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ssp_23 - 7 Continued 7-11 Paramagnetism Occurence Free...

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7-11 7 Continued
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7-12 Paramagnetism Occurence: Free atoms, molecules or lattice defects with odd number of electrons. total spin cannot be zero permanent magnetic moment. (few compounds with even electron number also paramagnetic.) Free atoms or ions of transition, rare eath and actinide elements with partially filled inner shells. permanent magnetic moment, often also paramagneic in solid state. E.g., rare eath and actinide salts (outer shells screen crystal field). Metals can be para- or diamagnetic.
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7-13 Atoms with N electrons Only total angular momentum J conserved. + = N i i N i i s l J (7.27) l i , s i : orbital and spin moment of i th J g B J J h µ µ = (7.28) general case: g J = g J ( l i , s i ). Interation energy in magnetic field B = (0,0, B ): E B ( J ) = - µ J B = B J g z B J µ (7.29) J z = m J h , m J = -J, ........ ,J (7.30) (cf. hydrogen atom: m l = -l, ..... ,l )
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7-14 Classical treatment: E B = - µ B = - µ B cos θ M = µ B <cos θ > n atoms all θ possible T 0 θ = 0 perfect alignment of µ parallel B . T statistical distribution <cos θ > = 0 Fig. 7.6 In the classical treatment the angular momentum, and hence the magnetic moment µ , can point to any direction. The magnetization results from the average of cos θ over all atoms in the sample. Quantum mechanical treatment (7.30) only 2 J + 1 discreate values of J z splitting of energy levels in 2 J + 1 levels. Simplest case: single electron in partially filled shell with l = 0 * J = s = 1/2, g J = g = 2 E B = ± µ B B (7.31) only 2 possible orientaions * In a closed shell there are always pairs of electrons with l i , s i and - l i ,- s i . Therefore the total angular spin and orbital momentum of closed shells are zero.
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7-15 Fig. 7.7 splitting of energy level of a simple spin in a magnetic field B = (0,0, B ). In the low energy state the magnetic moment is parallel to the magnetic field. Fig. 7.8 fractional population of a two-level system (Fig.7.7) in thermal equilibrium at temperature T in a magnetic field B . The magnetic moment is proportional to the difference between the two curves. Equilibrium populations of levels (see Fig. 7.8) for N = N 1 + N 2 atoms )] /( exp[ )] /( exp[ )] /( exp[ 2 1 2 , 1 kT B kT B kT B N N N B B B µ µ µ + + = + m (7.32) Resultant magnetization: x n e e e e n V N N M atoms x x x x B atoms B tanh 1 2 µ µ µ = + = = (7.33) x = µ B B/ ( kT )
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7-16 1) x <<1 tanh x x kT B n M B atoms 2 µ (7.34) χ C/T ( Curie law ) (7.35) k J J g C B J 3 ) 1 ( 2 2 0 µ µ + = ( Curie constant ) (7.36) for general case (see, e.g., Kittel ), here: J = s = 1/2 ,g J = g = 2 2) x >> 1 tanh 1 saturation magnetization M = n atoms µ B (7.37) Requires very strong fields and low temperatures! e.g., B = 5T, T = 300K x of order of 10 -2 : still χ C/T Fig. 7.9 Cource of magnetization as function of x = µ B B/ ( kT ). Note the linear range, which is obeyed for most practical cases, and the occurence of saturation at very low temperatures or extremely strong fields.
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7-17 Pauli paramagnetism of conduction electrons Magnetic moment of a conduction electron: (7.26) µ z = ±µ B (7.38) (only spin, no orbital angular momentum) Fig. 7.10 Pauli paramagnetism at about absolute zero for free electrons. (Left) for B = 0 the electrons with spin up and down have the same energy. (Middle) for
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