!
" #
#
$ "
"
" #
%
&
" '
()))
$
*
"
+ +
, +
-
- # . / 0
#
$
#
* !
'
1 , , 23 * 24 ()5
-
1 * E B . 6, 7# . 8 6,2 9
+
$
*
!
D . %,2 H . %, / #
+ #
!
.
D = D(E, B),
H = H(E, B).
2:
* #
* P M H = (1/µo)B − M + . . .
2(
D = o E + P + . . . ,
23
o = 8.85 · 10−12 %,6 µo = 4π · 10−7 6,% P M # E B * ,
#
" ;
% '
* α '
mag
Mα = (1/µo)
χαβ Bβ
24
Pα = o
β
χel
αβ Eβ ,
β
2<
χel/mag
,
* αβ
M = (1/µo )χmag B
2=
el
P = o χ E.
25
()2
+ + #" #,
/ #
D = o E
22
1
H =
2>
B,
µµ
o
= 1 + χel µ = 1 − χmag / χmag < 0 µ > 1 !
#
/ χmag > 0
µ < 1 #
" #
+
" %
V B "
/ " E mag = (1/2BH)V
⇒ 1/V dE mag = 1/2BdH + 1/2HdB.
2:)
/ ?+ 2> HdB = BdH 1/V dE mag = BdH = (µo H + µo M)dH = Bo dH + µo MdH.
2::
* µoH = Bo ; @ mag
dEsolid
= −µo MV dH.
2:(
A
m E = −mB # ?+ 2> M A
?+ 2:( #
+
1 ∂E(H) M(H) = −
µo V
∂H S,V
mag
χ
2:3
,
∂M(H) 1 ∂ 2 E(H) (H) =
= −
.
∂H S,V
µo V
∂H 2 S,V
()>
2:4
% 1 ∂F (H, T ) M(H, T ) = −
µo V
∂H S,V
mag
χ
1 ∂ 2 F (H, T ) (H, T ) = −
.
µo V
∂H 2 S,V
2:<
+ +
% #
#
M χmag ? " +
* "
#
$
23: !
B 23( * 233 / 234 ; * + nlm n l ml / n + l "
) (n−1) / l = 0, 1, 2, 3 . . .
s p d f nl
ml (2l + 1) −l +l
% + + ms "
−1/2
+1/2 / #
1s 2s
# 2p l
+ + nlm ,m l
(:)
s
el.
:
(
3
4
<
=
5
2
>
:)
ml =
(
↓
↓
↓
↓
↓
↓↑
↓↑
↓↑
↓↑
↓↑
:
↓
↓
↓
↓
↓
↓↑
↓↑
↓↑
↓↑
)
↓
↓
↓
↓
↓
↓↑
↓↑
↓↑
−:
↓
↓
↓
↓
↓
↓↑
↓↑
−2
↓
↓
↓
↓
↓
↓↑
S
:,(
:
3,(
(
<,(
(
3,(
:
:,(
)
L=|
(
3
3
(
)
(
3
3
(
)
ml |
J
3,(
(
3,(
)
<,(
4
>,(
4
<,(
)
$
2
D3/2
3
F2
4
F3/2
5
D0
6
S5/2
5
D4
4
F9/2
3
F4
2
D5/2
1
S0
2:. & d
l = 2 B1 * *
L=
ml
and
S=
ms ,
2:=
+ A
$
* L S J = L + S B
B1 .
: C # S ( L #
3 J L S .
• < 2l + 1
J = L − S D
• > 2l+1
J = L + S D
• # = 2l + 1
L = 0, J = S B1 + #
@ LS L = 0, 1, 2, 3, . . . =
(::
S, P, D, F, . . . %
# (2S +1) J (2S+1) LJ 2: #
d
B "
z# B H e = T e + V e−ion + V e−e 9
H e !
;
9A #
$
V e−ion V e−e !
"
/ " p (= ih̄∇) p → p + eA / z# 1
A = − (r × B) ,
2
2:5
B = (∇ × A) ∇ · A = 0 "
1 1 e
T (B) =
[pk + eA]2 =
pk − (rk × B)
2m k
2m k
2
e
2
2:2
.
