!"#$%& ' !"#$%&'()' ! !"## $%& '(!) +& $&( ,)-.&/%!%&(0.- ~M %. ,)/&(%& -&.)0&( 1&/() +/&/ 2&($&.3 4("1 5).. 6).$%& 7("1&. %. 8 4(099&. &%./&%#&.: &./!9(& +&.$ $&6 ;&(/&1&(&% + $&( 6)-.&/%! +&. <0!=&9/%>1%#%/?/χm = µr − 1. @A3BC

• !"#"$%&'!(#)(* −1 < χm < 0χm → −1 2%($ %. <09()#&%/&(. 1&% -&.D-&.$ ! +2) +&( 6)-.&/%! +&( ((&-0.- &((&% +/3

• +","#"$%&'!(#)(* 0 < χm < 105E%&( !%.$ &%.& 2% +/%-& './&(-(099& F&.& <01!/).=&.: 2&# +& !9"./).& ,)-.&/%!%&(0.-=&%-&.: =3 G3 H&(("6)-.&/%!60!3I%& ;%(50.- )0J $%& ~B>H&#$K&(/&%#0.- 1&% L"(+).$&.!&%. &%.&( &##%9!"%$JM(6%-&. 7("1& =&%-/ $%&N11%#$0.- O3B3P

!!"#$%&' ()*+ ,"-.%&' /0& $"121'&34"5 732 8#"&.59 !:;) <1-121'&34"5 732 =##"<50"$ 1%> 3"&3~B?@3#$/3-43"#%&' "2 A1.%%2) QR

! "#$ %&'()*+,-., !" #$%&%'()*$+&,+/+&-&'()*+,-., 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)9 6)+ :&9&-&'()*+, D)(2*34)( E)97) =*#/&, =1&,,+, D) FD)39)- E3( 8&9-39GH)1&7)() F)+1 D)( -+* 8&7.(' ~q .(7 /9)D+-:.1, ~L >ID9)( +- ~B@A)17 )+() J9;0),,+3(,6)B)'.('-+* ~ωL &.,G~ωL = − q

2m· ~B; K"#LM

!!"#$%&' ()*+ ,"&- ". -#-/01"2 4-. 5-&016#780-&9"6# !-:-'0- ;6$%&' q <=410 ". >%?-1-& ~B@A-#$$"- ;61.8171>9-22"8& 6%2)#$%& () *)+", (+! -. N9)+,6)B)'.(' )+(), >9)+)( F)+1 D)(, -+* 8&7.(' q +- ~B@A)17# H1)+ D@')B+ D* E3( O)(*9+>.'&1=9&>* .(7 839)(*0=9&>*Gmrω2

c = qrωcB

ωc =q

m·B K"#PM

ωc +,* 7+) OQ=13*93(9),3(&(0>9)R.)(05 3>* &. D HQ93>9)R.)(0 ')(&((*# !"!" #$%&' )%*+,$+ -*& .'&/0&1&2-$3340+S+9 6)*9& D*)( 7+) DQ:3*D)*+, D) N9)+,6&D( )+(), ')6.(7)()( e− ,)(=9) D* 0. ~B# /.9 D R.&(@*)(-) D&(+, D) T)7+('.(')( +,* 7+),)9 A&11 E)963*)(5 -&( 6)=3--* &6)9 7&-+* 6)9)+*, 7)( S)9*E3( ωL# U, '+1* 7&, N9;>*)'1)+ D')B+ D*mrω2 =

e2

4πε◦r2+ erωB. K"#!M

Zentrifugalkraft = Coulombkraft + Lorentzkraft/+) 839)(*0=9&>* +,* +- V11')-)+()( )+() =1)+() N399)=*.9 0.9 )1)=*9+, D)( T+(7.('#

!" #$%&%'()*$+&,+ -./)$ ω = ω◦ 012 B = 0! #%(( '$3*mrω2 = mrω2

◦+ erωB

ω2 − eB

mω − ω2

◦ = 0

ω1,2 =eB

2m±

√(

eB

2m

)2

+ ω2◦. 4 !.56(*)2 7)2 8,29)3 +*):* 7)2 ;3)$() <=22);*,2*)2& (

eB2m

)2 9, ω2◦> ?)3 :)2 %,0 2) :*+A ,(7 3$(;+A72):)(7) e−(±) '3)$ : ?$2;*! #)2 7%B=2+*):)(7) C)2& eB

2m7%')')( +*):* 012 7)( )$(:)$*3$ :)(#2):+$(( 7)2 D2E9)++$=(+F)?)',('G

ωL =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(!

!!"#$%&' ()*+ ,%- .-"/&0"/-%&' 12& 3"42#/& "5 6%7/-/& ~B89/#$) :/ <0=+ .-"/&0"/-%&' 12& ~L%&$ ~µ >?- /"&/ 42="0"1/ @A$%&' q)D=*)(9$)33) L()2'$) )$()+ &%'(! #$H=3+ $& ~BAM)37 4=:() N)?)$+5GV = −~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)3 9, ~L.

