~B ~B m • 0 <χ < 10 5 m χ →− 1 m • − 1 <χ < 0 m r χ = µ − 1 . ~M

!"#$%&'()'

!!"##$%&'(!)+&$&(,)-.&/%!%&(0.-

M ~

^%.,)/&(%&-&.)0&(1&/()+/&/2&($&.34("15)..6).

$%&7("1&.%.8 4(099&.&%./&%#&.:&./!9(&+&.$ $&6;&(/&1&(&%+$&(6)-.&/%!+&.<0!=&9/%>

1%#%/?/

χ m = µ r − 1.

^@A3BC

•

!"#"$%&'!(#)(*

− 1 < χ m < 0

χ m → − 1

^2%($%.<09()#&%/&(.1&%-&.D-&.$!+2)+&(6)-.&/%!+&( ((&-0.-&((&%+/3

•

+","#"$%&'!(#)(*

0 < χ m < 10 ⁵

E%&( !%.$ &%.& 2%+/%-& './&(-(099& F&.& <01!/).=&.: 2&#+& !9"./).& ,)-.&/%!%&(0.-

=&%-&.:=3G3H&(("6)-.&/%!60!3

I%&;%(50.-)0J$%&

B ~

>H&#$K&(/&%#0.-1&%L"(+).$&.!&%.&%.&(&##%9!"%$JM(6%-&.7("1&=&%-/$%&

N11%#$0.-O3B3P

!!"#$%&'()*+ ,"-.%&'/0&$"121'&34"5732 8#"&.59 !:;)<1-121'&34"5732 =##"<50"$1%> 3"&3

B ~

?@3#$/3-43"#%&' "2A1.%%2)

!" #$%&%'()*$+&,+

/+&-&'()*+,-., 0)+')( &11) 2*34)5 6)+ 7)()( !" 7+) 8&9-39:9;0),,+3( &1, <)&=*+3( &.> )+(

;.?)9),

B ~

^{@A)17 )+(,)*0*#}/+&-&'()*+,-.,*9+**&1,3+--)9&.>5 B+97&6)96)+ :&9&-&'()*+,D)(

2*34)(E)97)=*#

/&,=1&,,+,D)FD)39)-E3(8&9-39G

H)1&7)()F)+1D)(-+*8&7.('

~q

^{.(7/9)D+-:.1,}

L ~

^>ID9)(+-

B ~

^@A)17)+()J9;0),,+3(,6)B)'.(' -+*

~ ω L

^&.,G

~ω L = − q

2m · B; ~

^K"#LM

!!"#$%&'()*+ ,"&-".-#-/01"24-.5-&016#780-&9"6#!-:-'0-;6$%&'

q

^<=410".>%?-1-&

B ~

^@A-#$

$"- ;61.8171>9-22"8&6%2)

#$%&() *)+",(+! -.N9)+,6)B)'.(')+(),>9)+)(F)+1D)(,-+*8&7.('

q

^+-

B ~

^@A)17#H1)+D@

')B+D*E3(O)(*9+>.'&1=9&>*.(7839)(*0=9&>*G

mrω ² _c = qrω c B ω c = q

m · B

^K"#PM

ω c

^+,*7+)OQ=13*93(9),3(&(0>9)R.)(053>*&.DHQ93>9)R.)(0')(&((*#

!"!" #$%&')%*+,$+ -*& .'&/0&1&2-$3340+

S+96)*9&D*)(7+)DQ:3*D)*+,D)N9)+,6&D()+(),')6.(7)()(

e ⁻

^,)(=9)D*0.

B ~

^#/.9DR.&(@

*)(-)D&(+,D)T)7+('.(')(+,*7+),)9A&11E)963*)(5-&(6)=3--*&6)97&-+*6)9)+*,7)(S)9*

E3(

ω L

^#U,'+1*7&,N9;>*)'1)+D')B+D*

mrω ² = e ²

4πε _◦ r ² + erωB.

^K"#!M

Zentrif ugalkraf t = Coulombkraf t + Lorentzkraf t

/)$

ω = ω _◦

⁰¹²

B = 0

^!#%(('$3*

mrω ² = mrω ² _◦ + erωB ω ² − eB

m ω − ω ² _◦ = 0 ω 1,2 = eB

2m ± s

eB 2m

²

+ ω ² _◦ .

^{4 !.5}

6(*)2 7)28,29)3 +*):*7)2 ;3)$() <=22);*,2*)2&

eB 2m

²

9,

ω ² _◦

^{> ?)3:)2 %,02):*+A ,(7 3$(;+A}

72):)(7)

e ⁻ ( ± )

^{'3)$:?$2;*! #)2} 7%B=2+*):)(7)C)2&

eB

2m

^{7%')')( +*):* 0127)(} )$(:)$*3$:)(

#2):+$((7)2D2E9)++$=(+F)?)',('G

ω L = eB

2m .

^{4 ! 5}

!"#$%&')$* +&,-.

~µ

^&/0$.1

B ~

#%+#$H=3&=&)(*B=()$(9)3()(C)$3:)(?$27&$*

µ

F)9)$:()*>,&I)2?):+3,(')(&$*J%++)

m

^F9?!K,%(*)(9%:3

m

^9,B)2:$(7)2(!

!!"#$%&' ()*+ ,%- .-"/&0"/-%&' 12& 3"42#/& "5 6%7/-/&

B ~

^{89/#$) :/<0=+} .-"/&0"/-%&' 12&

L ~

%&$

~ µ

^>?-/"&/42="0"1/@A$%&'

q

⁾

D=*)(9$)33)L()2'$))$()+&%'(!#$H=3+$&

B ~

^{AM)37 4=:()N)?)$+5G}

V = − ~ µ · B ~

^{4 !O5}

#2):&=&)(*G

D ~ = ~ µ × B ~

^{4 !P5}

~

µ = I · A ~ = q · ~ ω

2π · A. ~ µ

^{+*):*H%2%33)3 F9?!}%(*$H%2%33)39,

L. ~

L ~

= | ~r × ~ p | = rmv = mr ² ω = m · A π · ω

~

ω = ~ L · π

m · A ;

^{)$(')+)*9*$(=F$')Q3)$:,('012}

~ µ

^)2'$F*G

~

µ = q

2m · L; ~

!"#$%-/$)!#&')$* 2!*!..$.&'/3' 4 !-5

.*(/*1)-'&+2-%')(3-'&4(+,%&'".*563,6,(')+763&)%-+4(8/*1)-'&+2-%')(3-'&4(+.8(19

*,5-3+(+:

L ~ ² = < L ² >= ~ ² · l · (l + 1)

L z = m l · ~ ; l = 0, 1, 2 . . . ; m l = 0, ± 1, ± 2, . . . , ± l.

