The key step in the process is the on-substrate annealing at an intermediate film thickness (≈0.15 μm). These low dislocation density films have given X-ray rocking curves of less than 1.5' fwhm. Differences in rocking curve widths for Laue and Bragg geometry are consistent with the thickness effect expected from X-ray diffraction theory.

The Darwin theory for scattering by an ideal crystal in the asymmetric case

1. It has been shown that there was an error in the equations of [3] for the asymmetric Laue case. The corrected equations are given. 2. The systems of recurrence equations are converted to systems of homogeneous differential equations to simplify the solution. 3. If negative angles are taken into consideration, we can combine systems (7) and (12), as in the Ewald theory. Then system (12) describes both Laue geometry and Bragg geometry in the most general configuration.

Theory of X-Ray Diffraction by a Vibrating Crystal

The scattering amplitude for x-ray diffraction by a vibrating crystal is obtained. The Bragg diffracted propagators for the perfect crystal are replaced by new propagators in which the Fourier transform of the polarizability of the crystal, v(H), is multiplied by the Bessel function of zero order, J0(H·Aq), of the product of the vibrational amplitude Aq and a reciprocal lattice vector H, in a similar fashion to the correction by the Debye-Waller factor in the thermal problem. The scattering amplitude contains both dynamical and kinematical contribution. The case of ideal thickness vibration is studied in detail in the Laue geometry. The real part of the perturbed scattering amplitude is proportional to v(H) sin (H·Aq). This theory elucidates the transition from dynamical to kinematical scattering, that is, a decrease in extinction, as a function of the vibration amplitude.

Intensity of X Rays Diffracted from an Elastically Vibrating Single-Crystal Plate

The intensity of the x-ray beam diffracted from a single crystal changes when the crystal lattice is distorted. An elastically vibrating quartz plate driven piezoelectrically was used to study relationships between diffracted intensity and lattice distortions for the case of Laue geometry with μt∼1. The main advantages in studying the distortions of a plate vibrating at resonance are that the shape of the displacement of the lattice is amenable to mathematical analysis and that the amplitude of vibration can be varied continuously. Integrated intensity of the diffracted x-ray beam was measured as functions of the position in the crystal plate, the amplitude of vibration, and the phase of a vibration cycle. The following conclusions were drawn, which demonstrate and verify some predictions made by the dynamical theory of x-ray diffraction for a distorted single crystal for the nonabsorbing case: (1) For an intensity change to occur, the distortion in the net plane must have a component along the diffraction vector and have a nonzero second derivative in the plane of diffraction. (2) The measured integrated intensity is a monotonically increasing function of lattice distortions and closely follows the theoretical curve. (3) Increases in the diffracted intensity are independent of the sign of the second derivative of the displacement.

Quantum Theory of X-Ray Diffraction by a Crystal

A new formulation of the theory of X-ray diffraction by a crystal using techniques of quantum field theory has been developed. The classical field amplitudes of previous theories are replaced, in the present work, by scattering amplitudes for the transmitted and diffracted beams which are obtained from a photon Green's function. The case of a single-Bragg reflection for a parallel sided crystal has been treated in detail. The scattering amplitudes for this case give results which are essentially those of the semi-classical dynamical theory with absorption. The application of the theory to the diffraction of a beam of finite width for the Laue geometry provides an alternate demonstration that the beam emerges from the exit surface of the crystal at a point given by the direction of energy flow.

Absolute Intensities of X-Rays Anomalously Diffracted through Nearly Perfect Copper Crystals

Absolute integrated intensities of the x-ray beams anomalously transmitted through nearly perfect copper crystals have been measured for the (111) planes for values of $\ensuremath{\mu}t$ ranging from 10 to 90. Results obtained for both the symmetric and asymmetric Laue geometries are compared with theory. These comparisons yield a value of 0.965\ifmmode\pm\else\textpm\fi{}0.003 for the ratio f\ensuremath{''}(111)/f\ensuremath{''}(000) and indicate that the crystals investigated have a high degree of perfection. The measurements were made with Mo $K\ensuremath{\alpha}$ radiation using a double-crystal spectrometer arranged in the parallel condition and with the first crystal diffracting in the Bragg geometry. Measured profiles of the Bragg-Laue double-crystal rocking curves for $\ensuremath{\mu}t=10 \mathrm{and} 90$ are in semiquantitative agreement with the theory.