pê ø]  §T ° n ÞÀ W ¥ w Š ¹ ÅX ì Äß Ã Å Bloch U ê sX N ËÅ k Ä8 ý X N ˽  ʔ X ¢ A 0: € ¾M  ºü g Å 8 ýÇ X ØV R Ë

P c

pê ø]  §T  ° n ÞÀ W ¥ w Š ¹ ÅX ì Äß Ã Å Bloch U ê sX N ËÅ k Ä8 ý X N ˽  ʔ X ¢ A 0: € ¾M  ºü g Å 8 ýÇ X ØV R Ë



¡) o £ Ó

^∗

Ø



æ· ¡ ¤@ /† < Ɠ § Ó ü to † < Æõ  x 9  BK21 Ó ü to  á Ԗ ÐÕ ªÏ þ ›, ' õ AÅ Ò 361-763 (2008¸   4 Z 4 1{ 9  ~ à Î6 £ §)



©

œ  ñ Œ •6   x t  · ú §  H “ ¦„   Û ¼— 2 ;[ þ t\   r„    l  © œs     9 e ”   H — ¸4 S q`  ¦ ‚  × þ ˜ # Œ, y Œ ™ W† ½ Ós  \ O   H Bloch

~

½

Ó& ñ d ” `  ¦ Ä »• ¸ % i  . Õ ªo “ ¦ s  ~ ½ Ó& ñ d ” \ " f  l  o œ íl › ¸| \    " f # Q‹ "  % ò † ¾ Ó`  ¦ ~ à ΍  Ht \  ¦ › ¸



 % i  . Ó ü to & h Ü ¼– Ð z  ´‰ & ³ l  / 'î  r, ¢ - a„   & ñ § > = © œI , ¢ - a„   Á º| 9 " f  © œI , ¨ î + þ A © œI _  [ j œ íl › ¸| `  ¦

× þ

˜ # Œ › ¸ ô  Ç   õ  q ¨ î + þ A  l  o œ íl › ¸| \    " f ß ¼>  % ò † ¾ Ó`  ¦ ~ à ΍  H כ `  ¦ ˜ Ѐ Œ ¤ .

½

¨^ ‰& h Ü ¼– Ѝ  H # Œ Q ß ¼l _  – Ð~ ½ ӆ ¾ Ó (transverse)  r„    l  © œõ   r„   y Œ •”  1 l xà º\  @ / # Œ, q ¨ î + þ A



l  o œ íl › ¸| \  ß ¼>  _ ” > r† < Ê`  ¦ ˜ Ѐ Œ ¤ . œ íl › ¸| s  ¢ - a„   Á º| 9 " f  © œI “    â Ä º\   H  r„    l  © œ _

 ß ¼l 
 y Œ •”  1 l xà º\  Á º› ' a >   l  o_  ° ú כs  0`  ¦ Ä »t  % i  . Õ ª Q
 œ íl › ¸| s  ¢ - a„   & ñ § > = © œI 

“



  â Ä º\   H — ¸Ž  H › ¸|  A " f  l  o_  $ í ì  rs  r ç ß –\    " f ”  1 l x† < Ê`  ¦ ˜ Ð% i  . ¨ î + þ A © œI  œ íl › ¸

|

“    â Ä º\   Å Ò ± ú “ É r “ : r• ¸\ " f  H ¢ - a„   & ñ § > = © œI  œ íl › ¸| “    â Ä ºü < q 5 p wô  Ç ' Ÿ I \  ¦ ˜ Ð% i “ ¦, Õ ªo 

“

¦  Å Ò Z  }“ É r “ : r• ¸\ " f  H ¢ - a„   Á º| 9 " f  © œI  œ íl › ¸| “    â Ä ºü < q 5 p wô  Ç ' Ÿ I \  ¦ ˜ Ð% i  . ¢ ¸ô  Ç z-» ¡ ¤

&

ñ

 l  © œ\  q  # Œ  Œ •“ É r ß ¼l _   r„    l  © œ? /\ " f  H / B N”   y Œ •”  1 l xà º\ " f q ¨ î + þ A  l  o_  ”  1 l x”  ; Ÿ ¤ s

 Z  t³ 1 Ï ë ß –
p u ß ¼>  7 £ x† < Ê`  ¦ ˜ Ѐ Œ ¤ .Õ ªX O t ë ß –, & ñ  l  © œõ  – Ð~ ½ ӆ ¾ Ó  r„    l  © œ_   â Ô q t´ òõ M :ë  H\  y

Œ

•”  1 l xà º / B N”   y Œ •”  1 l xà º˜ Ð  & t >  | ¨ c  â Ä º\  &  ê ø Í  r„   l  © œ\  @ /K " f é ß –› ¸& h Ü ¼– Ð ”  1 l x; Ÿ ¤s  7

£

x # Œ  8s  © œ / B N”  ´ òõ \  ¦ % 3 `  ¦ à º \ O 6 £ §`  ¦ ˜ Ѐ Œ ¤ .

PACS numbers: 05.90.+m, 75.10.Hk, 75.74.Gb,

Keywords: q¨î+þA Û¼—2;1lx§4†<Æ, r„ýa³ð>, Bloch ~½Ó&ñd”, l/BN"î

I. " e  ] Ø

Ä

ºo     H  © œ ç ß –é ß –ô  Ç q ¨ î + þ A ë  H] j  H Hamilto- nians  Å Ò# Q”   Ó ü to > _  q ¨ î + þ A ' Ÿ I \  › ' aô  Ç  כ s   [1, 2]. Õ ªX O t ë ß – s    7 á xÀ Ó_  ë  H] j\  ¦  À Òl  0 Aô  Ç & ñ S X ‰ô  Ç + þ A d

”

 (formalism)s  Å Ò# Q4 R• ¸ î  r1 l x ~ ½ Ó& ñ d ” `  ¦ Û  ¦# Q    H

% 3

' õ A
>  # Q 9î  r ë  H] j\   Ò{ Œ •u >   ) a . z  ´] j– Ð & ñ S X ‰y  Û



¦o   H ë  H] j  _  \ O l M :ë  H\   Å Ò ç ß –é ß –ô  Ç Ó ü to > \  ¦

× þ

˜ # Œ s  כ `  ¦ & ñ S X ‰ >  Û  ¦“ ¦ Õ ª כ _    õ \  ¦  [ jy  › ¸



 # Œ s  Ó ü to > _  q ¨ î + þ A ' Ÿ I \  ¦ s K    H  כ “ É r q ¨ î +

þ

A : Ÿ x> % i † < Æ\ " f  © œ{ © œy  _ p e ”   H { 9 s  .

s

 Qô  Ç 3 l q& h `  ¦ 0 A # Œ  $ í Ó ü t| 9 _  Ù þ ˜ l / B N" î ‰ & ³ © œ ƒ  

½

¨\ " f l " é ¶   H  r„    l  © œ? /_   © œ  ñ Œ •6   x t  · ú §  H “ ¦

„



 Û ¼— 2 ;[ þ t_  — ¸4 S q`  ¦ › ¸  l – Ð ô  Ç  [3]. s  — ¸4 S q\ " f Û

¼— 2 ; $ í ì  r_  q ¨ î + þ A € © œ © œ^  ¦ ¨ î ç  H\  @ /ô  Ç î  r1 l x~ ½ Ó& ñ d ” “ É r y

Œ

™ W† ½ Ós  \ O   H ‰ & ³ © œ† < Æ& h “   Bloch ~ ½ Ó& ñ d ” õ  ° ú    [4–6].

