Aproton in a magnetic field has two spin states, one aligned with the field b and the other anti-aligned. we will refer to them as ”down” and ”up”, respectively. the energies of these spin states are: edown= −hγb/2 and eup=hγb/2, where γ= 26.8×107(tesla sec)−1.
(a) what are the probabilities pup and pdown of observing the proton in its up and down states? write your answer in terms of γ,b, temperature t, and fundamental constants. (hint: recall thatp(ν) =c×exp(−βeν), where c is a constant that ensures that probabilities add up to one. you will need to determine this normalization constant.)
(b) calculate the energy difference ∆e between up and down states, in units of kj/mol, for a field strength of 5 tesla. based on the magnitude of ∆e compared to kbt, comment on your expectation for the probabilities pup and pdown at room temperature.
(c) calculate numerical values of pup and pdown for b = 5 tesla and t = 298 k.
(d) for what field strength b would the up spin state be twice as probable as the down state (pdown = 2pup) at room temperature?
(e) nmr measurements rely on population differences between nuclear spin states. given that the
largest magnetic fields ever generated in a laboratory are on the order of 100 tesla, comment on
the sensitivity challenges inherent in nmr spectroscopy.