Diploma Thesis - Erich Schmid Institute

d
ϕψ
ϕψ,
hkl,
R
= d (1+
( x ε + ( 1−
x)
ε
0
33
Polycrystalline and monocrystalline Model
ϕψ,
V
33
Equation 7.11 will be used for nonlinear regression and d0 is substituted by:
d
ϕψ
a 0 ϕψ,
hkl,
R
ϕψ,
V
= (1+
( x ε33
+ ( 1−
x)
ε33
)) (equ 7.11)
2 2 2
h + k + l
7.2. Calculating stresses for a single crystalline material
The examination of isotropic polycrystalline material differs from anisotropic single
crystalline material in such a way that an untextured polycrystalline material fulfils the
Bragg condition at every ψ-ϕ tilt because there are always crystals with net plane
normals parallel to the measuring direction in the laboratory system. However Bragg
reflections of single crystals can only be detected at strict defined tilts [24].
ε = O O O O s σ
(equ 7.12)
ij
e
mi
e
nj
e
ok
Only strains can be measured, hence the stresses must be expressed with a linear
elastic anisotropic material model. The strains in the sample system are depending
on the crystal position therefore the orthogonal tensor O e , which is described by Euler
angles, is introduced to express the relation between the sample and the crystal
system.
ij
D
ki
e
pl
D
lj
kl
mnop
kl
))
ε = O O ε
ϕω (equ 7.13)
The strain measurement is done in the laboratory system, so the sample’s tilt is
defined by equation 7.13 which is similar to equation 7.6 except that the strain tensor
is rotated and ω−angle is used instead of ψ.
In the case of gallium nitride and gallium boron nitride on sapphire the basal plane of
the GaN crystal and BGaN crystal is perpendicular to the surface normal, so further
calculations are simplified in such a way that no orthogonal tensor O e is needed,
because of the transversal isotropy of the hexagonal lattice.
ε = O O s σ
ϕω (equ 7.14)
33
D
i3
D
j3
ijkl
kl
29