How to Draw 1 0 0 Plane
Lattice Planes and Miller Indices (all content)
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Contents
Main pages
Boosted pages
Aims
On completion of this TLP you should:
- Empathise 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 cell;
- Be able to visualise and describe a aeroplane when given its Miller indices;
- Be aware of how knowledge of lattice planes and their Miller indices can help to sympathise other concepts in materials science.
Before you start
You lot should sympathize the concepts of a lattice, unit of measurement cell, crystal axes, vrystal system and the variations, primitive, FCC, BCC which make upwards the Bravais lattice.
You might also like to look at the TLP on Diminutive Scale Construction of Materials.
Yous should sympathise the concepts of vectors and planes in mathematics.
Introduction
Miller Indices are a method of describing the orientation of a plane or gear up of planes within a lattice in relation to the unit prison cell. They were adult past William Hallowes Miller.
These indices are useful in understanding many phenomena in materials science, such every bit explaining the shapes of unmarried crystals, the grade of some materials' microstructure, the interpretation of 10-ray diffraction patterns, and the movement of a dislocation, which may determine the mechanical properties of the cloth.
How to index a lattice aeroplane
The side by side three animations take you through the basics of how to alphabetize a aeroplane. Click "Start" to begin each animation, then navigate through the pages using the buttons at the lesser right.
Parallel lattice planes
This blitheness explains the relationships between parallel planes and their indices. Click "Start" to begin and utilise the buttons at the bottom right to navigate through the pages.
Lattice planes tin can be represented by showing the trace of the planes on the faces of 1 or more than unit cells. The diagram shows the trace of the (23) planes on a cubic unit cell.
How to draw a lattice plane
Bracket Conventions
In crystallography there are conventions equally to how the indices of planes and directions are written. When referring to a specific plane, "round" brackets are used:
(hkl)
When referring to a gear up of planes related past symmetry, then "curly" brackets are used:
{hkl}
These might exist the (100) blazon planes in a cubic system, which are (100), (010), (001), (00),(00) and (00) . These planes all "await" the aforementioned and are related to each other past the symmetry elements present in a cube, hence their unlike indices depend only on the way the unit jail cell axes are divers. That is why it useful to consider the equivalent (010) ready of planes.
Directions in the crystal can be labelled in a like style. 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 as well be symmetrically equivalent, in which case a set of directions are written with "triangular" brackets:
<UVW>
Vectors and Planes
It may seem, after because cubic systems, that whatever lattice plane (hkl) has a normal direction [hkl]. This is not ever the case, equally directions in a crystal are written in terms of the lattice vectors, which are not necessarily orthogonal, or of the same magnitude. A uncomplicated instance is the example of in the (100) airplane of a hexagonal system, where the direction [100] is actually at 120° (or 60° ) to the plane. The normal to the (100) plane in this case is [210]
VR rotating image
Weiss Zone Law
The Weiss zone law states that:
If the direction [UVW] lies in the airplane (hkl), then:
hU +kV +lW = 0
In a cubic organisation this is exactly coordinating to taking the scalar product of the direction and the aeroplane normal, so that if they are perpendicular, the angle between them, θ, is 90° , and then cosθ = 0, and the direction lies in the plane. Indeed, in a cubic system, the scalar product tin can be used to determine the angle between a direction and a plane.
However, the Weiss zone police is more general, and tin can be shown to work for all crystal systems, to determine if a direction lies in a plane.
From the Weiss zone law the following rule can be derived:
The direction, [UVW], of the intersection of (h 1 yard 1 l one) and (h 2 k 2 fifty 2) is given past:
U =k ane l ii −k ii l 1
Five =fifty ane h two −l two h i
Due west =h 1 k two −h 2 k 1
As it is derived from the Weiss zone law, this relation applies to all crystal systems, including those that are non orthogonal.