?# +
e
T (B) =
k
e
p2k
e2
+
pk · (B × rk ) +
(rk × B)2
2m
2m
8m
= Toe +
k
e2 2 2
e
(rk × pk ) · B +
B
xk + yk2
2m
8m
k
2:>
.
* #
L #
(r
×
pk ) - µB = (eh̄/2m) = 0.579 · 10−4 6, k k
"
e2 2 2
∆T e (B) = µB L · B +
xk + yk2
.
2()
B
8m
h̄L =
k
* !
H e + - 0
$ B (:(
E + % where Sz =
∆H spin(B) = g0 µB BSz ,
sz,k
2(:
.
k
B
g0 g f actor α
+ O(α2) + . . . ,
2π
= 2.0023 . . . ,
g0 = 2 1 +
where α =
1
e2
≈
h̄c
137
2((
,
"
; (
B ∆H e (B) = ∆T e (B) + ∆H spin (B) = µB (L + g0 S) · B +
e2 2 2
B
xk + yk2 .
8m
k
2(3
?+ 2(3 +
# C
B A
$
n En → En + ∆En (B);
∆En = < n|∆H e (B)|n > +
| < n|∆H e (B)|n > |2
En − En
n =n
.
2(4
$ ?+ 2(3 + + ∆En = µB B· < n|L + g0 S|n > +
+
e2 2
B < n|
x2k + yk2 |n >
8m
k
| < n|µB B · (L + g0 S)|n > |2
En − En
n =n
.
2(<
$
|0> ?+ 2(< |0> (:3
< 0| k (x2k +yk2)|0 > ≈ 2/3 < 0| k rk2 |0 > #
∆E0dia ≈
e2 B 2 e2 B 2
2
Z r̄atom
< 0|rk2 |0 > ∼
12m k
12m
,
2(=
Z k
2
r̄atom
+
* "
Z ∼ 30
2
2
r̄atom F ∆E0dia ∼ 10−9 6 #
"
χmag,dia = −
2
µo ∂ 2 E0
µo e2 Z r̄atom
∼ −10−4 .
=
−
V ∂B 2
6mV
2(5
χmag,dia < 0 0 0
#
+
#
?+ 2(< J|0 >= L|0 >= S|0 > * |0> ;
?+ 2(< = #
* + ?+ 2:< χmag,dia
- " ?+ 2(< 2µo µ2B | < 0|(Lz + g0 Sz )|n > |2
χmag,vleck =
.
2(2
V
E −E
n
n
0
9
?+ 2(< χmag,vleck $
#
En > E0 χmag,vleck > 0 6 6
" * +
"
6 6
" (:4
?+ 2(< 0
6 6
" - / J = 0 % ?+ 2(< % J = 0 (2J + 1) ?+ 2(<
*
α = 1, . . . , (2J + 1) (2J+1)
∆E0,α = µB B
α =1
< 0α|Lz + g0 Sz |0α > = µB B
(2J+1)
α =1
Vα,α ,
2(>
z# # Vα,α # %,
# J Jz < JLS, Jz |Lz + g0 Sz |JLS, Jz > = g(JLS)Jz δJ ,J .
23)
JLS + |0> g(JLS) %# go ≈ 2 1 S(S + 1) − L(L + 1)
3
+
g(JLS) =
.
23:
2
2
J(J + 1)
z
z
* ?+ 23) ?+ 2(> 23(
(2J + 1) + g(JLS)µB B −g(JLS)µB JB, −g(JLS)µB (J−1)B, . . . , +g(JLS)µB (J−
1)B, +g(JLS)µB JB !
matom = −g(JLS)µB J.
233
?+ 2(< #
J = 0
∆EJLS,Jz = g(JLS)µB Jz B
,
-
matom ?+ 2(< (:<
/
2:. C - BJ (x) J !
B H spin = −matom · B + (2J + 1) * " #
H spin B - #
!
!
!
$ #
" #
%
#
B "
!
(2J + 1) g(JLS)µB B A
µB = 0.579 · 10−4 6, 10−4 6 kB T % #
+
?+ 2:< +
(2J +1) E
η = (g(JLS)µB B)/(kB T ) (:=
q =
J
e−ηJz =
Jz =−J
=
e−ηJ − eη(J+1)
1 − eη
e−η(J+1/2) − eη(J+1/2)
sinh [(J + 1/2)η]
=
−η/2
η/2
e
−e
sinh [η/2]
234
.
1 ∂(−kB T lnq)
1 ∂F
= −
23<
V ∂B
V
∂B
g(JLS)µB J
kB T ∂
[ln (sinh[(J + 1/2)η]) − ln (sinh[η/2])] =
BJ (η),
=
V ∂B
V
M(T ) = −
η
1
1
1
1
BJ (η) =
(J + ) coth (J + )η/J − coth
J
2
2
2
2J
.
23=
% / 2: BJ → 1 η 1 ?+ 23< matom ?+ 233 #
# M = matom /V η = (g(JLS)µB B)/(kB T ) % - BJ (η → 0) η
# BJ (η 1) ≈
J +1
η + O(η 3 ) .
3
235
* χmag,para (T ) = µo
µo µ2B g(JLS)2 J(J + 1) 1
∂M
=
∂B
3V kB
T
.
232
χmag,para > 0 ?+ 2(< J = 0 " χmag,para = CCurie /T CCurie '
$ % η '
* "
V ∼ F3 ?+ 232 χmag,para ∼ 10−2 % 6 6
" (:5
9
χmag,para 1 +
+
234 !
B #
#
$ ; #
. @
" + * #
" 0
C "
!
/ N V * (
E@$ 1
N() =
2π 2
2me
h̄2
3/2
√
23>
.
* /
∞
N
N() f (, T ) d,
=
V
0
24)
N/V +
/ G
/
#
/
/
$
rs =
50.1 eV
F = r .
24:
s
aB
E@$ /
+ rs
N(F ) =
1
20.7 eV3
rs
aB
−1
.
24(
H # $ # !
#
E@$ N ↑ N ↓ E@$ * (:2
/
2(. /
!
#
. 9↑ 9↓ z# 9 B0 z
+ E@$ E@$ ?+ 23>
N ↑ () = N ↓ () =
1
N() ,
2
243
/ 2(
% #
B !
$
# * !
+
?+ 2(: ∆H spin (B) = µB go B · s = +µB B (for spin up)
244
= −µB B (for spin down)
24<
#
go ( !
(:>
#
#
E@$
1
N( − µB B)
2
1
N( + µB B) ,
N ↓ () =
2
24=
N ↑ () =
245
/ 2( * /
!
# $
#
/ T ≈ 0 " / 2( ?
µB B ∼ 10−4 6 #
µB B 12 N(F ) /
E@$ N↑ =
N
1
+ N(F )µB B
2
2
242
N↓ =
1
N
− N(F )µB B
2
2
24>
$
µB 2<)
E@$ ∂M
= µo µ2B N(F ) .
χmag,Pauli = µo
2<:
∂B
H ?+ 24( /
E@$ $
rs
M = (N ↑ − N ↓ )µB = N(F )µ2B B
χmag,Pauli = 10−6
2.59
rs /aB
.
2<(
A
$
3< aB #
C χmag,Pauli > 0 " #
/ C # (()
* "
'
?+ 232 /
@
C '
#
TF T $
TF T C !
0 +
+ B " B 1
χmag,Landau = − χmag,Pauli ,
2<3
3
+ * "
" C ! ())) " * " +
% "
* ; . # #
• % .
C
0 •
χmag,para
χmag,dia
≈ 1/T ∼ 10−2 A
≈ const. ∼ −10−4
∆E0para ∼ 10−4 6
∆E0dia ∼ 10−9 6
/
.
C 0 χmag,Pauli ≈ const. ∼ 10−6
χmag,Landau ≈ const. ∼ −10−6
∆E0Pauli ∼ 10−4 6
∆E0Landau ∼ 10−4 6
* !
* #
,
((:
/
23. * 4f 5s 5p 4f !
* 3d ; % .
" J
= 0 * L =
"
" I @
6 6
" 0 * !
#
+
+
J = 0 A? f d $ '
# / A? '
?+ 232 / '
L = 0 S B1 " . % / 23 f A? s p !
@ d 7
% B1 @
S=0
(((
'
'
'
#
B
'
"
$ II # # * * "
" % J = L = S = 0 # C #
?+ 2<(
+
!
#
!
. . .
!
% 1 # !
. B
#
$
" " #$ #$ ((3
/
24. $
+
#
?#
* #
/ 24 * #
G
/ 24 B
!
!
* +
#
/ 24 $ $ # #
" " Ms # #
/ 24 %
$ / 24 * !
" Ms = 0 $
/ 2< # # B
#
((4
/
2<. # " % & '#( )*
?
%
Tc "
* Tc " #
TN 9
Tc I * #
Tc #
J
# ((<
/
2=. Ms cV
χo '
Tc % #
#
7
Ms (T ) / 2= K Tc Ms (T ) ∼ (Tc − T )β
(for T → Tc− ) ,
2<4
#
β 1/3 ' + T Tc
χo (T ) = χ(T )|B=0 ∼ (T − Tc )−γ
(for T → Tc+ ) ,
2<<
γ 4/3 +
7
+
"
cV (T )|B=0 ∼ (T − Tc )−α
(for T → Tc ) ,
2<=
α ): C
G * "
#
α, β γ /
C Tc ((=
µB matom µB Tc L Θc L
/
((
= 4
:)43
::))
'
:5
= 3
:3>4
:4:<
9
)=
< (
=(2
=<)
?
5:
5
(2>
:)2
&
2)
2
3)(
(2>
E
:)=
:)
2<
:<5
2(. +
. T = 0 L m̄s Tc '
Θc L / matom "
+
m̄s
$
/ 2=
χo (T ) ∼ (T − Θc )−1
(for T Tc ) ,
2<5
Θc '
$
γ ?+ 2<< #
: Θc '
Tc γ #
2( T → 0 L * Ms (T → 0 K) m̄s µB * matom = g(JLS)µB J ?+ 233
% 2( !
A
+
234 !
d M
L = 0 J = S g(JLS) = 2
2( * A? /
' 9 % # !
#
((5
'
!
# '
'
. *
#
-
Tc +
; $
C
C
"
d f B
A? # / A? f "
#
s d " +
%
H
A? # # !
"
G ;
'
"
*
+
"
# $#% ! :>)= !
#
/ !
" @ !
((2
! " $
!
M I "
M B mol = µo λM
2<2
,
λ / #
$
!
+
#
'
χmag,para = C/T ) M =
1 mag
1 mag ext
χ B =
χ (B + B mol )
µo
µo
1 C ext
=
(B + µo λM) .
µo T
2<>
$ M =
C
1
B ext
µo (T − λC)
2=)
,
∂M C
χo (T ) = µo
=
ext
∂B Bext =0
T − λC
(for T > Tc ) .
2=:
*
λ = Θc /C '
'
!
&
#
'
"
B
T > Tc
Θc = Tc γ = 1 #
&
+
* "
9 . #
" Ms Tc ((>
1 " / '
C ?+ 23< C =
Nµo µ2B g(JLS)2 J(J + 1)
≈
3kB V
N
V
µo m2atom
3kB
,
2=(
#
J(J + 1) ∼ J 2 matom = µB g(JLS)J T = 0 L
B
mol
Tc
(0 K) = µoλMs (0 K) = µo
C
N
3kB Tc m̄s
m̄s ≈
V
m2atom
2=3
2( m̄s matom
B mol ∼
[5Tc in K]
Tesla .
[matom in µB ]
2=4
0" 2( 103 % "
"
+
% / 2= Ms (T ) T = 0 L $
#
Ms (T ) +
; #
H 23( +
?+ 23<
Ms (T ) = Ms (0 K) BJ
g(JLS)µB B mol
kB T
,
2=<
- BJ (η) ?+ 23= % # # " +
; . * "
/ 2= $ '
+
Ms (T ) T → 0 L T → Tc− Ms (T → Tc− ) ∼ (Tc − T )1/2
2==
β = 1/2 :,3 * - #
T 3/2 " T 3/2 (3)
#
7 #
# " & &
%
%
$ * #
f A? i # mi %
!
i j coupling
Hi,j
= −
Jij
mi · mj
µ2B
.
2=5
B
Jij " #
. 6
E
"
−Jij mi mj /µ2B +Jij mi mj /µ2B 9
* +
# B
A
*
#
Jij Jij #
# + / +
# % / 25 B
" / 25 B B (3:
/
25. $
. #
#
#
!
H Heisenberg =
M
i,j=1
coupling
Hi,j
= −
M
Jij
mi · mj
2
i,j=1 µB
.
2=2
$ = ;
$
1 " +
* #
% B f B ?+ 2=2 " % %
% +
+
# "
B
B B ?#
B
B
& (3(
/
22. $
* B
B G
"
+
"
' * +
#
m ?+ 2=2 J > 0 J < 0 / | ↑↑↑ . . . ↑↑> @
# 7 | ↑↓↑ . . . ↑↑> $ B
B # "
/ 22 % $ B
B - T 3/2 Ms (T ) T → 0 L $ #
+
# * #
B
#
#
B
; (33
'
1/T '
' B
#
#
@
# #
Jij 10−2 6 10−3 − 10−4 6 A?
+
+ # B
?+ 2=2 H Heisenberg = −
M
Jij
mi · mj
2
i,j=1 µB
M
⎛
⎞
M
M
Jij
⎠ = −
= −
mi · ⎝
m
mi · Bmol
j
2
µ
i=1
j=1 B
i=1
.
2=>
* "
Bmol i B
!
!
# !
< Bmol >
/ <B
mol
M
Jij
>=
< mj > =
2
j=1 µB
M
j=1 Jij
µ2B
N
Ms (T ) ∼ λMs (T ) ,
V
25)
i %
?+ 25) * + Jij * ?+ 25) µ2 B mol
J = JijN N = B
.
25:
12m̄
s
H 2( 103 #
10−2 6 10−3 −10−4 6 A? (34
"" , B " J +
m1 m2
r / J mag dipole =
1
[m1 · m2 − 3(m1 · r̂)(m2 · r̂)]
r3
,
25(
& J mag dipole ≈
1
m1 m2
r3
.
253
*
µB 2( ( F
J mag dipole ∼ 10−4 6 :( #
B ' %
0
" #
"
'
!
! #
"
G
* +
C ' !
" . '( " ! ! ) #
B
: ( B A B * B ho
ho |φ > = o |φ > ,
254
|φ >= |r, σ > r σ
B
(3<
B
H|Φ > = (ho (A) + ho (B) + hint )|Φ > ,
25<
|Φ > A
B (1) (2) σ1 σ2 $
H #
H |Φ > = |Ψorb > |χspin > ,
25=
% S2 Sz |χspin
|χspin
|χspin
|χspin
>S
>T1
>T2
>T3
=
=
=
=
2−1/2 (| ↑↓> − | ↓↑>)
| ↑↑>
2−1/2 (| ↑↓> + | ↓↑>)
| ↓↓>
S
S
S
S
=0
=1
=1
=1
Sz
Sz
Sz
Sz
=0
=1
=0
= −1
/
C # $
|χspin >S |χspin >T
|Φ >singlet = |Ψorb,sym > |χS >
|Φ >triplet = |Ψorb,asym > |χT > ,
255
|Ψorb,sym(1, 2) > = |Ψorb,sym(2, 1) > |Ψorb,asym(1, 2) > = −|Ψorb,asym (2, 1) >
E
B % B2 * < Φ|hint |Φ >= 0 |φ > B +
?+ 255 |Ψ∞
orb,sym >
= 2−1/2 [ |φ(1A) > |φ(2B) > + |φ(2A) > |φ(1B) > ]
|Ψ∞
orb,asym >
= 2−1/2 [ |φ(1A) > |φ(2B) > − |φ(2A) > |φ(1B) > ]
252
|φ(1A) > #
(1) A B2 # 2o ,
B % (3=
$
|φ(1A) > |φ(1B) > +
|φ(2A) > |φ(2B) > S = < φ(1A)|φ(1B) >< φ(2A)|φ(2B) > = | < φ(1A)|φ(1B) > |2
.
