Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids using techniques like X-ray crystallography. X-ray crystallography works by firing X-rays at crystalline samples and analyzing the diffraction patterns to deduce the positions of atoms in the crystal lattice. Miller indices are used in crystallography to describe planes and directions in crystal lattices, with (hkl) denoting a family of planes and <hkl> denoting a family of directions related by symmetry. The reciprocal lattice represents the Fourier transform of the direct lattice and plays a fundamental role in theories of crystal diffraction.
An Introduction to Crystallography, Elements of crystals crystal systems: Cubic (Isometric) System,Tetragonal System, Orthorhombic System, Hexagonal System; Trigonal System, Monoclinic System, Triclinic System
An Introduction to Crystallography, Elements of crystals crystal systems: Cubic (Isometric) System,Tetragonal System, Orthorhombic System, Hexagonal System; Trigonal System, Monoclinic System, Triclinic System
Starobinsky Inflation and Dark Energy and Dark Matter Effects from Quasicryst...Klee Irwin
The goal of this work on mathematical cosmology and geometric methods in modified gravity theories, MGTs, is to investigate Starobinsky-like inflation scenarios determined by gravitational and scalar field configurations mimicking quasicrystal, QC, like structures. Such spacetime aperiodic QCs are different from those discovered and studied in solid state physics but described by similar geometric methods. We prove that an inhomogeneous and locally anisotropic gravitational and matter field effective QC mixed continuous and discrete "aether" can be modeled by exact cosmological solutions in MGTs and Einstein gravity. The coefficients of corresponding generic off-diagonal metrics and generalized connections depend (in general) on all spacetime coordinates via generating and integration functions and certain smooth and discrete parameters. Imposing additional nonholonomic constraints, prescribing symmetries for generating functions and solving the boundary conditions for integration functions and constants, we can model various nontrivial torsion QC structures or extract cosmological Levi--Civita configurations with diagonal metrics reproducing de Sitter (inflationary) like and other types homogeneous inflation and acceleration phases. Finally, we speculate how various dark energy and dark matter effects can be modeled by off-diagonal interactions and deformations of a nontrivial QC like gravitational vacuum structure and analogous scalar matter fields.
Crystallography and X-ray diffraction (XRD) Likhith KLIKHITHK1
Atoms in materials are arranged into crystal structures and microstructures.
Periodic arrangement of atoms depends strongly on external factors such as temperature, pressure, and cooling rate during solidification. Solid elements and their compounds are classified into amorphous, polycrystalline, and single crystalline materials. The amorphous solid materials are isotropic in nature because their atomic arrangements are not regular and possess the same properties in all directions. In contrast, the crystalline materials are anisotropic because their atoms are arranged in regular and repeated pattern, and their properties vary with direction. The polycrystalline materials are combinations of several crystals of varying shapes and sizes. The properties of polycrystalline materials are strongly dependent on distribution of crystals sizes, shapes, and orientations within the individual crystal. Diffraction pattern or intensities of X-ray diffraction techniques are used for characterizing and probing arrangement of atoms in each unit cell, position of atoms, and atomic spacing angles because of comparative wavelength of X-ray to atomic size.The X-ray diffraction, which is a non-destructive technique, has wide range of material analysis including minerals, metals, polymers, ceramics, plastics, semiconductors, and solar cells. The technique also has wide industry application including aerospace, power generation, microelectronics, and several others. The X-ray crystallography remained a complex field of study despite wide industrial applications.
Fully explained working principle of Powder X-Ray Diffraction, understanding the concepts of different orientation of atomic plane in the crystal. In addition to these, analysis of data using software is also presented in simple method.
Types of crystals & Application of x raykajal pradhan
some basic information:-
A crystal lattice is a 3-D arrangement of unit cells.
Unit cell is the smallest unit of a crystal, By stacking identical unit cells, the entire lattice can be constructed
A crystal’s unit cell dimensions are defined by six numbers, the lengths of the 3 axes, a, b, and c, and the three interaxial angles, α, β and γ.
