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Spin, Magnetic Moment and Hund's Rules

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Spin, Magnetic Moment and Hund's Rules Origins of Magnetic Moments and Magnetism Magnetic moments are always associated with angular momentum. • Orbital angular momentum of electrons in atoms • Spin angular momentum of electrons in atoms • Spin angular momentum of neutrons and protons (or alternatively quarks) in nuclei Origin of electron Magnetic Moment – Orbital z B q Current Loop: m=IA r0 Consider an electron in a circular orbit y of radius r around a nucleus dl f Angular Momentum: L = r ×p x e- 2 r 22mm L= rmv = rm =ee r2 = A e TTT + q Magnetic Moment: m = IA = e A T m q gyromagnetic ratio: =e Lm2 e This classical picture gives the correct gyromagnetic ratio for electron orbital angular momentum. Orbital Magnetic Moment L= L2 = l( l +1) ; Lmz = orbital magnetic moment is e m ==m; 2me e m ==mBB; 2me e B = Bohr Magneton 2me http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html Spin moment • Electrons, protons and neutrons have spin ½ • z-component of angular momentum can be ± ½ • Classically, a rigidly rotating body with mass m and charge q distributed in the same way would give =q/(2m). • For an electron e=ge e/(2me) where ge= 2.0023 • Dirac: ge=2, corrections come from QED • For a nucleus or nucleon: = g e/(2mp) where g~1 • Nuclear moments are about 1000 times smaller than atomic and electronic magnetic moments. Summary Magnetic moment of atoms and solids are associated with angular momentum (similar to classical picture) 2 Angular momentum hasL= two contributors: L = l( orbital l +1) ; and spin Angular momentum is quantized in units of ℏ, magnetic moments are quantized in units of Bohr magneton. Lmz = Orbital quantum numberorbital l = 0,1,2…….(n magnetic-1) moment is e m ==m; Spin (S) quantum number = 1/2 m =-1 to +1 in integer steps2me S = ℏ 1ൗ (1ൗ + 1) e 2 2 m ==mBB; 1 S = ± ℏ 2me z 2 e m = ±휇 = Bohr Magneton s B B 2m e Total angular momentum quantum number , J = L+S J=ℏ 퐽(퐽 + 1) -24 2 B = 9.27 X 10 Am Hund’s Rules of electronic configuration • Rule 0: L=0, S=0 for filled shells or subshells. • Rule 1: State with highest S has lowest energy. (Most important for solids). S =S Sz • Rule 2: State with highest L has lowest energy(subject to Pauli exclusion principle). L =S Lz • Rule 3: If the subshell is less than or half filled, J=|L-S|; if more than half-filled J=L+S. This is due spin-orbit coupling. http://quantummechanics.ucsd.edu/ph130a/130_notes/node387.html Hund’s First Rule (result of coulomb repulsion and Pauli exclusion principle) • The Hamiltonian of a multi-electron atom must also include 2 electron-electron repulsion energy term: +e /rij where rij is the distance between electrons labeled by i and j. • To minimize coulomb repulsion, the electrons will try to stay as far away as possible. This is allowed when electrons occupy different orbitals. Electrons on same orbital are on top of each other which maximizes repulsion. • Same spin orientation makes sure that the probability of two electrons occupying the same orbital is zero (Pauli exclusion), thereby minimizing Coulomb repulsion at the same time. So, electrons will prefer to occupy different orbital with spins pointing in the same direction. • Therefore, the total spin is such that S =S Sz State with highest S has lowest energy S=1, Lower energy S=0, Higher energy S=1, but disallowed by Pauli exclusion principle Is this a low energy state? It is quantum-mechanically possible for an electron to jump into the vacant orbital (if allowed by pauli exclusion). So there are two states with S=0, one has higher energy than the other. So on an average it will be a high energy state. Why are half-filled orbitals preferred (lower energy). Answer: It minimizes coulomb repulsion (Hunds first rule). Electrons occupying same orbital with opposite spins raises the energy. Half-filled configuration partially-filled configuration p p p px py pz x y z S=3/2, lower energy since S=1/2, Higher energy. electrons occupy different some electrons occupy orbital same orbitals Again, state with highest S has lowest energy Hund’s Second Rule (minimizes Coulomb energy) • For large L value, some or all of the electrons are orbiting in the same direction. • That implies that they can stay a larger distance apart on the average since they could conceivably always be on the opposite side of the nucleus. • For low L value, some electrons must orbit in the opposite direction and therefore pass close to each other once per orbit, leading to a smaller average separation of electrons and therefore a higher energy. Hund’s Second Rule l = 0 More lobes make it easier for electrons to avoid one another l = 1 l = 2 l = 3.
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