9/11 Acquisition Reform Advertising Alaway Alcohol Ale Allergies Antisemitism Barack H. Obama Beer Billiards Biology Books Budget Bureaucracy California Capitalism Carbohydrates Carcinogen CDC Chemical Warfare Chemistry Chemophobia Chirality Climate Science Colonial Pines Computers Conservation Laws Constitution Consumerism Cosmology CPT Invariance Creationism Customer Service Daesh David Irving Dead End Defense Dinosaurs Disasters Economic Energy English Ethics Evolution Fluoride Food FTL Garden Care George W. Bush Gerlich and Tscheuschner GISS Glaciers GMOs HadCRU Haiti Health Himalayan Rock Salt HITRAN Holocaust Denial Home Brewing How It Looks From Here html Humor Information Infrared Spectroscopy IPCC Iran ISIS Islam Islamophobia Israel Ketotifen Fumarate Law Lawn Care Leibniz Lisbon Magnetism Math Medco Medicine Modeling Molecules Monopoly Monsanto Naphazoline hydrochloride Neutrinos Nietzsche NIH NIST Noether's Theorem Non-hazardous Norton Ghost Nuclear Warfare Oil Oil Spill Olopatadine hydrochloride Opinion Orson Scott Card Parody Pataday Patanol Pesticides Pheneramine maleate Physics Plumbing Politics Poll Pope POTUS Prescriptions Prop 65 Psychology Quantum Mechanics Quiz Racism Radiative Transfer Relativity Religion Respiration Senior Housing Signs Smoking Specific Gravity Statistics Stock Market Sugars Sun Tzu Surface Temperature Surgeon General Symantec Target Temperature Terrorism The Final Solution The Holocaust History Project Thermodynamics Time Trains Units Voltaire von Clausewitz Weather White House Wine Yeast

Friday, July 9, 2010

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.


Note: I am using the symbol:  ν ˜ 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 (ν ˜) in vacuum  is simple.  We need to start with the following equation:

      ν ˜ = 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:

      ν ˜ (cm-1)  = 104/λ(μm)

And conversely:

     λ(μm) = 104 /ν ˜ (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:

     ν ˜ (cm-1)  = 104/λ(μm)

     ν ˜ (cm-1)  = 104/ 14.0

     ν ˜  =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   ν ˜ 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 = hcν ˜

Planck's constant (h) and the speed of light (c) are both constants; so it is appropriate to treat ν ˜ 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 ν ˜ 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 = cν ˜

In air:

     λ-air x cν ˜ = c/n


     λ-air = 1/nν ˜    

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

      λ-air(μm) = 104 /nν ˜ (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.



Seth said...

Clear, simple explanation of inverse centimeters and wave numbers. Thanks for the help.

Hide said...

Thanks you very much!!


Pls am Working on lossless transmission lines, i want to convert 1metre to wavelength.kindly help me out