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Effective Hamiltonian method for S = 1 Ising ferromagnet
0 0 3 8 - 1 0 9 % ' 8 1 , 0 3 0 2 1 9 - 0 2 S02 00 '0
Sohd State Communicauons. Vol 38. pp 2 1 9 - 2 2 0 Pergamon Press Ltd 198I Pnnted m Great Bntam
EFFECTIVE HAMILTONIAN METHOD F O R S = 1 ISING F E R R O M 4 G N E T -~ Komoda and A Pqkalsk~ lnsntute of Theoretical Ph} stcs. Umvers~ty of '~,roc't'av.. ~,~,rod'aw, Poland { R e c e u ' e d 4 , V o v e m b e r 1980 b y S
4melmck_'~)
The effectwe Harrultontan method introduced for S = 1 2 b~ Ogucha and Ono, Is here extended for S = 1 lsmg ferromagnets The expressions for the magnenzatton, parallel suscepub[ht.,, near the cnt,cal point and the cnncal temperature, are derived The results for T c sho~. an impro~,ement over the MFA, and come close to the results from the h a ~ temperature expansmns and the renormahzat,on group
IN THIS NOTE vm report an extension for S = 1 ot the effecw, e Harmltontan method Introduced b', Ogucha and Ono [1] f o r S = 1/2 I t p r o ~ , e d u s e f u l m d e t e r m m m g cnncal temperatures for homogeneous and mhomogeneous magnets [2], as well as other thermodynarmc quantttms m the vlmmt.,, of the cnncal point The method ts attractwe as ~t is qmte smtpIe, ,,et tt gwes correct (at least qualltatxvel,, ) results, pred~ct,ng t a no long-range ordering for t~o-dtrnens,onal He,senberg and one-dimensmnal ismg systems It can also be eas@ adapted to cope wtth more comphcated, than treated here. Harmltomans Since the method for spin one follows, m general, the approach for S = 1/2, we shall mdtcate the mare points onl,, As usual, we start w~th one-spin. -fit, and two-spin. "Kn, Harrultomans
The mgen,,ectors for "Ki[ are la~', 11331, I"/"/). I ~ ) , 13~, [~"t~, I"ta). 1371 The baste equat,on oI the method is the self-cons~stenc~, c o n d m o n TrR.sPn
{3)
PI
=
where Pl and Pn are the normahzed denstt,, operators for the "K[ and K a, respectBel~ Switching oft" the external field we can obtain from equation {3') the equation todetermme Tc [thasthe form e "-~ smh ~,d(z - 2)', } - e -v'~ ~mh ~:,', ) -- slnh {/3,k) = 0 (41 Since close to Tc the effective field 3 is small, be can expand smh (j33) m powers of X and keep the hnear terms only This. apart from the paramagnenc solutton = 0. welds also e~ca(z- 2}--e-~¢az-
~i
= - z,XSa - (zS>, + t , ) S ~
1 = 0
151
tl)
Hence for the cnncal temperature v,e have and J.,'kTe = 7£ n = -- 2 J S ~ S ~
l/'~ha{1(
+ V I -'- 4 z l : -
2)1
+6 - [{.z -- 1)1,~ + 83) + It] ,.
{6}
t2) , ts~ + s~.s},
where z is the coordmanon number of the lattice, ~ is the effectwe field actmg on the spin S" at a w e n site, 8X is the change of that field due to the external field h, a n d J ts the exchange coupling between spins at the nearest nmghbounng sites R and R + 8 The elgenvectors of R'l, corresponding to the values + 1,0. - 1, are denoted by I~, IJ3)and I'9, respectwely
I
(I3X)" =
Slmflarl3, as done by Ogucha and Ono [I] ~e can get the expressions for the magnenzatton ~S-') and parallel suscepttbtht,, ,(", close to the crmcal point The magnetrzatmn l~ (S ") =
2 smh {z~> ) 2 cosh ( z ~ )
(7)
+ I
where
1212(=
-
2) -
e -~c'/]
4{--7 , 2 I ) ( : -- 2) -- {:2 _ 1)e-~e--~J2 25(-7.~ 219
2
4_" + 4 ) -
z2e -~¢J ] A11I
{s}
220
E F F E C T I V E H..L~,IlLTO\,I-L\ ~,IETHOD F O R S = [ ISING F E R R O b l A G N E T
Vot 3 S , N o 3
','. Ith 2u3 = 3 - 7~ For the suscept,b,ht3 above the Tc
