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Lecture notes on transportation pdf
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It has been known since antiquity that \loadstone" (magnetite, Fe 3 O 4 ) and iron attract each other. Plato (428/427{348/347 B.C.) and Aristotle mention permanent magnets. They are also mentioned in Chinese texts from the 4th^ century B.C. The earliest mention of a magnetic compass used for navigation is from a Chinese text dated 1040{1044 A.D., but it may have been invented there much earlier. It was apparently rst used for orientation on land, not at sea. Thus magnetism at rst referred to the long-range interaction between ferromagnetic bodies. Indeed, the present course will mainly address magnetic order in solids, of which ferromagnetism is the most straight- forward case. This begs the question of what it is that is ordering in a ferromagnet. Oersted (1819) found that a compass needle is de ected by a current-carrying wire in the same way as by a pemanent magnet. This and later experiments led to the notion that the magnetization of a permanent magnet is somehow due to pemanent currents of electrons. Biot, Savart, and Ampere established the relationship of the magnetic induction and the current that generates it. As we know, Maxwell essentially completed the classical theory of electromagnetism.
Can we understand ferromagnetism in terms of electron currents in the framework of Maxwellian classical electrodynamics? For N classical electrons with positions ri and moments pi, the partition function is
i
d^3 rid^3 pi exp
H(r 1 ; : : : ; p 1 ; : : : )
where H is the classical Hamilton function,
H =
2 m
i
pi + eA(ri)
Here, A(r) is the vector potential related to the magnetic induction B through B = ∇ A. B is presumably due to the currents (through the Biot-Savart or Ampere-Maxwell laws), in addition to a possible external magnetic eld. The electron charge is e. But now we can substitute pi! p~i = pi + eA(ri) in the integrals,
i
d^3 rid^3 p~i exp
2 m
i
p^ ~^2 i + V
Thus we have eliminated the vector potential A from the partition function. With the free energy F =
ln Z; (1.4)
this leads to the magnetization
M =
This is called the Bohr-van Leeuwen theorem. What have we shown? We cannot obtain equilibrium ferromagnetism in a theory that
(a) is classical and (b) assumes the magnetic eld to be due to currents alone.
Which of these two assumptions is to blame? Most books treat them as essentially equivalent. The usually offered solution is that we require an intrinsic magnetic moment carried by the electrons, which is then attributed to an intrinsic angular momentum, the spin. Text books on quantum mechanics usually claim that at least non-integer spins are only possible in quantum mechanics|there is some controversy on this, though. Most books thus drop (b) by assuming electrons to carry spin and state that (a) is then broken.
We know, however, that electrons carry spin S = 1=2 (e.g., from Stern-Gerlach-type experiments) and that they carry a magnetic moment. In the following, we review the relation between angular momenta aund magnetic moments. (a) Orbital motion: We know since the days of Oersted that moving charges generate magnetic elds. We consider an electron of charge e moving in a circle of radius R with constant angular velocity !.