;!"<#=

!"#$%$&#!#'

!!"#$%&'()*+ ,-".&/".-%&'01&

L ~

^%&$

~ µ

^"23%4.-.&5.#$

B z

⁾

•

>?86,%&'()*+1(+@(81A3)'*+

γ L

^7B8.8(1*,5-3+(:

µ = γ L · L ~

;!"<<=

•

^C618+1(+$%&'()6':

µ B = e · ~ 2m e

;!"<D=

;E-,@&3"F(8',%&'()6'

µ N = m e /m p · µ B

⁼

GB8H3(I)86'('(8&*J)+*1%-+K68+(*)*&(,H8&(J'*+

γ L = − e 2m e

; µ B = − γ L · ~ .

;!"<L=

.*(M%8,6858AE(++*6'3A++)+*1N()E)%-+4(,.8(1,6,(')(88(1'('OP(31(+

B ~

^%-7

~µ

^%-+BJ):

D ~ = ~ µ × B; ~

^%-Q(84(,&*3)

D ~ = d~ L dt

→ d~ L

dt = ~ µ × B; ~ | · γ L

γ L · d~ L

dt = γ L · (~ µ × B) ~

-'4,*)

µ = γ L · L ~

^763&)

d~ µ

dt = γ L · (~ µ × B) ~

;!"<R=

!!"#$%&'()*+,-".&/".-%&'01&

~ L

^%&$

~µ

^%&$

ω L

."&.2 3#.4/-1&2"56%7.-.&89'&./:.#$)

/,+ 0)121%345$)+*&%(%67

d~ µ

dt = µ sin Θ dϕ

dt ϕ ~ e = µ sin Θ ω L · ϕ ~ e

/(0)1)1+)$*+%,+86$')125)$:,('7

d~ µ

dt = − γ L · µB sin Θ ϕ ~ e

; !"<=

25)$:+)*>)( )1'$6*7

sin Θ · ω L = − γ L · B · sin Θ

~

ω L = − γ L · B ~

!"#$"%"&'(&)* ; !" =

#$)?)+:5),($',('+41@A*)>,1B$(+*)55,('0)1C1@>)++$8(+6)D)',('48&&)(%,+0)((%:0)&

E(0,4*$8(+')+)*>)(*+*):)(0)(F$16)5A)50)1(6)$& B$(+:%5*)(0)+

B ~

^GH)50)+!

!"!# $%&'()*&+,-(*/0&'

#$) I%(')J$(G25)$:,('6)+:1)$6* 0)( 0$%&%'()*$+:)( ?)$*1%'K D)5:)1 J8( 0)1 I%1&81L1@G

>)++$8(')6,(0)()1

e ⁻

^:)11M:1*7

C1@>)++$8(

→

N$('+*18&

→

$(0,>$)1*)+&%'(!O8&)(*

→

#$%&%'()*$+&,+!

#%+@,P)1)O%'()*A)50+)$

B z

^!Q,+@*>5$:)1R*18&6)$*1%'%,A'1,(0J8(

ω L

^)$()+)$(>)5()(

e ⁻

⁷

I e = − e · ω L

2π = − e ² · B z

4πm e

; !".=

?)D$14*)+#$L85&8&)(*7

δµ e = I e · A = I e · π · < x ² + y ² >

^{; !"S=}

/01,2341+,3)6*7-)'+8*9&((

< x ² + y ² > = 2

3 < r ² >

:"#; <

/01)+(,2341+,3),6*7--+*

Z

=8)>*17()('+8*9&((?

δµ = X Z i=1

δµ e,i = X Z i=1

e ² · B z

6m e

< r ² i >

δµ = − Z · e ² · B z · < r ² >

6m e

:"#@$<

< r ² >

^+,*0A)1&88)

e ⁻

^')-+**)8*

=(*,21)3)(9)1'+A*,+3B019+)-&'(#C.,D)2*+A+8+*4*

χ m = − µ 0 · M B z

:-+*

B z ≃ µ 0 H z

^<

χ m = − µ 0 · N · Z · e ² 6m e

< r ² >;

E&(')F+(GH8'#I; $J :"#@;<

N

^{,*)3*9&A)+B019+)}&*7-&1)K)+83)(9+3*)#

L)1)3(.('F7(

< r ² >

^{B019&,M)N)+8+')6*7-} !"#$%%& $')

e ⁻

*+,-./!-&"$01112

%)1>)?

•

⁶⁸⁸⁾=9)8'&,)D)+')(O+&-&'()*+,-.,:&A'),38#C3&8)(I,3N&3)PPDN+,3)(9)(6*7G -)(#<

•

Q7()(>1+,*&88)9)168>&8+G.(9=19&8>&8+3&87')(+9):D# L#

N a ⁺ , Cl ⁻

=9)8'&,>7(R'.1&*+7(<

D)+')()A)(B&88,O+&-&'()*+,-.,#

• χ Dia

^{+,*+()+()-N)+*)(L)1)+3}&!3,!"$&'"'.$/)4.0-0#

•

^O+)1),.8*+)1)(9)%&'()*+,+)1.(',>.1F)+,*8+()&1.(91)F)1,+A)8#

!" #$%$&$'()*+,&-,

!"!# $%&' (') *+,'! -.*/'0 )/1 23/405.'1

56,!"-3!.&!77!" 8!#'.9: O+) 6*7-,2)>*17,>72+) )1'&A N)+* -)31 S.&(*)(D.,*4(9)I &8,

), 9)1 C31T9+(')1'8)+3.(' )(*,21)3)( N019)# 6A ; @$ N+19 9)1

e ⁻

^{GC2+( &8,} =1>841.(' 9+,>.*+)1*# ; @J ,38&')( C# H7.9,-+* .(9 H# =# U38)(A)> )+()(

e ⁻

G=+')(91)3+-2.8, -+*

C2+(GS.&(*)(D&38

s = ¹ ₂

^F71#

+&!".*;!"7$)*<!"%') =>?=2

6A8)(>.(')+(),

Ag

G6*7-,*1&38,:

e ⁻

GV7(R'.1&*+7(

. . . 4d ¹⁰ 5s ¹ ; l = 0

^<

!!"#$%&' ()(* +,"- ./,0&12,0#3516,07%5 $%05#8%9/ ,"& /:-7/035# ,"& /03&7;,073#,7< 7/30=

"&5:-:',&,7>3'&,/9,#$)

!"#$"%#&&#'$#()%#*#+ ,-)$./!,#$"+0

V = − ~µ · B ~

/ !""0

1(-2"

F z

⁰

− ∂V

∂z = − ∂

∂z

− ~ µ · B ~

= ∂

∂z (µ z · B z ) = µ z · ∂B z

∂z

^{/ !"10}

2+) 34)+ ')5-(6)()( 7*%$89:)<)( $-5 6)% #8=*=>9$**) ?),*@*+')(A 6$,, ), ,+8 ?)+

~µ

^{-& 6)(}

7>+(8$(6)9*

m s = ± ¹ 2

!2%)8+&>-9,?)6+('*)%')?)(,+86$')')((-%

2l + 1

7*%$89:)<)(A6+) B$8934)++,*$-,'),89=,,)(!

C)+*)%)DE>)%+&)(*)A4)98)$99)%6+(',,84)%)%?3'9!6)&7>+(3-+(*)%>%)*+)%)(,+(6F

•

D+(,*)+(G6)H$$,GDE>)%+&)(*/I.IJ0

•

K$%()**GDE>)%+&)(*/I.IL0

K)+6)&),,)(6$,'M%=&$'()*+,8)N)%8@9*(+, +(D+,)(!OP%

µ z

^{)%'+?*,+8$-,6)%}Q-5,>$9*-(' 6)%?)+6)(R)+9,*%$89)(+&7*)%(GS)%9$8GN)%,-8<=%%)<*F

µ z = ± µ B

^{/ !"L0}

T(,'),$&*'+9*5P%6)(

e ⁻

^G7>+(F

S ~ ² = < S ² >= ~ ² s(s + 1); s = 1

2 ; m s = ± 1 2 S z = ~ · m s

| ~ µ s | = 2µ B | S ~ |

~ = µ B · √ 3 µ s,z = 2µ B · S z

~ = ± µ B

^{/ !"J0}

3(%#$"%#(4$)0

~

µ s = − 2µ B

~ · S ~

^{/ !" 0}

U)*3*9@,,*,+85P%6)(&$'()*=&)8$(+,8)(#$%$99)9+,&-,/V'9!L!I!I0$99')&)+(5P%)+()(2%)8G

+&>-9,

J ~

^A4)98)%$-,

~ L

^-(6

S ~

3-,$&&)('),)*3*+,*,8%)+?)(

~

µ = − g · µ B

~ · J. ~

^{/ !"W0}

.%,*)*+)/(0 !"#$%&!'()*

g

^/(1'*(0)#

230/('.0(4*,5-6+&*6)7

g = 1

^#

230/('

e ⁻

^895*'&*6)7

g s ≃ 2

^:;&('%-(07

g s ≃ 2.0023 . . .

^<

.(0 =(0)

g s = 2

(0&*>) +*4 %-+ /(0 .*0%(8@6(*4-'&# AB00(C)-0(' /%D- *, E0B,*66(8F(0(*4 GB6&('%-+H-%')('(6(C)0B/I'%,*+4('JK(C)('#

!"!" #$% &$'()*+%$,-)./0' $-1$' 2300$1$0$4*%51'

=*0 >()0%4)(' 4*(0 +," +,"-+."+/ L366('(6(C)0B'# M4'( =(4+(6N*0C-'& DN*+4('

~ L

^-'/

S ~

GB6&('%-+/(0.*0%8@6(*4-'&/*(O*(0P-%')('D%46('

n, l, m l , m s .

;"#QR<

S%)+T46*4N(64+(6N*0C('%>(0

~ L

^-'/

S ~

,%&'()*+4,*)(*'%'/(0#2B6&('/( U>+4T)D-'& D(*&) /('$(4%'*+,-+7

!!"#$%&' ()*+ ,-.-'%&' $-/

e ⁻

^"0 1-&234#562-&7"4# $-/ 8-3&/ !-."392 -"&

B ~

^{:;-#$ "0} <=%>-:

/?/2-0@$-/

e ⁻

⁾

U-G&0-'//(+.0(4*,5-6+(+>(N(&)+*4/%+

e ⁻

^,*)/(0@(+4N*'/*&C(*)

~v

^{-,/('A(0'V>(*(*'(,}

U>+)%'/

~r

;&0B>( WT4(0-'&<# .%'' >(N*0C) /%+ (6(C)0*+4( 2(6/ /(+ A(0'+ (*'( ,%&'()*+4(

2(6/CB,5B'(')+*,X-4(+I+)(,/(+J6(C)0B'+;O&6#A%5#Y<#

B ~ ^′ _{⊥~ v} = γ

!

B ~ − ~v × E ~ c ²

"

⊥~ v

B ~ ^′ _{⊥~ v} = − ~v × E ~

c ²

;"#QZ<

[,\B-6B,>5B)(')*%6/(+A(0'+&*6)

∂V

∂r = E ~ r = Z · e · ~r

4πε _◦ r ³

;"#]!<

0(*)%1)%2)(3-('3),45$,,+,7)(8%)7+&9-5,),

~ L = m · ~r × ~v

^)%'+:*,+7

B ~ _⊥ ^′ _{~ v} = − ~v × ~r

c ² · Z · e

4πε _◦ r ³ = Z · e

c ² · 4πε _◦ r ³ m · ~ L

^{; !<.=}

8$&+*)%'+:*,+7>?%3+)@)7,)52+%4-(',AB()%'+)C2+,7)(

L ~

^-(3

S ~

^D

W = − ~µ s · B ~ = λ · L ~ · S; ~

^&+*

λ = Z · e ²

4πε ◦ m ² _e c ² < r ³ >

; !<"=

E$*,F75+7+,*

λ

^-&3)(G$4*H%

¹ ₂

;E7H&$,AG$4*H%=45)+()%2)')(%)5$*+I+,*+,7)%JH%%)4*-%!

!"#!$

W

^2+%3&+(+&$5>?%

L ~

$(*+9$%$55)5C-

S ~

^!

JH(,)K-)(C3)%

L ~ ~ S

^AJH995-('D

#%FC),,+H(IH(

L ~

^-(3

S ~

^-&)+(,*$*+H(F%),

J ~

^{L>$55,4)+(F-M)%),}

B ~

^{AG)53$(5+)'*!}

!!"#$%&'()*+,"#$-&$-./-.012$3-4"15%#.-.

J ~

^0%.

L ~

^%&$

S ~

⁾

J ~ = L ~ + S ~

; !<<=

C!N!

l = 3 j = 7

2 ; m j = ± 7

2 , . . . ± 1 2 j = 5

2 ; m j = ± 5

2 , . . . ± 1 2

J ~ 7

2

= ~ r 7

2 · 9 2 ;

J ~ 5

2

= ~ r 5

2 · 7

2

^{; !<O=}

P-*)Q-$(*)(C$75)(:)+ 3)%

L~ ~ S

^{AJH995-(',+(3D}

n, l, j, m j

^!

0(*)%3)&B+(R-,,)+(),F-M)%)(

B ~

^{AG)53), C)+'*}

J ~

^3+):)4$((*)S$%&H%9%FC),,+H(!

