Soller Slits Research Articles

The role of Soller slits in small-angle scattering collimation systems

Theory is developed which allows the quantitative evaluation of an `effective' slit-length weighting function for small-angle diffractometers which make use of Soller slits. The theory is illustrated with an example and the disadvantages connected with the use of Soller slits in SAXS experiments are discussed.

Centroid shifts due to axial divergence and other geometrical factors in Seemann–Bohlin diffractometry

The axial (or vertical) divergence error associated with Seemann–Bohlin geometry is evaluated for arrangements containing one, two or no sets of Soller slits. It is found that the error resulting from measurements of high angle reflections amounts, typically, to about 0.0002 A in the lattice parameter and corrections for axial divergence are, therefore, necessary in precision diffractometry. It is also observed that little is gained in return for loss of intensity when Soller slits are used in the high angle region. In the lower angle regions, however, the effect of a set of Soller slits in the reflected beam is very marked. To provide a complete error analysis using a uniform system of presentation, all the other geometrical errors associated with the Seemann–Bohlin geometry, are evaluated or re-evaluated.

Rapid Recording of Powder Diffraction Patterns With Si(Li) X-Ray Energy Analysis System: W and Cu Targets and Error Analysis

X-ray diffraction patterns using continuous radiation from copper and tungsten target x-ray tubes and detected with a Si(Li) energy analysis system are presented. Errors caused by a misaligned diffractometer and x-ray penetration into the sample are shown to be more difficult to correct and larger in magnitude than errors arising from energy calibration. All these errors can be minimized by mixing a standard with the unknown sample.The energy resolution of the detector influences the breadth of the diffraction peaks more strongly than the standard slit systems available with commercial diffractometers. Thus, to reduce the recording time and maintain the same standard deviation for the data, one should increase the sizes of the front and receiving slits including the Soller slits. X-ray energy diffraction patterns can be recorded with standard deviations less than +0.001 Å in the d spacing with only 200 sec measurement time using the standard diffractometer slit system. Copper targets are probably as useful as tungsten even though the continuous intensity is about three times.less. Copper has fewer interfering characteristic lines, and its use permits convenient conversion to normal θ scanning diffractometer operation.

Neutron monochromator response based on the secondary extinction Green's function for Bragg geometry

A theoretical expression is derived for the differential output rate of a neutron monochromator system when used in the Bragg geometry. The derivation of this expression starts from the Green's function solution to the secondary extinction power equations for a thick mosaic slab. The combined effect of collimator and crystal positioning, beam divergence, total internal reflection off the collimator walls, and order contribution is included in the final expression. This expression has been programmed for numerical integration to yield a value for the absolute output counting rate in each order for arbitrary values of the two dozen or so independent parameters necessary to specify the system. The calculated output rate is compared with some preliminary experiments done with a single axis spectrometer equipped with a 2:1 half angling device. These experiments include rocking curves, tracking curves (movement of the crystal perpendicular to the diffracting planes), and 2:1 Bragg response curves, all of which can be generated for comparison within the program by proper choice of looped parameters. In addition, an example of a typical calculated radiation transfer matrix between sets of Soller slits is given. It is proposed that these calculations may prove useful in allowing the measurement of neutron cross sections in the sub-thermal energy range without a mechanical filter even though several orders are simultaneously present.

X-Ray Diffraction Topography by a Scanning Soller Slit Assembly

X-Ray Diffraction Topography by a Scanning Soller Slit Assembly

An x-ray monochromator for diffuse scatter measurements

A doubly bent LiF monochromator for CuKα radiation is described. Two mutually perpendicular sets of radially convergent Soller slits are used to produce a sharply defined intense beam 15 mm × 5 mm in cross section, suitable for thermal diffuse scattering measurements. Detailed design and construction are discussed and it is suggested that the Soller slit technique can be used to improve existing monochromators.