PPq
r
!; l
v
Its angular momentum is
l = r mev = mer (! r) = mer^2! me (r !) | {z } =
r = meR^2! = const:
if center in origin (1.6)
The magnetic eld, on the other hand, is very complicated and time-dependent. However, if the period T = 2=! is small on the relevant experimental timescale, we can consider the averaged eld. Since the Maxwell equations are linear, this is the magnetic eld of the averaged current. The averaged current is I =
e T
e! 2
The induction B can now be obtained from the Biot-Savart law,
B(r) =
dl′^ ∆r ∆r^3
PPq^1
r
r′ dl′
∆r = r r′
z
The Biot-Savart law can be rewritten as
B(r) =
dl′^ ∇
∆r
dl′^
∆r
! = ∇ A: (1.9)
Obviously, we can choose
A(r) =
dl′^
∆r
The induction B or the vector potential A can be evaluated in terms of elliptic integrals, see Jackson's book. We here consider only the limit R ≪ ∆r (the far eld). To that end, we perform a multipole expansion of 1 =∆r,
1 ∆r
jr r′j
r
jr r′j^3
r′=
r′^ =
r
r r′ jr r′j^3 r′=
r′
r
r r′ r^3
with g = 2:0023, in good approximation twice the expected moment. There is no natural classical explanation for g 2. On the other hand, the relativistic Dirac quantum theory does give g = 2. The solutions of the Dirac equations are 4-component-vector functions (\Dirac spinors"). In the non-relativistic limit v ≪ c, two of these components become small and for the other two (a "Pauli spinor") one obtains the Pauli equation
{[ (^1) 2 me
p + eA
eϕ
e me s B
j ⟩ = (E mec^2 )j ⟩; (1.22)
" ( 1 0 0 1
in spinor space
where p is the momentum operator and s is an angular momentum operator satisfying
s s =
Writing the Zeeman term as ms B, we nd
ms = e me
s = 2 B s ℏ
=^! g B
s ℏ
with g = 2: (1.24)
The interaction of the electronic charge with the electromagnetic eld it generates leads to small corrections (\anomalus magnetic moment"), which can be evaluated within QED to very high accurancy. One nds
g = 2 +
with the ne structure constant
=
4 ϵ 0
e^2 ℏc
e^2 c 2 h
The relevant leading Feynman diagrams are
s e
QQs
s
QQs
q q
q q q
In quantum physics, it is common to write angular momenta as dimensionless quantities by drawing out a factor of ℏ. Thus we replace s! sℏ, s=ℏ! s, ms! ms = gB s etc. We use this convention from now on. The derivation of the Pauli equation gives s s = 3=4. We know from introductory quantum mechanics that any angular momentum operator L has to satisfy the commutation relations [Lk; Ll] = i
m ϵklmLm, [Lk; L^2 ] = 0, which de ne the spin algebra su(2). (These commutation relations are to a large extend predetermined by the commutation relations of rotations in three-dimensional space|a purely classical concept. This does not course not x the value of ℏ, though.) It is also shown there that this implies that L^2 can have the eigenvalues l(l + 1) with l = 0; 12 ; 1 ; 32 ; 2 ; : : : and Lk, k = x; y; z, can than have the eigenvalues m = l; l + 1; : : : ; l 1 ; l. Thus the Dirac theory, and consequently the Pauli theory, describe particles with spin quantum number S = 1=2. As noted, experiments show this to be the correct value for electrons. It is very useful to introduce a representation of the spin algebra for S = 1=2. The common but by no means necessary choice are the Pauli matrices. We write sk = k 2
with the Pauli matrices
x :=
; y :=
0 i i 0
; z :=
One easily checks that [sx; sy ] = isz etc. are satis ed. Also,
s^2 =
(^2 x + ^2 y + z^2 ) =
is proportional to the unit matrix and thus commutes with everything. More generally, one can nd (2l + 1)- dimensional representations for total angular momentum l 1 =2.
We have established that electrons in solids carry magnetic moments. Quantum theory (and even QED) was needed to understand the size of the magnetic moment but not its existence. Now we know from electrodynamics that magnetic dipoles interact. The eld generated by a magnetic moment m 1 at the origin is
B(r) =
3(m 1 r)r r^2 m 1 r^5
The energy of another moment m 2 at r is then
Vdip = m 2 B(r) =
3(m 1 r)(m 2 r) r^2 m 1 m 2 r^5
This is clearly symmetric in m 1 and m 2. Can this dipole-dipole interaction explain magnetic order as we observe it? If it does, we expect the critical temperature Tc, below which the order sets in, to be of the order of the strongest dipolar interaction, which is the one between nearest neighbors. This follows within mean- eld theory, as we shall see, and is plausible in general, since the dipolar interaction sets the only obvious energy scale. A rough estimate is obtained by identifying the nearest neighbor separation with the lattice constant a and writing
kB Tc z
m^2 s a^3
= z
g^2 B 2 4 a^3
e^2 ℏ^2 4 m^2 ea^3
where z is the number of nearest neighbors. In the last step we have used g = 2. Taking z = 8 and a = 2:49 A (as appropriate for bcc iron), we get Tc 0 :3 K. But actually iron becomes ferromagnetic at 1043 K. Clearly the dipole-dipole interaction is too weak to explain this. Interestingly, it is thought that the dipolar interaction can lead to ferromagnetic long-range order on some crystal lattices, including bcc and fcc, even though the interaction is highly anisotropic. This has been predicted by Luttinger and Tisza in 1946. Experimentally, this type of order seems to be realized in Cs 2 NaR(NO 2 ) 6 with R = Nd; Gd; Dy; Er. In any case, we have to search for another, much stronger interaction. As we will discuss in chapter 4, this will turn out to be the Coulomb interaction in conjunction with the Pauli priniciple. Thus quantum mechanics is required for magnetic ordering at high temperatures but not for magnetic ordering per se.
