Problem_set_6

# Problem_set_6 - Physics 545 Problem Set 6 Due 1 Heat...

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Unformatted text preview: Physics 545 Problem Set 6 Due 1. Heat sapecity of ,onewdimeasicnal lattice..- Show that: the. heat. seesaw at e monstemie lattice in one diWETJSiOn in the Debt"e appraximssiss preportisfiafi to T/9D for 19w temperatures T << 90} where " is. the effective ' Debit: temretlatur‘e "in the.adimensionvdefinesi-“as One e ﬁwélkga; hére is t'heﬂisltzma‘ajnn gens-taut and a the in§'§¥.et9m19'ﬁepémtiws 2. Heat capacity from internal degrees of freedom. (3) Consider a two~level system with an energy splitting kBA between upper and lower states; the split- ting may arise from a magnetic ﬁeld or in other ways Show that the heat capac— ity per system is _ LU— "— (A/T)2 eMT C * (er)A “‘1‘” (1+ 8"”)2' The function is plotted in Fig. 12. Peaks of this type in the heat capacity are often known as Schottky anomalies. The maximum heat capacity is quite high, but for T < A and for T > A the heat capacity is low. (1)) Show that for T >> A we have C E kB(A/2T)Z+- - - . The hyperﬁne interaction between nuclear and electronic magnetic moments in paramagnetic salts (and in systems having electron spin order) causes splittings with A z i to 100 mK. These splittings are often detected experimentally by the presence of a term in l/T2 in the heat capacity in the region T> A. Nuclear electric quadrupole interactions (see Chapter 16) with crystal fields also cause splittings, as in Fig. 13. 0.5 0.4 9 w .0 to Heat capacity in units 128 0.1 x=T/A Figure 12 Heat capacity of a two—level system as a function of TIA, where A is the level splitting. The Schottky anomaly is a very useful tool for'deter- mining energy level splittings of ions in rare~earth and transition-group metals, compounds, and alloys. 3, Heat capacity of magnons. Use the approximate magnon dispersion relation to = Ak'2 to ﬁnd the leading term in the heat capacity of a three~dimensional fer- romagnet at low temperatures kBT < j. The result is 0.113 kn(knT/fz.A)3"2, per unit volume. The zeta Function that enters the result may be estimated numerically; it is tabulated in Jahnke-Emde. Heat capacity meaSurements on YIG are re— viewed by A. ]. Henderson, Jr, D. G. Onn, H. Meyer, and J. P. Remeika, Phys. Rev. 185, 1218 (1969). 4- Discuss the specifis heat of a twOedimensippal square lattice with a near-est neighbor separatinn '.a'- on the basis, of the -Debye'~appro2simationv Show that; at. low temperatures the spesifitz heat 'i.S_ propertional 1:0ng- [See 'G-F- Newell, J, Chely- Phys. 23;, .2341 (1955).] 5. Explain why the speeifio heat of "black body" radiation is always proportional to 1‘3 whereas for a solid in the Debye approximation this is true only at low temperatures. ...
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Problem_set_6 - Physics 545 Problem Set 6 Due 1 Heat...

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