∗E-mail: [email protected]

s

 Qô  Ç y Œ ™ W† ½ Ós  \ O   H Bloch~ ½ Ó& ñ d ” \  @ /K " f• ¸ – Ð

~

½

ӆ ¾ Ó_   l  © œs   Œ •“ É r  â Ä º\ ë ß –   H & h “   K  Å Ò# Q4 R e

”

  [2].   " f ‘ : r  7 Hë  H\ " f  H r ç ß –\    " f      H { 9  ì

ø

Í& h “   – Ð~ ½ ӆ ¾ Ó  l  © œ\  @ /ô  Ç y Œ ™ W† ½ Ós  \ O   H Bloch~ ½ Ó

&

ñ

d ” _  & ñ S X ‰ô  Ç K  ÷  rë ß –  m   Õ ª כ s  œ íl › ¸| \    

"

f # Q‹ "  % ò † ¾ Ó`  ¦ ~ à ΍  Ht \  ¦ ˜ Ðs “ ¦  ô  Ç .

II. { ¢¨ | Ê Ý Æ U ؎ Ò ÞU ê sX N ËÅ k Ä



r„    l  © œ? /_  N > h_  " f– Ð  © œ  ñ Œ •6   x`  ¦ t · ú §  H “ ¦

„



 Û ¼— 2 ;[ þ t“ É r  6 £ § Hamiltonian\  _ K " f l Õ ü t ) a .

H = − − → h (t) ·

X

N

i=1

−

→ s

i

= −h

z

X

N

i=1

s

^z_i

− h

0

X

N

i=1

(s

^x_i

cos ωt − s

^y_i

sin ωt) (1)

#

Œl " f − →

h (t)  H r ç ß –\    " f      H  l  © œ, h

z

  H   l

 © œ_  z−~ ½ ӆ ¾ Ó $ í ì  r, Õ ªo “ ¦ h

0

  H y Œ •5 Å q• ¸ ω– Ð r > ~ ½ ӆ ¾ Ó

-16-

Ü

¼– Ð(clockwise)  r„     H  l  © œ $ í ì  r_  ß ¼l s  . ¢ ¸ ô



Ç s

^x_i

= sin θ

i

cos φ

i

, s

^y_i

= sin θ

i

sin φ

i

Õ ªo “ ¦ s

^z_i

= cos θ

i

  H    0 Au  i\  e ”   H “ ¦„  & h  Û ¼— 2 ; − → s

i

_  $ í ì  r`  ¦

  · p . ô  Ǽ #  θ

i

  H F Gy Œ • (polar angle) Õ ªo “ ¦ φ

i

  H

~

½

Ó0 Ay Œ •(azimuthal angle)`  ¦ y Œ •y Œ •
  · p . 8 ú x Û ¼— 2 ;$ í ì  r S

α

= P

_N

i=1

s

^α_i

(α = x, y, z)`  ¦ s 6   xK " f 0 A_  Hamilto- nian`  ¦  6 £ § + þ AI – Ð ³ ð‰ & ³½ + É Ã º e ”  .

H = − − → h (t) · − → S

= −h

z

S

z

− h

0

(S

x

cos ωt − S

y

sin ωt) (2)

8 ú

x Û ¼— 2 ; 7 ˜' \  › ' aô  Ç î  r1 l x ~ ½ Ó& ñ d ”   6 £ § d ” `  ¦ s 6   xK " f

½

¨½ + É Ã º e ”  .

d − → S

dt = − − → h (t) · − →

S (3)

t

F K 8 ú x Û ¼— 2 ; 7 ˜' $ í ì  r\  @ /ô  Ç î  r1 l x~ ½ Ó& ñ d ” `  ¦ ½ ¨^ ‰& h  Ü

¼– Ð ³ ð‰ & ³ €    6 £ §õ  ° ú   .

dS

x

dt = h

z

S

y

+ h

0

S

z

sin ωt (4) dS

y

dt = −h

z

S

x

+ h

0

S

z

cos ωt (5) dS

z

dt = −h

0

S

y

cos ωt − h

0

S

x

sin ωt (6) s

p  ƒ  / å Lô  Ç ü < ° ú  s  0 A_  î  r1 l x~ ½ Ó& ñ d ” _  € © œ © œ^  ¦ ¨ î ç



H“ É r ‰ & ³ © œ† < Æ& h “   y Œ ™ W† ½ Ós  \ O   H Bloch ~ ½ Ó& ñ d ” õ  ° ú    [2, 4]. ‰ & ³ © œ† < Æ& h “   s ¢ - a† ½ Ós  [ þ t# Q e ”   H { 9 ì ø Í& h “   Bloch ~ ½ Ó

&

ñ

d ” “ É r Ù þ ˜ l  / B N" î ‰ & ³ © œs 
 „   l  ` O Û ¼_  & h 6   x`  ¦ : Ÿ x ô



Ç  l  o ÷ &| 9 l  (magnetization reversal) ‰ & ³ © œ`  ¦ [ O " î

  HX < $ í / B N& h Ü ¼– Ð  6   x ) a   e ”   [3,5].

y

Œ

™ W† ½ Ós  \ O   H  â Ä º • ¸ Bloch ~ ½ Ó& ñ d ” `  ¦ f ” ] X & h Ü ¼– Ð Û



¦l   H ~ 1 t  · ú §Ü ¼Ù ¼– Ð z−» ¡ ¤\  @ /K " f y Œ •5 Å q• ¸ ω– Ð r > 

~

½

ӆ ¾ ÓÜ ¼– Ð  r„     H ý a³ ð> \  ¦ Ò q ty Œ •K  ˜ Ð . s   â Ä º\  8 ú x Û

¼— 2 ; 7 ˜' _  r ç ß –\  @ /ô  Ç • ¸† < Êà º  H “ ¦& ñ (fixed) ý a³ ð>  ü

<  r„   (rotating) ý a³ ð> \ " f  6 £ §õ  ° ú  “ É r › ' a> \  ¦ ”  



.

d − → S

dt |

f ixed

= ∂ − → S

∂t |

rotating

+ − → ω × − → S = − − →

h (t) × − → S (7)



r„   ý a³ ð> \ " f_  î  r1 l x~ ½ Ó& ñ d ” \   H r ç ß –\  f ” ] X & h Ü ¼

–

Ð _ ” > r   H † ½ Ós  \ O  . ¼ # _  © œ d − →

S /dt|

f ixed

\  ¦ d − → S /dt – Ð , Õ ªo “ ¦ ∂ − →

S /∂t|

rotating

\  ¦ ∂ − →

S

^∗

/∂t  “ ¦ ³ ð‰ & ³ “ ¦  r„   ý

a³ ð> \  ¦ ∗ ý a³ ð>  “ ¦  ÒØ Ôl – Ð ô  Ç . s o  # Œ ∗ ý a³ ð

>

\ " f 8 ú x Û ¼— 2 ; 7 ˜' _  î  r1 l x~ ½ Ó& ñ d ” “ É r  6 £ §õ  ° ú  s  j þ t à º e

”

 .

