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Effect of Nuclear motion on Spectra of Hydrogen like Atom

Hello friends welcome once again on our blog. I’m back with a new topic on Atomic Physics  that is Effect of Nuclear motion on Spectra of Hydrogen like Atom. In my last post I’ve discussed Shortcomings of Bohr’s theory.

So without wasting your time, I’m going to today’s topic. So let us start…. 
In Bohr’s theory, we assume that nucleus of hydrogen atom is infinitely heavy in comparison to electrons so it remains stationary while electrons revolves around it.
The nucleus has a finite mass and both the electron & nucleus revolve about their common centre of mass C, with a common angular velocity ω.
Effect of Nuclear motion on Spectra of Hydrogen like Atom
Fig 1.Arrangement of Nucleus & electron
Let “m” be the mass of electron & “Mh” mass of Hydrogen nucleus & “r” distance between them.

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Let “x” be the distance of nucleusfrom common centre of nass,  then the distance of electron will be (r-x)At equilibrium the movement of “Mh” and “m” about “C” will be equal. Total angular momentum :

Momentum of atom about the centre of mass:

Taking nuclear motion into account ,Bohr’s first postulate that electron revolves around only those orbits whose angular momentum is nh/2π.

Hence, eq. (i)  is exactly equal to the eq. (ii)  except that “u” has replaced by m
Therefore, Bohr’s result can be written as the energy of electron in nth orbit is given by :
Since, “μ” is slightly less than “m“. Thus the electron energies are slightly less when nucleus is in motion.
We have :
Effect of finite nuclear mass on Rydberg Constant :
 
Taking nuclear motion in account, the Rydberg constant for Hydrogen atom is given by :

This shows that the Rydberg constant for hydrogen “Rh” depends upon the mass “Mh” of hydrogen nucleus. Thus, it is a constant only for a particular atom and varies slightly from atom to atom.

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This is how we calculate Rydberg constant for Hydrogen atom & also explained Effect of Nuclear motion on Spectra of Hydrogen like Atom. That’s all for today. In my next topic I’ll explain something about Heavy Hydrogen. So keep visiting. Still any query, suggestions, write to us in comment section. 

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