INFLUENCE OF FOREIGN IONS ON THE CRITICAL FIELD STRENGTH OF AN ANTIFERROMAGNETIC

Transcription

1 R 326. Philips Res. Rep. ~2, ,1951' NFLUENCE OF FOREGN ONS ON THE CRTCAL FELD STRENGTH OF AN ANTFERROMAGNETC hy K. F. NESSEN Summary The critical field strength is calculated for an anti-ferromagnetic in each of whose suhlattices a number of original magnetic ions has been replaced by foreign magnetic ions with different magnetic moments or simply by non-magnetic ions. Especially in the latter case the change in the critical field strength can be given by a simple formula (formula (14». \ Résumé L'intensité critique du champ magnëtique est calculëe pour un antiferromagnétique dans les deux sous-rëseuux duquel un nombre d'ions magnétiques a été remplacé par des ions magnétiques ayant un moment magnétique différent, ou simplement par des ions non-magnétiques. Dans e.dernier cas, 'intensitë-crltique du champ peut être donnée par une formule simple (formule (14». Zusammenfassung Es wird die kritische magnetische Feldstärke abgeleitet für einen antiferromagnetischen Werkstoff, in dessen beiden Teilgittem eine Anzahl magnetischer onen durch fremde magnetische onen oder einfaeh durch nicht-magnetische onen ersetst wurden. Für den letzten Fall kaun die kritische Feldstärke durch eine einfache Formol gegeben werden (Forme (14». 1. ntroduction n an antiferromagnetic inwhich a number of original ions with magnetic moment m are assumed to he replaced hy ions with moment m' =Fm the magnetic moment will devi~te from zero if the numher of replacing ions is different for the two suhlattices. ''his deviation from zero as a function of the temperature has heen investigated in a previous paper, to he referred to as 1). The magnetic energy in the presence of an external magnetic field will also he changed hy the ahove replacement m -7 m'. f the external field (which is always small with respect to the exchange field) is directed along the preferential axis, the moment of one suhlattice can he parallel and that of the other antiparallel to the preferential axis (the parallel case), hut hoth moments can also he directed nearly perpendicular to this axis and at the same timenearly opposite to one another (the perpendicular case). n these two cases the change in magnetic energy caused hy the replacement m --+ m' 'will he different and consequently the critical field strength (at which both cases have equal total energy) will he influenced hy the ahove replacement of ions. This influence will he studied in the present paper.

2 356, K. F. NESSEN 2. Magnetic energyin the parallel and in. the perpendicular case We consider an antiferromagnetic.with in its -j-attice and in its -lattice H ~ e+)n ions with moment m, te+ N ions with moment m' =1= m, H - e_)n ions with moment m, te_ N ions with moment m' =1= m, -M' + A'M~ /! 1Hz " // '1 1/ 1-,~:.~"-",-"-"""""",,,,,..._... :_,},_.._..._.''';y,j. -z Fig. 1. Parallel case. Construction of the effective fields H.+, Ho' +, H,,_, He'_ from, the antiferromagnetic field (with coefficientsa,a',a"), the external field H and the anisotropy fieldsha. The partial moments M+, M' +, M_, M'_ have the direction ofthe corresponding effective fields.

3 NFLUENCE OF FOREGN ONS ON THE CRTCAL FELD STRENGTH, etc N being Avogrado's number and 8+ ~ 1, 8_~1. For the calculation of the critical field strength we need the total energy in the // case and in the 1 case, both in the presence of the same external field Hz along the preferential axis (z-axis). Nevertheless in figs 1 and 2 the two cases are illustrated in the presence of a field (Hz, Hy) with a second component, Hy, perpendicular to the preferential axis. The effective fields He+, ~H~+,He- and H~_ acting on an m-ion and on an m'-ion in the +lattice and in the -lattice, respectively, are constructed by adding three fields: an "antiferromagnetic" field, the external field (Hz, Hy) and an anisotropic field directed perpendicular to the corresponding moment and called Ha+, H~+, Ha- and H~_ respectively. The antiferromagnctic field is due to the action of all ions of that sublattice to which the ion does not belong. t is composed of two components, there being two different kinds of magnetic ions in that sublattice. z Fig. 2. Perpendicular case. Construction of the effective fields Hc+,Ho' +.Hc-.Ho'_ from the antiferromagnetic field (with coefficients A. A'. A"). the external field H and the anisotropy fields Ha. The partial moments M+. M' +. M... M' _ have the direction ofthe corresponding effective fields. n the // case the magnetic energy will be and- in the 1 case. Er; = - J MZ/l dhz - J M Y dhy (1) where M z and My are- the two components of the moment of the whole crystal. f fig. 2-is turned through 90 so that the y-axis becomes vertical and the +z-axis horizontal (pointing to the left) and if moreover the new figure is (2)

