# Burgers vector

The Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted b, that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice.[1]

Burgers vectors

The vector's magnitude and direction is best understood when the dislocation-bearing crystal structure is first visualized without the dislocation, that is, the perfect crystal structure. In this perfect crystal structure, a rectangle whose lengths and widths are integer multiples of "a" (the unit cell length) is drawn encompassing the site of the original dislocation's origin. Once this encompassing rectangle is drawn, the dislocation can be introduced. This dislocation will have the effect of deforming, not only the perfect crystal structure, but the rectangle as well. The said rectangle could have one of its sides disjoined from the perpendicular side, severing the connection of the length and width line segments of the rectangle at one of the rectangle's corners, and displacing each line segment from each other. What was once a rectangle before the dislocation was introduced is now an open geometric figure, whose opening defines the direction and magnitude of the Burgers vector. Specifically, the breadth of the opening defines the magnitude of the Burgers vector, and, when a set of fixed coordinates is introduced, an angle between the termini of the dislocated rectangle's length line segment and width line segment may be specified.

The direction of the vector depends on the plane of dislocation, which is usually on the closest-packed plane of unit cell. The magnitude is usually represented by the equation:

$\|\mathbf{b}\|\ = \textstyle\frac{a}{2}\sqrt{h^2+k^2+l^2}$

where a is the unit cell length of the crystal, ||b|| is the magnitude of Burgers vector and h, k, and l are the components of the Burgers vector, b = <h k l>. In most metallic materials, the magnitude of the Burgers vector for a dislocation is of a magnitude equal to the interatomic spacing of the material, since a single dislocation will offset the crystal lattice by one close-packed crystallographic spacing unit.

In edge dislocations, the Burgers vector and dislocation line are at right angles to one another. In screw dislocations, they are parallel.[2]

The Burgers vector is significant in determining the yield strength of a material by affecting solute hardening, precipitation hardening and work hardening.

In materials science/engineering it is often useful to know the magnitude of the Burger’s vector in metres. This is easily done for BCC and FCC lattice materials using the previously mentioned equation as only the slip system and unit cell length 'a' need to be known. So for a FCC lattice where a = 2 R (2)^1/2 and the slip system is <110> the length of the Burger’s vector is simply b = 2 R, where R is the atomic radius. The HCP system is even simpler with the Burger’s vector simply equal to the unit cell length a, i.e. b =a, where a = 2 R for the HCP system. Hence the Burger’s vector of titanium would be 0.29 nm long.

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## References

1. ^ Callister, William D. Jr. "Fundamentals of Materials Science and Engineering," John Wiley & Sons, Inc. Danvers, MA. (2005)
2. ^ Kittel, Charles, "Introduction to Solid State Physics," 7th edition, John Wiley & Sons, Inc, (1996) pp 592-593.
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