The Larmor (or resonant) frequency ω0 is the frequency at which the nuclide precesses about the magnetic field. The resonant frequency is equal to the magnetogyric ratio γ (specific to the nuclide) times the magnetic field B (Brandolini, 2004):
the nuclide 13C at 75 MHz. From the equation above, the magnetogyric ratio γ is constant so that γ =ω0 /B = ω’0 /B’, where ω’0 is the resonant frequency when the magnetic field B’ = 1.5 T. Solving for ω’0 : ω’0 = (ω0 B’)/B what is the mean B and B
This is explained in the sentence directly above: the single prime corresponds to the resonant frequency when the magnetic field is 1.5 T. You are asking what is meant by B’, but if you look at the sentence above, it was just defined: B’ = 1.5T. It is the magnetic field at 1.5T. B’’ is just a different value of the magnetic field (in this case 4 T) where we are trying to find the frequency w0’’ that corresponds to it.
From this equation, if you know the frequency ω0 and the magnetic field B, then the ratio of these is the gyromagnetic ratio. We know the frequency at 6.9T from the reference cited above. Therefore, to find the frequency at a different magnetic field, we just use the equation w0/B = γ = constant. So another set of corresponding values of w0 and B, call these new values w0’’ and B’’, will also have the same ratio: w0’’/B’’ =γ constant = w0/B. Since we now have w0’’/B’’ = w0/B, we can multiply both sides by B’’ to get: w0’’ = w0 * B’’/B . Hopefully you can now see where that equation comes from.
The reason I didn’t put the calculation down in this case, is because it is EXACTLY the same as the calculation before it, but with different values. You can just follow the equations that were used in the example above it, putting in the