∣∣∣~L

∣∣∣ = |~r × ~p| = rmv = mr2ω =

m ·Aπ

· ω

~ω =~L · πm · 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+(+:~L2 = < L2 >= ~

2 · l · (l + 1)

Lz = ml · ~; l = 0, 1, 2 . . . ; ml = 0,±1,±2, . . . ,± l. ;!"<#= !"#$%$&#!#'

!!"#$%&' ()*+ ,-".&/".-%&' 01& ~L %&$ ~µ "2 3%4.-.& 5.#$ Bz)• >?86,%&'()*+ 1(+ @(81A3)'*+ γL 7B8 .8(1*,5-3+(:

µ = γL · ~L ;!"<<=• C618+ 1(+ $%&'()6':

µB =e · ~2me

;!"<D=;E-, @&3" F(8',%&'()6' µN = me/mp · µB=GB8 H3(I)86'(' (8&*J) +* 1 %-+ K68+(*)*&(, H8&(J'*+γL = − e

2me

; µB = −γL · ~. ;!"<L=.*( M%8,6858AE(++*6' 3A++) +* 1 N()E) %-+ 4(, .8(1,6,(') (88( 1'('O P(3 1(+ ~B %-7 ~µ %-+BJ):~D = ~µ× ~B; %-Q(84(, &*3) ~D =

d~L

dt

→ d~L

dt= ~µ× ~B; | · γL

γL ·d~L

dt= γL · (~µ× ~B)-'4 ,*) µ = γL · ~L 763&)

d~µ

dt= γL · (~µ× ~B) ;!"<R=

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

!!"#$%&' ()*+ ,-".&/".-%&' 01& ~L %&$ ~µ %&$ ωL ."&.2 3#.4/-1&2 "5 6%7.-.& 89'&./:.#$)/,+ 0)1 21%34 5$)+* &%( %67d~µ

dt= µ sin Θ

dϕ

dt~ϕe = µ sin Θ ωL · ~ϕe/(0)1)1+)$*+ %,+ 86$')1 25)$ :,('7

d~µ

dt= −γL · µB sin Θ ~ϕe ; !"<=25)$ :+)*>)( )1'$6*7

sin Θ · ωL = −γL · B · sin Θ

~ωL = −γL · ~B !"#$"%"&'(&)* ; !" =#$) ?)+ :5),($',('+41@A*) >,1 B$(+*)55,(' 0)1 C1@>)++$8(+6)D)',(' 48&&)( %,+ 0)( (% : 0)&E(0,4*$8(+')+)*> )(*+*):)(0)( F$16)5A)50)1( 6)$& B$(+ :%5*)( 0)+ ~BGH)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*7C1@>)++$8( → N$('+*18& → $(0,>$)1*)+ &%'(! O8&)(* → #$%&%'()*$+&,+!#%+ @,P)1) O%'()*A)50 +)$ Bz ! Q,+@*>5$ :)1 R*18&6)$*1%' %,A'1,(0 J8( ωL )$()+ )$(>)5()( e−7Ie = −e · ωL

2π= −e2 ·Bz

4πme

; !".=?)D$14*)+ #$L85&8&)(*7δµe = Ie · A = Ie · π· < x2 + y2 > ; !"S=

! "#$ %&'()*+,-.,/01 ,2341+, 3) 6*7-) '+8* 9&((< x2 + y2 > =

2

3< r2 > :"#; </01 )+( ,2341+, 3), 6*7- -+* Z =8)>*17()( '+8* 9&((?

δµ =

Z∑

i=1

δµe,i =

Z∑

i=1

e2 · Bz

6me

< r2i >

δµ =−Z · e2 · Bz· < r2 >

6me

:"#@$<< r2 > +,* 0A)1 &88) e− ')-+**)8*=(*,21) 3)(9 )1'+A* ,+ 3 B01 9+) -&'(# C.,D)2*+A+8+*4*

χm = −µ0 ·MBz

:-+* Bz ≃ µ0Hz<χm = −µ0 ·N · Z · e2

6me

< r2 >; E&(')F+(GH8'#I ; $J :"#@;<N ,*)3* 9&A)+ B01 9+) &*7-&1) K)+8 3)(9+ 3*)#L)1) 3(.(' F7( < r2 > B01 9&, M)N)+8+') 6*7- !"#$%%& $' ) e−*+,-./!-&"$01112%)1>)?• 688) =9)8'&,) D)+')( O+&-&'()*+,-., :&A'), 38# C 3&8)(I , 3N& 3) PP DN+, 3)( 9)( 6*7G-)(#<• Q7()(>1+,*&88) 9)1 68>&8+G .(9 =19&8>&8+3&87')(+9) :D# L# Na+, Cl− =9)8'&,>7(R'.1&*+7(<D)+')( )A)(B&88, O+&-&'()*+,-.,#• χDia +,* +( )+()- N)+*)( L)1)+ 3 &!3,!"$&'"'.$/)4.0-0#• O+) 1),.8*+)1)(9) %&'()*+,+)1.(',>.1F) +,* 8+()&1 .(9 1)F)1,+A)8# !" #$%$&$'()*+,&-, !"!# $%&' (') *+,'! -.*/'0 )/1 23/405.'156,!"-3!.&!77!" 8!#'.9: O+) 6*7-,2)>*17,>72+) )1'&A N)+* -)31 S.&(*)(D.,*4(9)I &8,), 9)1 C 31T9+(')1'8)+ 3.(' )(*,21) 3)( N019)# 6A ; @$ 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

2F71#+&!".*;!"7$ )*<!"%' ) =>?=26A8)(>.(' )+(), AgG6*7-,*1&38, :e−GV7(R'.1&*+7( . . . 4d10 5s1; l = 0<