4 358 K. F. NESSEN reflected in the vertical axis we obtain fig. 3, which shows a large resemblance with fig. 1 belonging to the parallel case. Only the fieldsperpendicular to the moments in fig. 3 cannot be interpreted as anisotropy fields, their directions being wrong. n all other respects fig. 3 can serve to illustrate the parallel case. ' y Fig. 3. This figure appears when fig: 2 is turn~d through 90 and then reflected in the vertical axis... The parallel case in an external field (Hz, Hy) has been investigated in in which the components MZ/l and M Y were found as u.n:ctionsof the temperature T and of Hz, Hy, 8+, 8_ and the anisotropy coefficients c+, e; for the two sublattices of the original uniaxial crystal,

5 NFLUENCE OF FOREGN ONS ON THE CRTCAL FELD STRENGTH, etc. 359 Comparison of fig. 3 with fig. 1 èads to the relations '...,Mzl = My" (with Hy -+ Hz, Hz -+ Hy, c -+,-c), } M n = Mzil (with Hy -+ Hz, H:s-+ (3) Hy~'c-+ -c), the above replacements to be executed in the formulae for My" and M zw These relations can also be found analytically, treating the 1case in a manner analogous to that in. On account of (1), (2) and (3) we also have ET = En (with Hy -+ Hz, u,-+ Hy, c± -+ -c:!:), (4) the latter replacements to be executed in the formula for En' The replacement c -+ -c has no effect sinée the anisotropy coefficients c+ and c_ appear only in My" and there only in the ratio v 2 = c+/c_. From (1) and the expressions 1(28),1(35) for M Z and My" we derive E,/!= -tn(e_-e+)mlrhz- (Hi/A)fL'(l+ fl')-l ~1+He+ + e_)s~- - (Hy2/2A)~1+ (e++ s.) W1+ (e+- 8_) W2~; (5) R, S, W 1 and W2are given in and are independent of Hz, Hy. Applying (4) we have: Ei = -tn(e_-e+)mlrhy-(hl/a)fl' (l+fl')-l ~l+t(e++ 8_)S~- -(H:N2A) ~1+ (e++ 8_)W 1 + (e+-8_)w2~'. (6) The quantity A is the antiferromagnetic coefficient occurring in the action of all m-ions of one suhlattice on each m-ion of the other suhlattice. n figs 1 and 2, A" appears also as the analogous coefficient for the action of all m'-. ions in one suhlattice on eachm' -ion in the other, and further A' in the action of all m(m')-ions in onesuhlattice on eachm'(m)-ion in the other suhlattice. A, A' and A" occur also in the definitions Nmm'A' f' = 2kT ' Nm'2A" f" = 2kT (7) n (5) and (6) L'and L' represent the Langevin function and.its derivative for a certain argument 1].: ' L.. L(1].), L'.= L'(1].)' Here 1]. is determined as the root of the equation jl(1].) =' 1]. '." (8)

6 360 K. F. NESSEN " 3. Anisotropy energy in the // and 1 eases The equation for the critical field strength is not based on equal magnetic energy, however, but on equal total energy in the two cases. Therefore we also need the values Er,n and E1 n of the anisotropy energies in the uniaxial crystal with different anisotropy coefficients for the two sublattices. n the parallel case the external field (Hz, Hy) causes the total moment M+ of all m-ions in the -l-latticc to deviate from the -l-z-axis over an angle fj+ and that of all m'-ions (i.e. M~) over fj~. The analogous deviationsfrom the -z-axis in the -lattice are called fj_ and fj'_for M_ and M'_, respectively. Rigorously we would have:.~ N N Ell = (1- s+) "2 ml(g)c+ fj+2/2 + (1- s.) "2 ml( 1])c_fJ_2/2 4- with g = (m/kt)he+, 1]= (m/kt)he-, N + s+"2m'l(g')c~ fj~2/2 + s_"2m'l(1]')c'_fj'.2j'j, g' = (m'/kt)h~+, 1'/ = (m'/kt)h~. For the sake of simplicity we calculate the anisotropy energies for T = 0, i.e. we take L = 1. Moreover, contributions containing sfj2 will he omitted. Thus we write N The first term of the approximation for fj+, found in, and that for fj_ are Hence: fj+ = (m/kt)hy/(v 2 + 1)1]*, fj_ = (m/kt)hyv 2 /(v 2 + 1)1]*. (9) (10) n the perpendicular case we are concerned with deviations l' +, 1'~ from the +y-axis for M + and M~, and l'_, 1''_ from the -y-axis for M_ and M'_. Again taking T = 0 we have ~ N' N El = (1- s+)"2 mc+ (1-1'+2)/2 + (-sj"2 mc_ (1-1'_2)/2 + N + s+"2m'(1-1'~2)/2 + s_"2m'(1-1''_2)/2. Contributions containing s1'2 will be omitted. Comparison of fig. 3 with fig. 1 shows that the y's can he ~educed from the fj's by interchanging Hy and Hz N

