What is Hartmann Dispersion Formula?

 

Hartmann dispersion formula is a semi-empirical formula. It is also referred to as Cornu Hartmann formula. It relates refractive index (n) with the wavelength (λ). It is given by: 

n (λ)= X + Y/( λ-Z)............................................................Equation 1 

X, Y, and Z are empirical constants while n (λ) is the refractive index of a material of light of wavelength λ. The constants derived as follows:

Z= [(n1-n2)/ (n2-n3) λ1 (λ2- λ3) - λ3- λ3 (λ1- λ2)]/ [(n1-n2)/ (n2-n3) (λ2- λ3)-(λ1- λ2)]-....................................................................................Equation 2 

Y= (n1-n2)/ [1/ (λ1-Z)-(1/ (λ2-Z)…………………………Equation 3 

X=n1-B/ (λ1-Z)…………………………………………...Equation 4 

Hartmann dispersion formula is applied in the measurements of spectra wavelengths taken on a prism instruments. 
Constant Z is the spectrometer optical function. In the laboratory, it usually measured by lab technicians for every spectrometer and it is labeled on the side of every spectrometer.
Equation 1 can be written as: 

y (n(λ))=Y/ ( λ-Z) + X …………………………………...Equation 5

By plotting values of y against (λ-Z) gives a straight line in which the values of constants Y and X can be obtained.  

The pros of using the Hartmann dispersion equation technique (plotting) to determine constants
  • ·         It requires fewer calibration points.
  • ·         It is stress-free to draw.
  • ·         It is accurate 

The cons of using the Hartmann dispersion equation technique (plotting) to determine constants
  • ·         You must rely on the observations made by someone else.
  • ·         Reading from the plotted graph’s straight line values does not give λ directly.

                                            References
Forziati, F., A.(1950). Refractive index as a function of wavelength for sixty APS-NBS hydrocarbons. Journal of Research, 44, 373-385.

Wood, A., L., & Tilton, W., L. (1949). Refractive index of natural rubber for different wavelengths. Journal of Research, 43, 57-64.

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