MFA is based on a few assuptions, some of them we have already discussed.

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1 Mean Feld Approxmaton (MFA) n the theory of magnetsm. MFA was constructed by P. Wess and P. Cure for explanng the behavor of ferromagnetc materals. Such systems behave le paramagnets at hgh temperatures, but below certan crtcal temperature T c they exhbt spontaneous magnetzaton.e., a szable macroscopc magnetc moment even n the absence of an external magnetc feld. In ts orgnal verson MFA was a phenomenologcal theory. Mcroscopc justfcaton for the model postulates was added later, when the nature of nteractons between ndvdual magnetc atoms (Hessenberg exchange) became understood on the grounds of quantum mechancs. MFA s often regarded as a qute prmtve model. In fact, some of ts predctons especally, concernng the system behavor at T 0 and n the regon close to T c (the so-called crtcal regon ) appear to be n less than perfect agreement wth the experment. It would be too much to say that the MFA results are completely wrong n these two regons; however, they defntely cannot be used for fne-tuned quanttatve data nterpretaton. The man reason for the poor model performance for T 0 s that smple averagng s not a good tool for descrbng propertes assocated wth propagatng exctatons, especally n the case of long wavelengths and t s the long-λ propagatng spn wave modes that play the prncpal role n the ferromagnet behavor at the lowest temperature regon 1. The same apples to fluctuatons: averagng, for obvous reasons, smoothens out any fluctuatons. Consequently, a method based on averagng cannot descrbe absollutely correctly any phase transston phenomena because, as we now, fluctuatons play a crucal role n them. Models that produce a much better agreement wth expermental observatons n those two regons have been developed. However, they were never able to fully remove the MFA from the stage because they all lac comprehensveness none of them wors n BOTH regons. MFA, n contrast, offers solutons for the entre regon between T 0andT. So, f used wth cauton, ths sple model may stll offer much help n understandng the propertes of magnetc materals. From the general vepont of statstcal physcs MFA represents a smple approach to systems consstng of strongly nteractng partcles. Hstorcally, t has been develpoped for studyng magnets however, the same approach can be used for modellng a number of other physcal effects not nvolvng magnetsm. MFA s based on a few assuptons, some of them we have already dscussed. 1. Ideal paramagnet. It conssts of non-nteractng magnetc atoms, each of whch has a magnetc moment: µ = gµ B J (1) where J s the total angular momentum, µ B s the Bohr magneton, and g, the Lande factor, s gven by: J(J +1)+S(S +1) L(L 1) g =1+ (2) 2J(J +1) where S, L and J are the spn, orbtal momentum, and total angular momentum numbers, respectvely. The magnetzaton M (.e., the magnetc moment per unt volume) of an dal paramagnet, 1 Note that the Ensten s model of sold s nether very successful for T 0 t s for analogous reasons, because the model gnores the propagatng nature of exctatons n crystals. 1

2 nduced by an external magnetc feld H s (see Notes, page 22 ): M = NgJµ B B J (x), wth x gjµ Bµ 0 H T where N s the number of atoms per unt volume, and B J (x) s the Brlloun functon: B J (x) = 2J +1 2J coth [ ] (2J +1)x 2J 1 ( ) 1 2J coth 2J (3) (4) The fact that B s the conventonally used symbol for the Brlloun functon forces us to use H, notb, for the magnetc feld. Also, t s not very convenent that J appears n the Brlloun functon because J s also the conventonally used symbol for the exchange parameter. We wll start usng the exchange parameter n our math pretty soon, and the exchange parameter and the total angular momentum number wll both appear n the same equatons. In order to avod confuson, one J has to be renamed. The smplest way seems to be startng usng S n the Brlloun functon nstead of J: B S (x) = 2S +1 2S coth [ ] (2S +1)x 2S 1 ( ) 1 2S coth 2S It maes much sense, because n many magnetc atoms the orbtal momentum s zero, so J = S. Hence, one can safely use the S symbol, only eepng n mnd that for certan atoms J S and n such cases S n the fnal formulae should be replaced by the total angular momentum number encoded by a symbol clearly dstngushable from the exchange parameter J One of the prncpal objects of nterest n the MFA modellng of magnetcs s the magnetc susceblty χ = dm/dh. Snce M(H), n general, s not a lnear functon, the value of χ s H- dependent. But f we do not specfy H, t means that we are talng about χ n the regon where M(H) s approxmately lnear.e., n the low feld regon. Then, ndeed, one can assume that χ χ(h). Tang the seres expanson of the coth x functon: coth x = 1 x + x 3 x (6) 45 and applyng t to Eq. (5), we obtan, after droppng x 3 /45 and hgher-order terms and some uncomplcated algebra: B S (x) = S +1 3S x (7) Combnng ths wth the x defnton n Eq. (3), we get: (5) B S (x) = NS(S +1)g2 µ 2 B µ 0 H (8) 3T whch s vald for x 1, or µ 0 H T/gµ B S (one can also wrte µ 0 H T/µ B, because the value of gs s always 1, but t s rarely larger than 10 rarely, as t happens only for some atoms belongng to the Rare Earth Metals famly). In ths regon χ s feld-ndependent: χ = C T where C = NS(S +1)g2 µ 2 Bµ (9)

