Effects of Higher-Order Laue Zone reflections on structure images

1993 ◽  
Vol 51 ◽  
pp. 450-451
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Wave Vector ◽  
Theoretical Calculations ◽  
Reciprocal Lattice ◽  
Image Plane ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Higher Order ◽  
Lattice Image ◽  
Lattice Vector ◽  
Image Position

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

scholarly journals Analysis of Contrast from Lattice Defects

1968 ◽  
Vol 26 ◽  
pp. 246-247
Author(s):  
W. L. Bell
Keyword(s):  
Reciprocal Lattice ◽  
Dynamical Theory ◽  
Displacement Function ◽  
Foil Thickness ◽  
Linear Combinations ◽  
Lattice Vector ◽  
Diffracted Waves ◽  
The Mean ◽  
Image Position ◽  
Standard Techniques

The differential equations of the two-beam dynamical theory of electron diffraction in an absorbing single crystal are, ignoring unimportant phase factors:where T(z) and S(z) are the transmitted and diffracted amplitudes, the mean and anomalous absorption distances; to is the extinction distance, z the foil thickness, Sg the macroscopic deviation parameter, ḡ the reciprocal lattice vector and R the displacement function of any defect present in the crystal. Occasionally, but not often, these equations can be handled by standard techniques; the wave vectors allowed in the crystal can be determined and the transmitted and diffracted waves expressed as linear combinations of waves with these wave vectors.

On interpreting weak beam images of microtwins

1996 ◽  
Vol 54 ◽  
pp. 132-133
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Theoretical Calculations ◽  
Reciprocal Lattice ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Image Contrast ◽  
Fundamental Importance ◽  
Weak Beam ◽  
The Matrix

The concept of a dynamically coupled set is of fundamental importance in carrying out theoretical calculations of image contrast of microtwins. Up to the present, considerations based on the dynamical theory have indicated that the diffracted beams taken into account in a particular calculation of image contrast must always reside in the same dynamically coupled set. The work presented in this paper indicates that for weak beam images this may not the case. The reason for this can be seen from Fig.1b in which g, h and (i), (j) are reciprocal lattice vectors and branches of the dispersion surfaces in the matrix and microtwin respectively. It can be seen that if the condition (h − g) = Δγnf is obtained then the Bloch wave vectors in the matrix and twin, and , are parallel. Under these circumstances will make a coherent contribution to the intensity of diffracted beam g, even though h does not reside in the same dynamically coupled set.

Temperature dependence of ECP contrast: independent or dependent Bloch wave model?

1978 ◽  
Vol 36 (1) ◽  
pp. 198-199
Author(s):  
H.-J. Kohl
Keyword(s):  
Temperature Dependence ◽  
Wave Field ◽  
Room Temperature ◽  
Reciprocal Lattice ◽  
Wave Model ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Storage Unit ◽  
Lattice Vector ◽  
Backscattered Electrons

The temperature dependence of the ECP contrast of thick Si, Cu, Ag, Au crystals has been investigated with the rocking crystal method /1,2,3/. The ECP is written into a storage unit. The contrast measurements could be performed by writing linescans of the ECP signal on an x-y recorder at a specimen temperature of 500 K and at room temperature. The results obtained with silicon and silver (fig. 2) show a decrease of contrast c with increasing value g of the reciprocal lattice vector belonging to the investigated band edge (c=(IBmax-IBmin)/(IBmax+IBmin), IB: intensity of the backscattered electrons).With the two beam case of the dynamical theory the temperature dependence of the ECP contrast could be calculated. The contrast is produced by a modulation of the Bloch wave field in the crystal. If absorption is ignored the modulation does not change with depth in the independent Bloch wave model, but in the dependent Bloch wave model it is developing with depth due to the interference between the Bloch waves.

Kinematical and Dynamical CBED Pattern Models: Increasing the Accuracy of the Unit-Cell Parameter Measurements Based on CBED Patterns

1990 ◽  
Vol 48 (2) ◽  
pp. 540-541
Author(s):  
I.N. Yadhikov ◽  
S.K. Maksimov
Keyword(s):  
Reciprocal Lattice ◽  
Unit Cell ◽  
Reciprocal Lattice Vector ◽  
Unit Cell Parameters ◽  
Lattice Vector ◽  
Cell Parameters ◽  
Accuracy Of Measurements ◽  
The Difference ◽  
Image Position ◽  
Convergent Beam