∂ − → S

^∗

∂t = −[ − →

h

^∗

(t) + − → ω

^∗

] × − → S

^∗

(8)

#

Œl " f − →

h

^∗

(t) = h

0

x ˆ

^∗

+ h

z

ˆ z

^∗

ü < − → ω

^∗

= −ωˆ z

^∗

  H y Œ •y Œ • ∗ ý

a³ ð> \ " f_   l  © œõ  y Œ •”  1 l xà ºs  9 ˆ x

^∗

, ˆ y

^∗

ü < ˆ z

^∗

  H ∗ ý

a³ ð> \ " f_  é ß –0 A 7 ˜' s  . s  ∗ ý a³ ð> \ " f_  Hamil- tonian “ É r

H

^∗

= −h

0

S

_x^∗

− (h

z

− ω)S

^∗_z

(9)

s

“ ¦,  r„   ý a³ ð> ü < “ ¦& ñ ý a³ ð> \ " f_  é ß –0 A 7 ˜'   H   6

£

§õ  ° ú  “ É r › ' a> d ” `  ¦ ë ß –7 á ¤ô  Ç .

ˆ

x

^∗

= ˆ x cos ωt − ˆ y sin ωt (10) ˆ

y

^∗

= ˆ x sin ωt + ˆ y cos ωt (11) ˆ

z

^∗

= ˆ z (12)

~

½

Ó& ñ d ”  (8)`  ¦ „  > h €   8 ú x Û ¼— 2 ;$ í ì  r_  î  r1 l x ~ ½ Ó& ñ d ” “ É r



6 £ §õ  ° ú   .

∂S

_x^∗

∂t = (h

z

− ω)S

_y^∗

(13)

∂S

_y^∗

∂t = −(h

z

− ω)S

_x^∗

+ h

0

S

_z^∗

(14)

∂S

_z^∗

∂t = −h

0

S

_y^∗

(15)

s

 î  r1 l x ~ ½ Ó& ñ d ” `  ¦  $  Û  ¦l  0 AK " f ∗ ý a³ ð> _  y

^∗

-» ¡ ¤\  @ /K " f y Œ •• ¸ γë ß –
p u  r„  r †   D h– Ðî  r ý a³ ð> 

\



¦ • ¸{ 9   . # Œl " f Ä »´ ò  l  © œ_  [ jl   H h

ef f

= p h

²₀

+ (h

z

− ω)

²

– Ð Å Ò# Qt “ ¦ D h– Ðî  r x−» ¡ ¤ ~ ½ ӆ ¾ Ó`  ¦ Ø Ô

†



 . Õ ªo “ ¦ tan γ = (h

z

− ω)/h

0

, sin γ = (h

z

− ω)/h

ef f

, cos γ = h

0

/h

ef f

s  . s  ý a³ ð> \  ¦ ∗∗ ý a³ ð>  “ ¦  ÒØ Ô



. s  ý a³ ð> \ " f_  Hamiltonian “ É r  6 £ §õ  ° ú  s  Å Ò# Q

”



 .

H

^∗∗

= −h

ef f

S

_x^∗∗

(16)

∗∗ ý a³ ð> ü < ∗ ý a³ ð> _  é ß –0 A 7 ˜' _  › ' a>   H  6 £ §õ  ° ú  



.

ˆ

x

^∗∗

= x ˆ

^∗

cos γ + ˆ z

^∗

sin γ (17) ˆ

y

^∗∗

= y ˆ

^∗

(18)

ˆ

z

^∗∗

= −ˆ x

^∗

sin γ + ˆ z

^∗

cos γ (19)

Õ

ªo  # Œ 8 ú x Û ¼— 2 ;$ í ì  r S

_α^∗∗

(α = x, y, z) _  î  r1 l x ~ ½ Ó& ñ d

”

“ É r  6 £ §õ  ° ú  >   ) a .

∂S

_x^∗∗

∂t = 0 (20)

∂S

_y^∗∗

∂t = h

ef f

S

_z^∗∗

(21)

∂S

_z^∗∗

∂t = −h

ef f

S

_y^∗∗

(22)

s

[ þ t î  r1 l x ~ ½ Ó& ñ d ” `  ¦ Û  ¦€  

S

_x^∗∗

(t) = S

_x^∗∗

(0) (23) S

_y^∗∗

(t) = S

_y^∗∗

(0) cos h

ef f

t + S

^∗∗_z

(0) sin h

ef f

t (24) S

_z^∗∗

(t) = −S

_y^∗∗

(0) sin h

ef f

t + S

_z^∗∗

(0) cos h

ef f

t (25)

s

 . t F K ~ ½ Ó& ñ d ”  (17), (18), (19), (23), (24) Õ ªo “ ¦ (25)\  ¦ ∗ ý a³ ð> _  8 ú x Û ¼— 2 ; 7 ˜' 

−

→ S

^∗

(t) = ˆ x

^∗

S

_x^∗

(t) + ˆ y

^∗

S

_y^∗

(t) + ˆ z

^∗

S

_z^∗

(t) (26)

= ˆ x

^∗∗

S

_x^∗∗

(t) + ˆ y

^∗∗

S

^∗∗_y

(t) + ˆ z

^∗∗

S

_z^∗∗

(t) (27)

\

 u ¨ 8 Š “ ¦

S

_x^∗∗

(0) = S

_x^∗

(0) cos γ + S

_z^∗

(0) sin γ (28) S

_y^∗∗

(0) = S

_y^∗

(0) (29) S

_z^∗∗

(0) = −S

_x^∗

(0) sin γ + S

_z^∗

(0) cos γ (30)



  H  z  ´`  ¦ s 6   x €  

S

_x^∗

(t) = S

^∗_x

(0)[cos

²

γ + sin

²

γ cos h

ef f

t]

+S

_y^∗

(0) sin γ sin h

ef f

t

+S

_z^∗

(0) sin γ cos γ(1 − cos h

ef f

t) (31) S

_y^∗

(t) = −S

_x^∗

(0) sin γ sin h

ef f

t + S

_y^∗

(0) cos h

ef f

t

+S

_z^∗

(0) cos γ sin h

ef f

t (32) S

_z^∗

(t) = S

_x^∗

(0) sin γ cos γ(1 − cos h

ef f

t)

−S

_y^∗

(0) cos γ sin h

ef f

t

+S

_z^∗

(0)[sin

²

γ + cos

²

γ cos h

ef f

t] (33)

\



¦ % 3   H . ~ ½ Ó& ñ d ”  (10), (11),(12), (31), (32)ü < (33)`  ¦   6

£

§ d ” \  ¶ ú š{ 9  “ ¦

−

→ S (t) = ˆ xS

x

(t) + ˆ yS

y

(t) + ˆ zS

z

(t) (34)

= ˆ x

^∗

S

_x^∗

(t) + ˆ y

^∗

S

_y^∗

(t) + ˆ z

^∗

S

_z^∗

(t) (35)



A _  › ' a> d ” 

S

_x^∗

(0) = S

x

(0) (36) S

_y^∗

(0) = S

y

(0) (37) S

_z^∗

(0) = S

z

(0) (38)

`



¦ s 6   x €  

S

x

(t) = S

x

(0)[{cos

²

γ + sin

²

γ cos(h

ef f

t)} cos(ωt) − sin γ sin(h

ef f

t) sin(ωt)]

+S

y

(0)[sin γ sin(h

ef f

t) cos(ωt) + cos(h

ef f

t) sin(ωt)]

+S

z

(0)[sin γ cos γ{1 − cos(h

ef f

t)} cos(ωt) + cos γ sin(h

ef f

t) sin(ωt)] (39) S

y

(t) = S

x

(0)[−{cos

²

γ + sin

²

γ cos(h

ef f

t)} sin(ωt) − sin γ sin(h

ef f

t) cos(ωt)]

+S

y

(0)[− sin γ sin(h

ef f

t) sin(ωt) + cos(h

ef f

t) cos(ωt)]

+S

z

(0)[− sin γ cos γ{1 − cos(h

ef f

t)} sin(ωt) + cos γ sin(h

ef f

t) cos(ωt)] (40) S

z

(t) = S

x

(0) sin γ cos γ{1 − cos(h

ef f

t)}

+S

y

(0) cos γ sin(h

ef f

t)

+S

z

(0){sin

²

γ + cos

²

γ cos(h

ef f

t)} (41)

\



¦ þ j7 á x& h Ü ¼– Ð % 3   H .

III. Bloch U ê sX N ËÅ k Ä8 ý – ¥ø m ÇA 0õ u §  Œ À W ¥6 K ý

m

Ç~ ¿” X ¢ ö n ÚP X ì Ä® Žz º W ì ÆM   üÆ U Ø € ¾M   ºü g Åô p §

#

Œ Q t _  0 p xô  Ç œ íl › ¸| [ þ t ×  æ\ " f Ó ü to & h Ü ¼– Ð % 3  l

 / 'î  r, +z−~ ½ ӆ ¾ ÓÜ ¼– Ð Û ¼— 2 ;[ þ ts  ¢ - a„  y  & ñ § > = ) a  © œI , Û ¼

— 2

;C \ P s  ¢ - a„  y  Á º| 9 " fô  Ç  © œI , Û ¼— 2 ;[ þ t_  ¨ î + þ A  © œI \  ¦ Bloch ~ ½ Ó& ñ d ” _  q ¨ î + þ A ' Ÿ I \  ¦ › ¸    HX <  6   x l – Ð

 .

(1) Û ¼— 2 ;[ þ ts  ¢ - a„  y  & ñ § > =ô  Ç œ íl  © œI 

+z−~ ½ ӆ ¾ ÓÜ ¼– Ð Û ¼— 2 ;[ þ ts  ¢ - a„  y  & ñ § > =ô  Ç œ íl  © œI \ " f



l  o $ í ì  r m

α

(0) ≡< S

α

(0) > /N (α = x, y, z ) “ É r

m

x

(0) = 0 (42)

m

y

(0) = 0 (43) m

z

(0) = 1 (44)

(2) Û ¼— 2 ; C \ P s  ¢ - a„  y  Á º| 9 " fô  Ç œ íl  © œI 

—

¸Ž  H Û ¼— 2 ;[ þ ts  ì ø Ít 2 £ §s  1“   / B N_  ³ ð€  `  ¦ Á º| 9 " f > 

o v   H ¢ - a„  y  Á º| 9 " fô  Ç œ íl  © œI \ " f  l  o_  € © œ © œ

^



¦ ¨ î ç  H“ É r # Q‹ "    à º A\  › ' aô  Ç € © œ © œ^  ¦ ¨ î ç  Hd ” 

< A >= Π

^N_i=1

R

_2π

0

R

_π

0

A sin θ

i

dθ

i

dφ

i

Π

^N_i=1

R

_2π

0

R

_π

0

sin θ

i

dθ

i

dφ

i

(45)

`



¦ s 6   xK " f > í ß –½ + É Ã º e ”   HX < Õ ª   õ   H  6 £ §õ  ° ú   .

m

x

(0) = 0 (46)

m

y

(0) = 0 (47) m

z

(0) = 0 (48)

(3) Û ¼— 2 ;C \ P s  ¨ î + þ A © œI “   œ íl  © œI 

œ

íl  © œI  ¨ î + þ A © œI “    â Ä º\   H Hamiltonian

H

0

= −h

z

S

z

− h

0

S

x

− h

1

S

y

= −h

z

X

i

s

^z_i

− h

0

X

i

s

^x_i

− h

1

X

i

s

^y_i

(49)

\



¦ s 6   x # Œ ì  rC † < Êà º (partition function)

Z = Π

^N_i=1

Z

_2π

0

Z

_π

0

sin θ

i

dθ

i

dφ

i

e

^−βH0

= [ Z

_2π

0

Z

_π

0

e

^{a cos θi+b sin θicos φi+c sin θisin φi}

sin θ

i

dθ

i

dφ

i

]

^N

= (4π)

^N

sinh

^N

( √

a

²

+ b

²

+ c

²

)

(a

²

+ b

²

+ c

²

)

^N/2

(50)

\



¦ % 3   H . # Œl " f y−~ ½ ӆ ¾ Ó_   © œ  l  © œ h

1

“ É r > í ß – © œ _

 3 l q& h Ü ¼– Ð • ¸{ 9 ÷ &% 3 “ ¦, a = βh

z

, b = βh

0

, c = βh

1

,

⍠ H “ : r• ¸_  % i à ºs  . 0 A_  > í ß –\ " f  H  6 £ §d ”  [7]

R

_2π

0

R

_π

0

f (a cos θ + b sin θ cos φ + c sin θ sin φ) sin θdθdφ

= 2π R

_π

0

f ( √

a

²

+ b

²

+ c

²

cos α) sin αdα

= 2π R

₁

−1

f (( √

a

²

+ b

²

+ c

²

z)dz (51)

\



¦  6   x % i  . Õ ªo “ ¦ ፠ H 3 " é ¶\ " f ‘ : rA  ý a³ ð> \  ¦ z−» ¡ ¤\  @ / # Œ  r„  r &  % 3 “ É r D h– Ðî  r ý a³ ð>   s _  y Œ •

•

¸s  .