25>
* +
* %
?+ 252 ; |ΨHL
orb,sym >
= (2 + 2S)−1/2 [ |φ(1A) > |φ(2B) > + |φ(2A) > |φ(1B) > ]
22)
|ΨHL
orb,asym >
= (2 − 2S)−1/2 [ |φ(1A) > |φ(2B) > − |φ(2A) > |φ(1B) > ]
.
E
ES =< Φ|H|Φ >singlet ET =< Φ|H|Φ >triplet ET − ES = 2
CS − A
1 − S2
22:
,
C = < φ(1A)| < φ(2B)| Hint |φ(2B) > |φ(1A) >
A = < φ(1A)| < φ(2B)| Hint |φ(2A) > |φ(1B) >
(Coulomb integral)
(Exchange integral) .
22(
223
' A #
* A = CS B
/
C # A = CS !
B2 * . # ' G
#
C # . ! ' /
"
/ " C
? C A ?+ 223 B (ET − ES) * B2 + B
(35
B ?+ 2=2 #
J B
0 # "
#
B
B B
" ; #
?+ 22) +
B
B % #
' N % U '( " &
' *
B
0
#
B B
+
#
/ #
;
" "
233 # #
"
E
!
" C * #
#
!
B
/" # #
* #
:> # # (E/N)jellium = Ts + E XC
HF
≈
30.1 eV
rs
aB
2
−
12.5 eV
rs
aB
,
224
233 "
#
#
(32
/
2>. C ∆E(P ) ?+ 22> P α % α > 0.905
#
# $
rs /aB =
1/3
3
(V /N)
4π
N
Ejellium,HF(N) = N 78.2 eV
V
2/3
N
− 20.1 eV
V
1/3 22<
.
spin
Ejellium,HF
(N ↑ , N ↓ ) = Ejellium,HF(N ↑ ) + Ejellium,HF(N ↓ ) =
⎧
⎨
=
⎩
⎡
2/3
N↑
⎣
78.2 eV
V
+
2/3 ⎤
N↓
⎦
V
−
⎡
1/3
N↑
⎣
20.1 eV
V
+
1/3 ⎤⎫
⎬
N↓
⎦
⎭
V
22=
,
N ↑ N ↓ N ↑ +N ↓ = N) !
P =
N↑ − N↓
N
225
,
P = ±1 P = 0 ?# N ↑ = N2 (1 + P ) N ↓ = N2 (1 − P ) ?+
22= Ejellium,HF(N, P ) =
!
"
"
1
5 !
5/3
5/3
4/3
4/3
NT
(1 + P ) + (1 − P )
− α (1 + P ) + (1 − P )
2
4
(3>
222
,
α = 0.10(V /N)1/3 /
P = 0 ∆E(N, P ) = E(N, P ) − E(N, 0) < 0 C #
B
/" # ∆E(N, P ) =
NT
!
"
!
22>
"
1
5
(1 + P )5/3 + (1 − P )5/3 − 2 − α (1 + P )4/3 + (1 − P )4/3 − 2 .
2
4
/
2> P α %
∆E(N, P ) < 0 α > αc =
2 1/3
(2 + 1) ≈ 0.905 ,
5
2>)
P = 1 H α #
* $
rs
>
∼ 5.45 ,
2>:
a
B
$
<= aB A
1.8 < rs < 5.6 ' ; B
/" #
%
' /
,',9 B
#
!
B
/" #
# ?+ 2>: +
" (rs /aB ) ≈ 50 ± 2 O/B P ' 0 E
'
C A
- ++ )3=5)3 ())(Q % B
/" +
!
#
rs M
#
1.8 < rs < 5.6 #
;
. '
/
,',9 # d
* #
" ;
. !
(4)
"- . ' * 35 G
# E/ B
E/N = Ts [n] + E ion−ion [n] + E el−ion [n] + E Hartree [n] + E XC [n] ,
2>(
+
n(r) +
# #
# 0E% R'
r +
n(r) A
E/ E/N = Ts [n↑ , n↓ ] + E ion−ion [n] + E el−ion [n] + E Hartree [n] + E XC [n↑ , n↓ ] .