If a unit cell has the same type of atom at the corners of the unit cell but not also in the middle of the faces nor in the centre of the cell, it is called primitive and given by symbol P
7 types of crystal system details
14 bravis lattice
APPLICATION X-RAY CRYSTALLOGRAPHY
1. Structure of crystals
2. Polymer characterisation
3. State of anneal in metals
4. Particle size determination
a) Spot counting method
b) Broadening of diffraction lines
c) Low-angle scattering
5.Applications of diffraction methods to complexes
a) Determination of cis- trans isomerism
b) Determination of linkage isomerism
6.Miscellaneous applications
The study of crystal geometry helps to understand the behaviour of solids and their
mechanical,
electrical,
magnetic
optical and
Metallurgical properties
An exact solution of einstein equations for interior field of an anisotropic ...eSAT Journals
Abstract
In this paper, an anisotropic relativistic fluid sphere with variable density, which decreases along the radius and is maximum at
the centre, is discussed. Spherically symmetric static space-time with spheroidal physical 3-space is considered. The source is an
anisotropic fluid.
The solution is an anisotropic generalization of the solution discussed by Vaidya and Tikekar [1]. The physical three space
constant time has spheroidal solution. The line element of the solution can be expressed in the form Patel and Desai [2]. The
material density is always positive. The solution efficiently matches with Schwarzschild exterior solution across the boundary. It is
shown that the model is physically reasonable by studying the numerical estimates of various parameters. The density vs radial
pressure relation in the interior is discussed numerically. An anisotropy effect on the redshift is also studied numerically.
Key Words: Cosmology, Anisotropic fluid sphere, Radial pressure, Radial density, Relativistic model.
X ray, invisible, highly penetrating electromagnetic radiation of much shorter wavelength (higher frequency) than visible light. The wavelength range for X rays is from about 10-8 m to about 10-11 m, the corresponding frequency range is from about 3 × 1016 Hz to about 3 × 1019 Hz.
Starobinsky Inflation and Dark Energy and Dark Matter Effects from Quasicryst...Klee Irwin
The goal of this work on mathematical cosmology and geometric methods in modified gravity theories, MGTs, is to investigate Starobinsky-like inflation scenarios determined by gravitational and scalar field configurations mimicking quasicrystal, QC, like structures. Such spacetime aperiodic QCs are different from those discovered and studied in solid state physics but described by similar geometric methods. We prove that an inhomogeneous and locally anisotropic gravitational and matter field effective QC mixed continuous and discrete "aether" can be modeled by exact cosmological solutions in MGTs and Einstein gravity. The coefficients of corresponding generic off-diagonal metrics and generalized connections depend (in general) on all spacetime coordinates via generating and integration functions and certain smooth and discrete parameters. Imposing additional nonholonomic constraints, prescribing symmetries for generating functions and solving the boundary conditions for integration functions and constants, we can model various nontrivial torsion QC structures or extract cosmological Levi--Civita configurations with diagonal metrics reproducing de Sitter (inflationary) like and other types homogeneous inflation and acceleration phases. Finally, we speculate how various dark energy and dark matter effects can be modeled by off-diagonal interactions and deformations of a nontrivial QC like gravitational vacuum structure and analogous scalar matter fields.
Crystallography and X-ray diffraction (XRD) Likhith KLIKHITHK1
Atoms in materials are arranged into crystal structures and microstructures.
Periodic arrangement of atoms depends strongly on external factors such as temperature, pressure, and cooling rate during solidification. Solid elements and their compounds are classified into amorphous, polycrystalline, and single crystalline materials. The amorphous solid materials are isotropic in nature because their atomic arrangements are not regular and possess the same properties in all directions. In contrast, the crystalline materials are anisotropic because their atoms are arranged in regular and repeated pattern, and their properties vary with direction. The polycrystalline materials are combinations of several crystals of varying shapes and sizes. The properties of polycrystalline materials are strongly dependent on distribution of crystals sizes, shapes, and orientations within the individual crystal. Diffraction pattern or intensities of X-ray diffraction techniques are used for characterizing and probing arrangement of atoms in each unit cell, position of atoms, and atomic spacing angles because of comparative wavelength of X-ray to atomic size.The X-ray diffraction, which is a non-destructive technique, has wide range of material analysis including minerals, metals, polymers, ceramics, plastics, semiconductors, and solar cells. The technique also has wide industry application including aerospace, power generation, microelectronics, and several others. The X-ray crystallography remained a complex field of study despite wide industrial applications.
Fully explained working principle of Powder X-Ray Diffraction, understanding the concepts of different orientation of atomic plane in the crystal. In addition to these, analysis of data using software is also presented in simple method.
Types of crystals & Application of x raykajal pradhan
some basic information:-
A crystal lattice is a 3-D arrangement of unit cells.
Unit cell is the smallest unit of a crystal, By stacking identical unit cells, the entire lattice can be constructed
A crystal’s unit cell dimensions are defined by six numbers, the lengths of the 3 axes, a, b, and c, and the three interaxial angles, α, β and γ.