The degeneracy found in the previous subsection is partially lifted by the Coulomb repulsion beyond the Hartree approximation. Note that the Coulomb interaction
Vc =
4 ϵ 0
i̸ =j
e^2 jri rjj
commutes with the total orbital angular momentum (of the shell) L :=
i li, since^ Vc^ is spherically sym- metric, and with the total spin (of the shell) S :=
i si^ and of course also with^ L
(^2) and S (^2). L and S also
commute since they describe completely different degrees of freedom.( Thus it is possible to classify the 2(2l + 1) nnl
many-particle states in terms of quantum numbers L; mL; S; mS. If we now have a state j ⟩ with quantum number mL < L then we can apply the raising operator L+^ := Lx + iLy to j ⟩ and obtain j ′⟩ / L+j ⟩ with m′ L = mL + 1. However, since [H; L+] = 0, this new state has the same energy as the old one. Since there are (2L + 1)(2S + 1) states that are connected by L
and S^ (L ^ := Lx iLy etc.), the
2(2l + 1) nnl
-fold degenrate state splits into multiplets with xed L and
S and degeneracies (2L + 1)(2S + 1). Typical energy splittings between multiplets ate of the order of 10eV. The ground-state multiplet is found from the empirical Hund rules:
1 st^ Hund rule: The ground state multiplet has the maximum possible S. (The maximum S equals the largest possible value of ⟨Sz ⟩.)
2 nd^ Hund rule: If the rst rule leaves several possibilities, the state with maximum L is lowest in energy. (The maximum L equals the largest possible value of ⟨Lz ⟩.)
These rules hold in most cases but not always. We will return to their origin later. A short qualitative explanation can be given as follows:
For the 1st^ rule: same spin and the Pauli principle result in the electrons being further apart, which leads to lower Coulomb repulsion.
For the 2nd^ rule: large L means that the electrons have aligned orbital angular momenta, i.e., rotate in same direction. They are thus further apart, which leads to lower Coulomb repulsion.
The multiplets ate labeled as 2 S+1L, where L is denoted by a letter according to
0 1 2 3 4 5 6... S P D F G H I...
We have already seen that a relativistic description is required to understand the magnetic moment of the electron. We will see now that the same holds for the many-particle states of ions. By taking the non-relativistic limit of the Dirac equation, one arrives at the Pauli equation mentioned above. By including the next order in v=c one obtains additional terms. Technically, this is done using the so-called Foldy-Wouthuysen transformation. We only consider the case of a static electric potential. The book by Messiah (vol. 2) contains a clear disscussion. In this case one obtains the Hamiltonian, in spinor space,
H = p^2 2 me
p^4 8 m^3 ec^2 | {z } correction to kinetic energy
2 m^2 ec^2
r
@r
s l | {z } spin-orbit coupling
8 m^2 ec^2
| {z } correction to potential
The last term is also called the \Darwin term". The only term relevant for us is the spin-orbit coupling
2 m^2 ec^2
r
@r
s l: (2.8)
For the Coulomb potential of the nucleus,
2 m^2 ec^2
Ze^2 4 ϵ 0
r
@r
r
s l =
2 m^2 ec^2
Ze^2 4 ϵ 0
s l r^3
g^2 B Z
s l r^3
; assuming g = 2: (2.9)
For several electrons in an incompletely lled shell, the operator of spin-orbit coupling is
g^2 B Z
i
si li r^3 i
In principle, we should include not only the nuclear potential but the full effective potential of the Hartree approximation. In practice, this is expressed by replacing the atomic number Z by an effective one, Zeff < Z. We now evaluate the contribution of spin-orbit coupling to the energy, treating HSO as a weak pertubation to H 0. Then
ESO := ⟨HsO⟩ =
g^2 B Zeff
i
⟨ (^) s i ^ li r^3 i
For free ions, the radial wave function Rnl(r) is the same for all orbitals comprising a shell. Thus
g^2 B Zeff
r^3
nl
i
⟨si li⟩ : (2.12)
We now call the electrons with spin parallel to S \spin up" (") and the others \spin down" (#). Furthermore, si and li commute. We can thus replace, in the expectation value, si by S= 2 S for spin up and by S= 2 S for spin down, respectively. (Note that si has magnitude 1/2.) Thus
g^2 B Zeff
r^3
nl
i spin up
⟨S li⟩ 2 S
i spin down
⟨S li⟩ 2 S
We have three cases:
If the shell is less than half lled, nnl < 2 l + 1, all spins are aligned and the spin-down sum does not contain any terms. Then
g^2 B Zeff
r^3
nl
i
li
g^2 B Zeff
r^3
nl
with =
g^2 B Zeff
r^3
nl
If the shell is more than half lled, nnl > 2 l + 1, the spin-up sum vanishes since it contains
∑^ l
ml= l
⟨lmljljlml⟩ = 0 (2.16)
and we obtain ESO =
g^2 B Zeff
r^3
nl
with =
g^2 B Zeff
r^3
nl
If the shell is half lled, nnl = 2l + 1, both the spin-up and the spin-down sum vanish and we get ESO = 0. Note that one does nd a contribution at higher order in pertubation theory.
We have found that the spin-orbit coupling in a free ion behaves, within pertubation theory, like a term HSO = LS in the Hamiltonian, where > 0 ( < 0) for less (more) then half lled shells. This LS-coupling splits the (2L + 1)(2S + 1)-fold degeneracy. We introduce the total angular momentum operator J := L + S. We can then write
The full Hamiltonian including HSO commutes with J, and thus with J^2. It does not commute with L or S because of HSO but it does commute with L^2 and S^2. Therefore, we can replace J^2 ; L^2 ; S^2 by their eigenvalues,
HSO!
Thus since J S = L,
mobs = B J
J =: gJ B J; (2.24)
where we have introduced the Lande g-factor
gJ = 1 +
Note that gJ sati es 0 gJ 2. It can actually ba smaller than the orbital value of unity.
Protons and neutrons are both spin-1/2 fermions. Both carry magnetic moments, which might be surprising for the neutron since it does not carry a net charge (the neutrinos, also spin-1/2 fermions, do not have magnetic moments). But the neutron, like the proton, consists of charged quarks. Due to this substructure, the g-factor of the proton and the neutron are not close to simple numbers:
Proton: mp = 5| {z }: 5856 =gp
N s
with the nuclear magneton N := eℏ= 2 mp and the proton mass mp. It is plausible that in the typical scale the electron mass should be replaced by the proton mass. B is by a factor of nearly 2000 larger than N , showing that nuclear magnetic moments are typically small. Incidentally, this suggests that the nuclear moments do not carry much of the magnetization in magnetically ordered solids.
Neutron: mn = | 3 :{z (^8261) } =gn
N s (note that this is antiparallel to s).
In nuclei consisting of several nucleons, the total spin I has contibutions from the proton and neutron spins and from the orbital motion of the protons and neutrons. The nuclei also have a magnetic moment mN consisting of the spin magnetic moments of protons and neutrons and the orbital magnetic moments of the protons only. The neutrons do not produce orbital currents. The orbital g-factors are thus glp = 1 and gln = 0. Due to the different relevant g-factors, the instantaneous magnetic moment is not alligned with I. In analogy to Sec. 2.4, only the averaged moment mN parallel to I is observable, leading to distinct muclear g-factors. We thus have mN = gN N I (2.26)
with nucleaus-speci c gN. Typically, jgN j is of the order of 1 to 10. gN is positive for most nuclei but negative for some.
