T i N H TOAN NANG L U d N G Ttf DO CUA H E SPIN TRONG MANG MONG Stf DtJNG P H l / d N G P H A P
TICH PHAN PHIEM HAM
PHAM HUONG THAO Trudn.g Dai hoc Su pham - Dai hoc Hue Tdm t a t : Mot bigu dign tich phan philm hara va md binh Heisenberg cho he spin dinh xii da dirdc ap dung di tmh toan nang ludng tir do cua he spin trong mang mdng. Cac tinh toan giai tich da dSn den viec bieu dien nang lirdng ti^ do ciia h$ nbir mot tich phan phiem hara. Sii dung bilu thiic nang lUdng tiJ do nay dg tim sir phu thuoc nhigt dp ciia do tir hda cua hg khi co tradng ngoai, kit qua tim ditdc khd phii hdp vdi ket qua ciia hf 2 chigu tira dttdi bang phildng phap ham Green.
I GlCl THIEU
Linh viic tir hpc da nhan diidc mot su thiic day to ldn do su xuat hien cac v§.t lieu mdi vli cac cong cu vdi nhflng thao tac tinh \ i Mo hinh Heisenberg, m6 t a mot tap hop cac momen tii dinh xii dUdc ghep cap b6i tiidng tac trao doi, la mot trong nhiing md hinh thich dang nhat trong hoan canh nay.
Tich phan phi#m ham Ian dau tien dUdc ap dung trong cd hpc lUdng tii bdi R.
Feynman va bay gi5 la mot trong nhiing phudng phap to^n hoc hiiu hi?u nhat trong vat ly lildng til dUdng thdi. Pham vi iing dung rgng rai cua cac tich phan philm ham [1 da khuyen khich su phat tnen cua chung. Cac phudng phap tich phan phiim ham dUdc su: dung rong rai trong vat ly ly thuylt hi?n dai [6]-[8]. Cu the cac phudng phap na\' dUdc su dung d l dat dUdc cac tien trinh quan trong vg cac hien tudng tdi han bdi phudng phap nhdm tai chuSn hoa [7]. Phudng phap nay ddn gian hdn so vdi phUdng phap toan tii. Trong cac vin de cua ly thuygt tdng quat ciia sU chuyin pha, ling dyng phUdng phap tich phan phi^m ham giiip x§,y dUng b^c tranh luong t ^ ciia cac hien tUdng va phat triln cac phudng phap tinh toan gan dung, trong mpt vai van de. no cho phep chiing ta chiing minh cac kit qua nhan dudc bdi cac phUdng phap khac, lam sang to cac kha nang iing dung cua chiiag.
T;ip chi Khoa hoc va Giao dye, Trudng D^i hoc Su ph9,m Hue ISSX 1859-1612. So 01{17)/2011: tr. 36-42
TLXH TOAX N A X G L C O X G T U DQ CUA HE SPIN TRONG MA.\G MOXG... 37 Trong bai bao nay, tac gia da bieu dign naag ludng t u do cua he spin trong mang mong n h u mot tich phan philm ham vh sau dd tinh toan nang lUpng tU do cua he trong phep g ^ dung Gaussian.
2 M O HiXH TIXH TOAX \"A K E T QUA
Xet mOt mang mong gom n ldp spin va gia sii r i n g co A" spin tren mdi ldp. Vi tri cua mdi spin trong mang dupc xac dinh bdi cac chi sQ v,j, vdi v = 1, ...,n la chi s6 ldp. Gpi Rj, mpt vectd hai chilu, la vecto bieu thi vi tri ciia spin thii j trong ldp u.
