Moment Magnetic in a Magnetic Field
=0
In a uniform magnetic field, a torque will force the magnetic moment to precess about the direction of B. In a non-uniform magnetic field, a force is generated along (or opposite to) the gradient of B.
Fz = z dBz dz
= B
S Ag, H Stern-Gerlach (1922), Phipps-Taylor (1927) N
l should lead to (2l+1) beams s should lead to 2 beams if s=1/2
Magnetic Moment and Angular Momentum
N S
B
Potential energy of magnetic moment in a magnetic field B
U m = B cos
I
A
Magnetic moment of a current loop
= IA
Magnetic moment of an electron in a circular motion
= efr
2
L = mvr = 2mfr 2
=
Um =
e L 2m
Potential energy of an atom in a magnetic field B Let B || z-axis
Lz = L cos = ml
e Um = ml B B ml B 2m
e LB cos 2m
Bohr magneton=5.79x10-5 eV/T
Electron Spin and Magnetic Moment
Electron spin is a quantum mechanical property. Electron spin quantum number s = 1 2 Spin angular momentum S = s(s + 1) = Space quantization of electron spin z-component of spin angular momentum s z = ms = 2 Spin magnetic moment
s =
e S m
3 2
z-component of spin angular momentum sz = Gyromagnetic ratio for orbital angular momentum Gyromagnetic ratio for spin angular momentum
e = B = 5.79 10 5 eV T 2m
L
= =
e 2m e m
S
Energy ordering of quantum states in multi-e atoms
The lower l states has more penetration of the inner electron cloud, hence the greater effective nuclear charge and lower the energy. f f d d p p s n=1 s 2 3 4 5 s p s d p s
Electron Configurations of the Atoms
Shells K (n=1) L (n=2) l 0 (s) 1 (p) 0 (s) 2 (d) 1 (p) 0 (s) 3 (f) 2 (d) 1 (p) 0 (s) 0 0, 1 0 0, 1, 2 0, 1 0 0, 1, 2, 3 0, 1, 2 0, 1 0 ml ms 1/2 (1/2)3 1/2 (1/2)5 (1/2)3 1/2 (1/2)7 (1/2)5 (1/2)3 1/2
Max.# of electrons
Elements H, He (1s2) Li, Be, B, C, N, O, F, Ne (1s22s22p6) Ar (1s22s22p63s23p6)
2 6 2 10 6 2 14 10 6 2
M (n=3)
N (n=4)
Kr (1s22s22p63s23p63d104s2 4p6)
Pauli exclusion principle: no two electrons in an atom can have the same set of quantum numbers (n, l, m, ms).
Ionization Energy
Ionization energy: the energy required to remove an electron from an atom For H, the ionization energy is E0=13.6 eV For He, without e-e interaction, electrons each binding energy is
Z 2 E0 E1 = 2 = 4(13.6 ) = 54.4eV n
But the experimental value is 24.6 eV.
Screening effect: the outer electrons only see partially the positive ion core Binding energies of the alkali metals (2s, 3s, ..) are the lowest, and the closed shell inert gases are the highest.
Total Angular Momentum for single-electron atoms
Classically, total angular momentum J=S+L can be in any direction. Quantum mechanically J has the magnitude (expectation value) where
j = l + s, l + s 1,...., l s
j ( j + 1)
z-components m j = j , j + 1,..., j 1, j
L and S cannot be parallel
J min > L S
J max < L + S
l=2, s =1/2
Spin-orbit Interaction
L p e In electrons frame p e B
B=
0 I
2r
=
0 ef
2r
=
0e L 4me r 3
0 = 4 10 7 Tm A
For n=2, l=1, B~0.4 T
The magnetic moment of the electron will interact with the magnetic B from the circulating proton to create a potential energy U = s B = sz B = B B S L
B L S
2p3/2 2p 1s Fine-structure splitting For n=2, l=1, U~0.045 meV
U=2B
external j j-1
2p1/2
B L S
-j+1 -j Zeeman effect
Correction of energy levels of H atoms
3d 5 2
E3
Including the spin-orbit interaction
3 p1 2
E = 1 1 dV 4m 2 c 2 r dr
2
3s, 3p, 3d
3 p3 2 3d 3 2 3s1 2
SL
=
E2
2 p3 2 2 s1 2 2 p1 2
4m 2 c
[ j ( j + 1) l (l + 1) s(s + 1)] 2...