Page 333 - Vibrational Spectroscopic Imaging for Biomedical Applications

Raman Imaging of Str ess Patterns in Biomaterials 307

Zirconia—The monoclinic phase of ZrO , with space group P2 /2,
2 1
can be represented with the point group C at Γ, consisting of opera-
y 2h
tions E, I, C , and M . The character table of this point group indi-
2 y
cates that there are four symmetry classes and thus four irreducible
representations, each of which is one dimensional. Standard group-
theoretical analysis indicates that the modes at the Γ point can be
decomposed as 24,25
m
Γ = 9A + 9A + 9B + 9B (10.3)
g u g u
where the superscript m refers to the monoclinic polymorph.
Among the total 36 modes, 18 modes 9A + 9B are Raman active and
g g
15 modes 8A + 7B are infrared active, the remaining three modes
u u
being the zero-frequency translational modes. Only the 15 infrared-
active modes contribute to the lattice dielectric tensor. In the case of
the tetragonal polymorph, the irreducible representations for the
optical modes in the crystal can be expressed as 24–26
t
Γ = 1A + 2A + 3E + 3E + 2B + 2B (10.4)
1g 2u g u u 1g
where the superscript t refers to the tetragonal polymorph. The E
u
and E representations are two-dimensional while all other modes
g
are one-dimensional. One A mode and one E pair are acoustic,
2u u
leaving one IR-active A and two IR-active E pairs; A , B , and E
2u u 1g 1g g
are Raman active, while B is silent. Raman spectra of tetragonal,
2u
monoclinic, and mixed tetragonal/monoclinic polymorphs are shown
in Figs. 10.4a, b, and c, respectively. Raman active modes are labeled
on the spectra in Figs. 10.4a and b for tetragonal and monoclinic
phases, respectively.
10.3.2 PS Behavior
In single-crystalline materials, the PS effect can be represented with a
linear (tensorial) relationship between the second-rank stress tensor σ
ij
and the observed Raman wavenumber shift Δν with respect to the
unstressed (reference) wavenumber characteristic of a selected Raman
band: 12,21
∗
Δν = Π σ = Π σ ∗ (10.5)
ij ij ij ij
where i, j = 1, 2, 3 and Π is the tensor of PS coefficients (given in units
ij
−1
of cm /GPa). Data for the matrix of PS coefficients of single-crystals
are typically reported with reference to the main crystallographic
axes of the crystal. In order to make this explicitly known a suffix “∗” is
usually added to all symbols referring to this special choice of coordi-
∗
∗
nate system (e.g., Π and σ ). The number of independent components
ij ij
of the PS tensor might eventually reduce according to crystal sym-
metry; for example, in the particular case of a single-crystal with full
∗
symmetry the PS tensor reduces to a scalar, Π , while in crystals with
∗
planar symmetry Π is purely diagonal.
ij