Md hinh Heisenberg cho hg spin ciia mang mong cd dang:
H = if„ 4- //,„. = -tiJ2 ''Si, - ^ E •'-•'(^ - 'iy)s:,si,,,, (1)
vdi HQ la Hamiltonian ciia he spin khong tUdng tac trong mot tii trudng diu dudc dinh hudng doc theo true z vdi cudng dd h. H^nt la Hamiltonian tUdng tac trao ddi Heisenberg, J^,,'{Rj — Hj') latudng tac trao ddi gifla spin S°_, \'aspin 3",^; ck = x,y,z va. (1 \a momen tu cua mpt nut mang. Sii dung phep biln doi Fourier cho cac toan tii spin, t a dupc:
«=-A" E ''^^i - ^ E •'«"'(*)S:JS2_E , (2)
Toan tii th6ng kg ciia he trong bilu dign tftdng tac:
(4)
(5) exp{-3H) = exp{^.SH„)tn^expi f i J ] J„^(fc)S°f(r)S; _ J ( T ) *
Sir dung phep biln d5i tich phan [4]
bien d6i bilu thftc dftdi dau T tich trong phUdng trinh (4), ta nhan dftdc exp(-0H) = exp{-/SHo) / (dip) exp I - - J2 P°(9)t='°(-?) J x
fexpi Y, ^Jt!MSiig)] , (6)
vdi (dip) la phep do ciia tich phan philm ham dirpc dinh nghia bSi [4], va
f=(*.-).E- = E E - -
Sii sung bilu thflc (6), chiing ta bilu dign nang lupng tu do cua he nhu mpt ti'ch phau philm ham:
F = -l}-Hn(Spe-^«) = F„~ i / n | (d.p) eip | - 5 E <(9)^°(-9) " ^...MJ ,
vdi (7)
/3 ^ ' 0 sh{0^h/2) '•^>
la nang luong tu do cua he khong tuong tac va /S"' = ksT. Va
-F.„,IP] = - E fi'"'M = (m!)-' Z • E iTp'i...i^-=,)S^'i...^',l , (9)
" 2 ' ii.ii i«.l„
^* i^Pii-P,^)'o ' i cac gia tri trung binh riit gon. Ta dat
^ M = 2 E 'Pii'l)'Pi(~g) + F,„,[,p] = Foiv) + AFlp]. (10)
- 2'' E ^--'iy)Kv)J..Ak)<p:{g)^;(-g) ill)
AfM = ^ f N [ ^ , ,
(12)
TiXH TOAN X.4NG L C Q N G T U DO CUA HE SPIN TRONG MANG MONG. - 39
vdi K^^iy) = - - ] ^ ; y == I3fih; b{y) va t/(y) la ham BriUouin va dao ham bac 1 ciia no [8 Xang lupng tU do trong (7) co the dupc bilu diln theo cac thang giang Gaussian:
F = FO + FG^AF, (13)
vdi gan diing bac khong Fo va na,ng lupng tU do FQ trong phep g i n dung Gaussian FG = -13-Hn j (d^)exp(-FGM) =
^ Y, indet ( / - 0b'(y)Ji,y(k)) -K | ^ Indet ( / - l3K.Uy)b{y)J.yik)) (.14) ky '^ qy
Cac hieu chinh Ai^ co the dupc bilu diln dudi dang cac khai triln theo cac thSng giang Gaussian rut gpn:
AF =Yi{n\)-'{(FTWTo , (15)
n > l
Cl day chi so G d l chi gia tr; trung binh rut gpn theo phan b l Gaussian
{(...))o = I(dp)e-''''''(...)/ j(dv)e-''M (16) Sfl dung bieu thdc nang lupng t u do (13)-(15) d l tfnh toan dd tfl hda cua he spin
trong mang mong 4 ldp, n = 4, d l ddn gian t a xet trong g i n diing thap nh4t A F = 0.
Ta CO, dp tfl hoa cua he dupc tinh theo cong thflc:
M = ~ = Ma - Ml + M2 (17) vdi M„ = iinN(((S+ll2)exp(y(S+ll2))-{-S-ll2)exp(-y(S+ll2)))l(exp(yl2)-
exp(-y/2)) - (exp(y(S + 1 / 2 ) ) - exp(-y(S^l/2)))/{exp(y/2)-exp{-y/2))\l/2 x exp(y/2) + 111 X exp(-yl2)))l(exp(y(S + 1/2)) - exp(~y(S + l/2)))(eip(!//2) -
exp(-yl2));
Ml = -^l3ph'{y)b{y){AJ, + 6J^) = \l3tib\y)b(y)Uie + <i)
= j/3M6'WK.)55(|fyj(4e+l). (18)
d day T^ tuong iing la nhiet dp Curie cua ban d i n khoi, e = ^ la ti so ciia cac tich phan trao ddi gifla cac spin lan can gan nhat trong cting mpt mat phang va trong hai mat phang g ^ nhit;
1 a(det{I-0{y)J(k))) 1
M2 = - ^ X -!^ i- X ^ , (19) 0 dli det(I - l3V(y)J(k))
PHAM HUO.NG THAO
det{I - 3b'(y)J{k)) =
(l3^y\y)J.^ + 1 - ZI3b'(y)J, + •fii'(y)J, + fb%)J.J^ 0'h'\y)J,^)x
{0'b%)J.' - 1 - 0b'(y)J, + 3Pb'(y)J, - fb''(v)J.J^ - il'b'\y)j/), (20) Dl tinh toan s6, ta thilt lap cac thong s6 S = 1/2, ^ = 0,927.10"^'', i-jj = 1,38.10"^, n = 4, -Y = 10^, e = Js/Jp = 0.4. T^ = 1043. Hinh 1 bilu diSn su phu thupc nhiet dQ ciia do tit hoa cua mang mong ttt trong ttf tnidng h, la kit qua cua bai bao. Hinh 2 bieu dign stt phu thuoc vao nhiet dp ciia dp ttt hoa trong he 1 chilu v^ 2 chieu diing phuong phap ham Green [3].
