Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

• According to statistical mechanics (Ch. 15), if a particle is in thermal equilibrium at temperature T and it can exist in one of two quantum states with energies E1 and E2, then the probabilities P(E1) and P(E2) that the particle will occupy each of the two levels are related by

where k is Boltzmann’s constant. For a proton in a magnetic field B = 1 T at temperature T = 300 K, compute the ratio P(higher energy)/P(lower energy) using the result of Problem 1. Note how close to 1 this result is. Because of thermal agitation, a collection of protons is only very weakly magnetized when placed in a strong magnetic field; that is, there is only a very slight excess of moments aligned with the field over those anti-aligned. Consequently, the proton radio signal in MRI is exceedingly weak, and several minutes of signal averaging are required to produce a high-quality image.

Problem 1

• Just as the electron magnetic moment is given approximately by the Bohr magneton (9.17), the nuclear magnetic moment is conveniently expressed in terms of the nuclear magneton,

where mp is the proton mass. (a) Find the value of μN in eV/T. Compare this with the Bohr magneton. (b) The proton’s magnetic moment is found to be 2.793μN. (This common — and somewhat ambiguous — statement means that the two values of μz are ±2.793 μN.) Find the energy-level separation for a proton in a magnetic field B = 1 T.