Observation of Dislocation in α-Arsenic Single Crystals by X-Ray Diffraction Topography*

AbstractX-ray diffraction topographs were obtained from large arsenic single crystals. The camera employed copper Kα, radiation from a microfocus tube and an oscillating assembly of Soller slits limited the beam divergence. Reflections of the type (11) and (20) (primitive rhombohedral cell) were used to characterise dislocation Burgers vectors. The technique has been applied to arsenic single crystals grown from the vapour and from the melt. The majority of dislocations were found to belong to Burgers vectors <10>. Comparison has been made between dislocation etch pit patterns on (111) surfaces and X-ray topographs.

Measurement of Parallel (Soller) Slits with a Modified Weissenberg Goniometer

Measurement of Parallel (Soller) Slits with a Modified Weissenberg Goniometer

X-Ray Source-Image Distortion Technique for the Study of Crystal Distortion and Vibration

A technique is described which permits graphic display and quantitative determination of ultrasonic oscillation amplitudes, and of warping and other macroscopic strains, in crystals. An x-ray ``source image'' appears superimposed on the crystal image formed by one diffracted beam from a stationary crystal. ``Source-image'' distortion (SID) occurs on reflection from distorted regions of the crystal. X-ray sources ranging from 100-μ spots to 15-mm line sources interrupted by Soller slits have been used. The latter experimental arrangements bear some similarities to those described by Cullity and Julien, by Barth, and by Carlson and Wegener. Lattice tilts of ∼1′ and lattice strain variations (Δd/d) of ∼10−4 can be routinely detected and separately identified. The underlying crystal image often contains useful topographic information of limited quality. SID photographs showing distortion produced in quartz plates by mounting clips, cements, cracks, and piezoelectric oscillations are presented. An example of direct measurement and mapping of vibrational amplitudes, e.g., 650±25 Å in one case, in piezoelectrically oscillating crystals is given. Oscillation mode characterization has been aided by the SID demonstration of a 180° phase relation in some cases and by both SID and topographic display of the pattern of vibrational activity.

Effect of Soller Slits on X-Ray Intensity in a Modern Diffractometer

The properties of Soller slits in a modern diffractometer are discussed in relation to the effect they have on the x-ray intensity distribution within the plane that contains the line focus and diffractometer axis. On the basis of an idealized Soller slit collimator, equations are derived for computing relative x-ray intensity in terms of the distance from the edge of the specimen. An independent mathematical derivation substantiates these equations and also shows the effect of varying the number of slits while keeping other properties of the collimator unchanged. When the equations are used to calculate the over-all intensity profiles of two diffractometers at the specimen position, the results are found to be in reasonable agreement with photographically determined profiles. Similar equations are used to explain the effect of the receiving Soller slits on the diffracted rays.

Counter diffractometer - the effect of axial divergence on the breadth of powder lines

The axial divergence of the beam in a counter diffractometer causes a broadening of diffraction lines. The use of the variance as a measure of this broadening is reviewed and the work of previous authors is extended to include a diffractometer having two sets of Soller slits of any aperture. For an aperture commonly used in commercial diffractometers, the variance is given as a function of 2θ.

Counter diffractometer - the effect of vertical divergence on the displacement and breadth of powder diffraction lines

The divergence of the beam in a counter diffractometer causes a broadening, and a shift of centre of gravity, of diffraction lines. The effects do not disappear as 2 theta tends to 180°, and hence present extrapolation methods fail to remove the errors introduced. Experimental results are quoted that are not explained by previous treatments, and a theoretical investigation is made of the diffractometer with no Soller slits, one set of slits, and two sets of slits. Results show a marked asymmetry about 2 θ = 90°, both in shift of centre of gravity and in line broadening, the maximum effects occurring at low values of 2 θ. The use of Soller slits for various practical applications of the diffractometer is discussed. For accurate lattice-parameter measurement it is shown that the fractional error in spacing introduced by vertical divergence may be written in the form Δd/d = - a + b cos2 θ where a and b are positive constants depending on goniometer radius, specimen height, and Soller-slit aperture (if used). Extrapolation against cos2 theta and addition of the constant a will therefore give the true lattice spacing. The magnitude of the correction for a Norelco high-angle goniometer with two sets of Soller slits is approximately 4 parts in 105.

