How To Convert To and From Wavenumbers

The question of how to convert from one set of units to another comes up from time-to-time, and I think it might be helpful to have a few short posts that simply address unit conversion.  This post addresses conversion to and from wavenumbers (cm^-1) (also called reciprocal centimeters, inverse centimeters or  Kaisers). A previous post What is Infrared Radiation (IR)? addresses the concepts behind this unit.  The unit is proportional to frequency, and can be considered a unit of frequency or of energy.

Symbol

Note: I am using the symbol:  for wavenumber.  Because it is not a true font, it appears to be elevated in some contexts.  It is not meant to appear to be an exponent anywhere.

In Vacuum


Converting to and from wavenumber () in vacuum  is simple.  We need to start with the following equation:


      = 1/λ

in which λ is the wavelength.   In the IR, wavelength is commonly reported in microns (μm), and wavenumber is reported in inverse centimeters (cm^-1).  We need to know that:

      1 m = 100 cm = 10⁶ μm

It follows that:

     1 cm =  10⁴ μm

Putting it together, I  get:

      (cm^-1)  = 10⁴/λ(μm)

And conversely:

     λ(μm) = 10⁴ / (cm^-1)

 So for example, suppose there is a source of 14.0 μm radiation in vacuum, how many wavenumbers is that radiation? 

Start with the equation:

     (cm^-1)  = 10⁴/λ(μm)

     (cm^-1)  = 10⁴/ 14.0

       =714 cm^-1

Vacuum Wavenumber as a Unit of Energy

     Recall from What is Infrared Radiation (IR) that in vacuum the frequency times the wavelength is equal to the speed of light:

     λν = c


Here ν is the frequency in Hz, not to be confused with   in cm^-1 . Recall also that frequency is proportional to energy.

     E = hν


I can replace ν with c/λ:

     E = hc/λ

And keeping everything in SI units, I get:

     E = hc

Planck's constant (h) and the speed of light (c) are both constants; so it is appropriate to treat as a unit of energy.

1 electron volt (eV)  = 8065.47 cm^-1
1  Joule(J)  =  6.242 x 10¹⁸ eV = 5.034 x 10²²  cm^-1

In Air


Because it is convenient to keep as a unit of energy, it is commonly reported in vacuum units, even if the radiation is propagating through air.  To convert to and from vacuum wavenumbers for radiation propagating through air, strictly speaking, one needs to keep track of the fact that the speed of light in air is different than the speed of light in vacuum.  This difference is in the real part of the index of refraction.

      s = c/n

The speed of light in air (s) is equal to the speed of light in vacuum divided by the real part of the index of refraction (n).

In vacuum:

     ν = c

In air:

     λ-air x c = c/n

Or

     λ-air = 1/n    

Instead of SI units, one may wish to have  λ-air in μm and in cm^-1:

      λ-air(μm) = 10⁴ /n (cm^-1)

At 0 °C and 1 atm, n = 1.000293.  In many cases, it is justifiable to neglect correcting for the refractive index.  It depends, on the wavelength region and the accuracy required. The index of refraction depends on pressure.  As pressure decreases, n approaches its vacuum value of 1.

Sources

• IUPAC Goldbook
• UVA: Constants, Units & Conversions 
• Wikipedia: List of refractive indices