Examples of lattice planes
The (100), (010), (001), (00), (00) and (00) planes class the faces of the unit 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 back of the unit of measurement cell, both indices refer to the same family of planes, every bit explained in the blitheness Parallel lattice planes. Information technology should be noted that these half dozen planes are not all symmetrically related, equally they are in the cubic organization.
The (101), (110), (011), (10), (i0) and (01) planes form the sections through the diagonals of the unit prison cell, along with those planes whose indices are the negative of these. In the paradigm the planes are shown in a different triclinic unit of measurement prison cell.
The (111) type planes in a confront centred cubic lattice are the close packed planes.
Click and drag on the image below to see how a shut packed (111) aeroplane intersects the fcc unit of measurement cell.
VR rotating image
Draw your own lattice planes
This simulation generates images of lattice planes. To see a plane, enter a set of Miller indices (each index between 6 and −6), the numbers separated by a semi-colon, then click "view" or printing enter.
Applied Uses
An understanding of lattice planes is required to explain the form of many microstructural features of many materials. The faces of single crystals form on certain lattice planes, typically those with depression indices.
In a similar way, the form of the microstructure in a polycrystalline material is strongly dependent on lattice planes. When a new phase of material forms, the surfaces tend to be aligned on low index planes, as with single crystals. When a new solid phase is formed in another solid, the interfaces occur on along the most energetically favourable planes, where the two lattices are most coherent. This leads to plate-similar precipitates forming, at specific angles to each other.
Section through an Fe-Ni meteorite showing plates at 60° to each other
DoITPoMS standard terms of use
I method of plastic deformation is by dislocation slip. Understanding lattice planes, and directions is essential to explain why dislocations move, combine and tangle in the observed way. More data can be obtained in the TLP - 'Skid in Single Crystals'
A scanning electron micrograph of a single crystal of cadmium
deforming by dislocation slip on 100 planes, forming steps
on the surface
DoITPoMS standard terms of use
Twinning is where a part of the crystal is "flipped" to course a mirror image of the rest of the crystal, reflected in a detail lattice aeroplane. 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 material. By interpreting the diffraction patterns equally reflections from lattice planes in the material, the construction tin exist determined. More information tin be obtained in the TLP - 'X-ray diffraction '
Apparatus for carrying out single crystal 10-ray diffraction.
Worked examples
Example A
The effigy below is a scanning electron micrograph of a niobium carbide dendrite in a Fe-34wt%Cr-5wt%Nb-four.5wt%C alloy. Niobium carbide has a confront 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 i another (one set pointing out of the plane of the image, the other ii sets, to a good approximation, lying in the plane of the image), and each arm has a pyramidal shape at its end. Information technology is known that the crystallographic directions along the dendrite artillery correspond to the < 100 > lattice directions, and that the management ab labelled on the micrograph is [ten] .
sourced from Dendritic Solidification
1) If point c (non shown) lies on the axis of this dendrite arm, what is the management cb ? Index face C , marked on the micrograph.
The diagram shows the [10] direction in cerise. The [100] management is a < 100 > type direction that forms the observed acute angle with ab, and tin be used as cb. Of the < 100 > type directions, we could likewise have used [00] .
Using a correct handed set of axes, we then have z-centrality pointing out of the plane of the prototype, the x-axis pointing along the direction cb, and the y-axis pointing towards the top left of the image.
Confront C must incorporate the direction cb, and its normal must point out of the aeroplane of the paradigm. Therefore confront C is a (001) plane.
2) The four faces which lie at the stop of each dendrite arm have normals which all make the same bending with the direction of the arm. Observing that faces A and B marked on the micrograph both incorporate the direction ab , and noting the general directions forth which the normals to these faces indicate, 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 confront B has a negative k alphabetize.
The morphology of the ends of the arms is that of half an octahedron, suggesting that the faces are (111) type planes. This would make face up A, in green, a (111) airplane, and face B, in blue, a (ane1) plane. As required, they both incorporate the [10] management, in ruby.
