Unit of Chemical Shift:
You have learnt that in a spectrum the X-axis of the spectrum refers to the energy of the EM radiation in terms of wavelength, frequency or wave number. In that case of NMR the X- axis should, in principle that should be the frequency of the radiation being absorbed or the field at which the resonance is achieved for a predetermined (fixed) radiofrequency. However, using the frequency or the field is quite inconvenient. More so since the frequency of absorption depends on the applied field, the position of the signals for a given analyte would depend on the applied field. That is the position of the signal obtained for a given analyte on different instruments would be different. Therefore, we need a parameter that is independent of the applied field. Such a parameter is called δ or ppm which is a dimensionless quantity. Now let us try to know the meaning and the genesis of δ.
Assume we measure a test sample and the reference, TMS using the same magnetic field B0, the resonance conditions for the two would be given as below.
hvTest = g N µ N B0 (1 - σ Test )
hvTMS = g N µ N B0 (1 - σ TMS )
Thus, the shift in resonance frequency would be
hvTest - hvTMS = g N µ N B0 (1- σ Test ) - g N µ N B0 (1- σ TMS )
h (vTest - vTMS ) = g N µ N B0 [1 - σ Test - (1 - σ TMS )]
(vTest - vTMS ) = g N µ N B0 (1- σ Test ) /h ... σ TMS =0
Suppose we measure the same two samples (Test and TMS) at double the field then, the shift in resonance frequency would be
(vTest - vTMS ) = 2g N µ N B0 (1- σ Test ) /h
That is the frequency difference between test and reference signal gets doubled. Since the shift in resonance position is due to the chemical environment this must be independent of the applied field. This could be achieved through dividing the shift in field (for constant frequency measurement) or shift in frequency (for constant field measurement) by the respective spectrometer field or frequency.
= (vTest - vTMS )/vop
where, vop is a operating frequency of the spectrometer.