0

A_  d ” \ " f & h ] X ô  Ç ¼ # p ì  r`  ¦ ô  ÇÊ ê\  c = 0 “ ¦ Z  ~Ü ¼

€



  6 £ §õ  ° ú  “ É r ¨ î + þ A © œI \ " f_   l  o $ í ì  r m

^eq_α

≡<

S

α

>

eq

/N (α = x, y, z )

m

^eq_x

= < S

x

>

eq

N = b

√ a

²

+ b

²

coth p

a

²

+ b

²

− b

a

²

+ b

²

(52)

m

^eq_y

= < S

y

>

eq

N = 0 (53)

m

^eq_z

= < S

z

>

eq

N = a

√ a

²

+ b

²

coth p

a

²

+ b

²

− a

a

²

+ b

²

(54)

`



¦ % 3   H .

IV. # b [ € ¾M  ºü g Å; c 6 ” X ¢ R ç g Ë] k ù  M × D8 ý X

N

˽  ʔ X ¢ ƒ »”  ôÅ k Ä

II] X \ " f ½ ¨ô  Ç ~ ½ Ó& ñ d ”  (39), (40), (41)_  q ¨ î + þ A € © œ © œ

^



¦ ¨ î ç  H\  III] X _  œ íl   l  o › ¸| `  ¦ @ /{ 9  €    A ü <

° ú

 “ É r q ¨ î + þ A  l  o\  › ' aô  Ç & ñ S X ‰ô  Ç d ” `  ¦ % 3 `  ¦ à º e ”  .

(1) Û ¼— 2 ;[ þ ts  ¢ - a„  y  & ñ § > =ô  Ç œ íl  © œI  Û

¼— 2 ;[ þ ts  ¢ - a„  y  & ñ § > =ô  Ç œ íl  © œI \ " f_   l  o $ í ì



r_  r ç ß –\    É r ”   o  H

m

x

(t) = sin γ cos γ{1 − cos(h

ef f

t)} cos(ωt) + cos γ sin(h

ef f

t) sin(ωt) (55) m

y

(t) = − sin γ cos γ{1 − cos(h

ef f

t)} sin(ωt) + cos γ sin(h

ef f

t) cos(ωt) (56) m

z

(t) = sin

²

γ + cos

²

γ cos(h

ef f

t) (57)

ü

< ° ú   .

(2) Û ¼— 2 ; C \ P s  ¢ - a„  y  Á º| 9 " fô  Ç œ íl  © œI  Û

¼— 2 ; C \ P s  ¢ - a„  y  Á º| 9 " fô  Ç œ íl  © œI \ " f_   l  o

$ í

ì  r_  r ç ß –\    É r ”   o  H

m

x

(t) = 0 (58) m

y

(t) = 0 (59)

m

z

(t) = 0 (60)

s

  ) a .

(3) Û ¼— 2 ;C \ P s  ¨ î + þ A © œI “   œ íl  © œI  Û

¼— 2 ;C \ P s  ¨ î + þ A © œI “   œ íl  © œI \ " f_   l  o $ í ì  r _

 r ç ß –\    É r ”   o  H

m

x

(t) = m

^eq_x

[{cos

²

γ + sin

²

γ cos(h

ef f

t)} cos(ωt) − sin γ sin(h

ef f

t) sin(ωt)]

+m

^eq_z

[sin γ cos γ{1 − cos(h

_{ef f}

t)} cos(ωt) + cos γ sin(h

_{ef f}

t) sin(ωt)] (61) m

y

(t) = m

^eq_x

[−{cos

²

γ + sin

²

γ cos(h

ef f

t)} sin(ωt) − sin γ sin(h

ef f

t) cos(ωt)]

+m

^eq_z

[− sin γ cos γ{1 − cos(h

ef f

t)} sin(ωt) + cos γ sin(h

ef f

t) cos(ωt)] (62) m

_z

(t) = m

^eq_x

sin γ cos γ(1 − cos(h

_{ef f}

t)) + m

^eq_z

(sin

²

γ + cos

²

γ cos(h

_{ef f}

t)) (63)

ü

< ° ú   .

V. À X Ø 8 ý

IV] X \ " f % 3 “ É r & ñ S X ‰ô  Ç d ” _  r ç ß –\    É r ”   o\  ¦ à ºu 

&

h

Ü ¼– Ð ¶ ú ˜( R ˜ Ð . Û ¼— 2 ;s  ¢ - a„  y  & ñ § > = ) a œ íl › ¸| \ " f





H  l  o_  z−$ í ì  rs  0s   m Ù ¼– Ð — ¸Ž  H  l  o $ í ì  r s

  ™ è t  · ú §“ É r r ç ß –& h  ”   o\  ¦ ˜ Ð{ 9  כ s    H \ V © œ`  ¦

½ +

É Ã º e ”  . s  Qô  Ç œ íl  © œI   H ] X @ /“ : r• ¸ 0 K M :_  ¨ î + þ A



©

œI \  K { © œô  Ç . s  כ “ É r B Ä º  Œ •“ É r  r„    l  © œs     2 ; +

'\  • ¸ B Ä º ± ú “ É r “ : r• ¸ü < B Ä º ± ú “ É r y Œ •”  1 l xà º\ " f ¢ - a

„



y  & ñ § > = ) a œ íl  © œI ü < ¨ î + þ A œ íl  © œI \ " f q ¨ î + þ A   l

 o q 5 p wô  Ç ' Ÿ I \  ¦ ˜ Ðs >  H † d`  ¦ _ p ô  Ç .



6 £ §Ü ¼– Ð Û ¼— 2 ;C \ P s  ¢ - a„  y  Á º| 9 " fô  Ç œ íl  © œI \ " f





H — ¸Ž  H  l  o $ í ì  rs  0s Ù ¼– Ð Õ ª כ [ þ t“ É r — ¸Ž  H r ç ß –\ 





5 g" f 0_  ° ú כ`  ¦ t >  ÷ &Ù ¼– Ð  8 s  © œ_  K $ 3 “ É r € 9 כ ¹

\ O

>   ) a . Õ ªo “ ¦ s  Qô  Ç œ íl  © œI   H ∞ K_  ¨ î + þ A © œI 

\

 K { © œô  Ç .