Convergent beam electron diffraction (CBED) is widely used as a microanalysis tool. By the relative position of HOLZ-lines (Higher Order Laue Zone) in CBED-patterns one can determine the unit cell parameters with a high accuracy up to 0.1%. For this purpose, maps of HOLZ-lines are simulated with the help of a computer so that the best matching of maps with experimental CBED-pattern should be reached. In maps, HOLZ-lines are approximated, as a rule, by straight lines. The actual HOLZ-lines, however, are different from the straights. If we decrease accelerating voltage, the difference is increased and, thus, the accuracy of the unit cell parameters determination by the method becomes lower.To improve the accuracy of measurements it is necessary to give up the HOLZ-lines substitution by the straights. According to the kinematical theory a HOLZ-line is merely a fragment of ellipse arc described by the parametric equationwith arc corresponding to change of β parameter from -90° to +90°, wherevector, h - the distance between Laue zones, g - the value of the reciprocal lattice vector, g‖ - the value of the reciprocal lattice vector projection on zero Laue zone.

scholarly journals Applications of dynamical theory of X-ray diffraction by perfect crystals to reciprocal space mapping

2017 ◽  
Vol 50 (5) ◽  
pp. 1256-1266 ◽  
Author(s):  
Vasily I. Punegov ◽  
Konstantin M. Pavlov ◽  
Andrey V. Karpov ◽  
Nikolai N. Faleev
Keyword(s):  
Reciprocal Lattice ◽  
Dynamical Theory ◽  
Two Dimensional ◽  
Instrumental Function ◽  
X Ray Diffraction ◽  
Reciprocal Space Mapping ◽  
Lattice Vector ◽  
Reflected Waves ◽  
X Ray ◽  
Special Case

The classical dynamical theory of X-ray diffraction is expanded to the special case of transversely restricted wavefronts of the incident and reflected waves. This approach allows one to simulate the two-dimensional coherently scattered intensity distribution centred around a particular reciprocal lattice vector in the so-called triple-crystal diffraction scheme. The effect of the diffractometer's instrumental function on X-ray diffraction data was studied.

scholarly journals Spatial aliasing and distortion of energy distribution in the wave vector domain under multi-spacecraft measurements

2009 ◽  
Vol 27 (8) ◽  
pp. 3031-3042 ◽  
Author(s):  
Y. Narita ◽  
K.-H. Glassmeier
Keyword(s):  
Energy Distribution ◽  
Brillouin Zone ◽  
Wave Vector ◽  
Flow Direction ◽  
Reciprocal Lattice ◽  
Periodic Pattern ◽  
Lattice Vector ◽  
Model Calculations ◽  
Spatial Aliasing ◽  
Spacecraft Data

Abstract. Aliasing is a general problem in the analysis of any measurements that make sampling at discrete points. Sampling in the spatial domain results in a periodic pattern of spectra in the wave vector domain. This effect is called spatial aliasing, and it is of particular importance for multi-spacecraft measurements in space. We first present the theoretical background of aliasing problems in the frequency domain and generalize it to the wave vector domain, and then present model calculations of spatial aliasing. The model calculations are performed for various configurations of the reciprocal vectors and energy spectra or distribution that are placed at different positions in the wave vector domain, and exhibit two effects on aliasing. One is weak aliasing, in which the true spectrum is distorted because of non-uniform aliasing contributions in the Brillouin zone. It is demonstrated that the energy distribution becomes elongated in the shortest reciprocal lattice vector direction in the wave vector domain. The other effect is strong aliasing, in which aliases have a significant contribution in the Brillouin zone and the energy distribution shows a false peak. These results give a caveat in multi-spacecraft data analysis in that spectral anisotropy obtained by a measurement has in general two origins: (1) natural and physical origins like anisotropy imposed by a mean magnetic field or a flow direction; and (2) aliasing effects that are imposed by the configuration of the measurement array (or the set of reciprocal vectors). This manuscript also discusses a possible method to estimate aliasing contributions in the Brillouin zone based on the measured spectrum and to correct the spectra for aliasing.

The dynamical theory of a double crystal electron interferometer

1978 ◽  
Vol 36 (1) ◽  
pp. 188-189 ◽  
Author(s):  
B.F. Buxton ◽  
G.M. Rackham ◽  
J.W. Steeds
Keyword(s):  
Reciprocal Lattice ◽  
Plane Waves ◽  
Dynamical Theory ◽  
Double Crystal ◽  
Exit Surface ◽  
Incident Plane ◽  
The Many ◽  
Image Position ◽  
Crystal Electron ◽  
Convergent Beam

Recently, Rackham et al.(1977) reported the observation of fine fringes in convergent beam patterns obtained from double crystal interferometers consisting of two perfectly aligned crystals separated by a gap D of about 1 μm (Fig. 1). If we assume for simplicity that the reciprocal lattice vectors G of the operating reflections are parallel to the crystals, the many-beam wavefunction Ψ(R,t1) on the exit surface of crystal I due to an incident plane wave eix.r. (Fig. 2), given by(1)can be matched onto plane waves propagating in directions K+G between the crystals. In the gap therefore, the wavefunction is(2)where(3)since x ≫ |K+G| for all reflections of interest. The wavefunction (2) is diffracted by the second crystal, from which emerge waves in directions K+H with amplitudes.

