Chapter 8: Magnetism

Transcription

1 Chapter 8: Magnetism Holstein & Primakoff March 28, 2017 Contents 1 Introduction The Relevance of Magnetostatics Non-interacting Magnetic Systems Coulombic Correlation Effects Moment Formation Magnetism and Intersite Correlations The Exchange Interaction Between Localized Spins Exchange Interaction for Delocalized Spins Band Model of Ferromagnetism Enhancement of χ Finite T Behavior of a Band Ferromagnet Effect of B Mean-Field Theory of Magnetism Ferromagnetism for localized electrons (MFT) Mean-Field Theory of Antiferromagnets

2 5 Spin Waves Second Quantization of Ferromagnetic Spin Waves Antiferromagnetic Spin Waves Criticality and Exponents 44 2

3 1 Introduction Magnetism is one of the most interesting subjects in condensed matter physics. Magnetic effects are responsible for heavy fermion behavior, ferromagnetism, antiferromagnetism, ferrimagnetism and probably high temperature superconductivity. Unlike our previous studies, most magnetic systems are not well described by simple models which ignore intersite correlations. The reason is simple: magnetism is inherently due to electronic correlations of moments on different sites. As we will J Figure 1: Both moment formulation and the correlation between these moments (J) are due to Coulombic effects see, systems without these inter-spin correlations (or those without well defined moments to be correlated) have uninteresting and unimportant (energetically) magnetic properties. 1.1 The Relevance of Magnetostatics Perhaps the term magnetism is a misnomer, or rather describes only the external probe (B) which we use to study magnetic behavior. Magnetic effects are mainly due to electronic correlations, those mediated or due to Coulombic effects, and not due to magnetic correlations between moments (these are smaller by orders of v/c, ie., they are relativistic corrections). For example, consider the magnetic correlation between 3

4 two moments separated by a couple of Angstroms. U = 1 r 3 [m 1 m 2 3(m 1 n)(m 2 n)] (1) then, If we let, U dipole dipole m 1m 2 r 3 (2) m 1 m 2 gµ B e h m r 2 A then, U (gµ B) 2 r 3 ( e 2 hc ) 2 ( a0 ) 3 e ev (3) r a 0 Or roughly one degree Kelvin! Magnetic correlations due to magnetic interactions would be destroyed by thermal fluctuations at very low temperatures. 1.2 Non-interacting Magnetic Systems We will define non-interacting magnetic systems as those for which the independent moments do not interact with each other, and only interact with the probing field. For the moment let s consider only the magnetic moments due to electrons (as we will see, since they can interact with each other, they are by far the most important moments in the system). They have a moment µ B (L + 2S) (4) The system energy will change by an amount E gµ B B(L + 2S) µ B B, µ B = e h 2m ev T e (5) in an external field. The largest field which can regularly be produced in a lab is 10T e (100T e or more can be produced at LANL by blowing things up), thus E < 10 3 ev or k B (10 o K) (6) 4

5 This is a very small energy. