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17-6. Use either the expression for(E) n Example 17-1 or the one in Problem 17-5 to show that hy B T→0 1 (E) → as and that E) → 0 as T→00



problems it references

EXAMPLE 17-1 Derive an equation for (E) for the simple system of a (bare) proton in a magnetic field B SOLUTION: According to Equation 14.16, the energy can take on one of the two values where γ is the magnetogyric ratio. The partition function consists of just two terms: The average energy is obtained from either Equation 17.20 or 17.21: (E) =-(aane), =-Q(12) (a) 1 (ag This expression for (E) (in units of hy B,/2) is plotted against 7 (in units of hy B,/2k,) in Figure 17.2. Note that ( E) -h γ B1/2 as T-+0 and that (E)-+ 0 as T oo. As T0, there is no thermal energy, so the proton orients itself parallel to the magnetic field with certainty. As T → oo, however, the thermal energy of the proton increases to such an extent that the proton is equally likely to point in either direction.


17-5. Show that the partition function in Example 17-1 can be written as Q (B, B) = 2 cosh ( phy4) = 2 cosh (ma 2k3T hy B Use the fact that d cosh x/dx sinh x to show that 2 2 2k, T

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