Which Specific Members of the âå¸â¨110ã¢å¸â© Family of Directions Lie Within the (111) Plane?
Lattice Planes and Miller Indices (all content)
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Contents
Master pages
Boosted pages
Aims
On completion of this TLP y'all should:
- Understand the concept of a lattice airplane;
- Be able to determine the Miller indices of a plane from its intercepts with the edges of the unit jail cell;
- Be able to visualise and depict a airplane when given its Miller indices;
- Be aware of how knowledge of lattice planes and their Miller indices can help to understand other concepts in materials science.
Before you start
Y'all should empathize the concepts of a lattice, unit prison cell, crystal axes, vrystal organisation and the variations, primitive, FCC, BCC which brand up the Bravais lattice.
You might as well like to look at the TLP on Diminutive Scale Structure of Materials.
Yous should understand the concepts of vectors and planes in mathematics.
Introduction
Miller Indices are a method of describing the orientation of a plane or set of planes inside a lattice in relation to the unit jail cell. They were adult by William Hallowes Miller.
These indices are useful in agreement many phenomena in materials science, such as explaining the shapes of single crystals, the form of some materials' microstructure, the interpretation of X-ray diffraction patterns, and the movement of a dislocation, which may determine the mechanical properties of the cloth.
How to index a lattice plane
The next three animations take you through the basics of how to alphabetize a airplane. Click "Start" to brainstorm each animation, and then navigate through the pages using the buttons at the bottom right.
Parallel lattice planes
This blitheness explains the relationships between parallel planes and their indices. Click "Starting time" to brainstorm and use the buttons at the bottom right to navigate through the pages.
Lattice planes can be represented by showing the trace of the planes on the faces of one or more unit of measurement cells. The diagram shows the trace of the (2three) planes on a cubic unit jail cell.
How to draw a lattice aeroplane
Bracket Conventions
In crystallography at that place are conventions every bit to how the indices of planes and directions are written. When referring to a specific aeroplane, "circular" brackets are used:
(hkl)
When referring to a prepare of planes related by symmetry, then "curly" brackets are used:
{hkl}
These might be the (100) type planes in a cubic system, which are (100), (010), (001), (00),(00) and (00) . These planes all "expect" the same and are related to each other by the symmetry elements present in a cube, hence their different indices depend only on the way the unit jail cell axes are divers. That is why it useful to consider the equivalent (010) set of planes.
Directions in the crystal can be labelled in a similar fashion. These are effectively vectors written in terms of multiples of the lattice vectors a, b, and c. They are written with "square" brackets:
[UVW]
A number of crystallographic directions tin also be symmetrically equivalent, in which instance a set of directions are written with "triangular" brackets:
<UVW>
Vectors and Planes
It may seem, after because cubic systems, that any lattice airplane (hkl) has a normal direction [hkl]. This is non always the example, as directions in a crystal are written in terms of the lattice vectors, which are non necessarily orthogonal, or of the same magnitude. A elementary example is the example of in the (100) aeroplane of a hexagonal system, where the direction [100] is really at 120° (or 60° ) to the airplane. The normal to the (100) plane in this case is [210]
VR rotating image
Weiss Zone Law
The Weiss zone police force states that:
If the direction [UVW] lies in the plane (hkl), and so:
hU +kV +lW = 0
In a cubic organization this is exactly analogous to taking the scalar product of the direction and the aeroplane normal, and so that if they are perpendicular, the angle betwixt them, θ, is 90° , so cosθ = 0, and the direction lies in the aeroplane. Indeed, in a cubic system, the scalar product can be used to determine the angle between a direction and a plane.
Nevertheless, the Weiss zone constabulary is more full general, and can be shown to work for all crystal systems, to determine if a direction lies in a plane.
From the Weiss zone police force the following rule can exist derived:
The direction, [UVW], of the intersection of (h ane chiliad 1 l 1) and (h two k two l 2) is given past:
U =k one fifty two −k 2 l 1
V =50 1 h ii −l 2 h i
W =h 1 k two −h 2 g one
As it is derived from the Weiss zone police force, this relation applies to all crystal systems, including those that are not orthogonal.
Examples of lattice planes
The (100), (010), (001), (00), (00) and (00) planes form the faces of the unit prison cell. Here, they are shown as the faces of a triclinic (a ≠ b ≠ c, α ≠β ≠γ) unit cell . Although in this image, the (100) and (00) planes are shown as the front and dorsum of the unit cell, both indices refer to the same family of planes, as explained in the animation Parallel lattice planes. It should be noted that these six planes are not all symmetrically related, as they are in the cubic system.