The Synthesis of X-Ray Spectrometer Line Profiles with Application to Crystallite Size Measurements

An improved convolutional method is used to synthesize theoretical line profiles generated by the joint action of seven instrumental weight functions characteristic of the x-ray spectrometer. These include in addition to the five geometrical factors described in a previous publication: (VI) a so-called misalignment function and (VII) a spectral impurity function. The method is applied to the new spectrometer design incorporating a narrow x-ray source, Soller slits, and an angular range 2θ=0 to 165° or higher. Good agreement between theoretical and experimental line profiles is achieved for quartz reflections at 2θ=24.0 and 90.7°. The improved treatment is used to define new curves for correcting experimental spectrometer line breadths for instrumental broadening effects. It is demonstrated that spectrometers of the new design are capable of measuring crystallite dimensions as large as 3000A.

The Effect of a Soller Slit on the Diffraction of X-Rays by Deformed Crystals

In the application of the Guinier-Tennevin focusing Laue method to the study of crystal distortion, it is found that the use of a Soller slit in the incident beam produces striations in reflections from certain crystals. The diffraction geometry of a beam so modified is examined in detail, and it is shown that the striations are due to torsion of the crystal. The relation between the amount of torsion and the slope and spacing of the striations is treated quantitatively. There results a useful extension of the focusing Laue method by which not only the bending, but also the torsion, of a crystal may be measured.

Effect of vertical divergence on the displacement and breadth of X-ray powder diffraction lines

A treatment is given of the aberrations caused by the extension of the X-ray source, specimen and receiving slit normal to the plane of the camera or spectrometer. A detailed examination is made of the aberrations which occur: (a) when the height of the specimen is the sole cause, and (b) when the vertical divergence of the X-ray beam is limited by Soller slits, the displacement and profile of the line being determined in each case. It is shown that vertical mis-alinement of a spectrometer can cause an additional broadening of the lines, especially in the back-reflexion region.

Geiger-Counter X-Ray Spectrometer - Influence of Size and Absorption Coefficient of Specimen on Position and Shape of Powder Diffraction Maxima

In powder spectrometers of modern design the `vertical' divergence of the beam is limited by Soller slits, and has little influence on the position and shape of the diffraction maxima. `Horizontal' divergence cannot be considered apart from absorption and thickness of specimen. These three effects shift the centre of gravity of the maximum toward lower values of θ. 2A is the illuminated length of the specimen, t is its thickness, R is the spectrometer radius, and μ is the linear absorption coefficient of the specimen. In most spectrometers A is limited by slits to αR csc θ, where 2α is the horizontal divergence of the incident beam, but in some the specimen is irradiated over its full length L, and A is constant at ½L. The shift of the peak is more complicated, and requires a consideration of the line profile. In the limiting case of theta large a first approximation gives the same displacement for the peak as for the centre of gravity; in the limiting case of θ small the peak displacement is the smallest of the three expressions A2sin 2θ/2R2, 0.214 sin 2θ/µR, t cos θ/R No simple expression has been found for intermediate values of theta, Except for weakly absorbing substances and low Bragg angles the displacements are very small. The error in the spacings calculated from high-angle lines is discussed; there seems to be no reason why Geiger-counter spectrometry should not reach the same accuracy as the powder camera in the measurement of spacings.

Scattering of X-rays from Gases

Measurement of the intensity of scattering of x-rays by various gases has been made for scattering angles between 10\ifmmode^\circ\else\textdegree\fi{} and 90\ifmmode^\circ\else\textdegree\fi{}. Soller slits were used to get a well-defined scattering angle and matched filters of Zr${\mathrm{O}}_{2}$ and SrO were used to isolate the ${K}_{\ensuremath{\alpha}}$ lines of molybdenum. Intensity measurements have been made for hydrogen, helium, oxygen, neon and argon. All the measurements were put on the same scale by comparison of the intensity of scattering from the various gases with the scattering from hydrogen at 90\ifmmode^\circ\else\textdegree\fi{}. This is equivalent to putting the intensities on an absolute scale since the scattering from hydrogen at 90\ifmmode^\circ\else\textdegree\fi{} can be calculated with considerable certainty. Values of the structure factors for neon and argon were calculated and comparison made with those determined from wave mechanics for similar atoms.