scholarly journals A Positivity Method for the Determination of Complete Orientation Distribution Functions

1988 ◽  
Vol 10 (1) ◽  
pp. 21-35 ◽  
Author(s):  
M. Dahms ◽  
H. J. Bunge
Keyword(s):  
Expansion Method ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Positivity Condition ◽  
Odd Order ◽  
Pole Figures ◽  
Zero Range ◽  
Even Order ◽  
Series Expansion Method

A refinement of the zero-range method, a procedure to calculate the odd order coefficients in the series expansion method of texture analysis, is presented. The only assumption in this procedure is the positivity condition. In this respect, it is comparable to the quadratic method. Contrary to this method, however, the even order coefficients are not changed. No zero range in the pole figures and no shape of the existing texture is to be assumed.

scholarly journals Approximation of the Orientation Distribution of Grains in Polycrystalline Samples by Means of Gaussians

1992 ◽  
Vol 19 (1-2) ◽  
pp. 9-27 ◽  
Author(s):  
D. I. Nikolayev ◽  
T. I. Savyolova ◽  
K. Feldmann
Keyword(s):  
Rolling Texture ◽  
Expansion Method ◽  
Operating Mode ◽  
Orientation Distribution ◽  
New Method ◽  
Gaussian Distributions ◽  
Positivity Condition ◽  
The Central Limit Theorem ◽  
Pole Figures ◽  
Series Expansion Method

The orientation distribution function (ODF) obtained by classical spherical harmonics analysis may be falsified by ghost influences as well as series truncation effects. The ghosts are a consequence of the inversion symmetry of experimental pole figures which leads to the loss of information on the “odd” part of ODF.In the present paper a new method for ODF reproduction is proposed. It is based on the superposition of Gaussian distributions satisfying the central limit theorem in the SO(3)-space as well as the ODF positivity condition. The kind of ODF determination offered here is restricted to the fit of Gaussian parameters and weights with respect to the experimental pole figures. The operating mode of the new method is demonstrated for a rolling texture of copper. The results are compared with the corresponding ones obtained by the series expansion method.

The iterative series-expansion method for quantitative texture analysis. II. Applications

1992 ◽  
Vol 25 (2) ◽  
pp. 259-267 ◽  
Author(s):  
M. Dahms
Keyword(s):  
Distribution Function ◽  
Series Expansion ◽  
Expansion Method ◽  
Orientation Distribution ◽  
Polycrystalline Materials ◽  
Pole Density ◽  
General Outline ◽  
Positivity Condition ◽  
Pole Figures ◽  
Series Expansion Method

The orientation distribution function (ODF) of the crystallites of polycrystalline materials can be calculated from experimentally measured pole density functions (pole figures). This procedure, called pole-figure inversion, can be achieved by the series-expansion method (harmonic method). As a consequence of the (hkl)-({\bar h}{\bar k}{\bar l}) superposition, the solution is mathematically not unique. There is a range of possible solutions (the kernel) that is only limited by the positivity condition of the distribution function. The complete distribution function f(g) can be split into two parts, \tilde {f}(g) and \tildes {f}(q), expressed by even- and odd-order terms of the series expansions. For the calculation of the even part \tilde {f}(g), the positivity condition for all pole figures contributes essentially to an `economic' calculation of this part, whereas, for the odd part, the positivity condition of the ODF is the essential basis. Both of these positivity conditions can be easily incorporated in the series-expansion method by using several iterative cycles. This method proves to be particularly versatile since it makes use of the orthogonality and positivity at the same time. In the previous paper in this series [Dahms & Bunge (1989) J. Appl. Cryst. 22, 439–447] a general outline of the method was given. This, the second part, gives details of the system of programs used as well as typical examples showing the versatility of the method.

scholarly journals Determination of the Complete Orientation Distribution Function by the Zero-Range Method

1986 ◽  
Vol 6 (4) ◽  
pp. 289-313 ◽  
Author(s):  
H. P. Lee ◽  
H. J. Bunge ◽  
C. Esling
Keyword(s):  
Distribution Function ◽  
Pole Figure ◽  
Orientation Distribution Function ◽  
Statistical Error ◽  
Orientation Distribution ◽  
Positivity Condition ◽  
Additional Information ◽  
Pole Figures ◽  
Zero Range

Because of the superposition of pole figures corresponding to symmetrically equivalent crystal directions, only the reduced orientation distribution function f∼(g) can be obtained directly by pole figure inversion. The additional information contained in the positivity condition of the ODF allows, however, the determination of an approximation to the “indeterminable” part and hence of the complete ODF f(g), if the texture has sufficiently large zero-ranges. The application of the method and the accuracy of the results was tested using two theoretical and one experimental textures. The accuracy of the complete ODF depends on the size of the zero-range, the errors in its determination, and on the errors, experimental and truncational, of the reduced ODF. The “physical zero” used in order to determine the zero-range is defined according to the statistical error of the pole figure measurement.

scholarly journals The Use of a Quadratic Form for the Determination of Non-negative Texture Functions

