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When we calculate the Raman scattering, we usually derive the imaginary part of the Raman response function. The general formula of the imaginary part is given by \begin{align} \text{Im}[I(\omega)] \propto \sum_{f}\left|\langle f|R|i \rangle\right|^2 \delta(\hbar \omega - (E_f-E_i)), \end{align} where $|i\rangle$ and $|f\rangle$ denote the initial and final states of the system. The delta function relates to Fermi's golden rule and therefore this imaginary part means the energy dispersion (the other words, the energy absorption (emission)). In this case, what is the physical meaning of the real part? Naively, the real part is given by \begin{align} \text{Re}[I(\omega)] \propto \mathcal{P}\sum_{f}\left[\frac{\left|\langle f|R|i \rangle\right|^2}{\hbar \omega - (E_f-E_i)}-\frac{\left|\langle f|R|i \rangle\right|^2}{\hbar \omega - (E_i-E_f)}\right]. \end{align}

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This can be understood on general grounds through the theory of linear response. In Raman scattering and many other cases, we're typically interested in the response of a Hermitian operator to a real-valued perturbation, for which the time-domain response function (or generalized susceptibility) $\chi(t)$ is itself real, which upon Fourier transforming leads to a complex response function in frequency space, $$ \chi(\omega) = \mathrm{Re} \chi(\omega) + i\,\mathrm{Im}\chi (\omega). $$ The real and imaginary parts are related by the Kramers-Kronig relations, in a way that enforces causality in the sense that the response for $t<0$ (i.e. before the scattering event) vanishes. The imaginary part (odd in $\omega$) is often called the dissipative (or absorptive) part of the response since, as your equation shows, the delta function involves an energy transfer of $\hbar\omega$ due to the scattering process.

The real part (even in $\omega$) is in-phase with the perturbation and is known as the reactive part of the response. It is the Fourier transform of the time-reversal symmetric part of $\chi(t)$, which corresponds to reversible processes and does not know about the arrow of time. There is a heuristic argument relating $\mathrm{Re}\chi(\omega)$ in a scattering system to virtual transitions in J. S. Toll., Phys. Rev. 104, 1760 (1956).

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  • $\begingroup$ Thank you for your response. I had not considered the behavior of the response function with respect to time reversal. I would like to know if there is an experimentally observable physical quantity that corresponds to the real part. $\endgroup$
    – rtrtw
    Oct 29, 2023 at 11:15
  • $\begingroup$ @rtrtw I'm not aware of one in scattering experiments, since the cross section is entirely related to the imaginary part. Reconstructing the entire $\chi$ from scattering is sometimes known as the "phase problem", and I've only seen it done using the Kramers-Kronig relations. $\endgroup$
    – Anyon
    Oct 29, 2023 at 16:28
  • $\begingroup$ Thank you so much! $\endgroup$
    – rtrtw
    Nov 1, 2023 at 0:41

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