the molecular field approxamauon [2], h a ~ temperature expansions [41 and renormahzatton grou~ [5] for square (-" = 41 stmple cubic ~_ = 61 and bod,, centred cubic 2 + e -:''c'l r " re ~-- 1 3 \ lga) (..- = '3llattlce~ -ks can be seen, the results of the present '~.lr [21-- - 2 1 - e - = 4 J l J ~ r - T~I'Te ,~ork are not tar from those obtained b,. more sop~sttand below,, the 7". cared (and also much more comphcated ) methods The 2 4- e-ZdcJ comparison ,a,ath the experimental value ot k T c 7 = o 9 7", T,. 63, (9b } ',1 - -" I [2~:_ 2)_el~calj(T_ rc~ T 69r FeCI, ~hach is ~.ell descnbed b,. the apm one lsmg f e r r o m a ~ e t with simple cubic structure [ro], sho~.s a Of course the critical e x p o n e n t s ha',e the claas~cal values good a ~ e e m e n t wath our value of 6 6" The lov~ temperature behavmur ~s hov, ever correct, ~ e FmaLl,,, tt ma,, be noted that the present approach ma,, be utthzed m calculating the c n n c a l eroperne~ of a 1,m ,'S" = 1 (10) T--O dilute b m a ~ allo,, ,a,lth the vector lu..' descnbtn~ a site C o m p a n n g the expresstons for T c for a sx stem ',*.lth occupted bx an atom 4, [I,~ - a site occupted b.,, an atom S = 1.2 [I] and eq 16), one can easfl', sho~, that B, and I,.b - an erupt,, one Te(S - 1) > T ~ S = 1 2), and the difference grows udth increasing coordination n u m b e r : Table 1
REFERENCES Coordination number \lethod
4
\IFA HTSE RG Present v.ork
5 3 3 3
333 430 2S0 0028
I
o
8
8 o 3oi 5 050 o 6"0
I0 o 8 0
2 6~ 032 "0o 2nl
3 4 5
In Table 1 ~e present the values ot the crxt~cal ten> perature calculated t o r S = I Ismg Ierromagnet, uamg
o
T Ogucha & l O n o , J Ph~s Soc J a p a n 2 1 , 2 1 7 8 (1966) T Ogucha & T O b o k a t a , J Phas Snc Japan 27. 1111 (1969) K H a t t o n , J Phvs Soc Japan 42, 1525 (19~7), S Katsura, Phys Stares Sohdl ( b J 9 7 , 6 6 3 (1980), M Ausloos & 4. Pe,kalskk S o h d State C o m m u n 2 6 , 9 7 7 (1978) J S Smart, E f f e c m ' e Field Theones o] Magnensm Saunder~, Ph.tladelphaa (1o6o) A P~kalskl 4 c t a P h y s Polon 4.54, 23 ( 1 9 " 8 ) T B u r k h a r d t & R H Swendsen, P h t s Rev BI3, 30"1 119"o) k J DeJongJa& 4.R t~hedema, q&, P h j s 23,1 I 1974;
Sohd State Communicauons. Vol 38. pp 2 1 9 - 2 2 0 Pergamon Press Ltd 198I Pnnted m Great Bntam
EFFECTIVE HAMILTONIAN METHOD F O R S = 1 ISING F E R R O M 4 G N E T -~ Komoda and A Pqkalsk~ lnsntute of Theoretical Ph} stcs. Umvers~ty of '~,roc't'av.. ~,~,rod'aw, Poland { R e c e u ' e d 4 , V o v e m b e r 1980 b y S
4melmck_'~)
The effectwe Harrultontan method introduced for S = 1 2 b~ Ogucha and Ono, Is here extended for S = 1 lsmg ferromagnets The expressions for the magnenzatton, parallel suscepub[ht.,, near the cnt,cal point and the cnncal temperature, are derived The results for T c sho~. an impro~,ement over the MFA, and come close to the results from the h a ~ temperature expansmns and the renormahzat,on group
IN THIS NOTE vm report an extension for S = 1 ot the effecw, e Harmltontan method Introduced b', Ogucha and Ono [1] f o r S = 1/2 I t p r o ~ , e d u s e f u l m d e t e r m m m g cnncal temperatures for homogeneous and mhomogeneous magnets [2], as well as other thermodynarmc quantttms m the vlmmt.,, of the cnncal point The method ts attractwe as ~t is qmte smtpIe, ,,et tt gwes correct (at least qualltatxvel,, ) results, pred~ct,ng t a no long-range ordering for t~o-dtrnens,onal He,senberg and one-dimensmnal ismg systems It can also be eas@ adapted to cope wtth more comphcated, than treated here. Harmltomans Since the method for spin one follows, m general, the approach for S = 1/2, we shall mdtcate the mare points onl,, As usual, we start w~th one-spin. -fit, and two-spin. "Kn, Harrultomans
The mgen,,ectors for "Ki[ are la~', 11331, I"/"/). I ~ ) , 13~, [~"t~, I"ta). 1371 The baste equat,on oI the method is the self-cons~stenc~, c o n d m o n TrR.sPn
{3)
PI
=
where Pl and Pn are the normahzed denstt,, operators for the "K[ and K a, respectBel~ Switching oft" the external field we can obtain from equation {3') the equation todetermme Tc [thasthe form e "-~ smh ~,d(z - 2)', } - e -v'~ ~mh ~:,', ) -- slnh {/3,k) = 0 (41 Since close to Tc the effective field 3 is small, be can expand smh (j33) m powers of X and keep the hnear terms only This. apart from the paramagnenc solutton = 0. welds also e~ca(z- 2}--e-~¢az-
~i
= - z,XSa - (zS>, + t , ) S ~
1 = 0
151
tl)
Hence for the cnncal temperature v,e have and J.,'kTe = 7£ n = -- 2 J S ~ S ~
l/'~ha{1(
+ V I -'- 4 z l : -
2)1
+6 - [{.z -- 1)1,~ + 83) + It] ,.