The (101), (110), (011), (10), (one0) and (01) planes grade the sections through the diagonals of the unit of measurement cell, along with those planes whose indices are the negative of these. In the image the planes are shown in a different triclinic unit prison cell.
The (111) type planes in a face centred cubic lattice are the shut packed planes.
Click and drag on the image below to see how a close packed (111) airplane intersects the fcc unit cell.
VR rotating image
Describe your own lattice planes
This simulation generates images of lattice planes. To run across a airplane, enter a set of Miller indices (each index between vi and −six), the numbers separated by a semi-colon, and then click "view" or printing enter.
Applied Uses
An agreement of lattice planes is required to explicate the form of many microstructural features of many materials. The faces of single crystals grade on certain lattice planes, typically those with low indices.
In a like fashion, the class of the microstructure in a polycrystalline textile is strongly dependent on lattice planes. When a new phase of material forms, the surfaces tend to be aligned on depression alphabetize planes, as with single crystals. When a new solid phase is formed in some other solid, the interfaces occur on forth the most energetically favourable planes, where the ii lattices are well-nigh coherent. This leads to plate-like precipitates forming, at specific angles to each other.
Section through an Fe-Ni meteorite showing plates at sixty° to each other
DoITPoMS standard terms of use
1 method of plastic deformation is past dislocation slip. Agreement lattice planes, and directions is essential to explain why dislocations motion, combine and tangle in the observed manner. More information can be obtained in the TLP - 'Skid in Single Crystals'
A scanning electron micrograph of a single crystal of cadmium
deforming past dislocation sideslip on 100 planes, forming steps
on the surface
DoITPoMS standard terms of use
Twinning is where a part of the crystal is "flipped" to form a mirror image of the residuum of the crystal, reflected in a detail lattice airplane. This can either occur in annealing, or as a mechanism of plastic deformation.
Annealing twins in brass (DoITPoMS micrograph library)
X-ray diffraction is a method of determining the crystal structure of a cloth. Past interpreting the diffraction patterns every bit reflections from lattice planes in the fabric, the structure can exist adamant. More information can be obtained in the TLP - 'X-ray diffraction '
Apparatus for conveying out single crystal Ten-ray diffraction.
Worked examples
Example A
The figure below is a scanning electron micrograph of a niobium carbide dendrite in a Fe-34wt%Cr-5wt%Nb-4.5wt%C alloy. Niobium carbide has a face centred cubic lattice. The specimen has been deep-etched to remove the surrounding matrix chemically and reveal the dendrite. The dendrite has 3 sets of "arms" which are orthogonal to one another (ane set pointing out of the plane of the image, the other two sets, to a good approximation, lying in the plane of the image), and each arm has a pyramidal shape at its cease. It is known that the crystallographic directions along the dendrite arms correspond to the < 100 > lattice directions, and that the direction ab labelled on the micrograph is [10] .
sourced from Dendritic Solidification
1) If bespeak c (not shown) lies on the centrality of this dendrite arm, what is the direction cb ? Alphabetize face C , marked on the micrograph.
The diagram shows the [10] management in red. The [100] direction is a < 100 > type direction that forms the observed astute bending with ab, and can be used as cb. Of the < 100 > blazon directions, nosotros could also accept used [00] .
Using a right handed set of axes, we then have z-axis pointing out of the plane of the prototype, the x-centrality pointing forth the management cb, and the y-axis pointing towards the elevation left of the image.
Face C must comprise the direction cb, and its normal must point out of the aeroplane of the image. Therefore face C is a (001) plane.
2) The four faces which lie at the end of each dendrite arm have normals which all make the aforementioned angle with the direction of the arm. Observing that faces A and B marked on the micrograph both contain the direction ab , and noting the full general directions along which the normals to these faces signal, index faces A and B .
Both faces A and B have normals pointing in the positive x and z directions, i.e. positive h and 50 indices. Face A has a positive k index, and face B has a negative k index.
The morphology of the ends of the arms is that of half an octahedron, suggesting that the faces are (111) type planes. This would brand face A, in green, a (111) plane, and face B, in blue, a (anei) plane. As required, they both contain the [ten] management, in cerise.
Example B
1) Work out the common direction betwixt the (111) and (001) in a triclinic unit cell.
The relation derived from the Weiss zone law in the section Vectors and planes states that:
The direction, [UVW], of the intersection of (h i k 1 l one) and (h two k 2 50 2) is given by:
U =k i l 2 −k two fifty i
V =l 1 h 2 −l 2 h i
Due west =h i k 2 −h 2 k ane
We can use this relation equally it applies to all crystal systems, including the triclinic organisation that we are because.