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

!!"#$%&' ()(* +,"- ./,0&12,0#3 516,07% 5 $%0 5#8%9/ ,"& /:-7/035# ,"& /03&7;,073#,7< 7/30="&5:-:',&,7 >3'&,/9,#$) !"#$"%#&&# '$#()%# *#+ ,-)$. /!,#$"+0V = −~µ · ~B / !""01(-2" Fz0

−∂V

∂z= − ∂

∂z

(

−~µ · ~B)

=∂

∂z(µz · Bz) = µz ·

∂Bz

∂z/ !"102+) 34)+ ')5-(6)()( 7*%$89:) <)( $-5 6)% #8=*=>9$**) ?),*@*+')(A 6$,, ), ,+ 8 ?)+ ~µ -& 6)(7>+( 8$(6)9* (

ms = ± 1

2

)! 2%)8+&>-9,?)6+('* )%')?)( ,+ 8 6$')')( (-% 2l + 1 7*%$89:) <)(A 6+)B$89 34)+ +,* $-,'), 89=,,)(!C)+*)%) DE>)%+&)(*)A 4)9 8) $99)%6+(', , 84)%)% ?3'9! 6)& 7>+( 3- +(*)%>%)*+)%)( ,+(6F• D+(,*)+(G6) H$$,GDE>)%+&)(* /I.IJ0• K$%()**GDE>)%+&)(* /I.IL0K)+6) &),,)( 6$, 'M%=&$'()*+, 8) N)%8@9*(+, +( D+,)(! OP% µz )%'+?* ,+ 8 $-, 6)% Q-5,>$9*-('6)% ?)+6)( R)+9,*%$89)( +& 7*)%(GS)%9$ 8GN)%,- 8 <=%%)<*F

µz = ±µB / !"L0T(,'),$&* '+9* 5P% 6)( e−G7>+(F~S2 = < S2 >= ~

2 s(s + 1); s =1

2; ms = ±1

2Sz = ~ ·ms

|~µs| = 2µB

|~S|~

= µB ·√

3

µs,z = 2µB ·Sz

~= ±µB / !"J03(%#$"%#(4$)0

~µs = −2µB

~· ~S / !" 0U)*3* 9@,,* ,+ 8 5P% 6)( &$'()*=&) 8$(+, 8)( #$%$99)9+,&-, /V'9! L!I!I0 $99')&)+( 5P% )+()( 2%)8G+&>-9, ~J A 4)9 8)% $-, ~L -(6 ~S 3-,$&&)('),)*3* +,* , 8%)+?)(

~µ = −g · µB

~· ~J. / !"W0

!! "#! $%&'()*+,-+.%,*) *+) /(0 !"#$%&!'()* g /(1'*(0)#230 /(' .0(4*,5-6+ &*6)7 g = 1#230 /(' e−895*' &*6)7 gs ≃ 2: ;&('%-(07 gs ≃ 2.0023 . . .<.(0 =(0) gs = 2 (0&*>) +* 4 %-+ /(0 .*0%(8@6(* 4-'&# AB00(C)-0(' /%D- *, E0B,*66(8F(0(* 4GB6&(' %-+ 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 -'/ ~SGB6&(' %-+ /(0 .*0% 8@6(* 4-'& /*( O*(0 P-%')('D%46('n, l, ml, ms. ;"#QR<S%)+T 46* 4 N(6 4+(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

c2

)

⊥~v

~B′⊥~v = −~v × ~E

c2;"#QZ<[, \B-6B,>5B)(')*%6 /(+ A(0'+ &*6)

∂V

∂r= ~Er =

Z · e · ~r4πε◦r3

;"#]!<

!" #$%$&$'()*+,&-, ./.0(*)% 1)%2)(3-(' 3), 45$,,+, 7)( 8%)7+&9-5,), ~L = m · ~r × ~v )%'+:* ,+ 7~B′⊥~v = −~v × ~r

c2· Z · e4πε◦r3

=Z · e

c2 · 4πε◦r3m· ~L ; !<.=8$&+* )%'+:* ,+ 7 >?% 3+) @) 7,)52+%4-(',AB()%'+) C2+, 7)( ~L -(3 ~SD

W = −~µs · ~B = λ · ~L · ~S; &+* λ =Z · e2

4πε◦m2ec

2 < r3 >; !<"=E$*,F 75+ 7 +,* λ -& 3)( G$4*H% 1

2;E7H&$,AG$4*H%= 45)+()% 2)')( %)5$*+I+,*+, 7)% JH%%)4*-%! !"#!$ W 2+%3 &+(+&$5 >?% ~L $(*+9$%$55)5 C- ~S!JH(,)K-)(C 3)% ~L~SAJH995-('D#%FC),,+H( IH( ~L -(3 ~S -& )+( ,*$*+H(F%), ~J L >$55, 4)+( F-M)%), ~BAG)53 $(5+)'*!

!!"#$%&' ()*+ ,"#$-& $-. /-.012$3-4"15%#.-. ~J 0%. ~L %&$ ~S)~J = ~L + ~S ; !<<=C! N! l = 3

j =7

2; mj = ±7

2, . . .± 1

2

j =5

2; mj = ±5

2, . . .± 1

2∣∣∣ ~J

∣∣∣7

2

= ~

√

7

2· 9

2;∣∣∣ ~J

∣∣∣5

2

= ~

√

5

2· 7

2; !<O=P-*) Q-$(*)(C$75)( :)+ 3)% ~L~SAJH995-(' ,+(3D n, l, j, mj!0(*)% 3)& B+(R-,, )+(), F-M)%)( ~BAG)53), C)+'* ~J 3+) :)4$((*) S$%&H%9%FC),,+H(!

