Question

Why are chemical shift spectra “Lorentzian” in shape? Derive mathematically.

Answer 1

The line shapes in NMR spectrum is determined by the process of free induction decay, which is almost exponential, hence the line shape Lorentzian. This validates because the Fourier transform of an exponential function in the time domain is a Lorentzian in the frequency domain. Further more, the lifetime of the excited states is relatively long in the NMR spectrum, hence very sharp lines are observed in the NMR spectrum, i.e. high-resolution spectrum.

A few information about the Lorentzian function is given below.

A Lorentzian line shape function can be represented as shown below.

L = 1/(1 + x²), where L = Lorentzian function standardized for spectroscopic purposes to a maximum value of 1 and x is a subsidiary variable.

i.e. x = (p⁰ - p) / (w/2), where p⁰ = the position of the maximum, corresponding to the transition energy, p = position, and w = full width at half maximum, i.e. FWHM

The width of the curve when the intensity is half of the maximum intensity that occurs at the points p = p⁰ ± w/2

The units of p⁰, p, and w are same as that of frequency.

The variable x is dimensionless and x = 0 at p = p⁰