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The document discusses texture measurement using X-ray diffraction. It describes the Schulz reflection method for measuring pole figures using a 5-axis goniometer. The key steps are to first identify diffraction peaks via a Bragg scan, then fix the Bragg angle and measure intensity while rotating the sample through angles φ and ψ to map the intensity distribution on a pole figure for the chosen diffraction plane. Intensity variations reveal the texture components present in the material.
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g.pdf
The document discusses the difference between bulk/macrotexture and microtexture. Bulk texture provides information about the overall orientation distribution in a material but not the spatial location of orientations. Microtexture provides information about individual grain orientations and neighbors from EBSD mapping. Bulk texture is represented by pole figures and ODFs from XRD, while microtexture uses orientation image maps from EBSD. The document also explains how pole figures are obtained from XRD by rotating the sample to satisfy Bragg's law for different crystal planes and orientations.
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1. There are several types of X-ray cameras used to analyze crystal structures, including Laue, rotating crystal, and powder methods.
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Microtexture refers to the crystallographic orientation of individual grains within a polycrystalline material. It is determined through spatially resolved microstructural analysis techniques like electron backscatter diffraction (EBSD) in SEM or microdiffraction in TEM. These techniques produce diffraction patterns from each sampled volume, allowing the crystallographic orientation to be determined on a grain-by-grain basis. Interpreting the Kikuchi patterns generated in these techniques involves identifying the crystallographic indices of bands in the pattern and determining their relative position to deduce the orientation of each sampled volume.
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1) Using four laser beams, researchers generated a three-dimensional optical lattice that traps 490nm polystyrene spheres in solution, forming a face-centered orthorhombic crystal structure.
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The document discusses limitations of texture measurement procedures using X-ray diffraction and ways to overcome them. It describes issues like defocusing error that occurs at large tilt angles, limitations of the Schultz method related to divergence slit width and receiving slit width, and how to perform corrections for defocusing error, background subtraction, and normalization. It also provides example questions to test understanding of pole figures and related concepts.
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The document discusses the difference between bulk/macrotexture and microtexture. Bulk texture provides information about the overall orientation distribution in a material but not the spatial location of orientations. Microtexture provides information about individual grain orientations and neighbors from EBSD mapping. Bulk texture is represented by pole figures and ODFs from XRD, while microtexture uses orientation image maps from EBSD. The document also explains how pole figures are obtained from XRD by rotating the sample to satisfy Bragg's law for different crystal planes and orientations.
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Microtexture refers to the crystallographic orientation of individual grains within a polycrystalline material. It is determined through spatially resolved microstructural analysis techniques like electron backscatter diffraction (EBSD) in SEM or microdiffraction in TEM. These techniques produce diffraction patterns from each sampled volume, allowing the crystallographic orientation to be determined on a grain-by-grain basis. Interpreting the Kikuchi patterns generated in these techniques involves identifying the crystallographic indices of bands in the pattern and determining their relative position to deduce the orientation of each sampled volume.
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The document discusses limitations of texture measurement procedures using X-ray diffraction and ways to overcome them. It describes issues like defocusing error that occurs at large tilt angles, limitations of the Schultz method related to divergence slit width and receiving slit width, and how to perform corrections for defocusing error, background subtraction, and normalization. It also provides example questions to test understanding of pole figures and related concepts.
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X-ray diffraction is the most common method for measuring macrotexture, providing information about the average texture over penetration depths of micrometers. Neutron diffraction can measure average bulk texture over centimeters by penetrating thicker samples. Electron backscatter diffraction has the best spatial resolution for local microtexture over nanometers but is limited to surface analysis. These techniques work by fixing the diffraction condition for crystal planes and rotating the sample to measure variations in intensity, which are represented as pole figures mapping orientations.
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This document summarizes an experiment that uses four laser beams to trap thousands of sub-micron polystyrene particles in water, forming an optically induced crystal. Bragg scattering patterns from the crystal agree with the calculated lattice structure and polarization dependence. By observing the decay and rise of Bragg scattering intensity when turning the lattice on and off, the researchers study the Brownian motion dynamics of particles in the periodic potential, finding agreement with simulations based on the Langevin equation.
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Here are the key points about representing orientations in ODF vs pole figures:
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- Measure multiple pole figures experimentally
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Every crystalline phase of a given substance produces a characteristic X-ray diffraction pattern.
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Essentially three types of information can be derived from a powder diffraction pattern:
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This document summarizes an experiment that uses four laser beams to trap thousands of sub-micron polystyrene particles in water, forming an optically induced crystal. Bragg scattering patterns from the crystal agree with the calculated lattice structure and polarization dependence. By observing the decay and rise of Bragg scattering intensity when turning the lattice on and off, the researchers study the Brownian motion dynamics of particles in the periodic potential, finding agreement with simulations based on the Langevin equation.