~µj = −gj ·µB

~· ~J

~ωL = +gj ·µB

~· ~B ; !<T=

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

!!"#$%&' ()*+ ,"&- .-&-/'-0"1 3 %&'4&10"'-5 67'#" 38-"0 $-/ 9/"-&0"-/%&' :;& ~L, ~S, ~J %&$ ~B)/)0) 2(.(' 34( gj 5~µL = −µB

~· ~L; ~µs = −2µB

~· ~S6., ')4-)*0+, 2)( /)*0& 2*.(')( 73'8$ 699$ :$ !; )0'+9* ,+ 2

~µJ = −gj · µB

~

~J = −µB

~

(

~J + ~|S| · cos(~S, ~J) ·~J

| ~J |

)

=

= −µB

~

~J

(

1 +|~S|| ~J |

·~S · ~J

|~S| · | ~J |

)

= −µB

~

~J

(

1 +~S · ~J

J2

) 7#$<#;

!!"#$%&' ()<=+ >-;?-0/"1 3- @-0/A 30%&' B%/ $$"0";& :;& ~J %&$ ~µJ )~L = ~J − ~S; L2 = J2 + S2 − 2 · ~J · ~S; 2 ~J · ~S = J2 + S2 − L2

→֒ ~µj = −µB

~

~J

(

1 +J2 + S2 − L2

2J2

)

︸ ︷︷ ︸

gj

7#$<=; !"#$%&!'()*+gj = 1 +

j(j + 1) + s(s + 1)− l(l + 1))

2j(j + 1)7#$<>;

!" #$%$&$'()*+,&-, ./0 !"#$%&'%()$*+%,- ./#!,(+0%1+%0$%2()$*+%,-312& , 45$ 4)( ~B67)89 %+ 4*)* ,+ 4 ~J $-,: 9$, +,* 9)% 4##5$,&67#1+ &+* 9)% ;%+)(*+)%-(',<-$(6*)(=$48 mj -(9 >()%'+)?)%, 4+)@-('WB = mj · gj · µB ·B. A !0BC2& ,*$%D)( ~B67)89 )(*DEFF)8( ~L -(9 ~S A8$( :#,&;$ 1&67#1+C G$88, ~µj · ~B ≫ λ · ~L · ~S! H$((,+(9 n, l, ml, ms '-*) I-$(*)(=$48)(1

WB = (ml + 2ms)µB ·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(, µn = me

mp· µB +,*9+),)% >N)D* ?+)8 D8)+()% $8, 9+) 7)+(,*%-D*-%$-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)%>()%'+)(+?)$-, )+( AE(n, j, l) 6= E(n, j, l ± 1) +& Q)')(,$*= =- 9)& >%')@(+, 9)% H+%$ 6Q8)+ 4-('C! R$ 4,*)4)(9) S@@+89-(')( ')@)( )+()( L+(5)+, $-G 9)( >+(T-,, 9)% ?)%, 4+)9)()(>N)D*)!

!!"#$%&' ()**+ ,-"./0'- 1%2 H34"5-6%7 9-26 :9&- 0%;-/-7 ~B3<-#$ 5:& #"&=7 &6 9 /- 9.7+ ,:9/3>:$-##? @"/6 3A#'?) B62!3C9"D.? EFG-/D-"&6%D7G6#.%&' $%/ 9 H-/&7G"&)

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

!!"#$%&' ()*+, %-./0#1%&' $23 405657"&"2& "8 . :;0 :2& <#"&=.> %&$ .103=2& <32 :1.> ?%@232&A2#$) !"!# $%& '%()*+,&%-.*/01( ,%( 2+3*(/01')(2) /3&'),*)11.(')( 4)32)( +( 2+),)- 5&6+*)1 7)8&(2)1*9• :+) 4)32)( ;*0-, 8&1)( -+* =1)>*30()( 7),)*?*@• :&, +,* 2)3 3),.1*+)3)(2) A),&-*23)8+-6.1, 2)3 Z =1)>*30()(@B)2), CD11)()1)>*30( 4+32 2.3 8 E+)3 F.&(*)(?&81)( Gn, l, j, mjH 7?4$ Gn, l, ml, msH ')>)((?)+ 8I()*$F.&(*)(-) 8$ /032)3.('9 A1)+ 8) J)+1 8)( ,+(2 .(.(*)3, 8)+27&39

|φ(1, 2)|2 = |φ(2, 1)|2 ; φ = :)11)(K.(>*+0( 2), LI=1)>*30()(,M,*)-, G#$" HN)+- ;.,*&., 8 ?4)+)3 e− O3$ .(2 L 2&3K ,+ 8 2+) ;.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)+1 8)(I&.,*&., 8$ R&(( K01'* &., 07+')3 /032)3.(' G"$" H9φ(1, 2) = Ψ(1, 2)±Ψ(2, 1). G#$"LH=S6)3+-)(*)11 ?)+'* ,+ 89/D3 N0,0()( -+* +83)- '&(??&81+')( Q6+( '+1*9

φ(1, 2) = φ(2, 1); + I T)+ 8)( +( 07+')3 A1)+ 8.(' G#$"UH

!" #$%$&$'()*+,&-, ./012% 3)+4 6)( &+* 6$478$64+')& 9:+( '+4*;φ(1, 2) = −φ(2, 1); − < =)+ 6)( +( >7+')% ?4)+ 6-(' @ !AABC)*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-,*$(D n, l, j, mj P+%D 6H 6,*)(, &+* !" # e− !"!#$#% &'"('"#!' !)*+ # "+ -.!+'! /(' 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 =