‰

&

³ © œ† < Æ& h “   Bloch ~ ½ Ó& ñ d ” _  É Òo \    ¨ 8 Š ) a (Fourier-

transformed) K   H ω = h

z

– Ð Å Ò# Qt   H / B N”   y Œ • ”  1 l xà º\ 

t

0 100 200 300 400 500 600

mx(t)

-0.10 -0.05 0.00 0.05 0.10

ordered disordered equilibrium

t

0 50 100 150 200 250 300

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium

t

0 10 20 30 40 50

-0.10 -0.05 0.00 0.05 0.10

ordered disordered equilibrium

(a)

(b)

(c)

Fig. 1. m

x

vs. time at a dimensionless inverse tem- perature of βh

z

= 100 for dimensionless magnetic fields h

0

/h

z

= 0.01 with angular frequencies ω = (a) 0.01, (b) 1.0, (c) 1.5 in units of rad/s, respectively.

"

f  @ /ô  Ç ”  ; Ÿ ¤`  ¦ t >   ) a  [2].

ô



Ǽ #   l  o_  y−$ í ì  r“ É r x−$ í ì  rs  0 A © œ…  ;s  ) a כ õ 

° ú

 Ü ¼Ù ¼– Ð ‘ : r  7 Hë  H\ " f  H  l  o_  x− x 9  z−$ í ì  rë ß – Ò q t y

Œ

• l – Ð ô  Ç .



l  o\  ¦ à ºu & h Ü ¼– Ð > í ß –   HX < e ” # Q" f z  ´] j  l / B N

"

î

z  ´+ « >\ " fü < ° ú  s  z−~ ½ ӆ ¾ Ó_  & ñ  l  © œs  B Ä º ß ¼“ ¦, 

"

é

¶s  \ O   H  r„    l  © œ h

0

/h

z

≤ 1.0s  “ ¦ & ñ l – Ð ô

 Ç .

Fig. 1õ  Fig. 2  H 3> h_  " f– Ð   É r œ íl › ¸| \  @ / # Œ,

" é ¶s  \ O   H “ : r• ¸ βh

z

= 100{ 9 M :, " é ¶s  \ O   H  l 



©

œ_  [ jl  h

0

/h

z

= 0.1s “ ¦ y Œ •”  1 l xà º ω = (a) 0.01, (b)1.0, (c) 1.5 (rad/s_  é ß –0 A){ 9 M :  l  o_  x−$ í ì  r_  q

¨ î + þ A ' Ÿ I \  ¦ ˜ Ð# ŒÅ ҍ  HX <  6 £ §õ  ° ú  s  K $ 3 ½ + É Ã º e ”  .

t

0 100 200 300 400 500 600

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium

t

0 100 200 300 400 500 600

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium (a)

(b)

(c)

ordered disordered equilibrium

t

0 100 200 300 400 500 600

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium (c)

Fig. 2. m

z

vs. time at a dimensionless inverse tem- perature of βh

z

= 100 for dimensionless magnetic fields h

0

/h

z

= 0.01 with angular frequencies ω = (a) 0.01, (b) 1.0, (c) 1.5 in units of rad/s, respectively.

¢ -

a„  y  & ñ § > = ) a  â Ä º
  Å Ò ± ú “ É r “ : r• ¸_  ¨ î + þ A © œI  œ í l

 › ¸| s €  , z−~ ½ ӆ ¾ Ó_   H  l  © œõ  xy−¨ î €   © œ_  B Ä º



Œ

•“ É r  r„    l  © œ\ " f y Œ •”  1 l xà º ω = 0.01 rad/s{ 9 M :,



Œ

•“ É r  r„    l  © œ“ É r Ä »´ ò  l  © œ ~ ½ ӆ ¾ Ó_  í  Hç ß –  r„  » ¡ ¤ Å Ò 0

A– Ð  l  o_  [ j î  r1 l x (precession)`  ¦ Ä »• ¸ô  Ç . Õ ªA 

"

f  l  o\  ¦ xy−¨ î €  \  È Ò% ò €   ×  æ^ o ?÷ &  H  " é ¶s  ÷ &

“

¦  l  o_  x−$ í ì  r (Fig. 1(a))“ É r r ç ß –\  @ / # Œ ”  ; Ÿ ¤ s

  Œ •t ë ß – 4 Ÿ ¤¸ ú šô  Ç ”  1 l x`  ¦
 ? />  ÷ &  H  כ s  . ô  Ǽ # ,



l  o_  z−$ í ì  r (Fig. 2(a))“ É r xy−¨ î €  A á ¤Ü ¼– Ð €  •ç ß – l  Ö



¦# Q t t ë ß –,  r„    l  © œ\  q K " f z−~ ½ ӆ ¾ Ó_  & ñ  l  © œ s

  -Á º ß ¼Ù ¼– Ð  _  “ ¦& ñ  ) a  © œI – Ð e ”  .

/ B

N”   y Œ •”  1 l xà º ω

R

= h

z

= 1.0 rad/s{ 9 M :  H z−» ¡ ¤ © œ _

 î  r1 l xõ  xy−¨ î €   © œ_  î  r1 l xs    ´ ú >  ƒ    ÷ &# Q, ° ú š 

t

0 50 100 150 200 250 300

mx(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium

t

0 10 20 30 40 50

mx(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0

1.5 ordered

disordered equilibrium (a)

(b)

(c)

t

0 10 20 30 40 50

mx(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0

1.5 ordered

disordered equilibrium (c)

Fig. 3. m

x

vs. time at a dimensionless inverse tem- perature of βh

z

= 100 for dimensionless magnetic fields h

0

/h

z

= 1.0 with angular frequencies ω = (a) 0.01, (b) 1.0, (c) 1.5 in units of rad/s, respectively.

l

  r„    l  © œs  Ä »´ ò  l  © œ ~ ½ ӆ ¾ Ó_  í  Hç ß –  r„  » ¡ ¤\  @ /

# Œ  • 2 ; [ j î  r1 l x`  ¦ { 9 Ü ¼†   .  l  o_  x−$ í ì  r (Fig.

1(b))“ É r  Ø Ô>    › ¸ (modulated) ”  1 l x t ë ß –, Õ ª כ _   

>

h‚   (envelope)“ É r ω = h

0

_  Rabi ”  1 l xà º– Ð ”  1 l xô  Ç . ô  Ç

¼

#

  l  o_  z−$ í ì  r (Fig. 2(b))“ É r / B N”  \  _  # Œ  H ”  

;

Ÿ

¤`  ¦ t “ ¦ Rabi ”  1 l xà º– Ð ”  1 l x >   ) a . . /

B

N”   y Œ •”  1 l xà º\ " f # Á # Qè ß – ω = 1.5rad/s{ 9 M :  H z−~ ½ Ó

†

¾

Ó î  r1 l xõ  xy−¨ î €   © œ î  r1 l xs    s  ´ ú t  · ú §>  ƒ    ÷ &

Ù

¼– Ð,  l  o  H  Œ •“ É r ì ø Í â `  ¦ t “ ¦  Ø Ô>  [ j î  r1 l x

`



¦ ô  Ç . Õ ªA " f  l  o_  x−$ í ì  r (Fig. 1(c))“ É r  Œ •“ É r

”