7 NFLUENCE OF FOREGN ONS ON THE CRTCAL FELD STRENGTH, etc. 361 in (9). The same result follows from an analytical treatment. at Thus we arrive Er =i Nm(c+ + c_)-l Ne ; (mc+-m' c~)-inê_(mc_-m' c~)- -1 Nm (m 2 /k 2 T2) Hz 2 c+/(v 2 + 1)1]2*. (11) 4. On the critical field strength For the calculation of the critical field strength Hz = He we have to take Hy = 0 in (5), (6), (10), and (ll). The introduetion of Hy in the parallel case was merely to facilitate the calculation of El for an external field Hz from Er; for an external Hy. Further on we use only an external field Hz and its critical value He. Having put Hy = 0 we equate the sum of (5) and (6) to the sum of (10) and (ll) in order to find an equation for the critical field strength He: The case ê+ = e: = 0 For ê+ = e: = 0 this equation becomes -NmL'He 2!Nmc+ m 2 2 He 2 (1 +fl')kt = inm(c+ +c_)- (v2 + 1)1].2 k2t2 He -t A' n the left-hand side we substitute i/a for Nm 2 J2kT and in the right-hand side we transform: 1]* 2 by using (8) and the approximation (mjkt)he+ for 1]*:. 1]* 2 = 1]. fl = 1]. LNm 2 Af2kT = (mjkt)hc+ LNm 2 AJ2kT. The equation for He now takes the form Since the anisotropy forces are small with respect to the effective fields, for L = 1 the second term in [] can be dropped. The ordinary parallel and perpendicular suscep'tihilities (both defined by means of the parallel case), were discussed in 1. Using their values we finally find for the critical field strength Hz = (He)ooin the original antiferromagnetic (i.e. for s., = e: = 0) the well-known value (12)

8 362 K. F. NESSEN The case 8+ > 0,' 8_ > Our purpose, however, is to investigate the influence of 8+ and 8_ on the critical field strength. Thus we now keep the terms with 8+ and 8_ in (5), (6), (10), and (11), in which of course Hy =,0, and we neglect once more the term with c+he+ just as before. The condition for equal total energies in the and 1cases leads then to the following equation for Hc 2 : Hc2 fl'".' A.'~!- + fl'~ = l~m(c++c_) ~ (8++ 8_)Ql + (8+-8-,)Q2' () with the abbreviations Ql and Q2 containing Hc2:... ~ _ Hc 2,Sl- f'2l'l'o m' (2 f'l' 0 L' 0), Ql - - AfL ( (1 + fl')2 + m 1 + fl' - 2 f LL' + m' 2 L O'.L' 0 ' + ( m) (2LL'f''_:V )~-:-in (mc+-m'c~ +mc_-m'c~), (a) _ Hc 2 v 2-1 Sf'L'LO + L m' LOl Q2--tAv2+«(jL'+)L --;;L~-.' L+f'L'LO -in(mc+-m'c~-:-mc_+m'c~) -tnhcm ~ 1 +fl',. : LO~. (Th) Here LO and L' 0 represent the Langevin function and its derivative both with (j'!)'yj* as argument: ' From (7) we havef'lf= m'a'ma whereas A'A can he determined in the manner discussed in 1. Thus Ql and Q2 can be evaluated and with them the influence of 8+ and 8_ on the critical field strength. 5. Case of non-magnette foreign ions f in. the original antiferromagnetic t8+n and t8_n original ions have been replaced by non-magnetic ions, i.e. if we 'consider the case m' = 0, the formulae' (), (a), and (b) become much simpler. The value' m' = 0 implies also f' = and therefore LO = O. The condition for equal energy in the and 1 cases now i~ Hc2 1- fl' ', ft'. t. ' v2-1 1 A[2(1 + fl') + (8+ + 8_) (1 + fl')2:+ (8+-8_) 2(v2+) fl' + 1] =.' ':' L = -lnm (c., + c_) - (8+ + 8_) inm (c++ c_)- (8+-8_)!NHcm fl'+', (13)

9 , j. V. PHJlPS' GLOELAMPENFABREKEN NFLUENCE OF FORECN ONS ON THE CRTCAL FELD STRENCTH, etc.'. 363 where in the right-hand member the term - (8+ - 8_) tnm(c+ - c_), which is small with respect to the last term in (13) because Hc~ c+ and ~ c_, has been omitted. When Hc 2 is multiplied by 8+ or 8_ we may replace Hc 2 by (12). Furthermore fl' can he expressed in X and Xl of the original (unperturbed) crystal since according to Nagamiya 2) and Van Vleck 3)' we have [see (33) and (39)]: X = (1/A)2fL'/(1 +fl / ), Xl = /A. n the case m' = 0 we thus find the followingratio between (Hc2)s+,8_ for the perturbed crystal and (Hc2)O.O (given by (12)) for the original one: with where X and Xl belong' to the unperturbed crystal. f therefore (Hc2)o.O for the original crystal has been determined or calculated by means of (12), from the known values Xl and X the value (Hc2)s+,e_ for the crystal with non-magnetic foreign ions can be predicted if the ratio v 2 = c+/c_ between the anisotropy coefficients of the + and - suhlattices of the original crystal is known. Eindhoven, October 1956 REFERENCES 1) K. F. Niessen, Philips Res. Rep. 12, 69-81, 1957., 2) T. N agamiya, Progr. theor. Phys., Osaka 6, , ) J. H. van Vleck, J. chem. Phys. 9, 85-90, 1941.