~

µ j = − g j · µ B

~ · J ~

~

ω L = +g j · µ B

~ · B ~

^{; !<T=}

!!"#$%&'()*+,"&-.-&-/'-0"13%&'4&10"'-567'#"38-"0 $-/9/"-&0"-/%&':;&

L, ~ ~ S, ~ J

^%&$

B ~

⁾

/)0)2(.('34(

g j

⁵

~

µ L = − µ B

~ · L; ~ ~µ s = − 2µ B

~ · S ~

6.,')4-)*0+,2)(/)*0&2*.(')(73'8$699$:$ !;)0'+9*,+2

~

µ J = − g j · µ B

~ J ~ = − µ B

~

!

J ~ + | S ~ | · cos( S, ~ ~ J) · J ~

| J ~ |

"

=

= − µ B

~ J ~

! 1 + | S ~ |

| J ~ | · S ~ · J ~

| S ~ | · | J ~ |

"

= − µ B

~ J ~

!

1 + S ~ · J ~ J ²

"

7#$<#;

!!"#$%&'()<=+ >-;?-0/"13- @-0/A30%&' B%/ $$"0";&:;&

J ~

^%&$

~ µ J

⁾

L ~ = J ~ − S; ~ L ² = J ² + S ² − 2 · J ~ · S; 2 ~ J ~ · S ~ = J ² + S ² − L ²

֒ → ~ µ j = − µ B

~ J ~

1 + J ² + S ² − L ² 2J ²

| {z }

g _j

7#$<=;

!"#$%&!'()*+

g j = 1 + j(j + 1) + s(s + 1) − l(l + 1))

2j(j + 1)

7#$<>;

!"#$%&'%()$*+%,-./#!,(+0%1+%0$%2()$*+%,-31

2&,45$4)(

B ~

^{67)89%+4*)*,+4}

J ~

^{$-,:9$,+,*9)%}4##5$,&67#1+&+*9)%;%+)(*+)%-(',<-$(6

*)(=$48

mj

^-(9>()%'+)?)%,4+)@-('

W B = m j · g j · µ B · B.

^{A !0BC}

2&,*$%D)(

B ~

^{67)89)(*DEFF)8(}

L ~

^-(9

S ~

A8$(:#,&;$1&67#1+CG$88,

~ µ j · B ~ ≫ λ · ~ L · S ~

^!H$((

,+(9

n, l, m l , m s

^'-*)I-$(*)(=$48)(1

W B = (m l + 2m s ) µ B · B.

^{A !J/C}

;#5#01%,-#,<

H+) >()%'+)(+?)$-, 5)%9)( =-& >+()( $-G'%-(9 9)% KEFF8-(' &+* 9)& K)%(,F+( 5)+*)%

$-G'),F$8*)(1 LMF)%G)+($-G,F$8*-(': $-G'%-(9 9), D8)+()( K)%(&$'()*E(,

µ n = ^m _m ^e

p · µ B

^+,*

9+),)%>N)D*?+)8D8)+()%$8,9+)7)+(,*%-D*-%$-G,F$8*-('!

O)+*)% 5+%D* 9+) P$&@,4+G* A<-$(*)()8)D*%E9M($&+,4)% >N)D*C 9)-*8+4 $-G 9+) P$') 9)%

>()%'+)(+?)$-, )+( A

E(n, j, l) 6 = E(n, j, l ± 1)

^{+& Q)')(,$*= =- 9)&} >%')@(+, 9)% H+%$6 Q8)+4-('C!R$4,*)4)(9)S@@+89-(')(')@)()+()(L+(5)+,$-G9)(>+(T-,,9)%?)%,4+)9)()(

>N)D*)!

!!"#$%&'()**+,-"./0'-1%2

H

34"5-6%79-26:9&-0%;-/-7

B ~

^3<-#$5:&#"&=7&69/-9.7+,:9/3

>:$-##? @"/63A#'?) B62!3C9"D.?EFG-/D-"&6%D7G6#.%&'$%/9H-/&7G"&)

!!"#$%&'()*+, %-./0#1%&'$23405657"&"2&"8.:;0:2&<#"&=.>%&$.103=2&<32:1.>?%@232&

A2#$)

!"!# $%& '%()*+,&%-.*/01( ,%( 2+3*(

/01')(2)/3&'),*)11.(')(4)32)(+(2+),)-5&6+*)17)8&(2)1*9

•

^:+)4)32)(;*0-,8&1)(-+*=1)>*30()(7),)*?*@

•

^:&,+,*2)33),.1*+)3)(2)A),&-*23)8+-6.1,2)3

Z

=1)>*30()(@

B)2),CD11)()1)>*30(4+322.38E+)3F.&(*)(?&81)(G

n, l, j, m j

^H7?4$G

n, l, m l , m s

^H')>)((?)+8I ()*$

F.&(*)(-)8$/032)3.('9A1)+8)J)+18)(,+(2.(.(*)3,8)+27&39

| φ(1, 2) | ² = | φ(2, 1) | ² ; φ =

^:)11)(K.(>*+0(2),LI=1)>*30()(,M,*)-, G#$" H N)+-;.,*&.,8?4)+)3

e ⁻

^{O3$ .(2L2&3K,+82+)};.K)(*8&1*,4&83,8)+(1+8>)+*(+8*P(2)3($

Q)+)(

Ψ

^(+8*,M--)*3+,+)3*) /.(>*+0()(2), 7)*3&8*)*)(LI=1)>*30()(,M,*)-, 7?'1$J)+18)(I

&.,*&.,8$R&((K01'*&.,07+')3/032)3.('G"$" H9

φ(1, 2) = Ψ(1, 2) ± Ψ(2, 1).

^G#$"LH

=S6)3+-)(*)11?)+'*,+89

/D3N0,0()(-+*+83)-'&(??&81+')(Q6+('+1*9

φ(1, 2) = φ(2, 1); +

^{IT)+8)(+(07+')3A1)+8.(' G#$"UH}

12%3)+46)(&+*6$478$64+')&9:+('+4*;

φ(1, 2) = − φ(2, 1); −

^{<=)+6)(+(>7+')%?4)+6-(' @ !AAB}

C)*8*)%) 1>%D)%-('E (F&4+6 D+) $(*+,G&&)*%+,6) CH,-('

φ(1, 2)

^{78'4!D)% I)%*$-,6-(' J>(}

K4)L*%>()(+(I+)4)4)L*%>()(,G,*)&)(M26%*8-&#$-4+<#%+(8+:;

N)D)% O-$(*)(8-,*$(D

n, l, j, m j

^{P+%D 6H6,*)(, &+* !" #}

e ⁻

!"!#$#% &'"('"#!'!)*+ # "+- .!+'! /('

φ = 0

/!)"-+!0!'! 12"3'*4 5$*64 0!) 7)!-+8936"! 3'0 :9+'" &66!)

e ⁻

^{+' ;#(8!'}

<!)0!'$<!+=(9963'*"8!-&'+"8!' !( &-#!#>

•

!""#$%&'!()#*"%+,--$!(./

~ L~ S

%+,--$!(.0>

~ L = X

i

L ~ i ; S ~ = X

i

S ~ i ;

J ~ = L ~ + S. ~

^?@4ABC

7)!-+8936"!

L ~ i

^{0!) !+'$!6'!'}

e ⁻

^{.(99!6' $3}

~ L

^% !'#"9)!-!'0 0+! :9+'"

S ~ i

^$3

S ~

^D 0&)&3"

<+)0

J ~

^{*! +60!#4 7+!"! =(9963'* #)+## !+ 6!+-#!' 3'0} 8+##!6"-<!)!' =!)'!'%&3E $<4

<!''0+!+''!)!':-&6!'8+#.6!+'!8=!)'& "#&'0&3E*!EF66#<!)0!'?1&'#-&'+0!'%

4f

^C4

G+!)F !)<+!*#0+!!6!.#)("#&#+"-!

e ⁻ e ⁻

^HI!-"!6<+).3'*0!'

~ L i S ~ i

^{HJ!)8K!0!"!+'!$6!'!'}

e ⁻

^%"(0&"".!+'

J ~ i

^{*! +60!# <+)04}

• J ~ ~ J

^H=(9963'*

J ~ i = L ~ i + S ~ i

J ~ = X

i

J ~ i

^?@4A@C

7+!"!"=(9963'*""-!8&<+)0 !+8 ;3LF66!' 0!) M3N!)!' :-&6!'/(' "-<!)!' =!)'!'

!( &-#!#4 7!)

L ~ i S ~ i

^J!)8F !)<+!*#+'0+!"!8O&66?/*64=&94A4P4P%

| B ^′ | ∼ Z

^4C4

7+!1!()"3#( #.#$( !"-)!+ !'0!'!6!.#)('+"-!'Q)3'0$3"#&'0!+'!"R+!6!6!.#)('!'"S"H

#!8"47+!"!T!*!6'#)!L!' +'" !"('0!)! !+

L~ ~ S

^H=(9963'*$3>

U4 OF)& *!"-6(""!'!:-&6!'*+6#>

L = S = J = 0

P4 OF)9&)#+!66*!EF66#!:-&6!'*+6#>

4*"5#6*7,*75859

Q)3'0$3"#&'0 $!+*# 8&V+8&6!"

S ~

^{?0&8+# <+)0 0+!} !6!.#)("#&#+"-! W'!)*+! 8+'+8+!)#%

<!*!'5!/26.!)3'*3'#!)"-+!06+-!)5&-'!'

m l,i

^{0!)!+'$!6'!'W6!.#)('!'}

i

^C

:;#75# 6*7,*75859

7!)Q)3'0$3"#&'0$!+*#8&V+8&6!"

L ~

^{?04 -4}

P

i m l,i

8&V+8&6D0+! W6!.#)('!' !/()$3*!'

*6!+-!'X86&3E"+''%38"+-'+-#$3Y !*!*'!'ZC4

<*755# 6*7,*75859OF)0+!!)"#!:-&6!'-M6E#!*6# !+8;3LF66!'8+#W6!.#)('!'>

J = | L − S |

^?@4A[C

7&8+#<+)00+!:9+'H5&-'H=(9963'*"!'!)*+!8+'+8+!)#4OF)0+!$<!+#!:-&6!'-ME#!*+6#>

J = | L + S |

^?@4A\C

8+#

J ~ ² = ~ ² J(J + 1); ~ L ² = ~ ² L(L + 1); S ~ ² = ~ ² S(S + 1)

^?@4A]C

.%/%-+012&)-',*))(23%/45%++%3&(+721++('(87%2(''-/9-,.*%,%&'()*+,-+3(*)/%&('4:%/;

)*(22&(0<22)(87%2('5%&(&(''(3('5(,5*%,%&'()*+7('=')(*2(*'('&/1>(':%/%,%&'()*+7('

?(*)/%&9(*&('#?(*+:*(29-/?(+()9-'&+/(*7('012&((*'(/5;87%2(,*)

e ⁻

^@

!!"#$%&'()*+, -/0123"4/0526430##%&'7%14%87044"90& %:;##0&0"&06$<-/2#01"3=#0836><

&0&0&34?60/0&$$0&@%&$4/0&A0'0#&)

!"!# $%&'()+() ,()(-. /0& 12&23245(-')+( 627.( 3'- '85')+(& 9'5:

;%54

=)1,(,*)(*'(,A(+%,)5/(7*,:-2+

J ~ = L ~ + S ~

^{/*7)('+*7*'(*'(,B->(/('C(25%-+4+15%++}

5*(D-%')('9%72

m j

^5*(E(/)(

j, j − 1, . . . , − j

^%''*,,)#.*(F'(/&*('*G(%-++*'5%-0&(+:%2)('@

V = − ~ µ · B ~ = m j · g j · µ B · B

^H"#I!J

?(*+:*(2@

L ~ = ~ 0; | S ~ | = ~ 2

√ 3; m s = ± 1 2

!!"#$%&'()*B,C0630"#%&'0"&04D<E0"#/0&<=&401!#042%F$"0!0"$0&-?"&7%43G&$0

m s = ± 1/2

⁾

1)(( 2+)3)+*%$')(2)(&$'()*+,5)(67()()+(),8%+,*$99,:;$9<),=(>5)%-(',?)+,)-($35>('+'

@7()+($(2)%+5%)A-,%+5*-('

m s

^{)%5$9*)(B,7'+9*2+)}C79*<&$((D;*$*+,*+EB-(2FG%

m s = ± 1/2

)%'+3*,+5H

N 2

N 1

= e ^{− ∆ kT V} = e ⁻

2 µB ·B

kT ; N = N 1 + N 2

^{: !I.=}

JG%2+)K$'()*+,+)%-(')%'+3*,+5H

M = (N 1 − N 2 ) · µ B = N · µ B · tanh µ B · B

kT

^{: !I"=}

J$99,

µ B ·B kT ≪ 1

^H

tanh µ B · B

kT ≃ µ B · B

kT

^{: !IL=}

M$&+*F79'*FG%73+'),C)+,N+)9H

M ≃ N · µ ² _B k · B

T

^{: !IO=}

K$'()*+,5);-,<)N*+3+9+*>*

χ m = ^M _H ≃ ^{µ ◦} _B ^M

^!

K+*2)%P9)+5-(':O!IO=)%'+3*,+52$%$-,2$, !"#$%&$'$()H

χ m ≃ C

T ;

^{: !II=}

JG%$99')&)+(),

J

^{'+9*2$3)+FG%2+)}Q-%+)D87(,*$(*)H

C = µ _◦ · N j(j + 1)g _j ² µ ² _B 3k

M$%,*)99-('H

1 χ m

= 1 C · T

!!"#$%&'()*+, -./012##%&'$203%/"2452021620

R)5(+,5) C)2)-*-('H S%%)+5)( *+)F)% R)&N)%$*-%)( 2-%5*+#*,*(#'.$' /0(1*20$(#'#$%

"$0@7(N$%$&$'()*+,5)(;$9<)(?+)

Gd 2 (SO 4 ) 3 · 8H 2 O

^72)%

F eN H 4 (SO 4 ) 2 · H 2 O

^{B ?73)+2+)}

&$'()*+,5)(67()((-%G3)%P+**)%,5?+('-(')(:#57(7()(= ')E7NN)9*,+(2!

3"#0)#45 6& @7%')EG59*)( 8%+,*$99 ?)%2)( &+**)9,

B ~

^{DJ)92 2+) ;N+(,} $-,')%+5*)*! M$($5

?+%22)%8%+,*$99*5)%&+,5+,79+)%*B2+);N+(,@)%9+)%)(3)+&T-(*)%F$5%)(2),

B ~

^{DJ)92),*)+9?)+,)}

!!"#$%&' ()*(+ ,-./01##%&' $1/ 234#5.671//1/ 8 $"-!-0"/41/ :&0;-'&10"/"1.1&< "; =>?

,"-'.-;;)

+/0)102(.('32+)4()0'+))(*5+)/)(,+)2&6)+2)(7+**)0,/9+('.(')(

→

:6;</=.('$

>)')( 2)0 5.0<;6=)+6)(2)( ?@+(A02(.(' +-

mK

^{BC)0)+/ ;&(( 2)0 D0A5),, (+/* 5E;=+,/}

')F&/0)(9)02)($

4(*,/)+2)(2G >)/,)=9+0;.(' 2)0 &.,5.0+/*)(2)( H+@A=) .(*)0)+(&(2)0 -.,, ;=)+( ,)+(3

2&-+* 6)+ *+)F)0 I)-@)0&*.0

T

!"#!

(A/ )+() .(')A02()*) 10+)(*+)0.('JA0 2)- :(=)')( JA(

B ext = B 2

^JA0=+)'*$

!"!#$%&'( H+) )*+),)-+.0! 1&-")'&!-+.+!#%&' 23& 4!#&.5+&. F</0* (&/ 2)-

'=)+/)(KA(5)@*9)+*)06+,

T end ≃ µK

^$

!"!# $%&'()*%+%( ,%&'-.*/%0%1'(2.%.3-456*.%'&/4-/ 7%& 5%'600%.

6)#)")'&!-+.0!43"53&!&-!76)%8+ 96)#)")'&!-+."%.:

:.F'0.(22)0 ?@+(A0+)(*+)0.(' +-%&'()*F)=2

B

^{)0'+6* ,+/ )+(:(/)6)( 659$:6,)(;)(L)2),}

e ⁻

B4()0'+)(+J)&.,.-

± µ B · B

^$H&-+*,@&=*)*2+)M.,*&(2,2+/*)

D(E)

^2)0F0)+)(

e ⁻

^NJ'=$K&@$

O$ P+(59)+CQ(2)0&.FG

:( 2)0 R)0-+;&(*) '+6* ), )+()( C)+*0&' 5.- D&0&-&'()*+,-., +(()0/&=6 2)0 4()0'+)60)+*)

± µ B · B

^{$R<02)(S(*)0,/+)2+(2)0}4=);*0A()(;A(5)(*0&*+A()0'+6*,+/2&((6)+

T ≪ E F /k B

.(2

µ B · B ≪ E F

^{.(*)0T)09)(2.('JA(7='$NO$ UPG}

N ( ↑ ) − N( ↓ ) = D(E F )

2 · 2 · µ B · B = 3N

2E F · µ B · B

^N#$U#P

;)'&!-+.+!#%&'(

M p = µ B (N ( ↑ ) − N( ↓ )) = 3N

2E F · µ ² _B · B

D&.=+B?.,5)@*+6+=+*Q*G

χ p = µ _◦ · M p

B = 3N

2E F · µ _◦ µ ² _B

^N#$UVP

!!"#$%&'()*+, %-./0#12& $234%.10&$.$"612-32"23

e ⁻

^"7 8%9232&:0'&21-2#$)

!"#"$%&'!(*&+,#-,%&%'& .,% /&!'0%$(&1&2'3,%&%0%4 5,%&%36#-7&%

1+) 2%)+)( 34)5*%6()( 7)%8)( $-2'%-(8 8)% 96%)(*:5%$2* +& ;-<)%)(

B ~

^{=>)48 $-2} ,?+%$42@%&+') A$B()('):7-(')(C9$(8$-DE!#BF,+5 GD.0H/DI! "0℄!

K6(,)L-)(:MN-2'%-(8 8)% 9)(:,B)( P)')4 )%'+Q* ,+B )+() 8+$&$'()*+,B) R+%5-('D 7)4B)

8+)#$-4+=I-,:)?*+Q+4+*;*')%$8)-&

1

3

^{%)8-:+)%*!N4,}I-&&)$-,8)&9)+*-(',)4)5*%6()(Q)+*%$' -(8 8)& 8+$&$'()*+,B)( A)+*%$' S6( $Q'),B46,,)()( IB$4)( -(8 ')2T44*)( A;(8)%( &+*

Z i

=34)5*%6()(US'4!K$?!G!.!"D9$(')S+(=V4'!W)%'+Q*,+B8$((M

χ tot = N

E F · µ _◦ µ ² _B − µ _◦ N a · Z i · e ² 6m e

< r ² >

^{U !XYW}

1$Q)+Q)8)-*)(M

N =

9)+*-(',)4)5*%6()(8+B*)

N a =

^1+B*)8)%N*6&%T&?2)

Z i =

^{E$B48)%+(()%)(}

e ⁻

^?%6N*6&

8&!(-!&1&9

38)4&)*$44)

Cu, Ag, Au

^,+(88+$&$'()*+,BD"!Z)%&TQ)%7+)'*

[)*$44)8)%.!D"!-(8H!V%-??),+(8?$%$&$'()*+,B!

