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Exercises
Appendix B Exercises
Chapter 1 1. Verify that the magnetic field H, far from the source, due to a ring (Eq. (1.6)), is the same as the dipolar field due to a permanent magnet (Eq. (1.8)). 2. Consider an uniformly magnetized sphere of radius R and magnetization M along the z axis. Obtain the demagnetization factor for this case, using for this purpose Eq. (1.18).
Chapter 2 φ} and { A , φ }, given by Eqs. (2.A.13) and 1. Considering equivalent gauges { A, (2.A.14), shows that these potentials create the same magnetic field and electric E}, i.e., field { B, × A = B, × A = ∇ ∇ − ∂ A = E. − ∂ A = −∇φ −∇φ ∂t ∂t
(B.1) (B.2)
Chapter 3 1. Write the operators Jx , Jy , and Jz on the matrix form, considering j = 1. 2. Obtain the Zeeman energy due to a single electron, considering only the spin contribution.
Chapter 4 1. Prove the following: CB = CM + T
∂ M 2 ∂ M −1 , ∂ T B ∂ B T
(B.3)
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Fundamentals of Magnetism
where CB and CM are the specific heat with constant magnetic field and magnetization, respectively. 2. Prove the following: ∂M CB ∂B T = ∂M , (B.4) CM ∂B S
where CB and CM are the specific heat with constant magnetic field and magnetization, respectively.
Chapter 5 1. Considering the mean number of particles as a sum of the mean occupancy number of each level, i.e., N=
n¯ k
(B.5)
k
obtain the mean occupancy number n¯ k for those three different statistics (FD, BE, and MB).
Chapter 6 1. Evaluate the density of states for a 1D and 2D Fermions gas. 2. Evaluate the Grand canonic potential of an electron gas in the high temperature limit (i.e., Maxwell–Boltzmann framework), using, however, Eq. (5.74) instead of Eq. (5.72) into Eq. (6.21). Recover Eq. (6.41). 3. Show that for high temperature the fugacity tends to zero, i.e., T → ∞ ⇒ z = eμβ → 0.
Chapter 7 1. Obtain the amplitude of the oscillations found on the magnetization of a diamagnetic gas under high magnetic field, i.e., the de Haas–van Alphen effect. 2. Consider the hydrogen atom discussed in Complement 3.A and evaluate the diamagnetic susceptibility and magnetization of a 1s electron. 3. Derive the magnetization and magnetic susceptibility for the low magnetic field and high temperature regime (Eqs. (7.47) and (7.48)), but starting from Eq. (7.57). 4. Evaluate the first term of the integral of Eq. (7.28) to obtain the result of Eq. (7.29).
Appendix B Exercises
231
Chapter 8 1. Is CM = 0 expected? Verify this result considering the thermodynamic relationship between CB and CM CM = CB − T
∂ M 2 ∂ M −1 . ∂ T B ∂ B T
(B.6)
2. Verify, using the relation CM
3. 4. 5.
6.
∂ S =T , ∂ T M
(B.7)
that the specific heat at constant magnetization CM is zero. Since it is not possible to do it analytically for the whole range of x = b/t, do it for the x → 0 limit for both, quantum and classical cases. Show that Brillouin function B j (x) recovers the Langevin function L(x) for the j → ∞ limit. Prove that the total magnetization of an electron gas with spins is given by M = (N⇑ − N⇓ )μB . Evaluate the Pauli susceptibility, i.e., the paramagnetic contribution to the susceptibility at low temperature limits (F kB T ). However, instead of using Eq. (8.112), start from q⇑ (T, B, z) and q⇓ (T, B, z) (Eqs. (8.105) and (8.109), respectively), evaluate the corresponding Grand canonic potential (T, B, z) = − β1 q(T, B, z) and, finally, the total magnetization M = M⇑ + M⇓ . Obtain the non-perturbed energy, as well as the first- and second-order energy corrections of the Hamiltonians presented below. (a) Nondegenerated case: ⎛
1 H0 = ⎝ 0 0
0 3 0
⎞ ⎛ 0 0 0 ⎠, W = ⎝ 1 −2 0
1 0 0
⎞ 0 0 ⎠. 1
1 0 1
⎞ 0 1 ⎠. 1
(B.8)
(b) Degenerated case: ⎛
1 H0 = ⎝ 0 0
0 1 0
⎞ ⎛ 0 0 0 ⎠, W = ⎝ 1 −2 0
(B.9)
7. Obtain χ0 (Eq. (8.B.18)) and show that this term ( j = 0) is responsible for the first term (24) of Eq. (8.B.22).
232
Fundamentals of Magnetism
Chapter 9 1. Prove that symmetric (antisymmetric) spatial wave functions lead to antiparallel (parallel) alignment of the spins. 2. Obtain the Hamiltonian of a s = 1/2 spin with local magnetocrystalline anisotropy.
Chapter 11 1. Show that the maximum at ξ+− goes to zero at T0 .
Chapter 12 1. Consider a dimer with s1 = s2 = 1/2 and write the following basis for this system: |m s1 , m s2 and |s, m s , where |s1 − s2 | s s1 + s2 . 2. Derive the magnetic susceptibility due to a s1 = s2 = 1/2 dimer with an isotropic Heisenberg interaction added to a dipolar one.