∑

i

~Li; ~S =∑

i

~Si;

~J = ~L + ~S. ?@4ABC7)!-+8936"! ~Li 0!) !+'$!6'!' e− .(99!6' $3 ~L% !'#"9)! -!'0 0+! :9+'" ~Si $3 ~SD 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!'% 4fC4G+!) F !)<+!*# 0+! !6!.#)("#&#+" -! e−e− H I! -"!6<+).3'* 0!' ~Li~Si H J!)8 K!0!" !+'!$6!'!'

e−% "( 0&"" .!+' ~Ji *! +60!# <+)04• ~J ~JH=(9963'*

~Ji = ~Li + ~Si

~J =∑

i

~Ji ?@4A@C7+!"!" =(9963'*"" -!8& <+)0 !+8 ;3LF66!' 0!) M3N!)!' : -&6!' /(' " -<!)!' =!)'!' !( & -#!#4 7!) ~Li~Si J!)8 F !)<+!*# +' 0+!"!8 O&66 ?/*64 =&94 A4P4P% |B′| ∼ Z4C47+! 1!()" 3#( #.#$( !" -)!+ !' 0!' !6!.#)('+" -!' Q)3'0$3"#&'0 !+'!" R+!6!6!.#)('!'"S"H#!8"4 7+!"! T!*!6' #)!L!' +'" !"('0!)! !+ ~L~SH=(9963'* $3>U4 OF) & *!" -6(""!'! : -&6!' *+6#> L = S = J = 0P4 OF) 9&)#+!66 *!EF66#! : -&6!' *+6#>4*"5# 6*7,*75859Q)3'0$3"#&'0 $!+*# 8&V+8&6!" ~S ?0&8+# <+)0 0+! !6!.#)("#&#+" -! W'!)*+! 8+'+8+!)#%<!*!' 5!/26.!)3'* 3'#!)" -+!06+ -!) 5&-'!' ml,i 0!) !+'$!6'!' W6!.#)('!' iC:;#75# 6*7,*758597!) Q)3'0$3"#&'0 $!+*# 8&V+8&6!" ~L ?04 -4 ∑i ml,i 8&V+8&6D 0+! W6!.#)('!' !/()$3*!'*6!+ -!' X86&3E"+''% 38 "+ - '+ -# $3 Y !*!*'!' ZC4<*755# 6*7,*75859 OF) 0+! !)"#! : -&6!'-M6E#! *6# !+8 ;3LF66!' 8+# W6!.#)('!'>

J = |L− S| ?@4A[C7&8+# <+)0 0+! :9+'H5&-'H=(9963'*"!'!)*+! 8+'+8+!)#4 OF) 0+! $<!+#! : -&6!'-ME#! *+6#>J = |L + S| ?@4A\C8+# ~J2 = ~

2J(J + 1); ~L2 = ~2L(L + 1); ~S2 = ~

2S(S + 1) ?@4A]C

!" "#! $%&'()*+,-+.%/%-+ 012&) -',*))(23%/4 5%++ %3&(+ 721++('( 8 7%2(' '-/ 9-, .*%,%&'()*+,-+ 3(*)/%&('4 :%/;)*(22 &(0<22)( 8 7%2(' 5%&(&(' '(3(' 5(, 5*%,%&'()*+ 7(' =')(*2 (*'(' &/1>(' :%/%,%&'()*+ 7('?(*)/%& 9(*&('# ?(*+:*(2 9-/ ?(+()9-'&+/(*7('012&( (*'(/ 5;8 7%2( ,*) e−@

!!"#$%&' ()*+, - /0123"4 /0 526430##%&' 7%1 4%87044"90& %:;##0& 0"&06 $<- /2#0 1"3 =#0836><&0& 0&34?60 /0&$ $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 +1 5%++5*( D-%')('9%72 mj 5*( E(/)( j, j − 1, . . . ,−j %''*,,)# .*( F'(/&*('*G(%-+ +*'5 %-0&(+:%2)('@V = −~µ · ~B = mj · gj · µB ·B H"#I!J?(*+:*(2@~L = ~0; |~S| = ~

2

√3; ms = ±1

2

!!"#$%&' ()*B, C0630"#%&' 0"&04 D<E0"# /0&<=&401!#04 2%F $"0 !0"$0& -?"&7%43G&$0 ms = ±1/2)

!" #$%$&$'()*+,&-, ./01)(( 2+) 3)+*%$')(2)( &$'()*+, 5)( 67()( )+(), 8%+,*$99, :;$9<),= (>5)%-(',?)+,) -($35>('+'@7()+($(2)% +5%) A-,%+ 5*-(' ms )%5$9*)(B ,7 '+9* 2+) C79*<&$((D;*$*+,*+EB -(2 FG% ms = ±1/2)%'+3* ,+ 5HN2

N1

= e−∆VkT = e−

2µB ·B

kT ; N = N1 + N2 : !I.=JG% 2+) K$'()*+,+)%-(' )%'+3* ,+ 5HM = (N1 −N2) · µB = N · µB · tanh