!"! #$%&'($)$ *+,(-)&./).&$( .(' -+%()0($ 102($),-,$&.(2

1+)\%8(-('8)%I?+()+(,*)44-('7+%8Q)+ 2)%%6=D$(*+2)%%6=-(82)%%+&$'()*+,B)(I*6])(8-%B

8+)[+(+&+)%-('8)%,6'!N-,*$-,B)()%'+)Q)7+%5*!1+),)3()%'+))(*,?%+B*8)%1+])%)(:+(8)%

)4)5*%6,*$*+,B)(R)B,)47+%5-(')()%'+):7+,B)(E-,*;(8)(&+*-(*)%,B+)84+B)%I?+(6%+)(*+)=

%-('8)%34)5*%6()(! N4,A)+,?+)42T%8+)N-,*$-,B7)B,)47+%5-('7+%8:-(;B,*8+)56S$4)(*)

A+(8-('8),

H 2

^{=[64)5T4,Q)*%$B*)*!}

!!"#$%&'()*+, -."/01'. 2%034%#45!67&.0'". "589::.0:/4;54#.<=#)

!"#$!"&'(')*+','"

H 2

^-./0'120"

.%,*/)0'12-'3)*0'4-5(/(3)5*+7('8(7+(/9*53-'&:

H = e ² 4πε _◦

1 R − 1

r α1 − 1 r α2 − 1

r β1 − 1 r β2

+ 1 r 12

;"#<=>

3'+#0')-4/(,/(-5')6$&)'(7?-+&%'&+@-'3)+*'AA*(8(//('B-'3)*0'('A(5(*'4(/'('?)0,051

C*)%/(DA*(C(* &50E(,F(5'%C+)%'A4-)5(G('#

•

^H*(8(//('B-'3)*0'

Ψ

^A(+$(75)(*/7('+I+)(,+*,J5)+5%-,(5&*C)+*7A%''%/+K50A-3) A(5L*'4(/)(*/7('B-'3)*0'('#

•

^H*(M'-')(5+7(*AC%53(*)A(5N(*/7('A5O3)+*7*'A(5PI,,()5*( A(54-)5(G('A('Q(1 +%,)9(//('B-'3)*0'

φ

^%-+:

8/"/('(7PI,,()5*+7&(&('OC(5N(*/7('R(5)%-+7-'&

φ(1, 2, . . . i, j . . .) = + φ(1, 2, . . . j, i . . .)

;"#"!>

9'):+/('(7?')*+I,,()5*+7(8(//('B-'3)*0'

φ(1, 2, . . . i, j . . .) = − φ(1, 2, . . . j, i . . .)

;"#" >

H*( Q(+%,)9(//('B-'3)*0'*+) A%+K50A-3) R0'

Ψ

^{,*) A(5}P@*'B-'3)*0'

χ

^{#S(*&50E(,F(5'%C1}

+)%'A

R

^{49*+7('A('.1?)0,('/%-)()A%''A*(}8(//('B-'3)*0'BO5A%+T9(*1L/(3)50'('+I+)(,

;2(5,*0'('+I+)(,>:

φ as = N a [Ψ α (1) · Ψ β (2) − Ψ β (1)Ψ α (2)] · χ s

;"#"U>

Ψ α (1) =

⁸(//('B-'3)*0'R0'L/(3)50'V5# C(* F(5'

α Ψ β (1) =

⁸(//('B-'3)*0'R0'L/(3)50'V5# C(* F(5'

β

N a =

V05,*(5-'&+B%3)05

χ s =

PI,,()5*+7(P@*'B-'3)*0'

( ↑↑ ) χ a =

?')*+I,,()5*+7(P@*'B-'3)*0'

( ↑↓ )

?/)(5'%)*RC(30,,),%'(*'%')*+I,,()5*+7(+

φ sa

^,*)

φ sa = N s [Ψ α (1) · Ψ β (2) + Ψ β (1)Ψ α (2)] · χ a .

;"#"W>

/+)% +,* $0,1 2+) 3%*,4-(5*+1( ,6&&)*%+,8-(2 2+) 9:+(4-(5*+1( $(*+,6&&)*%+,8! ;+) <$8%=

,8)+(0+85)+*>'0)+8?)+*+'+(2)(3%*,@10-&+($

dV 1

^-(2

dV 2

^{)+(A0)5*%1($(?-*%)B)(C+%22-%8}

2+)D-,2%E5)

ρ as = | Ψ α (1)Ψ β (2) − Ψ β (1)Ψ α (2) | ² dV 1 dV 2

^F?C!

ρ sa = | Ψ α (1)Ψ β (2) + Ψ β (1)Ψ α (2) | ² dV 1 dV 2

F),*+&&*

.

^{G ! HI}

!!"#$%&' ()*+, -%.#"/./"01 2.34/1##%&' 05&

φ sa

^%&$

φ as

^{673 '3581} 9#"&:4; %&$ :#1"&1 931=/4;

>13&.!4/?&$1

α, β

^{)@.& 4"1=/A$.44 !1"}

φ sa

1"&1=B=131C#1:/35&1&.%61&/=.#/4D.=34=1"&#"=:1"/

ED"4=1&$1&!1"$1&F54"/"01&>13&1&!14/1=/9G"&$%&';)

!"#$% ;)% 2-%8

φ sa

F),8%+)F)() J-,*$(2 +,* ,*$F+0> 2$ )% 2-%8 )+() 8K8)%) D-4)(*=

8$0*,C$8%,8)+(0+85)+*

ρ sa

^{2)% ()'$*+@)( A0)5*%1()( ?C+,8)( 2)( :1,+*+@ ')0$2)()( L)%()(}

)()%')*+,8'E(,*+')%+,*!;+),)%'+F*2+)&!'("$)*$+,)-.)#!

/+)%C+%2$0,1 2+) D-,%+8*-('2)% F)+2)(

e ⁻

^=9:+(,

( ↑↓ )

^2-%82+) M+(+&+)%-('2)% N1-01&F=

A()%'+)+&

H 2

^{=M10)5E0)%?C-(')(!;+)}A()%'+)2+B)%)(? ?C+,8)(2)( J-,*O(2)(

( ↑↓ )

^-(2

( ↑↑ )

+,*+(2+),)&P$002+)/.0*(.02$)$3#,$!

$33!4(#)$*,04.0

Q)+&D-BE00)(2)%R2=98$0)*%+**P)%%1&$'()*+,&-,F)+2)(A0)&)(*)(P)G

3d ⁶ 4s ²

^I>N1G

3d ⁷ 4s ²

^I

-(2S+G

3d ⁸ 4s ²

^I$-4!

+$5.)-$%

•

^{;+) 2=3%F+*$0) ,:$0*)( $-4'%-(2 +8%)%} -(*)%,8+)20+8)( 3%+)(*+)%-(' +( Q)?-' $-4 2+) T+**)%($8F$%( $-4 GN1-01&F:1*)(?+$0I! ;$&+* 51&&* ), ?- DFC)+8-(')( @1( 2)(

/-(2,8)(U)')0(!

•

^{;$,)0)5*%+,8)} L%+,*$004)02 @)%-%,$8*)+( 67.$)2$) 8 2)% R2=Q$8(2%)8+&:-0,)V;+),) Q$8()(0+)')(C)+*$-W)(-(2C)%2)((+8*&)8%C+)+&?)(*%$0)(N1-01&F4)02+(Q)?-'

$-4)+()M$'()*4)02%+8*-(',*$F+0+,+)%*V

< L z > → 0

^{G ! XI}

!!"#$%&'()*+,-#./01"24.5"&$%&'2.&.1'".&671$".8%209&$.