Chapter 1 1. Verify that the magnetic field H, far from the source, due to a ring (Eq. (1.6)), is the same as the dipolar field due to a permanent magnet (Eq. (1.8)). 2. Consider an uniformly magnetized sphere of radius R and magnetization M along the z axis. Obtain the demagnetization factor for this case, using for this purpose Eq. (1.18).
Chapter 2 φ} and { A , φ }, given by Eqs. (2.A.13) and 1. Considering equivalent gauges { A, (2.A.14), shows that these potentials create the same magnetic field and electric E}, i.e., field { B, × A = B, × A = ∇ ∇ − ∂ A = E. − ∂ A = −∇φ −∇φ ∂t ∂t
(B.1) (B.2)
Chapter 3 1. Write the operators Jx , Jy , and Jz on the matrix form, considering j = 1. 2. Obtain the Zeeman energy due to a single electron, considering only the spin contribution.
Chapter 4 1. Prove the following: CB = CM + T
∂ M 2 ∂ M −1 , ∂ T B ∂ B T
(B.3)
230
Fundamentals of Magnetism
where CB and CM are the specific heat with constant magnetic field and magnetization, respectively. 2. Prove the following: ∂M CB ∂B T = ∂M , (B.4) CM ∂B S
where CB and CM are the specific heat with constant magnetic field and magnetization, respectively.
Chapter 5 1. Considering the mean number of particles as a sum of the mean occupancy number of each level, i.e., N=
n¯ k
(B.5)
k
obtain the mean occupancy number n¯ k for those three different statistics (FD, BE, and MB).
Chapter 6 1. Evaluate the density of states for a 1D and 2D Fermions gas. 2. Evaluate the Grand canonic potential of an electron gas in the high temperature limit (i.e., Maxwell–Boltzmann framework), using, however, Eq. (5.74) instead of Eq. (5.72) into Eq. (6.21). Recover Eq. (6.41). 3. Show that for high temperature the fugacity tends to zero, i.e., T → ∞ ⇒ z = eμβ → 0.
Chapter 7 1. Obtain the amplitude of the oscillations found on the magnetization of a diamagnetic gas under high magnetic field, i.e., the de Haas–van Alphen effect. 2. Consider the hydrogen atom discussed in Complement 3.A and evaluate the diamagnetic susceptibility and magnetization of a 1s electron. 3. Derive the magnetization and magnetic susceptibility for the low magnetic field and high temperature regime (Eqs. (7.47) and (7.48)), but starting from Eq. (7.57). 4. Evaluate the first term of the integral of Eq. (7.28) to obtain the result of Eq. (7.29).
Appendix B Exercises
231
Chapter 8 1. Is CM = 0 expected? Verify this result considering the thermodynamic relationship between CB and CM CM = CB − T
∂ M 2 ∂ M −1 . ∂ T B ∂ B T
(B.6)
2. Verify, using the relation CM
3. 4. 5.
6.
∂ S =T , ∂ T M
(B.7)
that the specific heat at constant magnetization CM is zero. Since it is not possible to do it analytically for the whole range of x = b/t, do it for the x → 0 limit for both, quantum and classical cases. Show that Brillouin function B j (x) recovers the Langevin function L(x) for the j → ∞ limit. Prove that the total magnetization of an electron gas with spins is given by M = (N⇑ − N⇓ )μB . Evaluate the Pauli susceptibility, i.e., the paramagnetic contribution to the susceptibility at low temperature limits (F kB T ). However, instead of using Eq. (8.112), start from q⇑ (T, B, z) and q⇓ (T, B, z) (Eqs. (8.105) and (8.109), respectively), evaluate the corresponding Grand canonic potential (T, B, z) = − β1 q(T, B, z) and, finally, the total magnetization M = M⇑ + M⇓ . Obtain the non-perturbed energy, as well as the first- and second-order energy corrections of the Hamiltonians presented below. (a) Nondegenerated case: ⎛
1 H0 = ⎝ 0 0
0 3 0
⎞ ⎛ 0 0 0 ⎠, W = ⎝ 1 −2 0
1 0 0
⎞ 0 0 ⎠. 1
1 0 1
⎞ 0 1 ⎠. 1
(B.8)
(b) Degenerated case: ⎛
1 H0 = ⎝ 0 0
0 1 0
⎞ ⎛ 0 0 0 ⎠, W = ⎝ 1 −2 0
(B.9)
7. Obtain χ0 (Eq. (8.B.18)) and show that this term ( j = 0) is responsible for the first term (24) of Eq. (8.B.22).
232
Fundamentals of Magnetism
Chapter 9 1. Prove that symmetric (antisymmetric) spatial wave functions lead to antiparallel (parallel) alignment of the spins. 2. Obtain the Hamiltonian of a s = 1/2 spin with local magnetocrystalline anisotropy.
Chapter 11 1. Show that the maximum at ξ+− goes to zero at T0 .
Chapter 12 1. Consider a dimer with s1 = s2 = 1/2 and write the following basis for this system: |m s1 , m s2 and |s, m s , where |s1 − s2 | s s1 + s2 . 2. Derive the magnetic susceptibility due to a s1 = s2 = 1/2 dimer with an isotropic Heisenberg interaction added to a dipolar one.