µB · BkT

: !I"=J$99, µB ·BkT

≪ 1Htanh

µB · BkT

≃ µB · BkT

: !IL=M$&+* F79'* FG% 73+'), C)+,N+)9HM ≃ N · µ2

B

k· B

T: !IO=K$'()*+, 5) ;-,<)N*+3+9+*>* χm = M

H≃ µ◦M

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

χm ≃ C

T; : !II=JG% $99')&)+(), J '+9* 2$3)+ FG% 2+) Q-%+)D87(,*$(*)H

C =µ◦ ·N j(j + 1)g2

j µ2B

3kM$%,*)99-('H1

χm

=1

C· T

!!"#$%&' ()*+, -./012##%&' $20 3%/"2452021620R) 5(+, 5) C)2)-*-('H S%%)+ 5)( *+)F)% R)&N)%$*-%)( 2-% 5 *+#*,*(#' .$' /0(1*20$(#'#$%"$0 @7( N$%$&$'()*+, 5)( ;$9<)( ?+) Gd2(SO4)3 · 8H2O 72)% FeNH4(SO4)2 ·H2OB ?73)+ 2+)&$'()*+, 5)( 67()( (-% G3)% P+**)%, 5?+('-(')( :#57(7()(= ')E7NN)9* ,+(2!3"#0)#45 6& @7%')EG59*)( 8%+,*$99 ?)%2)( &+**)9, ~BDJ)92 2+) ;N+(, $-,')%+ 5*)*! M$($ 5?+%2 2)% 8%+,*$99 *5)%&+, 5 +,79+)%*B 2+) ;N+(, @)%9+)%)( 3)+& T-(*)%F$5%)( 2), ~BDJ)92), *)+9?)+,)

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

!!"#$%&' ()*(+ ,-./01##%&' $1/ 234#5.671//1/ 8 $"-!-0"/ 41/ :&0;-'&10"/"1.1&< "; =>?,"-'.-;;)+/0) 102(.('3 2+) 4()0'+) )(*5+)/)( ,+) 2&6)+ 2)( 7+**)0, /9+('.(')( → :6;</=.('$>)')( 2)0 5.0< ;6=)+6)(2)( ?@+(A02(.(' +- mKBC)0)+ / ;&(( 2)0 D0A5),, (+ /* 5E;=+, /')F&/0)( 9)02)($4(*, /)+2)(2G >) /,)=9+0;.(' 2)0 &.,5.0+ /*)(2)( H+@A=) .(*)0)+(&(2)0 -.,, ;=)+( ,)+(32&-+* 6)+ *+)F)0 I)-@)0&*.0 T !"#! (A / )+() .(')A02()*) 10+)(*+)0.(' JA0 2)- :(=)')( JA(Bext = B2 JA0=+)'*$ !"!#$%&'( H+) )*+),)-+. 0! 1&-")'&!-+.+!#%&' 23& 4!#&.5+&. F</0* (& / 2)-'=)+ /)( KA(5)@* 9)+*)0 6+, Tend ≃ µK$ !"!# $%&'()*% +%( ,%&'-.*/%0%1'(2.%. 3-4 56*.%'&/4-/ 7%& 5%'600%.6)#)")'&!-+. 0! 43"53&!&-! 76)%8+ 9 6)#)")'&!-+."%.::.F'0.(2 2)0 ?@+(A0+)(*+)0.(' +- %&'()*F)=2 B )0'+6* ,+ / )+( :(/)6)( 659$ :6,)(;)( L)2),e−B4()0'+)(+J)&., .- ±µB ·B$ H&-+* ,@&=*)* 2+) M.,*&(2,2+ /*) D(E) 2)0 F0)+)( 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)+*)±µB ·B$ R<0 2)( S(*)0, /+)2 +( 2)0 4=);*0A()(;A(5)(*0&*+A( )0'+6* ,+ / 2&(( 6)+ T ≪ EF /kB.(2 µB ·B ≪ EF .(*)0 T)09)(2.(' JA( 7='$ NO$ UPG

N(↑)−N(↓) =D(EF )

2· 2 · µB ·B =

3N

2EF

· µB ·B N#$U#P;)'&!-+.+!#%&'(Mp = µB(N(↑)−N(↓)) =

3N

2EF

· µ2B ·BD&.=+B?.,5)@*+6+=+*Q*G

χp =µ◦ ·Mp

B=

3N

2EF

· µ◦µ2B N#$UVP

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

!!"#$%&' ()*+, %-./0#12& $23 4%.10&$.$" 612 -32"23 e− "7 8%9232& :0'&21-2#$) !"#"$%&'!( *& +,#-,%&%'& .,% /&!'0%$(&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$(8$-D E! #BF,+5 GD .0H/D I! "0℄!K6(,)L-)(:M N-2'%-(8 8)% 9)(:, B)( P)')4 )%'+Q* ,+ B )+() 8+$&$'()*+, B) R+%5-('D 7)4 B)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,,)()( I B$4)( -(8 ')2T44*)( A;(8)%( &+*

Zi=34)5*%6()( US'4! K$?! G!.!"D 9$(')S+(=V4'!W )%'+Q* ,+ B 8$((Mχtot =

N

EF

· µ◦µ2B − µ◦

Na · Zi · e2

6me

< r2 > U !XYW1$Q)+ Q)8)-*)(MN = 9)+*-(',)4)5*%6()(8+ B*)

Na = 1+ B*) 8)% N*6&%T&?2)Zi = E$B4 8)% +(()%)( e− ?%6 N*6&8&!(-!&1&938)4&)*$44) Cu, Ag, Au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

