The Optics Laboratory
Group of Hans Hallen, Physics Department, North Carolina State University

GFR: Quantum Derivation

 

Transitions in vibration levels due to coupling with a radiation field are described by the perturbation Hamiltonian , where μ is the dipole moment and E is the electric field. The dipole moment can be written as

where the {a,b,c} are a permutation of the coordinates {x,y,z} (summing over repeated indices is implied), μp is the permanent dipole moment, B is the magnetic field, and α is the polarizability tensor. The α, A, and G are given by: (Sass, et al., 1981)

where ; and where μp and θ, m are the electric dipole, quadrupole, and the magnetic dipole operators, respectively, and and are the initial and final states of the system, respectively. The derivation of the spectroscopic signals proceeds with a first order expansion of μ in the coordinate of vibration q:

The terms without q dependence (1st, 3rd, etc.) can be removed since they will not couple adjacent vibration states. The second term yields the direct photon absorption (infrared spectroscopy). Raman spectroscopy derives from the fourth term. Usually the electric field is assumed to be independent of q and hence is removed from the derivative. Since the field can vary very rapidly near a metal surface, we do not remove it from the derivative. The extra term that results is our ‘gradient field Raman’ term. The sixth term has been discussed before, and can also be important when the field varies rapidly, such as near a metal surface. The remaining terms are small even in high field-gradient regions and can be neglected. The relevant dipole terms can thus be written as:

The four terms result in IR absorption, Raman, GFR, and quadrupole-Raman. The ratio of the GFR term to the Raman term depends upon the field gradient and the polarizability gradient, which we approximate as α/a, where a is close to an atomic dimension. In vacuum, the field gradient yields terms of the order i(2πλ)Eb. Near a metal surface, the jellium approximation of Feibelman indicates that the electric field varies by nearly its full amplitude over a distance of 0.2 nm. The derivative is then approximately Eb/0.2 nm. The ratio of the GFR term/Raman term in vacuum is of order 2πa/λ. For 500 nm light and a = 0.2 nm, this is ~10-3, so that the GFR contribution is insignificant. The situation is different near a metal surface, where the ratio of the GFR term/Raman term is a/0.2 nm, or ~1. We thus expect to find a measurable GFR signal near metal surfaces.

The GFR differs appreciably from Raman spectroscopy in selection rules. These rules result from the requirement that be nonzero. The q dependence of μ means that this expectation will be nonzero if the ψ differ by one vibrational quantum. In addition, the coefficient of q must be nonzero. The Raman selection rules are determined by the requirement

This is equivalent to the condition that α and the vibration belong to the same symmetry species (Ferraro & Nakamoto, 1994). Conversely, the GFR selection rules require that E belong to the same symmetry species as the vibration, or

This will be true if the vibration has a component normal to the surface, since that is the direction in which E varies most rapidly. The polarizability must also be nonzero. For example, if z is normal to the surface, then αaz and Ea must be non-zero. This is the case for NSOM, in which all components of E are present near the probe, but it is not necessarily the case in far-field measurements. In summary, the selection rules for GFR include some Raman active modes and some IR active modes, since bond orientation is dominant.

The GFR effect should scale with polarizability, and should thus be stronger for ionic systems. This is in contrast to Raman spectroscopy, which typically is stronger for covalent bonding. Vibration modes with strong infrared absorption tend to have large polarizabilities, so we can expect the GFR intensity to reflect the IR intensity for those modes allowed by both GFR and IR. That is, the GFR spectra will complement the Raman spectra in many materials, particularly centro-symmetric materials. The extra requirement for observation of the GFR effect is a strong field gradient along the vibrating bond.

The effects of a strong electric field gradient (in the quadrupole term above) on Raman spectra has been discussed previously as a mechanism for some of the observed spectral lines in SERS (Sass, et al., 1981). A field gradient effect was also sought in the Raman spectra of microparticles suspended in laser traps (Knoll, et al., 1988). NSOM-Raman, with its mobile probe, offers the first opportunity to test the distance dependence of the GFR and this effect.

  •  More info is in the papers.

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