2>3
n↓ (r) * "
!
#
/
! 0$E% R'
; * n↑ (r)
m(r) =
n↑ (r) − n↓ (r) µB
,
2>4
n(r) m(r) $
L$ + '()* %
0$E% /
' 9 A
E@$ /
9 #
/ 2:) %
# ;
E
!
!
L$ !
/
!
J
0$E% m̄s = 2.1 µB
/
1.6 µB ' 0.6 µB 9 #
#
T = 0 L 2( E/ 0$E% +
N ; #
"
(4:
/
2:). $
E@$ " /
9 E/ 0$E% O
60 K/ K" %A C
C
:>52Q
# , " / " E@$ ; / 2:) !
" E@$ !
/ R'
# ↑↓
VXC
(r) =
↑↓
δEXC
[n(r), m(r)]
o
(r) ± Ṽ (r)m(r) ,
≈ VXC
δn(r)
(4(
2><
o
VXC
(r) R'
* !
R'
Ṽ (r)m(r) * ) #
1
↑↓
o
VXC,Stoner
(r) = VXC
(r) ± IM
2
2>=
,
I $
#
#
M = unit−cell m(r)dr M = (m̄s /µB )× I $
!
R'
. IM L$ +
!
R'
↑↓
i ↑↓
= oi ± 1/2IM .
2>5
i
E@$ ↑↓
N (E) =
i
BZ
o
δ(E − ↑↓
k,i )dk = N (E ± 1/2IM) ,
2>2
$
# # E@$
M
E@$ !
S $
$
" "
# E@$
N o (E) N * N M * $
!
I + M I I M N M E@$ /
N =
M =
F
F
2>>
[N o (E + 1/2IM) + N o (E − 1/2IM)] dE
[N o (E + 1/2IM) − N o (E − 1/2IM)] dE
.
2:))
$
N o(E) N #
+ F M B
+ M M =
F (M )
[N o (E + 1/2IM) − N o (E − 1/2IM)] dE ≡ F (M)
(43
2:):
/
2::. & M = F (M) ?+ 2:): F (M)
% #
M F (M) $
I E@$ N o (E) F N + M # "
N o (E) M
"
F (M) N o (E) > 0
F (M) = F (−M)
F (±∞) = ±M∞
F (0) = 0
F (M) > 0
B
M∞ ; F (M) +
. −∞ +∞ # F (M) +
"
/ 2:: * % M = 0 M = F (M)
?+ 2:): * - # M = 0
M = Ms = 0 !
F (M) . * : F (M) M % G
#
Ms F (0) > 1
" ?+ 2:): F (0) = IN o (F ) ) IN o (F ) > 1 .
2:)(
% G
# /
$
E/ " Vxc↑↓ (r)
D K/ K" C A
- -+ (<< :>55 0
(44
" /
* #
N o (F ) '
+
$
. * #
/
2:( +
$
E/ 0$E% & /
' 9 $
. / /
+
* N o (F ) d * /
+
$
%
3d 4d 5d ' /
' 9 M
' $ C +
$
". E@$ +
W . * K W * $
'
BT $
" E
# " " A?
+
#
"
f /
' 9 #
"
d s
B
B
#
B
0 0E%UH "
E/ 0$E% E
+
(4<
/
2:(. $
I E@$
/
D̃(EF ) $
/
' 9 O B * B 0V )
)
! $
:>>)Q
(4=
'
* #
" /
2:3. $
# !
% +
#
$
"D #
@ ! '
+
* / 2:3 !
""
!
#
-
#
#
!
C
#
. ?
#
# #
* " *
/ 2:4 #
% #
!
7 (45
/
2:4. $
#
" #
/ 2:< +
% +
# ! !
#
/ 2:= $ #
Ms @ " MR @
Bc E
. % Bc # / 2:5 $ +
"
#
/
/
9 /
' +
/ 2:5 # % ; /
'@#
/ 2:2 (42
/
2:<. * #
% #
*
/
2:=. (4>
/
2:5. B
/
2:2. * !
B
) :
% W W A
. * * :>22 #
"
/
' !
" #
(<)