3 Ths relaton expresses the Cure Law for paramagnets, and C s the Cure-Wess constant. 2. The Cure-Wess Law. Wess based hs model on the emprcal observatons that n the regon above the crtcal temperature T c a ferromagnet behaves very much as a paramagnet, except that ts magnetc susceptblty does not obey the χ 1/T law, but rather: χ 1 T + T c. (10) In order to explan ths behavor, Wess assumed that the feld seen by an average atom n a ferromagnet s the sum of the external feld H and some sort of nternal or effectve feld H E that he also called the molecular feld. One crucal pont n ths assumpton was that the effectve feld s proportonal to the magnetzaton: H E = λ M (11) where λ s the effectve feld constant. Next, he reasoned that f the magnetzaton of a paramagnet depends on H as: M = χ H = C T H, (12) then, by replacng the external feld by a sum of the external feld and the effectve feld, one should obtan the magnetzaton law for a ferromagnet: M = C T ( H + H E )= C T ( H + λ M). (13) By solvng that for M, Wess obtaned the magnetzaton formula: M = CH T Cλ (14) (snce M and H are parallel vectors, we dropped the vector symbols), and the suceptblty law (commonly, referred to as the Cure-Wess Law): χ = C T Cλ. (15) By comparng that wth the emprcal formula gven n Eq. (10), one can wrte: χ = C T T c wth T c = Cλ. (16) Usng Eq.(9), we get the followng formula for the effectve feld constant λ: λ = T c C = 3T c Ng 2 µ 2 Bµ 0 S(S +1). (17) 3

4 3. Mcroscopc justfcaton for the Wess effectve feld. Of course, n hs consderatons Wess could get only as far as to Eqs. (15) and (16), but not to Eq. (17). In 1907, he could not now anytng yet about the Brlloun functon. Hs theory was phenomenologcal because he dd not have enough nformaton for explanng the mcroscopc nature of the effectve feld. Ths could be done only years later, after Hessenberg formulated the famous law of nteracton between two magnetc atoms wth spns S 1, S 2 : H = 2J S 1 S 2 (18) where J s the exchange constant. Let s consder an ndvdual atom wth magnetc moment µ n a ferromagnetc crystal. The moment nteracts wth the external magnetc feld and wth the Wess effectve feld, so that the total nteracton energy s: E = µ (µ ) 0H + µ0heff, (19) From the vepont of the Hessenberg model, however, the energy E s the sum of nteracton energy wth the external feld, and of the exchange nteractons wth all relevant neghbors to atom. Let s assume that each atom has z such relevant neghbors. Also, for smplcty, let s assume that all those z neghbors nteract wth spn wth the same force,.e., that exchange parameter has the same value for all z neghbors. Then, the energy E can be wrtten as: E = µ (µ 0H) 2JS S, (20) where the ndex n the sum runs over all z neghbors of the atom. It wll be convenent to express the Hesenberg energy n the above equaton n terms of the atomc moments µ rather than n terms of the spns S. It can be readly done, consderng that the relaton between the spn and the atomc magnetc moment s µ = gµ BS. So, Eq. (20) can be rewrtten as: ( ) E = µ µ 0H 2J + µ (gµ B ) 2, (21) Note that the two expresson for the energy E, Eq. (19) and Eq. (21), are pretty smlar. However, the sum term n Eq. (21) s not the same as µ 0 H eff, because the former s the nteracton experenced by the ndvdual spn, whereash eff n the sprt of the Wess model s the tme average of all such nteracton terms n the entre system. So, n order to obtan the proper H eff, we should perform such averagng over all N atoms: µ 0Heff 2J = µ (gµ B ) 2 = 2J (gµ B ) 2 1 N N µ The queston of the ndces n the above expresson s lttle trcy : at the frst glance, t may loo formally ncorrect because the moments that are summed are not labelled wth the ndex. Actually, they are but n a hdden way. To explan more clearly what s gong on, let s rewrte the whole thng n a more elaborate fashon: µ 0Heff 2J = µ (gµ B ) 2 = 2J (gµ B ) 1 N µ z 2 N (23) 4 wth labelng all the z neghbors of the atom (22)

