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Home , Ferrimagnetism

Assignment 4 Due Friday, April 27, 2018 Problems from Orchard’s : Chapter 2: Exercises 2, 3, 5 Chapter 3: Exercises 1a,b,c,f, 3a,b,c, 8, 9a,b, 14a,b Chapter 4: Exercises 4, 6 Chapter 5: Exercises 1, 4, 7, 8, 11, 15

(1) Where possible, give definitions/explanations for each of the following. If not possible explain what is wrong with the terminology. (a) and ferromagnetically coupled – and the distinction between the two terms. (b) and antiferromagnetically coupled – and the distinction between the two terms. (c) Ferrimagnetism and ferrimagnetically coupled – and the distinction between the two terms. (2) In class, the susceptibility of a system with S = 1 and zero-field splitting was considered. In that treatment, second-order effects were neglected. Consider the following system of energy levels for a system with S = 1, including second-order terms (they are added as the last term in each expression – See the class notes): 2 gµ H E + = + 1 D + gµ H + ( B ) ( ) 3 B 6D 2 gµ H E − = + 1 D − gµ H + ( B ) ( ) 3 B 6D 2 gµ H E 0 = − 2 D − ( B ) ( ) 3 3D Note that the second-order shifts correspond to pushing both of the upper levels up half as much as the lower level is pushed down – this happens because the field mixes the upper two levels into the lower level and vice versa (the wavefunctions also mix). (a) Typically, temperature-dependent susceptibility data are collected by choosing a field strength and collecting data over a range of temperatures, say 4 K to 350 K. If one plots χT vs. T (as in the class notes) for this system and attempts to fit the data using the first- order expression given in the notes, what value of D would you obtain from the fit if g = 2.0, the field chosen is 1.0 T, and the true value of D/kB is 5.0 K? (b) How would you determine the true value of D? (c) How might EPR spectroscopy be used to probe the second-order effects? (Hint: What is the selection rule for EPR transitions and how will second-order mixing affect the observability of transitions? (3) We have discussed measurements of paramagnetic compounds, which are carried out at high fields and low temperatures (µBH >> kBT). For cases where spin-only applies, like octahedral Cr3+ complexes, high-spin Fe3+ (Mn2+) complexes, or Gd3+ complexes, the saturation are respectively 3.0 µB, 5.0 µB, and 7.0 µB per . If ferromagnetic gadolinium metal is placed in high at low temperature, conditions where the f7 magnetic moments are maximally aligned, the is also close to 7.0 µB per Gd . However, if one calculates the effective magnetic moments of these in the usual higher temperature regime where the Curie law applies (kBT >> µBH), one will obtain ge[S(S+1)]1/2µB = 3.87 µB, 5.91 µB, and 7.94 µB, respectively. What explains the apparent discrepancies; why are the effective magnetic moments larger than the saturation magnetizations per ion/atom? (4) In lecture, the effects of orbital contributions for ‘T-terms’ in octahedral complexes were discussed (and some graphs from Figgis & Hitchman, “Ligand Field Theory and its Applications”, Chapter 9 were presented). The ‘E-terms’ were not discussed – and for good reason. Figgis discusses such cases qualitatively and explains why this is so (Section 9.3). (a) Show whether a 2E term split in first order by spin-orbit coupling. (b) In your own words, supply an explanation for why Figgis & Hitchman don’t treat ‘E- terms’. (Hint: What is the angular momentum z-component for the two eg orbitals?)

(5) In class, zero-field splitting was been discussed only for positive values of D, but it is quite possible for a magnetic molecule to exhibit a negative ZFS parameter, opposite the sense you have seen in our EPR and magnetic discussions. Such molecules are of special interest as potential single-molecule . (a) Make a good drawing of an energy level diagram that shows how the ZFS and Zeeman splittings look for an axially symmetric molecule (E = 0) with S = 9/2 and D/ kB = –6 K. (b) Use Van Vleck’s equation to derive a formula for the susceptibility for this case. (c) Using a graphing program (Excel will do), plot χT vs. T from 0 to 300 K. Label the values at the high and low-temperature extremes (just as was done in plots supplied in your notes) and explain the significance of those values in terms of the values expected for Curie-Law cases, and the ZFS/Zeeman splitting diagram you drew in part (a).