1. Haidong Liu Mentor: Dr. Peter C Collins, Matt Kenney
Department of Materials Science and Engineering
Reconstruct the full elements of Nye tensor by coupling precession electron diffraction (PED) measurements with computation method in simulated Nano-scaled crystal
Haidong Liu
Mentor: Dr. Peter C Collins, Matt Kenney

2. Dislocation Density
Dislocation describes the line defects in crystal where the atoms are dislocated from lattice position.[1]
The study of dislocations is fundamental to the understanding the mechanical properties of materials, and is especially useful in metallurgy.
Dislocation line direction and Burgers vector describe a dislocation
Dislocation density is around 1012 m-2 for annealed steel
Dislocation density could be calculated by using Nye tensor
Example of edge dislocation
[1] Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, Print.

3. Nye tensor and Measurable
Nye proposed a method to characterize the state dislocation into second rank tensor
Second rank tensor is a 3x3 matrix [2]
𝛼 𝑖𝑗 Ω = 𝐾 𝑏 𝑖 𝐶,Ω 𝑑 𝑙 𝑗
Schematic of stress tensor in cartesian coordinate
[2] Meyers, Marc A., and Krishan Kumar Chawla. Mechanical Behavior of Materials. Cambridge: Cambridge UP, Print.

4. Transmission electron microscopy (TEM) method
TEM data is limited in 2 dimensions, the volume data is missing
Only five element of Nye tensor could be reconstructed. [5]
𝛼 12 , 𝛼 13 , 𝛼 21 , 𝛼 23 ,and 𝛼 33
Schematic of TEM [6]
[5] Pantleon, W. "Resolving the Geometrically Necessary Dislocation Content by Conventional Electron Backscattering Diffraction." Scripta Materialia (2008): Web.
[6] "What Is an Electron Microscope? - Definition, Types & Uses." Study.com. N.p., n.d. Web. 09 Apr

5. Precession Electron Diffraction (PED) technical
PED rotates the incident electron beam around central axis. The measurement of PED contains the spatial orientation information as the average of a volume in the certain direction.
The sample is separated into 3-D grid. Each slice of the grid is represented by a matrix
The measurement is taken at the first row of each column
Schematic of PED[3]
[3] Nicolopoulos, Stavros. "Precession Electron Diffraction: Applications in TEM." Precession Electron Diffraction. Italy, Siena. 03 Dec Web.

6. Interpretation of the measurement
The orientation of the crystal is related to the lattice curvature [5]
𝜅 𝑘𝑙 = 𝜕 𝜃 𝑘 𝜕 𝑥 𝑙 ≈ ∆ 𝜃 𝑘 ∆ 𝑥 𝑙
The lattice curvature is related to Nye tensor [5]
𝜅 𝑖𝑗 = 𝛼 𝑗𝑖 − 1 2 𝛿 𝑖𝑗 𝛼 𝑘𝑘

7. Interpretation of the measurement
Orientations are represented in Euler notation< 𝜑 1 𝚽 𝜑 2 >
Orientations are transformed in {hkl} <uvw> notation as vector field for the matrix computation
The vector field is separated into matrix for each slice to reconstruct the tensor

8. Computation
B (n x #) is the measurement matrix of average orientation,
# is the number of different angles used in the reconstruction.
n is the number of column
C (# x m) is the operating matrix for B(n x #) to reconstruct matrix A(m x n). B1 B2 B3 B4
B5… Bn
B (n x #)
c1 c2 c3 c4 c5 … cm
C (# x m)

9. 𝐵× 𝑐 = 𝑎 ( 𝑎 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 𝐴 𝑇 ,𝑐𝑜𝑙𝑢𝑚𝑛 𝑜𝑓 𝐴)
Computation
B1c1 B2c1 B3c1 B4c1 … Bnc1
B1c2 B2c2 B3c2 B4c2 … Bnc2
B1c3 B2c3 B3c3 B4c3 … Bnc3
B1c4 B2c4 B3c4 B4c4 … Bnc4 …
B1cm B2cm B3cm B4cm … Bncm
A(m x n)
A (m x n) is the full orientation matrix of one slice of the material
Each vector in C corresponds to each row in the original matrix A
Each component of {hkl} <uvw> vector field is calculated separately
[B x C] (n x m)
B1c1 B1c2 B1c3 …B1cm
B2c1 B2c2 B2c3 …B2cm
B3c1 B3c2 B3c3 …B3cm
B4c1 B4c2 B4c3 …B4cm
B5c1 B5c2 B5c3 …B5cm …
Bnc1 Bnc2 Bnc3 …Bncm
𝐵× 𝑐 = 𝑎 ( 𝑎 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 𝐴 𝑇 ,𝑐𝑜𝑙𝑢𝑚𝑛 𝑜𝑓 𝐴)
[𝐵 𝑛 × # ×𝐶 #× 𝑚 ] 𝑇 =𝐴(𝑚×𝑛)

10. Results The method is verified on a simulated Nano-crystal
Prof. LeSar calculated Nye tensor data for 25x25x10 voxels.[7]
Nye tensor was converted to crystal curvature and then to orientation.
Simulated crystal was reconstructed.
Simulated nano-crystal by Prof. LeSar
[7] Zhou, Caizhi, and Richard Lesar. "Dislocation Dynamics Simulations of Plasticity in Polycrystalline Thin Films." International Journal of Plasticity (2012): Web.

11. Results
Comparison of original and reconstructed {hkl} <uvw> vector field

12. Results
Histogram of the residual

13. Summary
Precession electron diffraction (PED) measurements coupled with computation method could reconstruct the full element of Nye tensor
Since the ordination data is average through volume, the sample size is limited to Nano-scale to assure accuracy
The future objective is to obtain analytical solution of the reconstruction matrix

14. Acknowledgement Dr. Peter Chancellor Collins Dr. Richard Alan LeSar
Matt Kenney

15. Reference
Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, Print.
Meyers, Marc A., and Krishan Kumar Chawla. Mechanical Behavior of Materials. Cambridge: Cambridge UP, Print.
Nicolopoulos, Stavros. "Precession Electron Diffraction: Applications in TEM." Precession Electron Diffraction. Italy, Siena. 03 Dec Web.
Nye, J.f. "Some Geometrical Relations in Dislocated Crystals." Acta Metallurgica 1.2 (1953): Web.
Pantleon, W. "Resolving the Geometrically Necessary Dislocation Content by Conventional Electron Backscattering Diffraction." Scripta Materialia (2008): Web.
"What Is an Electron Microscope? - Definition, Types & Uses." Study.com. N.p., n.d. Web. 09 Apr
Zhou, Caizhi, and Richard Lesar. "Dislocation Dynamics Simulations of Plasticity in Polycrystalline Thin Films." International Journal of Plasticity (2012): Web.