Example B
1) Work out the common management betwixt the (111) and (001) in a triclinic unit of measurement cell.
The relation derived from the Weiss zone law in the section Vectors and planes states that:
The management, [UVW], of the intersection of (h ane k one fifty 1) and (h 2 g 2 l ii) is given past:
U =k 1 fifty 2 −m 2 l 1
5 =fifty one h two −fifty 2 h i
Due west =h 1 g 2 −h ii k 1
We tin use this relation as it applies to all crystal systems, including the triclinic system that we are considering.
We take h 1 = 1, k 1 = 1, l 1 = 1
and h 2 = 0, k 2 = 0, 50 2 = 1
Therefore
U = (1 × 1) - (0 × i) = 1
V = (one × 0) - (one × i) = −i
Due west = (i × 0) - (0 × 1) = 0.
And so the mutual management is:
[10] .
This is shown in the image below:
If we had defined the (001) airplane as (h 1 k i l 1) and the (110) aeroplane as (h ii k 2 50 2) then the resulting direction would have been, [10] i.e. anti-parallel to [10] .
2) Use the Weiss zone constabulary to testify that the direction [10] lies in the (111) plane.
We have U =one, V =−ane, West =0,
and h = 1, k = 1, l = one.
hU +kV +lW = (ane × 1) + (1 × −ane) + (ane × 0) = 0
Therefore the direction [10] lies in the airplane (111).
Summary
Miller Indices are the convention used to label lattice planes. This mathematical description allows us to define accurately, planes within a crystal, and quantitatively analyse many problems in materials science.
Questions
Game: Identify the planes
Quick questions
Yous should be able to answer these questions without besides much difficulty subsequently studying this TLP. If not, and then you should get through it again!
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Which 1 of the following statements about the (4) and (iione) planes is faux?
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Does the [one2] direction lie in the (30) aeroplane?
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When writing the index for a set of symmetrically related planes, which type of brackets should be used?
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Which of the <110> type directions lie in the (112) plane?
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What is the common direction betwixt the (1 ) and (33) planes?
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Which prepare of planes in a cubic-close-packed construction (such as copper) is shut packed?
Open-ended questions
The following questions are not provided with answers, but intended to provide food for idea and points for further discussion with other students and teachers.
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Exercise sketching some lattice planes. Make sure you can draw the {100}, {110} and {111} blazon planes in a cubic system.
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Depict the trace of all the (21) planes intersecting a block 2 × 2 × ii block of orthorhombic (a ≠ b ≠ c, α = β = γ = 90°) unit of measurement cells.
-
Sketch the organization of the lattice points on a {111} type plane in a confront centred cubic lattice. Do the same for a {110} type aeroplane in a trunk centred cubic lattice. Compare your drawings. Why exercise you think the {110} type planes are frequently described every bit the "most close packed" planes in bcc?
Going farther
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.
Affiliate 5 covers lattice planes and directions. The residual of the book gives an introduction to crystallography and diffraction in full general.
[3] B.D. Cullity, Elements of X-Ray Diffraction , Prentice Hall, 2003.
Covers X-Ray diffraction in detail. Affiliate ii covers the crystallography required for this.
[four] C. Kittel, Introduction to Solid State Physics, John Wiley and Sons, 2004.
Affiliate 1 covers crystallography. The book and so goes on to cover a broad range of more advanced solid land scientific discipline.
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Bookish consultant: Noel Rutter (Academy of Cambridge)
Content development: Peter Marchment
Photography and video: Brian Barber and Carol Best
Web development: David Brook and Lianne Sallows
Translation: Jing Qiu, Kansong Chen, Ana Tabalan-Bailey, Marta Sanchez, Juan Vilatela
DoITPoMS is funded past the Great britain Center for Materials Pedagogy and the Section of Materials Scientific discipline and Metallurgy, Academy of Cambridge
Additional support 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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