; Ÿ ¤Ü ¼– Ð  Ø Ô>  ”  1 l x   HX < q  # Œ,  l  o_  z−$ í ì  r (Fig.2(c))“ É r & ñ  l  © œ_  [ jl   H › ' a> – Ð  _  { 9 & ñ ô  Ç

° ú

כ`  ¦ Ä »t ô  Ç .

t

0 10 20 30 40 50

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium

t

0 10 20 30 40 50

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0

1.5 ordered

disordered equilibrium

t

0 10 20 30 40 50

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0

1.5 ordered

disordered equilibrium (a)

(b)

(c)

Fig. 4. m

z

vs. time at a dimensionless inverse tem- perature of βh

z

= 100 for dimensionless magnetic fields h

0

/h

z

= 1.0 with angular frequencies ω = 0.01, 1.0, 1.5 in units of rad/s, respectively.

Fig. 3õ  Fig. 4  H  r„    l  © œs  & ñ  l  © œõ  ß ¼l 

° ú

 “ É r h

0

/h

z

= 1.0“    â Ä º\  ¦ “ ¦ 9 “ ¦ e ”   HX <, œ íl    l

 o  H z−» ¡ ¤\ " f xy−¨ î €  A á ¤Ü ¼– Ð  _  45

^◦

l Ö  ¦# Q4 R e ” 



. z−~ ½ ӆ ¾ Ó_  & ñ  l  © œõ  xy−¨ î €  _   r„    l  © œs  " f

–

Ð  â Ô q t Ù ¼– Ð, Ä ºo  s K    H ~ ½ Ód ” _  / B N”  ‰ & ³ © œs   



t “ ¦ œ íl › ¸|  _ ” > r$ í s  B Ä º ß ¼ .  l  o  H  f ” • ¸ Ä

»´ ò  l  © œ~ ½ ӆ ¾ Ó_  í  Hç ß –  r„  » ¡ ¤\  @ / # Œ [ j  î  r1 l x`  ¦ ô

 Ç .

Õ

ªA " f  l  o_  x−$ í ì  r“ É r y Œ •”  1 l xà º ω = 0.01 rad/s (Fig. 3(a)){ 9 M :, ¢ - a„  y  & ñ § > = ) a œ íl  © œI \ " f  H





› ¸ ) a  © œI – Ð ”  1 l x   H   HX < ì ø Í # Œ, ¨ î + þ A © œI  œ í l

 © œI “    â Ä º  H ¢ - a„  y  & ñ § > = ) a œ íl  © œI _  ¨ î ç  Hu \  K  {

©

œ   H ° ú כ_  ß ¼l \  ¦ t “ ¦ ω = 0.01 rad/s– Ð ”  1 l xô  Ç .

t

0 100 200 300 400 500 600

mx(t)

-0.10 -0.05 0.00 0.05 0.10

ordered disordered equilibrium

t

0 50 100 150 200 250 300

mx(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium

t

0 10 20 30 40 50

mx(t)

-0.10 -0.05 0.00 0.05 0.10

ordered disordered equilibrium (a)

(b)

(c)

Fig. 5. m

x

vs. time at a dimensionless inverse tem- perature of βh

z

= 0.1 for dimensionless magnetic fields h

0

/h

z

= 0.01 and angular frequencies ω = (a) 0.01, (b) 1.0, (c) 1.5 in units of rad/s, respectively.

/ B

N”   ”  1 l xà º ω = 1.0 rad/s (Fig. 3(b))\ " f ”  1 l x ”  ; Ÿ ¤“ É r

¢ -

a„  & ñ § > = œ íl › ¸| õ  ¨ î + þ A œ íl › ¸| { 9 M : š ¸y  9 ×  ¦# Q[ þ t t

ë ß – t    >  ”  1 l x`  ¦ ô  Ç . Õ ªo “ ¦ ”  1 l xà º ω = 1.5 rad/s (Fig. 3(c))– Ð  8 7 £ x €   ”  1 l x”  ; Ÿ ¤“ É r  8 & t t ë ß –

”    † < ʓ É r \ O # Q”   . ô  Ǽ # ,  l  o_  z−$ í ì  r“ É r x−$ í ì



rõ   H ² ú ˜o  ”  1 l xà º ω = 0.01 rad/s(Fig. 4(a))\ " f ω = 1.0 rad/s (Fig. 4(b))– Ð Õ ªo “ ¦ ω = 1.5 rad/s (Fig.

4(c))– Ð 7 £ x† < Ê\    " f, ¢ - a„  & ñ § > = œ íl › ¸| õ  ¨ î + þ A œ í l

› ¸|  — ¸¿ º  r„   l  © œs  h

0

/h

z

= 0.01{ 9 M :ü <  ð ø Í

t

– Ð / B N”  ”  1 l xà º\  s \  ¦M : t  ”  ; Ÿ ¤s  7 £ x €  " f ”  1 l x

  / B N”  ”  1 l xà º\  ¦  Å # Q" f€   ”  1 l x”  ; Ÿ ¤s  „  & h  y Œ ™™ è 

>

  ) a .

Fig. 5ü < Fig. 6“ É r y Œ •y Œ • 3> h_  " f– Ð   É r œ íl › ¸| \ 

@

/ # Œ, " é ¶s  \ O   H “ : r• ¸ βh

z

= 0.1{ 9 M :, " é ¶s  \ O 

t

0 100 200 300 400 500 600

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium

t

0 100 200 300 400 500 600

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium

t

0 100 200 300 400 500 600

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium (a)

(b)

(c)

Fig. 6. m

z

vs. time at a dimensionless inverse tem- perature of βh

z

= 0.1 for dimensionless magnetic fields h

0

/h

z

= 0.01 and angular frequencies ω = (a) 0.01, (b) 1.0, (c) 1.5 in units of rad/s, respectively.





H  l  © œ_  [ jl  h

0

/h

z

= 0.1s “ ¦ y Œ •”  1 l xà º ω = (a) 0.01, (b) 1.0, (c) 1.5 (rad/s_  é ß –0 A)“    â Ä º  l  o_  x− x 9  z−$ í ì  r_  q ¨ î + þ A ' Ÿ I \  ¦ ˜ Ð# Œï  r . “ : r• ¸ B Ä º Z



}Ü ¼Ù ¼– Ð ¨ î  © œI _  œ íl › ¸| “ É r ¢ - a„  y  Á º| 9 " fô  Ç œ íl › ¸

|

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 â

Ä º  l  o_  r ç ß –& h  ”   o — ¸€ ª œ“ É r ”  ; Ÿ ¤“ É r  Œ •t ë ß – Fig.

