Mechanical Behavior · Dislocations
Dislocations and the Burgers vector, from scratch
Start with picture 1: press Play and watch one extra half-plane of atoms carry the top of the crystal across by one atom spacing.
1. Metals slip one row of bonds at a time
A dislocation is a line where the crystal has slipped on one side and not yet on the other. Here it is an edge dislocation: an extra half-plane of atoms (coral) ends at the slip plane.
Settings
Numbers: b from Frost and Ashby (1982); G is the modulus for a dislocation in the real crystal (Frost and Ashby for Cu and Al, worked out the same way for Fe, see the theory below); measured critical resolved shear stress (CRSS) of pure single crystals at room temperature from Dieter (1988). The picture is a simple square lattice; the numbers are for the real metal.
Try it: copper, τ = 1 MPa, press Play: the dislocation crosses and leaves a step one b high on the right. Switch to "Whole plane at once" at the same stress: the top barely moves. It only slides once τ reaches about 1400 MPa.
2. Draw a Burgers circuit and find b
Walk atom to atom around the dislocation until you are back where you started. Take the same steps in a perfect crystal: you do not get back. The gap from the finish F to the start S is the Burgers vector b.
Settings
Rule used here (FS/RH, as in Hull and Bacon and in Hirth and Lothe): the line direction t points out of the screen (edge) or up (screw), the circuit runs anticlockwise when you look along minus t, it is closed in the real crystal, and b points from F to S in the perfect crystal.
Try it: press "Draw it for me": b is one atom step, at right angles to the line (edge). Now click four atoms that do not surround the ⊥: the circuit closes in both crystals and b = 0. Switch to Screw: the same kind of loop climbs one layer, so b lies along the line.
3. The force on a dislocation (Peach-Koehler)
A stress pushes on a dislocation line with a force per unit length F = (σ·b) × t. The part in the slip plane moves it by glide. The part out of the plane (climb) needs atoms to diffuse.
Settings
Axes: t along z, the glide direction x lies in the slip plane at right angles to t, y is the slip-plane normal. Tension is positive.
Try it: edge (90°) with τ at 90°: glide force is largest. Turn τ to 0° (along the line): the glide force drops to zero. Now set the character to 0° (screw): the same τ along the line now pushes the line sideways, at right angles to b. Move σxx: only the climb force changes.
4. The stress around an edge dislocation, and where solutes go
The extra half-plane squeezes the crystal above the slip plane and stretches it below. Atoms larger than the host lower their energy on the stretched side, smaller ones on the squeezed side.
Settings
Isotropic elasticity, b along +x, t out of the screen, extra half-plane above. Poisson's ratio from Callister, Table 6.1.
Try it: misfit +20%, press "Let them diffuse": within a few seconds about 70% of the solutes are on the red (tension) side and a cluster sits just below the core. Set the misfit to -20%: most move to the blue side. At 100 K nearly all crowd at the core; at 900 K they spread out again.
What to remember
More detail: derivations, numbers and the limits of these models
Theoretical shear strength
Frenkel (1926) treated the stress to slide one plane over the next as a sine of the displacement u, with period b and planes a distance d apart:
Better force laws bring this down to about G/30 (Hull and Bacon, section 1.1). Picture 1 uses G/30 and the same sine shape for the elastic shift below it. The measured CRSS of pure copper is about 2000 times lower; for iron it is about 80 times lower. Taylor, Orowan and Polanyi (1934) explained the gap with dislocations.
Burgers vectors of the three metals
In FCC metals slip is on {111} along <110>, and b = (a/2)<110>, so |b| = a/√2. In BCC metals b = (a/2)<111>, so |b| = a√3/2.
| Metal | a (nm) | |b| (nm) | G (GPa) | ν | CRSS (MPa) |
|---|---|---|---|---|---|
| Cu, FCC | 0.3615 | 0.256 | 42.1 | 0.34 | 0.65 |
| Al, FCC | 0.4050 | 0.286 | 25.4 | 0.33 | about 1 |
| Fe, BCC | 0.2866 | 0.248 | 64.4 | 0.30 | 27.5 |
G here is not the handbook shear modulus of a polycrystal. It is the modulus that sets the energy of a screw dislocation in the real, anisotropic crystal at 300 K, the quantity Frost and Ashby use. Cu and Al: Frost and Ashby, Table 4.1. Fe: 64.4 GPa, worked out the same way for a <111> screw from the 300 K single-crystal constants of Rayne and Chandrasekhar (1961). The polycrystal values used in the strengthening lab are a little higher: 45 GPa for Cu and 26 GPa for Al (Hansen 2004), and 83 GPa for steel (Callister). The CRSS values are for 99.999% Cu and 99.96% Fe single crystals (Dieter). The value for aluminium is only an order of magnitude: CRSS depends strongly on purity and on how the crystal was grown and handled.
How much slip one dislocation gives
When one dislocation crosses a crystal of height h, the top moves by b, a shear strain of b/h. For 1% shear of a crystal 1 mm tall, 0.01 × 1 mm / b dislocations must cross it: 39,000 in copper.
The Burgers circuit convention
FS/RH: choose the line direction t, walk a right-handed circuit about t that closes in the real crystal, repeat it in a perfect crystal, and take b from the finish F to the start S. Reversing t reverses b. An edge dislocation with t out of the screen and the extra half-plane above has b along +x.
Peach-Koehler force
With t along z, b = (be, 0, bs) and the slip plane normal along y:
Picture 3 leaves out σxz. A screw part has no climb force from σxx because it has no extra half-plane.