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2. Models of elliptical disks and disks with spiral arms were able to reproduce some observed line profiles and velocity shifts over time.
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This document discusses texture formation processes in metals and alloys, including deformation, annealing, and transformation textures. It focuses on deformation textures and the mechanisms of slip and twinning. The key factors that determine the dominant deformation mechanism are crystal structure and stacking fault energy. Materials with high stacking fault energy deform primarily through slip, while those with low stacking fault energy experience more twinning. The document also examines differences in slip systems for BCC, FCC, and HCP crystals and how deformation leads to the development of crystallographic texture through grain rotation.
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Here are the key points about representing orientations in ODF vs pole figures:
- ODF describes the volume/number fraction of all possible orientations in 3D space using the orientation distribution function f(g).
- Pole figures only describe orientations along certain crystallographic directions (poles), not the full 3D orientation space.
To calculate ODFs:
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- Express pole figures and ODF using spherical harmonic functions
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Here are the answers to the questions:
1. A pole figure is a graphical representation of the orientation of crystallites in a polycrystalline material. It shows the distribution of crystal orientations with respect to sample axes.
2. The figures shown are:
- The first figure is a (111) pole figure. It represents a brass texture with preferred orientations around the (111) poles.
- The second figure is an inverse pole figure (IPF). It represents the texture of a drawn wire, with preferred orientations around the (110) direction.
3. Pole figures represent the orientation of crystals with respect to the sample frame of reference.
4. The statement is true. In a 111 pole figure
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Here are the answers to the questions:
1. a, b, d are correct. Grain boundary character cannot be measured using XRD.
2. d is incorrect. Crystallite sizes below 1000 Å have errors greater than 20% when using Scherrer equation to calculate size from XRD peak broadening.
3. FCC structure:
1) 2θ = 44.7°, (111)
2) 2θ = 51.8°, (200)
3) 2θ = 75.5°, (220)
4. Use Scherrer equation:
B = Kλ / t cosθ
For 10°: t = 880 Å
For 45°
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This document discusses evaluating regression models and regularization techniques. It introduces adjusted R-squared (Adj R2) as a metric to evaluate regression models in addition to R-squared and p-values from ANOVA. Adj R2 accounts for the number of predictors in the model. The document also discusses regularization techniques like ridge and lasso regression that perform variable shrinkage. Ridge regression minimizes residuals subject to a constraint on the sum of squared coefficients. Lasso regression similarly minimizes residuals but constrains the sum of the absolute values of coefficients.
13.pdf
This document provides a short introduction to regression analysis. It explains that regression analysis studies relationships between variables and can be used for interpretation and prediction. Examples given include how employee wages are affected by factors like experience and education, and how stock prices depend on past values. It also categorizes regressions as linear or non-linear, simple or multiple, and cross-sectional or time-series. Key concepts discussed include response and explanatory variables, scatter plots, correlations, and fitting linear regression lines of the form y=mx+c to identify the best fitting model.
12.pdf
- ANOVA and chi-square tests of independence can be used to analyze relationships between categorical variables that have more than two categories.
- ANOVA compares group means and determines if observed differences are larger than expected by chance. It tests if multiple group means are equal.
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- After rejecting the null hypothesis with ANOVA, Tukey's test identifies which specific group means are significantly different from each other.
11.pdf
This document provides an introduction to confidence intervals in inferential statistics. It discusses how confidence intervals estimate population parameters like the mean or proportion by providing a range of values that are likely to contain the true population parameter. The document explains that if confidence intervals were constructed repeatedly from many samples, the true population parameter would fall within the given confidence level, such as 95%, of the intervals. It also provides examples of common statistical tests like z-tests, t-tests, and chi-square tests that use confidence intervals and formulas to calculate the range of values for the estimated parameters.
10.pdf
This document discusses type 1 and type 2 errors in statistical hypothesis testing. Type 1 errors occur when a null hypothesis is incorrectly rejected, also known as a false positive. Type 2 errors occur when an alternative hypothesis is incorrectly not rejected, also known as a false negative. The probability of a type 1 error is controlled by the significance level alpha, while the probability of a type 2 error, beta, depends on factors like the effect size, sample size, and alpha level. Power is equal to 1 minus beta and represents the probability of correctly rejecting the null hypothesis.
8.pdf
This document provides an overview of single sample z-tests for inferential statistics. It discusses testing the population mean when the variance is known or unknown, and for small or large sample sizes. An example is given of testing if the average phosphate in blood is less than 4.8 mg/dl. The key steps of the z-test are outlined as defining hypotheses, calculating the test statistic, determining the p-value using z-tables or software, and rejecting the null hypothesis for low p-values. Formulas are presented for the z-test, t-test, chi-square test, and proportion z-test, along with considerations for each.