If a unit cell has the same type of atom at the corners of the unit cell but not also in the middle of the faces nor in the centre of the cell, it is called primitive and given by symbol P
7 types of crystal system details
14 bravis lattice
APPLICATION X-RAY CRYSTALLOGRAPHY
1. Structure of crystals
2. Polymer characterisation
3. State of anneal in metals
4. Particle size determination
a) Spot counting method
b) Broadening of diffraction lines
c) Low-angle scattering
5.Applications of diffraction methods to complexes
a) Determination of cis- trans isomerism
b) Determination of linkage isomerism
6.Miscellaneous applications
The study of crystal geometry helps to understand the behaviour of solids and their
mechanical,
electrical,
magnetic
optical and
Metallurgical properties
An exact solution of einstein equations for interior field of an anisotropic ...eSAT Journals
Abstract
In this paper, an anisotropic relativistic fluid sphere with variable density, which decreases along the radius and is maximum at
the centre, is discussed. Spherically symmetric static space-time with spheroidal physical 3-space is considered. The source is an
anisotropic fluid.
The solution is an anisotropic generalization of the solution discussed by Vaidya and Tikekar [1]. The physical three space
constant time has spheroidal solution. The line element of the solution can be expressed in the form Patel and Desai [2]. The
material density is always positive. The solution efficiently matches with Schwarzschild exterior solution across the boundary. It is
shown that the model is physically reasonable by studying the numerical estimates of various parameters. The density vs radial
pressure relation in the interior is discussed numerically. An anisotropy effect on the redshift is also studied numerically.
Key Words: Cosmology, Anisotropic fluid sphere, Radial pressure, Radial density, Relativistic model.
X ray, invisible, highly penetrating electromagnetic radiation of much shorter wavelength (higher frequency) than visible light. The wavelength range for X rays is from about 10-8 m to about 10-11 m, the corresponding frequency range is from about 3 × 1016 Hz to about 3 × 1019 Hz.
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the mean positions of the atoms in the crystal can be determined, as well as their chemical bonds, their crystallographic disorder, and various other information.
Solid state chemistry- laws of crystallography- Miller indices- X ray diffraction- Bragg equation- Spectrophotometer- Determination of interplanar distance- Types of crystal
The crystal structure notes gives the basic understanding about the different structures crystalline materials and their properties and physics of crystals. It also throw light on the basics of crystal diffraction
Crystal Structures and X-Ray Diffraction - Sultan LeMarcslemarc
Report on the investigation of the characteristics of X-rays by measuring the count rate of X-rays reflected off alkali halide crystals at varying angles of incidence and using the principles of Bragg’s law. The experiment probes into crystal structures using X-ray diffractometry and deduces the lattice constants and ionic radii using the Miller index notation. The experiment successfully computes the characteristic wavelengths of Copper and clearly demonstrates the effect of filters on spectrum intensities. The interpretation of Miller indices and diffraction patterns are effectively used to analyse crystalline structures and compare lattice arrangements. By Sultan LeMarc
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
1. CRYSTALLOGRAPHY
• Crystallography is the experimental science of determining the
arrangement of atoms in crystalline solids (see crystal structure). The
word "crystallography" is derived from the Greek words crystallon "cold
drop, frozen drop", with its meaning extending to all solids with some
degree of transparency, and graphein "to write". In July 2012, the United
Nations recognised the importance of the science of crystallography by
proclaiming that 2014 would be the International Year of
Crystallography.[1]
• Before the development of X-ray diffraction crystallography (see
below), the study of crystals was based on physical measurements of
their geometry using using a goniometer [2]. This involved measuring
the angles of crystal faces relative to each other and to
theoretical reference axes (crystallographic axes), and
establishing the symmetry of the crystal in question. The
position in 3D space of each crystal face is plotted on a
stereographic net such as a Wulff net or Lambert net. The pole
to each face is plotted on the net. Each point is labelled with its
Miller index. The final plot allows the symmetry of the crystal to
be established.
2. • X-rays interact with the spatial distribution of electrons in the sample.
• Electrons are charged particles and therefore interact with the total charge distribution of
both the atomic nuclei and the electrons of the sample.
• Neutrons are scattered by the atomic nucleiproduce diffraction patterns with high noise
levels. However, the material can sometimes be treated to substitute deuterium for hydrogen.