A variational theory for the wave-vector-dependent transverse spin susceptibility and spin-wave velocity of a two-sublattice antiferromagnet

1986 ◽  
Vol 64 (1) ◽  
pp. 27-35
Author(s):  
K. L. Liu ◽  
S. H. Vosko
Keyword(s):  
Spin Wave ◽  
Wave Velocity ◽  
Wave Vector ◽  
Correlation Energy ◽  
Reciprocal Lattice ◽  
Energy Functional ◽  
Spin Susceptibility ◽  
Transverse Spin ◽  
Variational Theory ◽  
Lattice Vector

Based on the Hohenberg and Kohn theorems, we derive a variational lower bound expression for the static wave-vector-dependent transverse spin susceptibility χ+−(q) of a two sublattice commensurate antiferromagnet that has a characteristic magnetic reciprocal lattice vector Q. The variational result is valid for a general exchange-correlation energy functional. When appropriate trial functions are adopted, the variational expressions for χ+−(q) and χ+−(Q + q) have the correct small-q behavior, the latter being divergent as q−2 in accord with Wagner's 1/q2 theorem for broken symmetry. For systems in which spin waves are the only low-lying excitations, these approximate susceptibilities can be used to yield an upper bound for the antiferromagnetic spin-wave velocity [Formula: see text].

Observation of faint contrast at planar faults in alloys

1978 ◽  
Vol 36 (1) ◽  
pp. 340-341
Author(s):  
K. Kuroda ◽  
Y. Tomokiyo ◽  
T. Kumano ◽  
T. Eguchi
Keyword(s):  
Reciprocal Lattice ◽  
Beam Theory ◽  
Al Alloys ◽  
Small Displacement ◽  
Electron Microscopic ◽  
Lattice Vector ◽  
Fringe Contrast ◽  
Planar Fault ◽  
Lattice Displacement ◽  
The Many

The contrast in electron microscopic images of planar faults in a crystal is characterized by a phase factor , where is the reciprocal lattice vector of the operating reflection, and the lattice displacement due to the fault under consideration. Within the two-beam theory a planar fault with an integer value of is invisible, but a detectable contrast is expected when the many-beam dynamical effect is not negligibly small. A weak fringe contrast is also expected when differs slightly from an integer owing to an additional small displacement of the lattice across the fault. These faint contrasts are termed as many-beam contrasts in the former case, and as ε fringe contrasts in the latter. In the present work stacking faults in Cu-Al alloys and antiphase boundaries (APB) in CuZn, FeCo and Fe-Al alloys were observed under such conditions as mentioned above, and the results were compared with the image profiles of the faults calculated in the systematic ten-beam approximation.

A Stationary Solution of High-Energy Electron Reflection (HEER) for An Arbitrary Surface

1990 ◽  
Vol 48 (2) ◽  
pp. 374-375
Author(s):  
YIQUN MA
Keyword(s):  
Stationary Solution ◽  
High Energy ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
High Energy Electron ◽  
Bulk Materials ◽  
Periodic Surface ◽  
Electron Transmission ◽  
Electron Reflection ◽  
Long Time

For a long time, the development of dynamical theory for HEER has been stagnated for several reasons. Although the Bloch wave method is powerful for the understanding of physical insights of electron diffraction, particularly electron transmission diffraction, it is not readily available for the simulation of various surface imperfection in electron reflection diffraction since it is basically a method for bulk materials and perfect surface. When the multislice method due to Cowley & Moodie is used for electron reflection, the “edge effects” stand firmly in the way of reaching a stationary solution for HEER. The multislice method due to Maksym & Beeby is valid only for an 2-D periodic surface.Now, a method for solving stationary solution of HEER for an arbitrary surface is available, which is called the Edge Patching method in Multislice-Only mode (the EPMO method). The analytical basis for this method can be attributed to two important characters of HEER: 1) 2-D dependence of the wave fields and 2) the Picard iteractionlike character of multislice calculation due to Cowley and Moodie in the Bragg case.

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