φ sa

^!:;)

φ as

^"& !49&'"'/."0<=>

?.1&@!20@&$"& -"&4."0.&$.25=411@$"%2 A

0.53 · 10 ⁻¹⁰ m

^B)

!!"#$%&'()*C, %6'1%&$$.2?1"20@##D=0.&:"@#21"40.&2"4$".E1.4">D%#2.&"40>.412= ;".

">8.&01@#D=0.&:"@#@&."&.1 42.@%2F 2=&$.1&GH%.&4.&I)

/&-+*0)1.2+)0* ,+4 1)05&0&-&'()*+,4) 6(*)+72.0%&'()*+,+)0.(' &.81)( 95+(

S ~

^1)0:7);<

*0=()(#67,-&'()*+,4),%=-)(*50=>+**)057&*2 -+,,*-&(?

F e : µ ef f,F e = 2.2µ B

@"#""A

µ ef f,N i = 0.6µ B

@"#"BA

/+)

3d

^.(1

4s

<:7);*0=()( 7+)')( +( CD(1)0(E F)74) )()0')*+,4 GH)07&55)(E 1+) :7);*0=()(<

7)+*.(' F+01 1&4)0 &., H)+1)( CD(1)0( '),5)+,*# /&, )0;7D0* 1+) =H)(,*)4)(1)( )I);*+J)(

/+5=7-=-)(*)50=>+**)057&*2#

/+) :0;7D0.(' 8G0 1+) !"#$%#& '%(#&$) )&*+#( @6.,0+4*.(' 1)0 95+(, =4() D.K)0),

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!!"#$%&' ()**+ ,-./1"-!%&' $-. 2%/34&$/$"13-&

( ↑ )

^%&$

( ↓ )

4%5'.%&$ $-. 6-"/-&!-.'/1-&

%/34%/178-1/-#9".:%&')

01'2! &+*3()%'+)4)%,6+)7-('8)%9)+*-(',)2):*%;()(0<$=!>!"!?!.@ 7)+

B ext = 1T : ∆E = 2µ B · B ≃ 10 ^{− 4} eV )

^!

A+)B-,*$-,6C)6,)2C+%:-('DE6%*6+)%F-)+()%G=+(HB-,%+6*-('IC)26)C+)8)%8+)J;2;-&7H

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,6%)+7-('8)%,=;(*$()(L$'()*+,+)%-(')+(')DE6%*U

∆E = 2µ B · B W ; → B W ∼ 10 ⁴ T

^{DE%3+,)(V 0 ! W@}

B-,*$-,6HK)6,)2C+%:-('FC+,6)(8)(32):*%;()(

i, j

^U

H = 2J ij · S ~ i · S ~ j

^{0 ! Q@}

A+)=$%$22)2)G=+($-,%+6*-('DE6%* F-)+()% &+(+&$2)( J;-2;&7)()%'+)$-D'%-(8 8)%%X-&2+6

'E(,*+')(1)%*)+2-('8)%32):*%;()(7F'2!8)%=;,+*+4')2$8)()(Y;()(%E&=D)!

A+)GX**+'-(',&$'()*+,+)%-(':$((&$(+(Z-,$&&)(6$('&+*8)&K)+,,HO)287%+(')(U

B W = λ · M s

^{0 !SR@}

ZC+,6)(

λ

^-(8

J ij

^{2X,,*,+6)+()T)F+)6-('6)%2)+*)(!}

→

^A+)T),6%)+7-('E7)%

B W

^{+,*+(,;D)%()%2$-7*!}

!"#$

B W

^{)[+,*+)%*%&()$2,}&$'()*+,6),O)28I ,;(8)%((-%$2,L;8)22'%\])!

^7)%6$27 )+()% &$*)%+$2,=)F+_,6)( `)&=)%$*-%

T c

0J-%+)H`)&=)%$*-%@ 4)%,6C+(8)( 8+) D)%%;&$'()*+,6)( 3+')(,6$D*)(! A)% a7)%'$(' +,* )+( #6$,)(E7)%'$(' FC)+*)% B%* 0), C+%8

:)+()a7)%'$(',)()%'+)7)(\*+'*I$7)%),'+7*,=%-('6$D*)b(8)%-(')(4;(

c p . . .

^@!OE%

T ≥ T c

7),*)6*=$%$&$'()*+,6),1)%6$2*)(0^%8(-('8)%G=+(,C+%8'),*\%*@!

*+#&",-"&..,/".")0$

χ = C T − T c

;

^DE%

T ≥ T c

^{0 !S.@}

T c,F e ≃ 1043K

^{c OE%}

T → T c

^')6*

χ → ∞ : M ~ = χ · H ~ →

^c,=;(*$()L$'()*+,+)%-('DE%

T ≤ T c

^@!

/0&1&2*)1+,+)1.('3)114-&'()*+,0)16*47)89':#;)-<)1&*.1=)10&:*)(+-/.1+)>?)+,,>@:4*A

!!"#$%&'()*+, -./0.123%12!45&'"'6."3 78&

M s

%&3.142#! 78&

T c

^{98:". 78&}

1/χ

^{8!.142#! 78&}

T c

^;7'#)2%4=%1".>?.9.3@A)

1 χ = 1

C (T − T c )

^3B1

T ≥ T c

^C"#DEF

GB1

C

^'+:*)(*,<1)0)(HI&<#!#E#!

C = µ _◦ N g ² _j · S(S + 1)µ ² _B 3k B

CGB1J+,)(A

g j ∼ 2; S ∼ 1

^{F C"#DKF}

J+(-L':+0),M.3*1)*)(=4(G)114-&'()*+,-.,2&((-+*34:')(H)1N)H+('.('')<1B3*O)1H)(A

J e · D(E F ) · V a ≥ 1;

^{C6*4()1 .(H}?40:3&1*0F C"#D!F

J e =

M.,*&.,0)()1'+)<14J:)2*14(

D(E F ) =

P.,*&(H,H+0*)&(H)1G)1-+2&(*)

V a = Atomvolumen

Q8+') R(':)+0.(' )12:S1*T O&1.- (+0* &::) %)*&::) G)114-&'()*+,-., 9)+')(A

D(E F )

^-.,,

'14U')(.',)+(#VH)&:,+(H,0-&:)T0&:8')3B::*)J()1'+)8S(H)1#

J+() %+(+-+)1.(' H)1 -&'()*+,0)( J()1'+) 8)+- J+(*1)*)( =4( ,<4(*&()1 %&'()*+,+)>

1.(')134:'*&.334:')(H)?)+,)A

W+) <&1&::):) M.,1+0*.(' H)1 6<+(, )134:'* +( W4-S()( C?)+,,,0)( N)9+12)(FT O):0)

H.10 XN:40,0) ?S(H) Y=4()+(&(H)1 X')*1)((*Y,+(H# N)+- Z403&01)( )+(), S.U)1)(G):H),

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W&8)+'+8*),9O)+O),)(*:+0)@149),,)A

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