Quantum Theory of Magnetism (Diamagnetism & Paramagnetism) B.Sc.-2/P.G. By Kamad Nath Shandilya Guest Assistant Professor Department of Physics B.N. College Patna University, Patna
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Introduction Advantage of development of quantum theory of magnetism is, diamagnetism and paramagnetism will appear as two cases of the general theory developed here. Even now ferromagnetism requires different approach but it is again quantum. Here we will consider diamagnetism and paramagnetism due to electrons which are bound to nucleus. Diamagnetism has no further issue but paramagnetism will also be considered due to free electrons known as Pauli’s paramagnetism. The previous one is called paramagnetism of insulators and the latter one is known as paramagnetism of conductors. Obviously conductors have free electrons insulators haven’t. Generally, Pauli’s paramagnetism is discussed under new topic so we too will skip it here. The theory will be developed in the following sequence • Kinetic momentum & field momentum • Magnetization density & Susceptibility • General formulation of atomic susceptibility ▪ Diamagnetism or Larmor Diamagnetism: Susceptibility of insulators with all shells filled (J=0) ▪ Ground state of atoms or ions with a partially filled shell: Hund’s rule ▪ Susceptibility of atoms/ions with a partially filled shell: Paramagnetism 1.Van-Vleck paramagnetism(J=0) 2.Langevin paramagnetism(J≠0) * Quantum mechanical itroduction of magnetic dipole moment * Magnetization & Magnetic susceptibility: Curie-Brillouin law Kinetic Momentum & Field Momentum
Langrangian of a charged particle in electromagnetic field is, 1 ℒ= m푞̇ 2 – QΦ(q) +Q풒̇ ⋅A(q), ….. (1) 2 where m, Q and q are the mass, charge and general notation for space coordinate respectively. Φ(q) and A(q) are scalar and vector potential respectively. Now, momentum 휕ℒ p= = m풒̇ +Q A(q) …… (2) 휕푞̇
(Here momentum has been calculated axis wise like px, py, pz and then assembled as p = px +py +pz). m풒̇ is defined as Kinetic momentum and Q A(q) is defined as Field momentum. So,
p or ptotal = pkinetic + pfield. ….. (3) Magnetization and Susceptibility • Magnetization Density; M(B) At absolute temperature T=0, 1 ∂E (B) M(B) = - 0 ….. (4) V ∂B V: Volume of quantum mechanical system B: Uniform magnetic field
E0(B): Ground state energy in presence of field B At temperature T when thermal equilibrium has attained, then magnetization density will be calculated as statistical average as following,
−E /k T ∑n Mn(B)e n B M(B,T)= −E /k T ….. (5) ∑n e n B
M(B,T): Thermal equilibrium average of the magnetization density of each excited state n of energy En(B). And,
1 ∂En Mn(B) = - ….. (6) 푉 ∂B • Susceptibility;휒 1 휕2E At T=0, 휒 =-μ 0 v 휕B2 ∂M(B,T) 1 휕2E At T, 휒 = μ = -μ n , μ: permeability of medium ∂B v 휕B2 퐆퐞퐧퐚퐫퐚퐥 Formulation of atomic Susceptibility The Hamiltonian of an ion or atom in the presence of a uniform magnetic field, e H = T + gs BSz ….. (7) 2m Let us explain the kinetic energy term, T and the term representing the energy due to interaction between spins of 푒 electrons bound to nucleus and magnetic field ,gs BSz 2푚 separately. 1 • T = ∑ p2 , summation runs over total number of 2m i i,kinetic electrons bound to nucleus and obviously i denotes ith electron. Now pi,kinetic = pi,total –pi,field, here pi,field = -eA(ri), ri : position of ith electron in X-Y plane. 1 2 So T = ∑ [퐩 + e퐀(퐫 )] , here pi denotes pi,total . 2m i i i 1 Now choosing A = - (r ×B), so that B = ∇ × A and ∇ ⋅ A = 0. 2 1 1 So T = ∑ [퐩 − e (ri ×B)]2 , on simplifying we will get the 2m i i 2 result, 2 e e 2 2 T = T0 + 퐋 ⋅ 퐁 + B2∑ (퓍 + 퓎 ); …… (8) 2m 8m i i i 1 2 where T0 = ∑ 퐩 , L = ∑ 퐫 × 퐩 ; total electronic orbital 2m i i i i i 2 2 2 angular momentum and 퓍i +퓎i =ri .
