Measuring Strain Using X-Ray Diffraction

By: James Belasco

What is Strain?• When a force is applied

materials deform in two ways– It will deform and then

revert to its original shape (reversible)

– Change its shape permanently (irreversible)

The Structure of Materials

• At the local level, solids are composed o crystalline lattices

• Atoms in these crystal structures are arranged in different crystal shapes depending on the material

• In reversible strain the bonds act like springs

• Strain “stretches” the bonds in the lattice without breaking them

Who’s More Flexible?

How does a polycrystalline material distribute strain?

or

Polycrystalline Materials

• Nanocrystalline structures are made of a large number of randomly oriented crystals

• How does the bulk property of the material relate to the individual crystallites?

d1*

Strain

Figure Compliments of Matt Bibee, SULI 2006

Diffraction SpaceNormal Space

d0

d0d0

d0

d2*

d2*

d1*

StrainedSample

Strain Equations

2sincos2sinsinsin2sin

cossinsinsincos

13232

12

233

2222

2211

0

0

d

dd

kljlikij aa

cossinsinsincos

0cossin

sincossincoscos

x y zx

y

z

a =

DetectorSample

Beam

Straining Iron

[110] [200]

[211]

• Sets of crystals oriented in a particular direction from their own ring

• Iron produces three rings

χ

Q

• Diffraction images are transformed to a different coordinate system or “caked”

Transformation

One Ring

Q

χ

Straining Iron

• The slopes of the lines give us the elastic modulus in a crystallographic direction

0

100

200

300

400

500

600

-5.00E-04 0.00E+00 5.00E-04 1.00E-03 1.50E-03 2.00E-03 2.50E-03 3.00E-03 3.50E-03

Strain

Str

ess

Strain Direction

[200]: 152.3

Error: 6.05

[211]: 207.3

Error: 9.2

[110]: 177.5

Error: 10.9

0

100

200

300

400

500

600

-5.00E-04

0.00E+00

5.00E-04 1.00E-03 1.50E-03 2.00E-03 2.50E-03 3.00E-03 3.50E-03

Strain

Str

ess

How far can you stretch?Single Crystal Values

110 210 GPa

200 125 GPa

211 210 GPa

Measured Values

110 177 ± 11 GPa

200 152 ± 6 GPa

211 207 ± 9 GPa

The polycrystalline sample exhibits properties of the single crystal

Acknowledgements

• Department of Energy, SLAC, and SSRL

• Apurva Mehta

• David Bronfenbrenner