• through the strong nuclear forces, but in addition, the magnetic moment of neutrons is non-
zero. They are therefore also scattered by magnetic fields. When neutrons are scattered
from hydrogen-containing materials, they
3. Coordinates in square brackets such as [100] denote a direction vecctor.
Coordinates in a ngle brackets or chevrons such as <100> denote a family of directions
which are related by symmetry operations. In the cubic crystal system for example, <100>
would mean [100], [010], [001] or the negative of any of those directions.
Miller indices in parentheses such as (100) denote a plane of the crystal structure, and
regular repetitions of that plane with a particular spacing. In the cubic system, the normal to the
(hkl)
plane is the direction [hkl], but in lower-symmetry cases, the normal to (hkl) is not parallel
to Indices in curly brackets or braces such as {100} denote a family of planes and their normals.
In cubic materials the symmetry makes them equivalent, just as the way angle brackets denote
a family of directions. In non-cubic materials, <hkl> is not necessarily perpendicular to {hkl}.
4. Some materials that have been analyzed crystallographically, such as proteins, do not occur
naturally as crystals. Typically, such molecules are placed in solution and allowed to slowly
crystallize through vapor diffusion. A drop of solution containing the molecule, buffer, and
precipitants is sealed in a container with a reservoir containing a hygroscopic solution. Water in
the drop diffuses to the reservoir, slowly increasing the concentration and allowing a crystal to
form. If the concentration were to rise more quickly, the molecule would simply precipitate out of
solution, resulting in disorderly granules rather than an orderly and hence usable crystal
• Once a crystal is obtained, data can be collected using a beam of radiation. Although many
universities that engage in crystallographic research have their own X-ray producing
equipment, synchrotrons are often used as X-ray sources, because of the purer and more
complete patterns such sources can generate. Synchrotron sources also have a much higher
intensity of X-ray beams, so data collection takes a fraction of the time normally necessary at
weaker sources.
5. Complementary neutron crystallography
techniques are used to identify the positions
of hydrogen atoms, since X-rays only interact
very weakly with light elements such as
hydrogen.
Producing an image from a diffraction pattern
requires sophisticated mathematics and often
an iterative process of modelling and
refinement. In this process, the
mathematically predicted diffraction patterns
of an hypothesized or "model" structure are
compared to the actual pattern generated by
the crystalline sample.
6. Ideally, researchers make several initial guesses, which through refinement all converge on
the same answer. Models are refined until their predicted patterns match to as great a degree
as can be achieved without radical revision of the model. This is a painstaking process, made
much easier today by computers.
• The mathematical methods for the analysis of diffraction data only apply to patterns, which
in turn result only when waves diffract from orderly arrays. Hence crystallography applies for
the most part only to crystals, or to molecules which can be coaxed to crystallize for the sake
of measurement. In spite of this, a certain amount of molecular information can be deduced
from patterns that are generated by fibers and powders, which while not as perfect as a solid
crystal, may exhibit a degree of order. This level of order can be sufficient to deduce the
structure of simple molecules, or to determine the coarse features of more complicated
molecules. For example, the double-helical structure of DNA was deduced from an X-ray
diffraction pattern that had been generated by a fibrous sample.
7. X-Ray Crystallography
• X-ray crystallography (XRC) is the
experimental science determining the atomic
and molecular structure of a crystal, in which
the crystalline structure causes a beam of
incident X-rays to diffract into many specific
directions. By measuring the angles and
intensities of these diffracted beams, a
crystallographer can produce a three-
dimensional picture of the density of electrons
within the crystal. From this electron density,
the mean positions of the atoms in the crystal
can be determined, as well as their chemical
bonds, their crystallographic disorder, and
various other information.
8. X-ray diffraction is the elastic
scattering of x-ray photons by atoms
in a periodic lattice. The scattered
monochromatic x-rays that are in
phase give constructive interference.
Figure 1 illustrates how diffraction of
x-rays by crystal planes allows one to
derive lattice spacings by using the
Bragg's law.
9. becasue electrons have wave properties they can be different by crystals. electrons will be
diffracted when the angle of incidence, θ on a crystal plane satisfies the bragg equation.
nλ=2 d sin θ
where λ is the wavelength of the electrons, d is the spacing of the crystal planes and is an
interger.
A simple way to derive the bragg equation is as follow. the path difference between electrons
scattered from adjucent crystal planes is 2d sin θ. for constructive interference between the two
scattered beams the difference must be an interger multiple of electron wavelengths, which
gives the bragg equation.
10. • Dual lattice" redirects here. For duals of order-theoretic lattices, see order dual.