• Interaction energy of a magnetic dipole of dipole moment μdip and a magnetic field B is given by –μdip⋅ B. Magnetic dipole e moment of ith electron due to its spin is given by –gs si , gs 2m =2.0023, electronic g-factor, which is calculated in relativistic quantum mechanics1. As usual, magnetic field is assumed along z- axis. The interaction energy of the field with all electrons bound e e to nucleus UB,spin = gs 퐒 ⋅ 퐁 =gs BS ….. (9) 2m 2m z i i where Sz =∑i 퐬z; summation of z- components of each spin s . Finally, adding (8) and (9), Hamiltonian will be,
2 e e 2 2 e H = T0 + 퐋 ⋅ 퐁 + B2∑ (퓍 + 퓎 ) + gs BS 2m 8m i i i 2m z e e2 e 2 2 2 Or, H = T0 + 퐋 ⋅ 퐁 + B ∑ (퓍 + 퓎 ) + gs 퐁 ⋅ 퐒 2m 8m i i i 2m 2 e e 2 2 Or, H = T0 + (퐋 + g 퐒) ⋅ 퐁 + B2∑ (퓍 + 퓎 ) 2m s 8m i i i Except T0 the terms in H are perturbations, that is, perturbation,
e e2 ΔH = (퐋 + g 퐒) ⋅ 퐁 + B2∑ (퓍2 + 퓎2) 2m s 8m i i i Now applying the time independent perturbation theory up to second order correction of energy3,
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1 휕2ΔE Therefore susceptibility, 휒 =-μ n ….. (11) v 휕B2
Here in expression of 휒, we have considered ΔEn, that is correction in energy eigenvalue not total energy because 휒 is measure of ease of magnetization and the magnetization is produced when the material or medium is exposed to an external field, which again causes perturbation ΔH. Due to this perturbation correction terms in energy are produced. Here we have considered it up to 2nd order.
Larmor Diamagnetism or Langevin Diamagnetism: Susceptibility of insulators with all shells filled(J=0) If all atoms or ions of a solid have all electronic shells filled, then each atom or ion has zero spin, S=0 & zero total angular momentum L=0 & hence total angular momentum, J=0. Notice that S, L & J are determined for last filled orbitals because inner orbitals are generally2 filled and hence possess S=0, L=0 & J=0.
As for example, For d10; ↑↓ ↑↓ ↑↓ ↑↓ ↑↓
For d-orbital l =2, therefore ml =2 1 0 -1 -2, ml values of electrons are written just below their boxes in which electrons are filled. 10 i i Here total orbital magnetic quantum number ml = ∑i=1 ml = 0, ml denotes magnetic orbital quantum number for ith electron. Ml denotes total magnetic orbital quantum number.
Since ml varies as -L, -L+1, -L+2…………L where L is orbital angular momentum quantum number, then L =0 1 1 As m i = for ↑ and m i = - for ↓, where m i denotes spin magnetic s 2 s 2 s quantum number.
10 ⇒ total spin magnetic quantum number ms =∑i=i msi =0 and since ms varies as -S, -S+1, -S+2…………S where S is spin angular momentum quantum number or spin, then S =0 Now total angular momentum quantum number J takes values as |L-S|, |L-S| +1, |L-S|+2,…………. L+S, then J =0 We will discuss Hund’s rule as our next topic where we will see rules for evaluation of S, L and J although one may also get much idea from above example. Again we will see in langevin’s paramagnetism that the ground state of an atom or ion will be represented by |JLSJz>. So for above example, ground state will be |0> because all S, L and J are zero. Then only the third term in equation (10) will retain,
2 e 2 2 2 ΔE0 = B < n| ∑(퓍 + 퓎 )|n > 8m i i From spherical symmetry of the closed- shell ion, 1 < 0| ∑ 퓍2 |0 > = < 0| ∑ 퓎2 |0 > = < 0| ∑ r2 |0 > i i 3 i 2 e 2 2 Therefore, ΔE0 = B < 0| ∑ r |0 > ….. (12) 12m i If there is negligible probability of being an atom or ion in higher states in thermal equilibrium, then the susceptibility of a solid composed of N such ions/atoms per unit volume, from (11) & (12),
∂2ΔE e2 휒 =-Nμ 0 = - N < 0| ∑ r2|0 > ….. (13) 휕B2 6m i This is known as Larmor diamagnetic susceptibility or Langevin susceptibility. –ve susceptibility implies induced moment is opposite to the applied field.
