Assignment 4 Due Friday, April 27, 2018 Problems from Orchard’s Magnetochemistry: 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) Ferromagnetism and ferromagnetically coupled – and the distinction between the two terms. (b) Antiferromagnetism 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 saturation measurements of paramagnetic compounds, which are carried out at high fields and low temperatures (µBH >> kBT). For cases where spin-only magnetism applies, like octahedral Cr3+ complexes, high-spin Fe3+ (Mn2+) complexes, or Gd3+ complexes, the saturation magnetizations are respectively 3.0 µB, 5.0 µB, and 7.0 µB per ion. If ferromagnetic gadolinium metal is placed in high magnetic field at low temperature, conditions where the f7 magnetic moments are maximally aligned, the magnetization is also close to 7.0 µB per Gd atom. However, if one calculates the effective magnetic moments of these ions 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 magnets. (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). .