! "#! $%&'()*+,-+ !!"#$%&' ()*+, -."/01'. 2%0 34%#45!67&.0'". "5 89::.0:/4;54#.<=#) !"#$!" &'(')*+' ,'" H2 - ./0'120".%,*/)0'12-'3)*0' 4-5 (/(3)5*+ 7(' 8( 7+(/9*53-'&:

H =e2

4πε◦

[1

R− 1

rα1

− 1

rα2

− 1

rβ1

− 1

rβ2

+1

r12

] ;"#<=>3'+#0')-4/(,/(-5')6$&)'(7 ?-+&%'&+@-'3) +*'A A*( 8(//('B-'3)*0'(' A(5 (*'4(/'(' ?)0,051C*)%/(D A*( C(* &50E(, F(5'%C+)%'A 4-)5(G('#• H*( 8(//('B-'3)*0' Ψ A(+ $(75)(*/ 7('+I+)(,+ *, J5)+5%-, (5&*C) +* 7 A%'' %/+ K50A-3)A(5 L*'4(/)(*/ 7('B-'3)*0'('#• H*( M'-')(5+ 7(*AC%53(*) A(5 N(*/ 7(' A5O 3) +* 7 *' A(5 PI,,()5*( A(5 4-)5(G('A(' Q(1+%,)9(//('B-'3)*0' φ %-+:8/"/('(7 PI,,()5*+ 7 &(&('OC(5 N(*/ 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' BO5 A%+ T9(*1L/(3)50'('+I+)(,;2(5,*0'('+I+)(,>:

φas = Na [Ψα(1) ·Ψβ(2)−Ψβ(1)Ψα(2)] · χs ;"#"U>Ψα(1) = 8(//('B-'3)*0' R0' L/(3)50' V5# C(* F(5' α

Ψβ(1) = 8(//('B-'3)*0' R0' L/(3)50' V5# C(* F(5' β

Na = V05,*(5-'&+B%3)05χs = PI,,()5*+ 7( P@*'B-'3)*0' (↑↑)χa = ?')*+I,,()5*+ 7( P@*'B-'3)*0' (↑↓)?/)(5'%)*R C(30,,) ,%' (*' %')*+I,,()5*+ 7(+ φsa ,*)

φsa = Ns [Ψα(1) ·Ψβ(2) + Ψβ(1)Ψα(2)] · χa. ;"#"W>

!" #$%$&$'()*+,&-, .../+)% +,* $0,1 2+) 3%*,4-(5*+1( ,6&&)*%+, 8 -(2 2+) 9:+(4-(5*+1( $(*+,6&&)*%+, 8! ;+) <$8%=, 8)+(0+ 85)+*> '0)+ 8?)+*+' +( 2)( 3%*,@10-&+($ dV1 -(2 dV2 )+( A0)5*%1( $(?-*%)B)( C+%2 2-% 82+) D-,2%E 5)ρas = |Ψα(1)Ψβ(2)−Ψβ(1)Ψα(2)|2 dV1dV2 F?C!ρsa = |Ψα(1)Ψβ(2) + Ψβ(1)Ψα(2)|2 dV1dV2 F),*+&&*. G ! HI

!!"#$%&' ()*+, -%.#"/./"01 2.34/1##%&' 05& φsa %&$ φas 673 '3581 9#"&:4; %&$ :#1"&1 931 =/4;>13&.!4/?&$1 α, β) @.& 4"1=/A $.44 !1" φsa 1"&1 =B=131 C#1:/35&1&.%61&/=.#/4D.=34 =1"&#" =:1"/ED"4 =1& $1& !1"$1& F54"/"01& >13&1& !14/1=/ 9G"&$%&';) !"#$% ;)% 2-% 8 φsa F), 8%+)F)() J-,*$(2 +,* ,*$F+0> 2$ )% 2-% 8 )+() 8K8)%) D-4)(*=8$0*,C$8%, 8)+(0+ 85)+* ρsa 2)% ()'$*+@)( 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-% 8 2+) M+(+&+)%-(' 2)% N1-01&F=A()%'+) +& H2=M10)5E0 )%?C-(')(! ;+) A()%'+)2+B)%)(? ?C+, 8)( 2)( J-,*O(2)( (↑↓) -(2 (↑↑)+,* +( 2+),)& P$00 2+) /.0*(.0 2$)$3#,$! $33!4(#)$*,04.0Q)+& D-BE00)( 2)% R2=9 8$0) *%+** P)%%1&$'()*+,&-, F)+ 2)( A0)&)(*)( P)G3d64s2I> N1G3d74s2I-(2 S+G3d84s2I $-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)( -(2 C)%2)( (+ 8* &)8% C+) +& ?)(*%$0)( N1-01&F4)02 +( Q)?-'$-4 )+() M$'()*4)02%+ 8*-(' ,*$F+0+,+)%*V

< Lz >→ 0 G ! XI

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

!!"#$%&' ()*+, -#./01"2 4. 5"&$%&'2.&.1'".& 671 $". 8%209&$. φsa !:;) φas "& !49&'"'/."0 <=>?.1&@!20@&$ "& -"&4."0.& $.2 5=411@$"%2 A0.53 · 10−10mB) !!"#$%&' ()*C, %6'1%&$ $.2 ?1"20@##D=0.&:"@#2 1" 40.& 2" 4 $". E1.4">D%#2. &" 40 >.41 2= ;"."> 8.&01@#D=0.&:"@# @& ."&.1 42. @%2F 2=&$.1& GH%.& 4.&I)/&-+* 0)1.2+)0* ,+ 4 1)0 5&0&-&'()*+, 4) 6(*)+7 2.0 %&'()*+,+)0.(' &.8 1)( 95+( ~S 1)0 :7);<*0=()(# 67, -&'()*+, 4), %=-)(* 50= >+**)057&*2 -+,,* -&(?