The hyper ne interaction is the interaction between the electrons and the nucleus beyond the Coulomb attraction, which we have already taken into account. The origin of the name is that this interaction leads to very small splittings in atomic spectra. The rst obvious contribution to the hyper ne interaction is the magnetic dipole-dipole interaction be- tween electrons and nucleus. Naively, we would write
Vdip =
3(me r)(mN r) r^2 me mN r^5
This leads to a divergence if the elctron can by at the position of the nucleus, which is the case for s-orbitals. We have to calculate the B- eld due to the nuclear moment more carefully. We have seen in Sec. 1. that the vector potential is
A =
mN
^r r^2
mN
r r^3
Thus
B = ∇ A =
mN
r r^3
mN ∇
r
r
mN
mN r
With ∇ (∇ F) = ∇(∇ F) ∇^2 F we nd
mN r
mN r
Now we use a trick: we split the second term into two parts and apply ∇^2 (1=r) = 4 (r) to the second:
∇∇|{z} dyad
mN r
0 mN (r): (2.31)
Why did we do this? The rst term is well de ned for r ̸= 0 but what happens at r = 0? We can show that the rst term does not have a singularity there and can thus be analytically continued to r = 0 by choosing an irrelevant nite value. Proof: consider a sphere S centered at the origin. We integrate ( ∇∇
mN r
over S: (^) ∫
S
mN r
S
d^3 r
r
| {z } =:Q
mN : (2.33)
The rst factor Q is a matrix acting on mN. But Q does not distinguish any direction in space and thus has to be of the form Q = q 1 with q 2 R. On the other hand, the trace of Q is
3 q = Tr Q = Qxx + Qyy + Qzz
S
d^3 rTr
2 3
@^2 @x^2 ^
1 3
@^2 @y^2 ^
1 3
@^2 @z^2
@^2 @x@y
@^2 @x@z @^2 @x@y ^
1 3
@^2 @x^2 +^
2 3
@^2 @y^2 ^
1 3
@^2 @z^2
@^2 @y@z @^2 @x@z
@^2 @y@z ^
1 3
@^2 @x^2 ^
1 3
@^2 @y^2 +^
2 3
@^2 @z^2
r
S
d^3 r
@x^2
@y^2
@z^2
@y^2
@x^2
@z^2
@z^2
@x^2
@y^2
r = 0: (2.34)
We have thus found that Q = 0. We now make the radius of S arbitrarily small and nd that there is no singularity at r = 0. For r ̸= 0 we can evaluate B as in Sec. 1.2 and nd
3(mN r)r r^2 mN r^5 r̸=
0 mN (r): (2.35)
What we have achieved so far is to make the singularity explicit as the second term. The resulting interaction energy between the nuclear moment and the spin moment of an electron is
E 1 = me;spin B
=
me;spin mN r^3
(r me;spin)(r mN ) r^5
0 me;spin mN r
gB s mN r^3
3 gB (r s)(r mN ) r^5
0 gB s mN (r): (2.36)
The last, -function term is called the Fermi contact interaction. There are two cases:
For s-orbitals the probability density is spherically symmetric. The expectation value of the normal dipole interaction then vanishes by the same argument as in the proof above. Only the contact term remains: E 1 =
g 0 B s mN j (^) s(0)j^2 ; (2.37)
where we have averaged over space but kept the spin degrees of freedom. Thus
ggN 0 B s I j (^) s(0)j^2 =: Jhyper s I: (2.38)
This term clearly leads to a splitting of ionic energy spectra.
In this chapter, we study magnetic ions in crystal lattices. The crystal breaks the isotropy of space, which mainly affects the spatial motion of the electrons, i.e., the orbital angular momentum and its contribution to the magnetic moment. We assume that all electrons remain bound to their ions. In particular, we do not yet consider metals|these will be discussed in chapters 9 and 10 below.
Crystal eld effects are, as the name implies, the effects of the crystal on an ion. We consider the most important cases of 3d (4d, 5d) and 4f (5f) ions. In stable states, these ions typically lack the s-electrons from the outermost shell and sometimes some of the d- and/or f-electrons. d- and f- ions behave quite differently in crystals. We consider the examples for Fe2+^ and Gd3+:
3d 4f e.g., Fe2+^ e.g. Gd3+
6
1s
2s, 2p
3s, 3p
3d
7
1s
2s, 2p
3s, 3p, 3d
4s, 4f, 4d
4f 5s,5p
partially lled d-shell on the outside of ion partially lled f-shell on the inside of ion
The d-shell overlaps strongly with sur- rounding ions, thus crystal eld effects are strong due to hybridization. We have to treat the crystal led rst, then LS-coupling as a pertubation.
