Page 162 - Vibrational Spectroscopic Imaging for Biomedical Applications

138 Cha pte r F i v e

angle in this expression) depends on the square of the derivative of
the invariants of the molecular polarizability α and γ ; the trace and
anisotropy, respectively, of the polarizability tensor):

∂ ⎞
∂ ⎛ σ ⎞ π 2 v ( − v ) 4 ⎡ ⎢ ⎛ ∂α ⎞ 2 ⎛ γ 2 ⎤ ⎥
⎜ ⎟ = = 0 i g 45⎜ ⎟ + ⎜ 7 ⎟ (5.3)
⎝ ∂Ω ⎠ i 90 ε 2 0 1 − e −v i he / kT i ⎢ ⎣ ⎝ ∂q i i ⎠ ∂ ⎝ qi ⎠ ⎥ ⎦

where ν and ν are the wavenumbers of incident light and the
0 i
molecular vibration, respectively; g is the degeneracy of the ith
i
vibrational mode; h, k, T, and ε have the usual meaning (see Refs. 57
0
to 62). This theory assumes that mechanical and electrical anhar-
monicity are not significant. Apart from their use as molecular
signatures, Raman spectra can provide extremely sensitive infor-
mation on molecular structure and conformation. See for example
the dependence of CH stretching frequency on location within a
molecular framework and on orientation to the applied field. 60,61
The Raman trace scattering intensity associated with stretch of
individual CH bonds had been calculated at the HF/D95 (d,p)
level, for straight chain alkanes to C16 and for the CH bonds in
bicyclo-[1.1.1]-pentane. The derivative for the CH bond attached
to the terminal methyl group and lying in the plane of the all-
trans hydrocarbon chain increases with chain length from α/∂r =
1.05 × 10 −30 Cm/V for the CH bond in methane to 1.35 × 10 −30 Cm/V
for the in-plane CH in hexadecane. The two identical CH lying
above and below the chain and the remaining CH methylene
bonds located down the length of the chain have derivatives that
are smaller (about 0.95 × 10 −30 Cm/V), and remain approximately
constant regardless of chain length. The calculated and experi-
mental derivatives for the CH methylene and bridgehead CH
bonds in bicyclo-[1.1.1]-pentane differ by 25 percent, illustrate
the enormous influence of strain and location on individual scat-
tering intensity, even for this simple structural unit. Normal
Raman scattering, which is also at the heart of the SERS effect
(see Sec. 5.6), depends on the square of this derivative. As is
known for SERS experiments, only those molecules in extremely
close proximity to the intense localized plasmon field, and usu-
ally only those within 10 to 15 nm of the surface, will experience
the intense SERS enhancement. The fundamental intensity associ-
ated with a particular vibration and the orientation of the functional
group relative to the SERS substrate both play a part in the observed
SERS intensity, along with enhancement factors that are still being
7
determined. 1–7,63,64 The review by Smith presents a good overview
of the state of understanding, without invoking any of the current
mathematical models, and serves as an excellent introduction to the
phenomenon.