1983 ◽  
Vol 6 (1) ◽  
pp. 1-19 ◽  
Author(s):  
P. Van Houtte
Keyword(s):  
Series Expansion ◽  
Quadratic Forms ◽  
Expansion Method ◽  
Experimental Methods ◽  
Classical Analysis ◽  
Pole Figures ◽  
Zero Range ◽  
Euler Space ◽  
Series Expansion Method

The classical analysis of measured pole figures of textured polycrystals by the series expansion method does not necessarily produce a non-negative texture function. The main reason for this is, that the method is unable to find the terms of odd rank l of the series expansion.A new method is proposed, which introduces the non-negativity condition into the series expansion method by the use of quadratic forms. The method is found to be successful when treating sharp textures, which have a considerable zero range in Euler space. The preliminary determination of this zero range by experimental methods is however not necessary.

scholarly journals Calculation of Yield Surfaces and Determination of Forming Limit Diagrams of Aluminium Alloys

1993 ◽  
Vol 21 (2-3) ◽  
pp. 93-108 ◽  
Author(s):  
J. J. Fundenberger ◽  
M. J. Philippe ◽  
C. Esling ◽  
P. Lequeu ◽  
B. Chenal
Keyword(s):  
Aluminium Alloys ◽  
Expansion Method ◽  
Strain Ratio ◽  
Orientation Distribution ◽  
Forming Limit ◽  
Deformation Model ◽  
Plastic Strain Ratio ◽  
Yield Loci ◽  
Series Expansion Method

In order to point out the influence of the crystallographic texture on the formability of 2 aluminium alloys, the orientation distribution function (ODF) will be carried out using the series expansion method. Combining the ODF with a Taylor plastic deformation model we are able to calculate the yield loci and to predict the plastic strain ratio which is of high interest in the formability.

scholarly journals ODF Calculation by Series Expansion from Incompletely Measured Pole Figures Using the Positivity Condition Part I—Cubic Crystal Symmetry

1987 ◽  
Vol 7 (3) ◽  
pp. 171-185 ◽  
Author(s):  
M. Dahms ◽  
H. J. Bunge
Keyword(s):  
Series Expansion ◽  
Iterative Procedure ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Crystal Symmetry ◽  
Cubic Crystal ◽  
Positivity Condition ◽  
Pole Figures ◽  
Orientation Distribution Functions

The calculation of orientation distribution functions (ODF) from incomplete pole figures can be carried out by an iterative procedure taking into account the positivity condition for all pole figures. This method strongly reduces instabilities which may occasionally occur in other methods.

X-Ray Diffraction Investigation of Electrochemically Deposited Copper

2004 ◽  
Vol 443-444 ◽  
pp. 201-204 ◽  
Author(s):  
Karen Pantleon ◽  
Jens Dahl Jensen ◽  
Marcel A.J. Somers
Keyword(s):  
Process Parameters ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Organic Additives ◽  
X Ray Diffraction ◽  
X Ray ◽  
Pole Figures ◽  
Electrochemically Deposited ◽  
Cu Interconnect

Copper layers were deposited from acidic electrolytes containing different amounts of organic additives, designed for the formation of Cu-interconnect structures. Amorphous Ni-P substrates allow to study the unbiased growth of the electrodeposits. The crystallographic texture was investigated by the determination of X-ray diffraction (XRD) pole figures and the calculation of the orientation distribution functions. XRD results are discussed in relation to the morphologies of the electrodeposits as investigated with light optical microscopy and correlated with the process parameters during electrodeposition.

Textures Applied to Mechanical Processing Technology Assessment in Ancient Bronze

2005 ◽  
Vol 495-497 ◽  
pp. 719-724
Author(s):  
R.E. Bolmaro ◽  
B. Molinas ◽  
E. Sentimenti ◽  
A.L. Fourty
Keyword(s):  
Orientation Distribution ◽  
Distribution Functions ◽  
Archaeological Sites ◽  
Mechanical Processing ◽  
Post Processing ◽  
X Ray Diffraction ◽  
Mechanical Process ◽  
X Ray ◽  
Pole Figures ◽  
Microscopic Inspection

Some ancient metallic art craft, utensils, silverware and weapons are externally undistinguishable from modern ones. Not only the general aspect and shape but also some uses have not changed through the ages. Moreover, when just some small pieces can be recovered from archaeological sites, the samples can not easily be ascribed to any known use and consequently identified. It is clear that mechanical processing has changed along history but frequently only a "microscopic" inspection can distinguish among different techniques. Some bronze samples have been collected from the Quarto d’Altino (Veneto) archaeological area in Italy (paleovenetian culture) and some model samples have been prepared by a modern artisan. The sample textures have been measured by X-ray Diffraction techniques. (111), (200) and (220) pole figures were used to calculate Orientation Distribution Functions and further recalculate pole figures and inverse pole figures. The results were compared with modern forging technology results. Textures are able to discern between hammering ancient techniques for sheet production and modern industrial rolling procedures. However, as it is demonstrated in the present work, forgery becomes difficult to detect if the goldsmith, properly warned, proceeds to erase the texture history with some hammering post-processing. The results of this contribution can offer to the archaeologists the opportunity to take into consideration the texture techniques in order to discuss the origin (culture) of the pieces and the characteristic mechanical process developed by the ancient artisan. Texture can also help the experts when discussing the originality of a certain piece keeping however in mind the cautions indicated in this publication.

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