A new model for the calculation of homogeneously broadened Raman spectral bandwidths and bandshapes

Abstract

Raman spectral bandwidths and bandshapes can provide detailed information on the physics and chemistry of a sample. The Raman band reported by a spectrometer (observed Raman band [ORB]) is a result of contributions from the true, intrinsic Raman band (IRB) and the spectrometer slit function. To obtain the IRB spectral width and shape, the effects of the spectrometer slit function must be removed from the ORB by measuring the spectrometer slit function and modeling the ORB as a convolution of the IRB and slit function. Though this model has proven to be useful, it is not based on physical parameters of the measurement and is limited. Thus, new ways to model Raman bandshape and bandwidth are needed. I previously developed a new method of modeling Raman bands using spectrometer geometric optics to examine the dependence of Raman band intensity on slit width. The model calculates the intensity of the ORB from the contributions of a continuum of dispersed slit functions. Here, I extend the use of this model for examining ORB spectral widths and shapes. I show that the model produces the same result as the convolution model when using Lorentzian Raman and Gaussian slit functions, confirming the validity of the model. However, in contrast to the convolution model, the new model is based on spectrometer geometric optics allowing one to account for various optical effects when modeling Raman spectra. This is demonstrated by modeling the effects of slit width on the Raman spectrum. I use the model to provide a detailed, physical understanding of the effects of slit width and IRB width on Raman bandshapes.

A Model for Inhomogeneously Broadened Raman Bands

“…In this work, the Voigt bands are created using the pseudo Voigt function and Lorentzian percent (Lor%) is determined by examining the ratio of the FWHH and full width at quarter height (FWQH) of the Voigt as described in previous work. 8 For situations where the spectrometer entrance slit is larger than the scatter light image and other optical effects are negligible, the canonical convolution model and the integral model produce the same result. 8 Thus, the results in this section are valid for both the convolution and integral method of modeling the HORB.…”

“…8 For situations where the spectrometer entrance slit is larger than the scatter light image and other optical effects are negligible, the canonical convolution model and the integral model produce the same result. 8 Thus, the results in this section are valid for both the convolution and integral method of modeling the HORB. The dependence of IORB FWHH, spectral irradiance, area, and Lor% on different Voigt HORBs were determined using Eqs.…”

“…As discussed by Asthana and Kiefer, 3,8 there is a limited range of Γ SF /Γ IRB values where the HORB Lor% is affected by Γ SF and Γ IRB . When Γ IRB >> Γ SF (Γ SF /Γ IRB < ∼0.1), HORB ≈ IRB and when Γ IRB << Γ SF (Γ SF /Γ IRB > ∼15), HORB ≈ SF.…”

“…The dependence of Γ IORB on Γ SF FWZI is very similar to that of Γ HORB previously published. 8

Figure 8.Dependence of slit width on (a) Γ IORB , (b) Ir IORB , and (c) A IORB . Slit width is reported as the ratio of the slit function FWZI (slit width in cm −1 ) and slit function FWHH (Γ SF FWHH = 5 cm −1 ).

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“…Equation 1 was derived and discussed in detail in previous publications. 7,8 Briefly, the Gaussian function is the slit function of the spectrometer and the Lorentzian function is the IRB where the peak spectral irradiance is normalized to 1. The spectral irradiance of the HORB at a given wavenumber is equal to the sum of the spectral irradiances of all the slit functions in the continuum that have intensity at that wavenumber and the intensity of a slit function in the continuum is proportional to the IRB spectral irradiance at that wavenumber.…”