1õ  Fig. 2_  ± ú “ É r “ : r• ¸  â Ä ºü < ° ú  “ É rX < Õ ª s Ä »  H “ : r• ¸

\

   " f ”  ; Ÿ ¤ë ß –      H ½ ¨› ¸\  ¦  l  o t l  M :ë  H s

 .

Fig. 7õ  Fig. 8“ É r y Œ •y Œ • 3> h_  " f– Ð   É r œ íl › ¸| \ 

@

/ # Œ " é ¶s  \ O   H “ : r• ¸ βh

z

= 0.1{ 9 M : " é ¶s  \ O   H



l  © œ_  [ jl  h

0

/h

z

= 1.0s “ ¦ y Œ •”  1 l xà º ω = (a)

t

0 50 100 150 200 250 300

mx(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium

t

0 10 20 30 40 50

mx(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium

t

0 10 20 30 40 50

mx(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0

1.5 ordered

disordered equilibrium (a)

(b)

(c)

Fig. 7. m

x

vs. time at a dimensionless inverse tem- perature of βh

z

= 0.1 for dimensionless magnetic fields h

0

/h

z

= 1.0 and angular frequencies ω = (a) 0.01, (b) 1.0, (c) 1.5 in units of rad/s, respectively.

0.01, (b) 1.0, (c) 1.5 (rad/s_  é ß –0 A)“    â Ä º  l  o_  x−

x 9

 z−$ í ì  r_  q ¨ î + þ A ' Ÿ I \  ¦ ˜ Ð# Œï  r .  r  ô  ǁ   Fig.

5ü < 6_   â Ä º\  › ' aô  Ç s Ä »ü <  ð ø Ít – Ð  l  o_  r ç ß –

&

h

Ü ¼– Ð ”   o   H — ¸€ ª œs  ”  ; Ÿ ¤“ É r  Œ •t ë ß – Fig. 3õ  4_  ± ú 

“ É

r “ : r• ¸ â Ä ºü < ° ú   . Õ ªo “ ¦ Õ ª s Ä »  H · ú ¡\ " fü < ° ú  s ,

“ :

r• ¸\    " f ”  ; Ÿ ¤ë ß –      H ½ ¨› ¸\  ¦  l  o t l  M

:ë  Hs  .

VI. + s Ç Â ] Ø

‘ :

r  7 Hë  H\ " f Ä ºo   H  r„    l  © œ? /_   © œ  ñ Œ •6   xs  \ O 





H “ ¦„  & h  Û ¼— 2 ;_  > \  ¦ › ¸  % i  . Õ ª כ Ü ¼– Ð Â Ò'  y Œ ™ W

†

½

Ós  \ O   H Bloch ~ ½ Ó& ñ d ” `  ¦ Ä »• ¸ “ ¦ s \  ¦ & ñ S X ‰y  Û  ¦% 3 

t

0 10 20 30 40 50

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium

t

0 10 20 30 40 50

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0

1.5 ordered

disordered equilibrium

t

0 10 20 30 40 50

mz(t)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ordered disordered equilibrium (a)

(b)

(c)

Fig. 8. m

z

vs. time at a dimensionless inverse tem- perature of βh

z

= 0.1 for dimensionless magnetic fields h

0

/h

z

= 1.0 and angular frequencies ω = 0.01, 1.0, 1.5 in units of rad/s, respectively.



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”



‰ & ³ © œ`  ¦ ˜ Ѐ Œ ¤ . Õ ª Q
  r„    l  © œs   Å Ò & t €   ”  

;

Ÿ

¤s  7 £ x  8 • ¸ Ä ºo     H 7 á xÀ Ó_  / B N”  ‰ & ³ © œs    m

“ ¦ ”  ; Ÿ ¤s  t    K t   H ‰ & ³ © œ`  ¦ ˜ Ѐ Œ ¤ .

P c

p 8 ý ò k >

s

  7 Hë  H“ É r 2007† < Ƹ  • ¸ Ø  æ· ¡ ¤@ /† < Ɠ § l $ í  r ƒ  ½ ¨q  t 

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Y c

p w Š à U Ø ”  ô

[1] R. Kubo, M. Toda and N. Hashitsume, Statisitical Physics II, (Springer-Verlag, Berlin, 1985).

[2] G. F. Mazenko, Nonequilibrium Statistical Mechanics (Wiley-VCH, Weinheim, 2006).

[3] C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, Berlin, 1978).

[4] F. Bloch, Phys. Rev. 70, 460 (1946); R. K. Wangsness and F. Bloch, Phys. Rev. 89, 728 (1953); F. Bloch and I. I. Rabi, Rev. Mod. Phys. 17, 237 (1945).

[5] H. C. Torrey, Phys. Rev. 76, 1059 (1949); F. T. Hioe, Phys. Rev. A 30, 2100 (1984); A. V. Alekseev and N.

V. Sushilov, Phys. Rev. 46, 351 (1992); H. K. Kim and S. P. Kim, J. Korean Phys. Soc. 48, 2006; C.

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Phys. 26, 167 (1954).

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Exact Solutions of the Classical Bloch Equations without Damping Terms: Initial-condition Dependence

Suhk Kun Oh

^∗

BK21 Physics Program and Department of Physics, Chungbuk National University, Cheongju 361-763

(Received 1 April 2008)

The classical Bloch equations without damping terms originating from a model of noninteracting classical spins in a time-dependent rotating magnetic field are investigated. The nonequilibrium magnetization is calculated to see its dependence on the initial conditions. We adopted three physi- cally realizable, but different, initial conditions, namely, completely ordered, completely disordered, and equilibrium states, and we found a distinct initial-condition dependence of the nonequilibrium magnetization. For different magnitudes and angular frequencies of a transverse rotating magnetic field, we found a distinct initial condition dependence of the nonequilibrium magnetization. For the completely disordered states as initial conditions, the magnetization remains zero, irrespective of both the magnitude and the angular frequency of the rotating field. For the completely ordered state as the initial condition, all the components of the magnetization oscillate as a function of time.

For the equilibrium states as initial conditions, at low temperatures, the nonequilibrium behaviors of the components of the magnetization are similar to these for the completely ordered initial state, whereas the nonequilibrium behaviors of the components of the magnetization at high tempera- tures are similar to that of completely disordered initial states. Especially, for small rotating fields, compared to a static field along the z-axis, the nonequilibrium magnetization shows a remarkable increase in the oscillation amplitude at the resonance frequency. However, due to the competition between the static field and the rotating field, the oscillation amplitude increases monotonically for large rotating fields as the angular frequency is increased beyond the resonance angular frequency, so no resonance phenomena will be seen.

PACS numbers: 05.90.+m, 75.10.Hk, 75.74.Gb

Keywords: Nonequilibrium spin dynamics, Rotating coordinate system, Bloch equation, Magnetic resonance

∗E-mail: [email protected]