Edge dislocation stress field
Isotropic elasticity, D = G b / (2π(1 − ν)), r² = x² + y²:
The mean stress is σm = −2(1 + ν) D y / (3 r²). A solute with misfit volume ΔV has energy E = −σm ΔV (Cottrell and Bilby, 1949). Picture 4 moves the solutes with a Metropolis Monte Carlo rule at the set temperature.
Limits
All pictures use isotropic linear elasticity and a simple square lattice. Inside about one b of the core the stresses are not valid. Real crystals are elastically anisotropic, FCC dislocations split into partials, and the real CRSS is set by impurities and by other dislocations, not by the lattice alone. Motion on screen is not to scale in time.
On this site: see dislocations in the TEM with the two-beam defect image simulator, make them sharp with weak-beam dark field and the weak-beam tilt simulator, and find b with the g·b invisibility planner. Count geometrically necessary dislocations from EBSD in KAM, GNDs and step size. See the theoretical strength reached under a nanoindenter in pop-in statistics and dislocation storage in the indentation size effect.
Next in this chapter: Schmid factor and slip systems · strengthening mechanisms · reading a tensile curve · creep and deformation maps.
Questions people ask
What is a Burgers vector?
It is the size and direction of the slip that a dislocation carries. Draw a closed loop of atom-to-atom steps around the dislocation, repeat the same steps in a perfect crystal, and the gap you are left with is b. In FCC metals |b| = a/√2 (0.256 nm in copper).
What is the difference between an edge and a screw dislocation?
In an edge dislocation b is at right angles to the line and there is an extra half-plane of atoms. In a screw dislocation b is parallel to the line and the atom planes form a spiral ramp around it. Most real dislocations are mixed.
Why is the real yield strength of metals so much lower than the theoretical strength?
Because slip does not happen all at once. A dislocation moves by breaking one row of bonds at a time. Pure copper crystals slip at about 0.65 MPa, while shearing a whole plane would need about G/30, 1400 MPa.
How do you find the Burgers vector of a dislocation in the TEM?
Image it with several two-beam reflections g. It almost disappears when g·b = 0. Two reflections where it is invisible fix the direction of b. The g·b planner picks the reflections for you.
What is the Peach-Koehler force?
It is the force per unit length that a stress puts on a dislocation line: F = (σ·b) × t. It is always at right angles to the line. Its part in the slip plane equals the resolved shear stress along b times b.
What is a Cottrell atmosphere?
A cloud of solute atoms that has gathered around a dislocation because their energy is lower there. Large solutes go to the stretched side, small ones to the squeezed side. The cloud pins the dislocation. In mild steel, carbon and nitrogen atmospheres cause the sharp yield point.
Can a dislocation end inside a crystal?
No. Because b is the same all along the line, a dislocation can only end at a surface, at a grain boundary, or at a node where it meets other dislocations. Otherwise it closes on itself as a loop.
What is the difference between glide and climb?
Glide is motion in the slip plane, which contains both the line and b. It needs no extra atoms. Climb moves an edge dislocation out of its slip plane by adding or removing atoms at the half-plane, so it needs vacancies to diffuse. It matters at high temperature, in creep.
References
Show the 11 references
- D. Hull and D. J. Bacon, Introduction to Dislocations, 5th ed., Butterworth-Heinemann (2011). Chapter 1 (theoretical strength, Burgers circuit), chapter 4 (stress fields), chapter 5 (forces on dislocations). ISBN 978-0-08-096672-4.
- J. P. Hirth and J. Lothe, Theory of Dislocations, 2nd ed., Wiley (1982). Chapters 1 to 3 (FS/RH convention, stress fields, Peach-Koehler force).
- H. J. Frost and M. F. Ashby, Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics, Pergamon (1982). Table 4.1 (Cu, Al): b and shear modulus at 300 K; chapter 8 (alpha-Fe): b, and μ0 taken as the modulus for the energy of a <111> screw dislocation.
- J. A. Rayne and B. S. Chandrasekhar, Elastic constants of iron from 4.2 to 300 K, Physical Review 122, 1714 (1961). doi:10.1103/PhysRev.122.1714. 300 K constants used for the Fe value of G.
- G. E. Dieter, Mechanical Metallurgy, SI metric ed., McGraw-Hill (1988). Chapter 4, table of critical resolved shear stresses of metal single crystals.
- W. D. Callister and D. G. Rethwisch, Materials Science and Engineering: An Introduction, 10th ed., Wiley (2018). Table 6.1, Poisson's ratios.
- M. Peach and J. S. Koehler, The forces exerted on dislocations and the stress fields produced by them, Physical Review 80, 436 (1950). doi:10.1103/PhysRev.80.436
- J. Frenkel, Zur Theorie der Elastizitätsgrenze und der Festigkeit kristallinischer Körper, Zeitschrift für Physik 37, 572 (1926). doi:10.1007/BF01397292
- G. I. Taylor, The mechanism of plastic deformation of crystals. Part I. Theoretical, Proceedings of the Royal Society A 145, 362 (1934). doi:10.1098/rspa.1934.0106
- A. H. Cottrell and B. A. Bilby, Dislocation theory of yielding and strain ageing of iron, Proceedings of the Physical Society A 62, 49 (1949). doi:10.1088/0370-1298/62/1/308
- D. B. Williams and C. B. Carter, Transmission Electron Microscopy, 2nd ed., Springer (2009). Chapter 26 (imaging dislocations, g·b analysis).
BibTeX
@misc{tripathy2026dislocations,
author = {Tripathy, Manisha},
title = {Dislocations and Burgers Vectors},
year = {2026},
howpublished = {\url{https://untetheredatom.com/mechanical-behavior/dislocations-and-burgers-vectors}},
note = {Interactive web tool}
}