{6}
t2) , ts~ + s~.s},
where z is the coordmanon number of the lattice, ~ is the effectwe field actmg on the spin S" at a w e n site, 8X is the change of that field due to the external field h, a n d J ts the exchange coupling between spins at the nearest nmghbounng sites R and R + 8 The elgenvectors of R'l, corresponding to the values + 1,0. - 1, are denoted by I~, IJ3)and I'9, respectwely
I
(I3X)" =
Slmflarl3, as done by Ogucha and Ono [I] ~e can get the expressions for the magnenzatton ~S-') and parallel suscepttbtht,, ,(", close to the crmcal point The magnetrzatmn l~ (S ") =
2 smh {z~> ) 2 cosh ( z ~ )
(7)
+ I
where
1212(=
-
2) -
e -~c'/]
4{--7 , 2 I ) ( : -- 2) -- {:2 _ 1)e-~e--~J2 25(-7.~ 219
2
4_" + 4 ) -
z2e -~¢J ] A11I
{s}
220
E F F E C T I V E H..L~,IlLTO\,I-L\ ~,IETHOD F O R S = [ ISING F E R R O b l A G N E T
Vot 3 S , N o 3
','. Ith 2u3 = 3 - 7~ For the suscept,b,ht3 above the Tc
the molecular field approxamauon [2], h a ~ temperature expansions [41 and renormahzatton grou~ [5] for square (-" = 41 stmple cubic ~_ = 61 and bod,, centred cubic 2 + e -:''c'l r " re ~-- 1 3 \ lga) (..- = '3llattlce~ -ks can be seen, the results of the present '~.lr [21-- - 2 1 - e - = 4 J l J ~ r - T~I'Te ,~ork are not tar from those obtained b,. more sop~sttand below,, the 7". cared (and also much more comphcated ) methods The 2 4- e-ZdcJ comparison ,a,ath the experimental value ot k T c 7 = o 9 7", T,. 63, (9b } ',1 - -" I [2~:_ 2)_el~calj(T_ rc~ T 69r FeCI, ~hach is ~.ell descnbed b,. the apm one lsmg f e r r o m a ~ e t with simple cubic structure [ro], sho~.s a Of course the critical e x p o n e n t s ha',e the claas~cal values good a ~ e e m e n t wath our value of 6 6" The lov~ temperature behavmur ~s hov, ever correct, ~ e FmaLl,,, tt ma,, be noted that the present approach ma,, be utthzed m calculating the c n n c a l eroperne~ of a 1,m ,'S" = 1 (10) T--O dilute b m a ~ allo,, ,a,lth the vector lu..' descnbtn~ a site C o m p a n n g the expresstons for T c for a sx stem ',*.lth occupted bx an atom 4, [I,~ - a site occupted b.,, an atom S = 1.2 [I] and eq 16), one can easfl', sho~, that B, and I,.b - an erupt,, one Te(S - 1) > T ~ S = 1 2), and the difference grows udth increasing coordination n u m b e r : Table 1
REFERENCES Coordination number \lethod
4
\IFA HTSE RG Present v.ork
5 3 3 3
333 430 2S0 0028
I
o
8
8 o 3oi 5 050 o 6"0
I0 o 8 0
2 6~ 032 "0o 2nl
3 4 5
In Table 1 ~e present the values ot the crxt~cal ten> perature calculated t o r S = I Ismg Ierromagnet, uamg
o
T Ogucha & l O n o , J Ph~s Soc J a p a n 2 1 , 2 1 7 8 (1966) T Ogucha & T O b o k a t a , J Phas Snc Japan 27. 1111 (1969) K H a t t o n , J Phvs Soc Japan 42, 1525 (19~7), S Katsura, Phys Stares Sohdl ( b J 9 7 , 6 6 3 (1980), M Ausloos & 4. Pe,kalskk S o h d State C o m m u n 2 6 , 9 7 7 (1978) J S Smart, E f f e c m ' e Field Theones o] Magnensm Saunder~, Ph.tladelphaa (1o6o) A P~kalskl 4 c t a P h y s Polon 4.54, 23 ( 1 9 " 8 ) T B u r k h a r d t & R H Swendsen, P h t s Rev BI3, 30"1 119"o) k J DeJongJa& 4.R t~hedema, q&, P h j s 23,1 I 1974;