The f-shell hardly overlaps, thus crystal- eld effects are due to electro-static potential and therefore weak. We have to treat LS-coupling rst, then the crystal eld as a pertubation.
We rst consider the electrostatic potential ϕcryst(r) due to the other ions acting on the electrons. As noted, this is most relevant for f-ions. The electrostatic potential leads to a potential energy Vcryst(r) = eϕcryst(r),
which we can expand into multipoles:
Vcryst(r) =
4 ϵ 0
d^3 r′^ e(r′) jr r′j
e 4 ϵ 0
d^3 r′^ (r′) 4
l=
∑^ l
m= l
2 l + 1
rl (r′)l+^
Y (^) lm(′; ϕ′)Ylm(; ϕ); (3.1)
where we have assumed that r (inside the ion) is smaller than r′^ (outside of the ion). With
Ylm(; ϕ) =
2 l + 1 4
(l m)! (l + m)!
P (^) lm (cos ) eimϕ; (3.2)
where the P (^) lm (x) are the associated Legendre functions, we can write
Vcryst(r) =
lm
Klm rl^ P (^) lm (cos ) eimϕ^ (3.3)
with the coefficients given by comparison with the previous expression,
Klm =
e 4 ϵ 0
(l m)! (l + m)!
d^3 r′^
(r′) (r′)l+^
P (^) lm (cos ′) e imϕ
′ : (3.4)
We note that for m ̸= 0
Klm =
e 4 ϵ 0
(l + m)! (l m)!
d^3 r′^
(r′) (r′)l+^
P (^) l m(cos ′)eimϕ
′
e 4 ϵ 0
(l + m)! (l m)! ( 1)
m (l m)! (l + m)!
d^3 r′^ (r′) (r′)l+^
P (^) lm (cos ′)eimϕ
′ = ( 1)m^ (l + m)! (l m)!
K lm (3.5)
so that
KlmP (^) lm (cos )eimϕ^ + Kl; mP (^) l m(cos )e imϕ
= KlmP (^) lm (cos )eimϕ^ + ( 1)m^ (l + m)! (l m)!
K lm( 1)m^ (l m)! (l + m)!
P (^) lm (cos )e imϕ
=
Klmeimϕ^ + K lm e imϕ
P (^) lm (cos ): (3.6)
This shows that the terms combine to make the potential energy real. Which coefficients Klm are non-zero depends on the symmetry of the crystal, i.e., of (r′), under (proper) rotations and rotation inversions about the ion position. For example, for a cubic lattice, one can easily check that K 2 m = 0, K 4 ; 1 = K 4 ; 2 = K 4 ; 3 = 0, and K 44 = K 4 ; 4 = K 40 =336 so that
Vcryst(r) = K 00 |{z} irrelevant constant
+K 40 r^4
P 40 (cos ) + e^4 iϕ^ + e ^4 iϕ 336
P 44 (cos )
= const + K 40 r^4
(35 cos^4 30 cos^2 + 3) +
cos 4ϕ 168
105(cos^4 2 cos^2 + 1)
= const +
x^4 + y^4 + z^4
r^4
after some algebra. We consider an ion with groundstate multiplet characterized by the total angular momentum J. We only operate within this (2J + 1)-dimensional subspace. The ion is subjected to the potential Vcryst as a weak pertubation. Within this subspace, Vcryst has the matrix elements ⟨mJ jVcrystjm′ J ⟩, mJ ; m′ J = J; : : : ; J. The main result is that xp; yp; zp^ have the same matrix elements within this subspace as Jxp ; Jyp ; Jzp up to a scalar factor. This factor contains the average ⟨rp⟩ for the relevant shell|this is already dictated by symmetry and dimensional analysis|and a number that depends an the power p and on the shell, i.e., on the quantum numbers n; l; S; L; J. This rule is ambiguous if we have products like xy, since Jx and Jy do not commute so that and JxJy and Jy Jx are both plausible but not identical. The rule here is to symmetrize such products: (JxJy + Jy Jx)=2. In particular, we get
r^2 = x^2 + y^2 + z^2! J x^2 + J y^2 + J z^2 = J^2 = J(J + 1) (3.8)
Over a larger range of J we nd the following spectra, where each dot now represents a multiplet without indication of its multiplicity:
0 5 10 15 20
angular momentum J
spectrum
J
4 Harb. units
L
If the symmetry is not cubic, anisotropy terms tend to appear already at second order in r and, conse- quently, in J. For an orthorhombic lattice the second-order terms are
3 z^2 r^2!