Fig 1 The tentpemure dependence of the mogaeUz&tiati of magoetic tbin film for e=0 4
'K-
So sanh hinh ve 1 .4 2. ta thiy ring, kgt qua cOa tac gia kha phii hop vdi kit qua siS dung phuong phap ham Green. Co mpt s6 sai lech la do trong qua tnnh tinh toan tac gia da su ditng mpt si phep gin dung va tinh toan cho tradng hdp cu thg do la
TINH TOAN NANG LUONG TU DO CUA HE SPIN TRONG MANG MONG.. 41
mang mong tfl co chilu day bkng 4 ldp nguyen tfl. Su phu thupc nhigt dp ciia dp tif hoa dupc chi ra khi co trudng ngoai h = 8T. Thay rang, khi nhiet dp tang thi dp tfl hoa giam tUdng iing.
3 KfiTLUAN
Bilu diin nhan dupc cua nang lupng tU do cho mo hinh Heisenberg lupng tfl nhU mpt tich phan phiem ham (13)-(15) thuan lpi cho viec tinh toan cdc tinh chat vat ly cua hg spin theo m5 hinh Heisenberg nhu la nhi?t dp Curie, momen tfl, do tfl hoa, song spin tren mang mong... Them vao do, bilu diln nay co the dupc ap dung de dua ra ham Green va cue ciia ham Green ngang xac dinh ph6 song spin. Ham Green trong cac gan diing bac cao hdn, xac dinh cac hieu chinh cua pho va su t a t dan cua phd, co t h i dupc tinh toan nhu trong trudng hdp ciia nang lUpng t u do sfl dung (15)-(16). Tac gia se co gang de dua ra cac kit qua ciia cac tinh toan nay trong mpt bai bao khac.
T A I L I E U THAM KHAO
[1] A.D. Egorov, P.I. Sobolevsky and L.A. Yanovich (1993), Functional Integrals: Approx- imate Evaluation and Applications (Kluwer Academic Publishers, Dordrecht).
[2] Ahmed S. Hassan and A. M. El-Badry (2009), Thermodynamic properties of quasi- ecpiilibrium magnons in crystalline bulk vfiatenals and thm films, Turk J Phys 33, 129 - 138.
[3] Ai-Yuan Hu, Yuan Chen, Li-Jun Peng (2007), The anisotropic Heisenberg ferromagnet in a magnetic field, Journal of Magnetism and Magnetic Material 313, 366 - 372.
[4] I. A. Varkarchuk and Yu. K. Rudavskii (1981), Teor. Mat. Fiz. 49 235 (in Russia) [5] M. H. Zaidi (1983), Functional method, Fortschr. Phys. 31, 408 - 411.
[6] N. N Bogolyubov and D. V. Shirkov (1959), Introduction to the Theory of Quantized Fields, Interscience,
[7] S.-K. Ma (1976), Modem Theory of Critical Phenomena, Benjamin.
[8] Yu, A. Izuymov. F. A. Kassanogly, and Yu. N. Skriabin (l974:)Field methods in the theory of Ferromagnets (Mir: Moscow),
PHAM HUONG THAO
Title: METHOD OF FUNCTIONAL INTEGRATION FOR CALCULATING THE FREE ENERGY OF THE SPEV SYSTEM LN THIN FILMS
Abstract: .\ functional integral representation and Heisenberg raodel for a localised spin sj'stem were proposed for the free energy of the spin system in thin films. The result of the calculations was to represent the free energy of the system as a functional integral.
Using this result to find the temperature dependence of the ra^netization of the system in the extemal magnetic field, this result is compared with the result of the two-dimension system using the Green fimction method.
ThS. PHAM HUONG THAO
Khoa \'^t ly. IVirdng Dai hpc Su pham - Dai hoc Hue Email: hthao82@'gmail.com