• The computer-generated reciprocal lattice of a fictional monoclinic 3D crystal.
• A two-dimensional crystal and its reciprocal lattice
• In physics, the reciprocal lattice represents the Fourier transform of another lattice
(usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented
by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known
as the direct lattice. While the direct lattice exists in real-space and is what one would
commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space
(also known as momentum space or less commonly as K-space, due to the relationship
between the Pontryagin duals momentum and position). The reciprocal of a reciprocal lattice
is the original direct lattice, since the two are Fourier transforms of each other.
11. • The reciprocal lattice plays a very
fundamental role in most analytic
studies of periodic structures,
particularly in the theory of
diffraction. In neutron and X-ray
diffraction, due to the Laue
conditions, the momentum
difference between incoming and
diffracted X-rays of a crystal is a
reciprocal lattice vector. The
diffraction pattern of a crystal can
be used to determine the reciprocal
vectors of the lattice. Using this
process, one can infer the atomic
arrangement of a crystal.
12. • Miller indices form a notation system in crystallography for planes in crystal (Bravais)
lattices.
• In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller
indices. They are written (hkℓ), and denote the family of planes orthogonal to {displaystyle
hmathbf {b_{1}} +kmathbf {b_{2}} +ell mathbf {b_{3}} }h{mathbf {b_{1}}}+k{mathbf
{b_{2}}}+ell {mathbf {b_{3}}}, where {displaystyle mathbf {b_{i}} }{mathbf {b_{i}}} are the
basis of the reciprocal lattice vectors (note that the plane is not always orthogonal to the
linear combination of direct lattice vectors {displaystyle hmathbf {a_{1}} +kmathbf {a_{2}}
+ell mathbf {a_{3}} }h{mathbf {a_{1}}}+k{mathbf {a_{2}}}+ell {mathbf {a_{3}}} because
the reciprocal lattice vectors need not be mutually orthogonal). By convention, negative
integers are written with a bar, as in 3 for −3. The integers are usually written in lowest terms,
i.e. their greatest common divisor should be 1. Miller indices are also used to designate
reflexions in X-ray crystallography.
13. • In this case the integers are not necessarily in lowest terms, and can be thought of as
corresponding to planes spaced such that the reflexions from adjacent planes would have a
phase difference of exactly one wavelength (2π), regardless of whether there are atoms on
all these planes or not.
• There are also several related notations:[1]
• the notation {hkℓ} denotes the set of all planes that are equivalent to (hkℓ) by the symmetry of
the lattice.
• In the context of crystal directions (not planes), the corresponding notations are:
• [hkℓ], with square instead of round brackets, denotes a direction in the basis of the direct
lattice vectors instead of the reciprocal lattice;
14. • similarly, the notation <hkℓ> denotes the set of
all directions that are equivalent to [hkℓ] by
symmetry.
• Miller indices were introduced in 1839 by the
British mineralogist William Hallowes Miller,
although an almost identical system (Weiss
parameters) had already been used by German
mineralogist Christian Samuel Weiss since
1817.[2] The method was also historically
known as the Millerian system, and the indices
as Millerian,[3] although this is now rare.
• The Miller indices are defined with respect to
any choice of unit cell and not only with respect
to primitive basis vectors, as is sometimes
state.
15. • In crystallography, crystal structure is a
description of the ordered arrangement of
atoms, ions or molecules in a crystalline
material.[3] Ordered structures occur from the
intrinsic nature of the constituent particles to
form symmetric patterns that repeat along the
principal directions of three-dimensional space
in matter.
• The smallest group of particles in the material
that constitutes this repeating pattern is the unit
cell of the structure. The unit cell completely
reflects the symmetry and structure of the entire
crystal, which is built up by repetitive translation
of the unit cell along its principal axes. The
translation vectors define the nodes of the
Bravais lattice.
16. • The lengths of the principal axes, or edges, of the unit cell and the
angles between them are the lattice constants, also called lattice
parameters or cell parameters. The symmetry properties of the
crystal are described by the concept of space groups.[3] All
possible symmetric arrangements of particles in three-
dimensional space may be described by the 230 space groups.
• The crystal structure and symmetry play a critical role in
determining many physical properties, such as cleavage,
electronic band structure, and optical transparency.
17. • Finally, we would like to emphasize that despite the
profusion of increasingly sophisticated methods of
crystallographic analysis, the simple optical microscope
remains an extraordinarily discerning and comparatively
inexpensive device for the study of crystalline matter and
its imperfections.