r^2
3 J z^2 J(J + 1)
x^2 y^2!
r^2
J x^2 J y^2 )
Only the rst term exists for tetragonal symmetry. A term proportional to J z^2 is the simplest and the most inportant anisotropy that can occur. For negative prefactor it favors large jmJ j and is an easy-axis anisotropy, whereas for positive prefactor it favors small jmJ j and is a hard-axis or easy-plane anisotropy. We note that J x^2 + J y^2 + J z^2 = J(J + 1) is a constant and thus does not introduce any anisotropy. Therefore J z^2 is equivalent to +J x^2 + J y^2. Where are the rules xp^! ⟨rp⟩ Jxp etc. coming from? The formal proof requires group theory and can be given, for example, in terms of irreducible tensor operators, see Stevens, Proc. Phys. Soc. A 65 , 209 (1952) for a discussion. Also using group theory, it can be shown that terms of order higher than 2l (l = 2 for d-shells, l = 3 for f-shells) in Vcryst do not contribute.
As noted, in transition-metal ions the hybridization between the d-orbitals and orbitals of neighboring ions dominates the crystal- eld effects. However, although the mechanism is thus different from the electrostatic potential discussed previously, the splitting of multiplets is only a consequence of the reduced symmetry. We
can therefore discuss hybridization in terms of an effective potential having the correct symmetry. We must keep in mind, though, that for transition-metal ions crystal- eld effects ars stronger than the LS coupling so that we should apply crystal- eld theory to multiplets ignoring the LS coupling. LS coupling can later be treated as a weak pertubation. For example, for an ion in a cubic environement we obtain the term ⟨ r^4
L^4 x + L^4 y + L^4 z
in the single-ion Hamiltonian. To calculate the coefficient , we reiterate that the matrix elements of (5=2) K 40 (x^4 + y^4 + z^4 3 = 5 r^4 ) are proportional to those of the above operator (3.15). We can thus calculate one non-zero matrix element of each operator to obtain. The result is a lengthy expression, see Yosida's book. In particular, we nd alternations in the sign of with the number nn; 2 of electrons in the d-shell. If Hund's 1st and 2nd rules win over the crystal eld, the only possible values for L are 0, 2, 3. Together with the schemes on p. 18 we obtain the following splittings:
con guration
L
d^1 , d^6
2
0
d^2 , d^7
3 < 0
d^3 , d^8
3
0
d^4 , d^9
2 < 0
d^5
0
3.3.1 The Jahn-Teller effect
If the symmetry is lower than cubic, the levels will be split further. For sufficiently low symmetry, all degeneracies (except for spin degeneracy) are lifted. In cases where the ground state would still be degenerate for a putative crystal structure, the crystal will often distort to break the degeneracy and lower the symmetry. This is the Jahn-Teller effect. For example, in the 3d^9 con guration of Cu2+^ or in the 3d^4 con guration of Mn3+, the ground state would be twofold degenerate in a cubic environement. A distortion of the crystal causes two energy contributions:
(a) an elastic energy, which is increasing quadratically with the distortion for small distortions if the cubic crystal was stable neglecting the Jahn-Teller effect,
(b) a linear splitting of the angular momentum doublet as given by rst-order pertubation theory.
Thus the total ground-state energy generically has a minimum for non-zero distortion:
distortion
minimum
ground state
purely elastic
3.3.2 Quenching of the orbital angular momentum
If the ground state is not degenrate with respect to the orbital angluar momentum L (i.e., is a singlet), we nd the phenomenom of quenching. To understand it, we calculate ⟨ 0 jLj 0 ⟩, where j 0 ⟩ is the non-degenerate
