^{-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 = 10**

^{6}μm

It follows that:

1 cm = 10

^{4}μm

Putting it together, I get:

(cm

^{-1}) = 10

^{4}/λ(μm

**)**

And conversely:

λ(μm

**) =**10

^{4}/ (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}) = 10

^{4}/λ(μm

**)**

(cm

^{-1}) = 10

^{4}/

**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 10

^{18}eV = 5.034 x 10

^{22}

**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**

**) =**10

^{4}/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**

## 3 comments:

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

Thanks you very much!!

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

Thanks

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