Lorentzian lineshape in Nuclear Magnetic Resonance

Nuclear Magnetic Resonance (NMR) is a biophysical technique that is frequently used to determine the structure and interactions of biomolecules. Generally, each nucleus will generate a signal in a form of a Lorentzian peak. This Lorentzian peak carries the information of the corresponding nucleus based on its x-axis position as well as its linewidth.

• The position of the peak describes the electron shielding around the nucleus.
• The linewidth of the peak describes the relaxation time of the nucleus.

A Lorentzian curve is a near-gaussian curve described by the following equation:

\[ L(x) = \frac{\frac{1}{2}\Gamma}{(x - x_0)^2 + (\frac{1}{2} \Gamma)} \]

where \( Gamma \) is the lineshape and \( x_0 \) is the central position.

A normalized Lorentzian curve has an additional factor of \( 1/\pi \) in front to control the area under the curve to be always equal to 1:

\[ L(x) = \frac{1}{\pi} \frac{\frac{1}{2}\Gamma}{(x - x_0)^2 + (\frac{1}{2} \Gamma)} \]

The concept of the application is a visualization on how the signal strength (amplitude) depends on the linewidth. Below are some demonstrations on the amplitude values (\( L(x_0) \)), given a series of linewidth values.

linewidth <- c(1:5)
amplitude <- (1/pi)*(linewidth/2)/(linewidth/2)^2

cbind(linewidth, amplitude)

     linewidth amplitude
[1,]         1 0.6366198
[2,]         2 0.3183099
[3,]         3 0.2122066
[4,]         4 0.1591549
[5,]         5 0.1273240

The amplitude decreases with increasing linewidth