Fe : µeff,Fe = 2.2µB @"#""Aµeff,Ni = 0.6µB @"#"BA/+) 3d .(1 4s<:7);*0=()( 7+)')( +( CD(1)0(E F)7 4) )()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),L)71A F.01) .- M!N 1.0 4 L0)(;)7 .(1 O)+,)(H)0' ')')H)(? :, +,* 1+) ,+ $%+ ./& . &01/)*2+#( < H)+ 1)( PH)0'&(',)7)-)(*)( 2F+, 4)( 1)( C&(1)7);*0=()(#

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

!!"#$%&' ()**+ ,-./ 1"-!%&' $-. 2%/34&$/$" 13-& (↑) %&$ (↓) 4%5'.%&$ $-. 6-"/-&!-.'/ 1-& %/34%/ 178- 1/-#9".:%&')01'2! &+* 3()%'+)4)%, 6+)7-(' 8)% 9)+*-(',)2):*%;()( 0<$=! >!"!?!.@ 7)+ Bext = 1T : ∆E =2µB · B ≃ 10−4eV )!A+) B-,*$-, 6C) 6,)2C+%:-(' DE6%* 6+)% F- )+()% G=+(HB-,%+ 6*-('I C)2 6) C+)8)% 8+) J;2;-&7H)()%'+) +& <%+,*$22 &+(+&+)%*! K)(( &$( 8+) '2)+ 6) B-,%+ 6*-(' 8-% 6 )+( L$'()*D)28 )%%)+ 6)(C;22*)I 7):;&&* &$( 8$, M6N=;*6)*+, 6) +(()%) K)+,,HO)28 P! A+),), 6$* #! K)+,, .QRS F-% T)H, 6%)+7-(' 8)% ,=;(*$()( L$'()*+,+)%-(' )+(')DE6%*U

∆E = 2µB · BW ;→ BW ∼ 104T DE% 3+,)(V 0 ! W@B-,*$-, 6HK) 6,)2C+%:-(' FC+, 6)( 8)( 32):*%;()( i, jUH = 2Jij · ~Si · ~Sj 0 ! Q@A+) =$%$22)2) G=+($-,%+ 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)28 7%+(')(U

BW = λ ·Ms 0 !SR@ZC+, 6)( λ -(8 Jij 2X,,* ,+ 6 )+() T)F+)6-(' 6)%2)+*)(!→ A+) T), 6%)+7-(' E7)% BW +,* +(,;D)%( )%2$-7*! !"#$ BW )[+,*+)%* %& () $2, &$'()*+, 6), O)28I ,;(8)%( (-% $2, L;8)22'%\])!^7)%6$27 )+()% &$*)%+$2,=)F+_, 6)( )̀&=)%$*-% Tc 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;( cp . . .@! OE% T ≥ Tc7),*)6* =$%$&$'()*+, 6), 1)%6$2*)( 0^%8(-(' 8)% G=+(, C+%8 '),*\%*@!*+#&",-"&..,/".")0$

χ =C

T − Tc

; DE% T ≥ Tc 0 !S.@Tc,Fe ≃ 1043Kc OE% T → Tc ')6* χ→∞ : ~M = χ · ~H →c ,=;(*$() L$'()*+,+)%-(' DE% T ≤ Tc@!

! "#$ %&'()*+,-.,/0&1&2*)1+,+)1.(' 3)114-&'()*+, 0)1 6*47) 89':# ;)-<)1&*.1=)10&:*)( +- /.1+)>?)+,,>@:4*A

!!"#$%&' ()*+, -./0.123%12!45&'"'6."3 78& Ms %&3.142#! 78& Tc 98:". 78& 1/χ 8!.142#! 78&Tc ;7'#) 2% 4 =%1".>?.9.3@A)

1

χ=

1

C(T − Tc) 3B1 T ≥ Tc C"#DEFGB1 C '+:* )(*,<1) 0)(H I&<# !#E#!

C =µ◦Ng2

j · S(S + 1)µ2B

3kB

CGB1 J+,)(A gj ∼ 2; S ∼ 1F C"#DKFJ+( -L':+ 0), M.3*1)*)( =4( G)114-&'()*+,-., 2&(( -+* 34:')(H)1 N)H+('.(' ')<1B3* O)1H)(AJe · D(EF ) · Va ≥ 1; C6*4()1 .(H ?40:3&1*0F C"#D!FJe = M.,*&., 0)()1'+) <14 J:)2*14(

D(EF ) = P.,*&(H,H+ 0*) &( H)1 G)1-+2&(*)Va = Atomvolumen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

• %+* 9.()0-)(H)- S.U)1)- G):H =)1, 0+)8)( ,+ 0 9.(S 0,* N:4 0, 0) ?S(H) +( H)1 ?)+,)TH&,, 'B(,*+' 41+)(*+)1*) W4-S()( ,+ 0 =)1'1LU)1(#

!" #$%$&$'()*+,&-, ../• 0)+ 1)+*)% 2-()3&)(4)& 5-6)%)& 7)84 1+%4 4+) 9$'()*+,+)%-(',%+ 3*-(' $88)% ;<&5()(2-()3&)(4 ($ 3 4)& 5-6)%)( 7)84 $-,')%+ 3*)*!

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