We have h 1 = 1, thou 1 = 1, l one = one
and h 2 = 0, yard ii = 0, l two = i
Therefore
U = (1 × 1) - (0 × ane) = i
V = (one × 0) - (1 × 1) = −1
Due west = (1 × 0) - (0 × i) = 0.
And so the common direction is:
[10] .
This is shown in the image below:
If we had defined the (001) plane equally (h 1 thousand 1 l one) and the (110) plane as (h 2 thou 2 l 2) then the resulting direction would take been, [ten] i.east. anti-parallel to [one0] .
2) Employ the Weiss zone constabulary to prove that the direction [10] lies in the (111) plane.
We accept U =1, Five =−1, Westward =0,
and h = 1, k = 1, l = 1.
hU +kV +lW = (ane × 1) + (1 × −ane) + (1 × 0) = 0
Therefore the direction [10] lies in the plane (111).
Summary
Miller Indices are the convention used to label lattice planes. This mathematical clarification allows us to define accurately, planes within a crystal, and quantitatively analyse many bug in materials science.
Questions
Game: Identify the planes
Quick questions
You should be able to answer these questions without too much difficulty later on studying this TLP. If not, and then y'all should get through it again!
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Which i of the post-obit statements about the (4) and (21) planes is false?
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Does the [12] direction lie in the (30) plane?
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When writing the index for a fix of symmetrically related planes, which type of brackets should be used?
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Which of the <110> blazon directions lie in the (112) aeroplane?
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What is the common direction between the (1 ) and (33) planes?
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Which fix of planes in a cubic-close-packed structure (such every bit copper) is shut packed?
Open-ended questions
The following questions are not provided with answers, but intended to provide nutrient for thought and points for further discussion with other students and teachers.
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Practice sketching some lattice planes. Brand certain you lot tin can describe the {100}, {110} and {111} type planes in a cubic organisation.
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Depict the trace of all the (21) planes intersecting a block ii × ii × 2 block of orthorhombic (a ≠ b ≠ c, α = β = γ = 90°) unit of measurement cells.
-
Sketch the arrangement of the lattice points on a {111} blazon plane in a confront centred cubic lattice. Practise the same for a {110} blazon plane in a torso centred cubic lattice. Compare your drawings. Why practice you think the {110} type planes are often described as the "most shut packed" planes in bcc?
Going further
Books
[1] D. McKie and C. McKie, Crystalline Solids , Thomas Nelson and Sons, 1974.
A very comprehensive crystallography text.
[2] C. Hammond, The Basics of Crystallography and Diffraction , Oxford, 2001.
Chapter 5 covers lattice planes and directions. The rest of the book gives an introduction to crystallography and diffraction in general.
[3] B.D. Cullity, Elements of 10-Ray Diffraction , Prentice Hall, 2003.
Covers Ten-Ray diffraction in detail. Chapter 2 covers the crystallography required for this.
[4] C. Kittel, Introduction to Solid State Physics, John Wiley and Sons, 2004.
Chapter 1 covers crystallography. The book then goes on to cover a wide range of more advanced solid state science.
Cómo indexar un plano de una red
Las siguientes tres animaciones muestran los fundamentos básicos para calcular los parámetros del red. Haz click en "Inicio" para que comience cada animación, y luego navega a través de las páginas usando los botones situados en la parte inferior derecha.
如何标注一个晶格面
以下的三个动画 课程将让你了解关于标注晶格面的基本知识。点击'开始'来开始每个动画课程，然后用右下角的按钮来进入下一页。
Как обозначать плоскость кристаллической решётки
Следующие три анимации покажут основы того, как обозначать плоскость. Нажмите кнопку "Пуск", чтобы запустить каждую из анимаций, а затем управляйте анимацией с помощью кнопок, расположенных в правом нижнем углу.
Academic consultant: Noel Rutter (University of Cambridge)
Content development: Peter Marchment
Photography and video: Brian Hairdresser and Carol Best
Spider web development: David Brook and Lianne Sallows
Translation: Jing Qiu, Kansong Chen, Ana Tabalan-Bailey, Marta Sanchez, Juan Vilatela
DoITPoMS is funded past the United kingdom of great britain and northern ireland Centre for Materials Teaching and the Department of Materials Science and Metallurgy, University of Cambridge
Boosted back up for the development of this TLP came from the Worshipful Company of Armourers and Brasiers'
Source: https://www.doitpoms.ac.uk/tlplib/miller_indices/printall.php
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