5 Calculatng the rght sde double-sum average s a very smple tas f we ntroduce z, whch denotes the number of neghbors wth whch each ndvdual atom nteracts. Conversly, we can say that each atom s a neghbor to z other atoms. Hence, durng the summaton process, the magnetc moment of each atom wll be taen z tmes; n other words, the value of the double sum s smply z tmes the vector sum of all ndvdual atomc magnetc moments n the system! And the tme average of ths sum, of course, s nothng else than the average magnetzaton of the system. The above conclusons can be wrtten down n the equaton language: N N = z µ. and µ N z µ = zm. Combnng the above results wth Eq. (21) leads to a new equaton for the magnetc energy of an ndvdual atomc moment: ( E = µ H ext + 2Jz ) M (gµ B ) 2 N. (24) Note that ths s exactly what we wanted the equaton contans a term proportonal to the magnetzaton, n perfect agreement wth the Wess postulate. We can wrte now: µ 0 Heff = 2Jz (gµ B ) 2 N M, (25) from whch we mmedatly obtan the formula for the Wess effectve feld constant : λ = 2Jz (gµ B ) 2 µ 0 N. (26) Comparng ths equaton wth Eq. (17),.e., the phenomenologcal expresson for λ, we obtan the soluton for the crtcal temperature: 2zJS(S +1) T c = (27) 3 Ths soluton, note, does not contan any phenomenologcal parameters. It s often used for gettng an estmate of J n a gven system. In order to mae the formula more realstc, one has to analyze the nteractons wth neghbors n greater detal because even n the case of the smplest magnetc materals t s not possble to descrbe these nteractons n terms of a sngle echange parameter. But t s rather a straghtforward tas to construct an mproved verson of the model that properly taes nto account all relevant neghbors wth dfferent Js. In many cases very accurate values of exchange constants for dfferent neghbors can be obtaned from dsperson relatons of spn wave exctatons measured by nelastc neutron scatterng. It offers a chance to test the MFA results; such test show that for many systems the T c values gven by MFA may be accurate wthn a percent or two, even though the model s so smple. 4. The regon below T c. Another advantage of the Wess theory s that t offers a soluton for M the regon below the transton temperature as well. But here M s no longer small! 5

6 We cannot use the truncated power seres for the Brlloun functon any more. Instead, we have to use the equaton for magnetzaton n ts full full form, as gven by Eq. (3): M = NgSµ B B S (x). (28) (agan, we have changed J to S). In the case of a paramagnet, the argument x was: x = gsµ Bµ 0 H. T To descrbe the stuaton n a ferromagnet n the sprt of the Wess theory, all we have to do s to replace the external feld by a sum of the external feld and the effectve feld. So, now the argument s: x = gsµ Bµ 0 (H + H eff ) = gsµ Bµ 0 (H + λm) (29) T T Eq. (28) wth the argument n the form of Eq. (29) descrbes the ferromagnet magnetzaton n all temperature regons for any external feld value. However, the most nterestng subject for us s the spontaneous magnetzaton, the one that occurs wthout any ad from the external feld. To obtan the spontaneous M, we smply put H = 0 n the equaton and obtan: ( ) gsµb µ 0 λm M = NgSµ B B S (30) T By puttng n λ gven by Eq. (17) we obtan the fnal equaton form: ( 3M M = NgSµ B B S T ) c. (31) (S +1)Ngµ B T We can now ntroduce the reduced temperature τ: τ T T c (32) and m M Ngµ B S. (33) Note that the expresson n the denomnator, Ngµ B S,sthemaxmum value the magnetzaton can attan, when all magnetc moments are completely parallel. So, m s the reduced magnetzaton whch can tae values between 0 and 1. Wth the new symbols, Eq. (31) taes the form: ( 3S m = B S S +1 m ) (34) τ The only thng that has to be done now s to solve ths equaton for M to obtan the M(T ) dependence! Well, but t s easy to say that, and more dffcult to do that because ths equaton s a nontrval transcendental one, whose solutons cannot be expressed n terms of elementary functons. The only way s to solve t numercally. Wth a computer t s not a very dffcult tas. It s nstructve to fnd the soluton for the smplest possble case of S = 1. Then the whole 2 equaton reduces to: ( ) m m =tanh (35) τ If one doesn t need a very hgh accuracy, ths equaton can be solved even wthout a dgtal computer, by a smple constructon on a pece of graph paper. 6