Fractal Theory
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Fractal theory describes objects that have self-similar patterns across different scales. Fractals are defined as geometric shapes that can be split into parts, each of which is a reduced-size copy of the whole. They often have non-integer dimensions and fine structures at arbitrarily small scales. Common examples include natural shapes like coastlines as well as mathematical constructs like the Cantor set and Koch curve. Fractals are created by recursive processes that repeat a simple pattern on decreasingly smaller scales. Their dimension can be calculated from how their size changes with scaling factor.
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The accurate determination of peak positions.
Determination of the unit cell parameters from the peak positions.
• Bragg’s Law
• Interplanar d-spacing
• Systematic Absences
Bragg's Law
When x-rays are scattered from a crystal lattice, peaks of scattered intensity are observed
which correspond to the following conditions:
1. The angle of incidence = angle of scattering.
2. The pathlength difference is equal to an integer number of wavelengths.
The condition for maximum intensity contained in Bragg's law above allow us to calculate
details about the crystal structure, or if the crystal structure is known, to determine the
wavelength of the x-rays incident upon the crystal.Bravais lattices
Crystals possess a regular, repetitive internal
structure. The concept of symmetry describes the
repetition of structural features.
Symmetries are most frequently used to classify the
different crystal structures. In general one can
generate 14 basic crystal structures through
symmetries. These are called Bravais lattices. Any
crystal structures can be reduced to one
of these 14 Bravias lattices.Powder Diffraction: Indexing
Indexing is the process of determining the unit cell parameters from the peak
positions. To index a powder diffraction pattern it is necessary to assign Miller
indices, hkl, to each peak.
A diffraction pattern cannot be analyzed until it has been indexed. It is always the first
step in analysis.
Unfortunately it is not just the simple reverse of calculating peak positions from the
unit cell dimensions and wavelength.
We will show how one can manually index diffraction patterns of high symmetry
structures. For lower symmetry structures (monoclinic, triclinic) it is
usually necessary to use a computer algorithm. This is called Autoindexing.
The information in an XRD pattern is a direct result of two things:
1. The size and shape of the unit cells, which determine the relative positions of
the diffraction peaks.
2. Atomic positions within the unit cell, which determine the relative intensities of
the diffraction peaks
Manual indexing of cubic unit cells is a reasonably
straightforward process.
• Tetragonal, trigonal and hexagonal cells can also be indexed
manually.
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Lec8
Lec8Rùa Con Con Rùa 1. Miller indices arise from finding the plane intercepts with the crystal axes, taking their reciprocals, and reducing to integers.
2. Miller indices represent the components of the plane equation and each set of indices corresponds to a set of parallel planes.
3. Important crystal planes and forms are often given special names based on their symmetry and Miller indices. The reciprocal lattice is used to relate planes and directions in a crystal lattice.
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2. Bragg's law relates the wavelength of X-rays, the distance between lattice planes, and the diffraction angle, and describes the conditions for constructive interference of X-rays reflected from crystal planes.
3. The reciprocal lattice is a mathematical construct where vectors normal to real lattice planes radiate from the origin at distances that are the reciprocal of the corresponding interplanar spacings, allowing the diffraction pattern to provide information about the crystal structure.
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Fractal Theory
- 1. Fractal Theory BY : EHSAN HAMZEI - 810392121
- 2. Fractal? 1 A fractal is generally "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole“ – (B. Mandlebrot). This property is called self-similarity.
- 3. Graph Theory History 2 Lewis Fry Richardson, “War & and Common Border” Benoit Mandelbrot, “How Long Is the Coast of Britain”
- 4. Properties 3 A fractal often has the following features: It has a fine structure at arbitrarily small scales. It is self-similar (at least approximately). It is too irregular to be easily described in traditional Euclidean geometric. It has a dimension which is non-integer. It has a simple and recursive definition.
- 5. Examples of Fractals: 4 Natural: Mathematical:
- 6. Example (Creating) 5 Example: Cantor set. The Cantor set is obtained by deleting recursively the 1/3 middle part of a set of line segments.
- 7. Example (Creating) 6 Properties of the Cantor set C: C has a structure at arbitrarily small scale. C is self-similar The dimension of C is not an integer! Length = limn->∞((2/3)n)=0. (But infinite points…)
- 8. Example 2 (Creating) 7 The Koch curve is obtained as follows: start with a line segment S0. delete the middle 1/3 part of S0 and replace it with two other 2- sides. Subsequent stages are generated recursively by the same rule But >> limit K=S∞ !!
- 9. Fractal Dimension 8 A mathematical description of dimension is based on how the "size" of an object behaves as the linear dimension increases. In 1D, we need r segments of scale r to equal the original segment. In 2D, we need r2 squares of scale r to equal the original square. In 3D, we need r3 cubes of scale r to equal the original cube.
- 10. Fractal Dimension 9 This relationship between the dimension d, the scaling factor r and the number m of rescaled copies required to cover the original object is thus: Rearranging the above equation:
- 11. Back to Examples M = 2 R = 3 D = ln(M)/ln(R) = ln2/ln3 = 0.63 10
- 12. Back to Examples 11 M = 4 R = 3 D = ln4/ln3 = 1.26
- 13. References 11 Didier Gonze, “Fractals: theory and applications” Lecture Note Libre de Bruxelles University Mandelbort, “Fractal Theory”.
- 14. Thanks 12