Ground state of atoms/ions with a partially filled shell: Hund’s rule (i) The electrons’ magnetic spin quantum number add up to give maximum possible spin quantum number S consistent with pauli’s principle. (ii) The electrons’ magnetic orbital quantum number add up to give maximum possible orbital quantum number L consistent with pauli’s principle. (iii) For an incompletely filled shell, we have J = |L-S|, for a shell less than half occupied J= L +S, for a shell more than half occupied Example 1. Cr2+ :1s22s22p63s23p63d4, here the last filled orbital is d which have four electrons, ↑ ↑ ↑ ↑
ml = 2 1 0 -1
4 1 ms =∑ m i =2, sum of four halves(+ ) ⇒S =2 i=i s 2 4 i ml = ∑i=1 ml = 2⇒ L = 2 As this is the case of less than half filled orbital, so, J = L-S=0 5 Example 2. Ce3+ : ~f1, J = 2 Example 3. Pr3+ : ~f2, J = 4
8 8 i 8 Example 4. f , ml = ∑i=1 ml =3 ⇒L= 3 and ms =∑i=i msi =3 ⇒S = 3 As it is the case of more than half filled orbital, so, J = L+S = 6. Susceptibility of atoms/ions with a partially filled shell: Paramagnetism 1. Van-Vleck Paramagnetism
If the shell has J=0, for the shells that are one electron short of being half filled, then ground state is non-degenerate and the first term in equation (10) vanishes, then
e |
4 ′ ′ Now < JLSJz| 퐋 + gs퐒|JLSJz > = g< JLSJz| 퐉|JLSJz >,
J(J+1)−L(L+1)+푆(푆+1) where g = 1+ , this is known as Lande g-factor. 2J(J+1)
′ ′ ⇒ < JLSJz| 퐋z + gs퐒z|JLSJz > = g< JLSJz| 퐉z|JLSJz >
′ A < JLSJ | 퐉 |JLSJ > 퐉 훿 ′ gain, z z z = z 퐉z퐉z, that is off diagonal elements of the matrix will be zero.5
e e ′ So, ΔE0 =g B J 훿 ′ = g Bmjℏ (훿 ′ =1, when J = J , so the 2m z 퐉z퐉z 2m 퐉z퐉z z z corresponding term will only retain.) Here we have used Jz= mjℏ, mj is total magnetic quantum number corresponding to total angular momentum quantum number j and mj varies from –J, -J+1, ……..J. Or, ΔE0 = mjgμBB, ….. (15) eℏ where μ = = 1 B.M., (B.M. is for Bohr- Magneton) &1 B.M. B 2m =0.927×10-23J/T. Or, ΔE0 = μz,푑ipoleB, where μz,푑ipole= mjgμB, this is resultant magnetic dipole moment of electron along z- axis or along magnetic field. As the component of magnetic dipole moment parallel to field or along +ve z- axis corresponds to lower state and that antiparallel to field or along –ve z-axis correspond to higher state and further –ve values of mj correspond to obtuse angle
with field and +ve values of mj correspond to acute angle. Assigning a negative sign in (15) in view of the above fact then, ΔE0 =Emj = -mjgμBB ….. (16)
Magnetization and Magnetic Susceptibility: Curie- Brillouin Law As Emj = -mjgμBB, means in a magnetic field an atom or ion with angular momentum J has 2J+1 equally spaced energy levels. Then according to statistical mechanicsthe magnetization M is given by, J −E푚푗/k퐵푇 ∑ 푚푗g휇Be M = N −J J −E푚푗/k퐵푇 ∑−J e where N is the number of atoms/ions per unit volume and k퐵 is Boltzmann’s constant. 푚 gμ B (i) When 푗 B ≪1, that is, at high temperature, k퐵T
푚 gμ B em푗gμBB/k퐵T ≈1+ 푗 B k퐵T and by writing out the sums, paramagnetic susceptibility,
M 2 χ =μ =Ng2J(J+1)μ /3kBT B B Np2 μ2 C Or, χ = μ eff B = μ …… (17) 3k퐵T T where peff: Effective number of Bohr Magnetons, defined as, peff =g√J(J + 1) ……. (18) Equation (17) is known as Curie’ law and C is called Curie’s constant.
Plot of 1/χ vs T for paramagnetic Gadolinium salt, Gd(C2H5SO4)3.9H2O.
3k퐵 1 Slope = 2 2 = NpeffμB C (ii) At low temperature and strong magnetic field that is 푚 gμ B 푗 B ≫1 then, k퐵T
M =NgJμBBJ(퓍) …….. (19)
where 퓍 = gJμBB/kBT and BJ(퓍) is called Brillouin function. Equation (19) is called Curie- Brillouin law.
(Source: Charles Kittel, solid state physics,7e)
In above plot only spin has been considered not orbital angular momentum. Remarks Paramagnetism • Atoms, molecules and lattice defects possessing an odd number of electrons, e.g. free sodium atoms, gaseous NO, organic free radicals, F- centers in alkali halides. • Free atoms and ions with a partly filled inner shell, e.g. transition elements, ions isoelectronic with transition elements, rare earth and actinide elements. • A few compounds with an even number of electrons e.g.
molecular o2, organic biradicals. • Metals, which show pauli paramagnetism too. Diamagnetism • Li+, Na+, Mg2+, Ca2+,F-, Cl-, Cu, Zn, Bi, Ag, Au, Sb, NaCl, water etc show diamagnetism. Note: B.Sc. students may start directly from equation (15) for Paramagnetism. Diamagnetism has been already uploaded, so they may skip it here. Superscript Remarks
1. Pauli equation and magnetic moment of electron, page no.-712 Quantum mechanics, Bransden & Joachain, 2e, Pearson Education, (Here gs has been predicted theoretically) 2. Cu is diamagnetic, whose electronic configuration is Ar4s13d10, here s-orbital is half filled, so it should be paramagnetic but here diamagnetism is dominant. 3. Time independent perturbation theory, page no-263 & 268, D. J. Griffiths, introduction to quantum mechanics,2e, pearson 4. Weak field Zeeman effect, page no-289 & 290, D. J. Griffiths, introduction to quantum mechanics,2e, pearson 5. Degenerate Perturbation theory, Theorem, page no-271, D. J. Griffiths, introduction to quantum mechanics,2e, pearson References 1. Ashcroft & Mermin, solid state physics, India edition 2. Charles Kittel, solid state physics, 7e 3. A.J. Dekker, solid state physics, Macmillan publishers India Ltd