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- 1. Today’s objective We have learnt • How do the crystallites arrange in a polycrystalline material • How to represent polycrystal information in stereographic projection • Macro- and micro- texture • The principles of macro or bulk texture measurements by X-ray diffraction • To know the texture measurement procedure by X-ray diffraction
- 2. Second step:Fix the Bragg angle for the peak for which the pole figure is to be measured. -Pole figure is measured with 5-axis goniometer – Schulz Reflection Method -X-ray beam must not be transmitted through the specimen thickness>~0.2 mm First step: Record the X-ray diffraction pattern (normal Bragg scan) to identify the peak position -Deviation of relative intensities in a θ/2θ scan from powder file indicate the presence of texture but it is not possible to identify the texture component by normal Bragg scan Measurement strategy:
- 3. • 2 axes used to set Bragg angle. One has to choose a specific crystallographic plane with /2θ axes, the plane for which the pole figure has be determined. • Third axis tilts specimen plane w.r.t. the focusing plane. This is actually the rotation about an axis perpendicular to the sheet surface (angle ) • Fourth axis spins the specimen about its normal. This rotation is about an orthogonal axis through angle . • Fifth axis oscillates the Specimen under the beam. This is a simple translation – to and fro – it improves the statistical averaging of the texture measurement by increasing the number of grains that are sampled What are the 5 axes of the goniometer?
- 4. • Diffractometer is set in Bragg-Brentano geometry This geometry also implies that the incident beam is divergent and diffracted beam is convergent • The combination of angles by which the sample rotates, leads to irradiation of almost all the crystallites. In Bragg-Brentano Geometry, Eulerian cradle is set in such a way that the circle coincides with the bisector of the angle between incident and diffracted beam, and any direction on the pole figure can be brought parallel to this direction by the two rotations 90- and corresponding to the pole figure coordinates and .
- 5. 1. Source;2. Divergence slit; 3. Narrow horizontal slit; 4. Specimen; 5. Major circle of the goniometer; 6. Receiving slit; 7. Counter. One of the oldest models of X-ray texture goniometer is shown here. The figure clearly depicts all the components of the diffractometer based on Schulz reflection geometry. Figure adapted from: “Introduction to texture” by M. Hatherly and W.B. Hutchinson
- 6. A modern texture goniometer at Indian Institute of Science, Bangalore Mark the difference!!
- 7. • The already set Bragg peak (for example, 111 or 200 for face centred cubic materials) is recorded for all the combinations of angles • Next, the specimen is arranged with RD pointing vertically. This leads to sheet normal bisect angle between incident and diffracted beams, ND coincides with diffracting vector K. In this case, measured intensity comes from the (hkl) planes || sheet plane • The specimen is rotated by = 90o (along Euler Cradle). Tphis leads to RD || K . In this situation, the measured intensity corresponds to RD. This type of rotation gives radial scan of the pole figure. • Next, the specimen is rotated by = 90o (about axis perpendicular to specimen surface). This situation corresponds to TD || K. The diffracted intesity is sampled around the periphery of the pole figure.
- 8. • A fast rotation through + a slow rotation through is equal to the diffracted intensity along the spiral trace Total intensity to the counter • This way for each combination of and , the same Bragg peak (hkl) is recorded. The intensity variation for different angles gives an account of texture present in the material, as shown below. • The intensity is normalised by recording a diffraction pattern from a powder (random) sample. • The intensity ratio is plotted on a polar graph to result in a pole figure.
- 9. Practical Aspects • Typical to measure three PFs for the 3 lowest values of Miller indices. • Why? – A single PF does not uniquely determine orientation(s), texture components because only the plane normal is measured, but not directions in the plane (2 out of 3 parameters). – Multiple PFs required for calculation of Orientation Distribution.
- 10. Questions 1. If you are given two identical looking samples of a metallic material, how will you decide whether this is a single crystal or a polycrystal? If you are given a single crystal and unable to get a particular diffraction pattern, what will you do to locate the same. 2. In the measurement complete (111) pole figure, if the data is recorded at 5 intervals in both and angles, how many times the (111) peaks are recorded? 3. Describe the role of all the axes in a five axis goniometer as used in Schulz reflection geometry. 4. Can you measure the (111) pole figure for a steel samples? If yes, which peak will be fixed for measurement of texture? 5. In the Schulz reflection method, which of the following is correct: (a) the tilt angle enables most of the grains to come under diffraction condition (b) the rotation angle enables most of the grains to come under diffraction condition (c) the tilt angle brings the diffraction vector coincident with the specimen axes alternately (d) the rotation angle brings the diffraction vector coincident with the specimen axes alternately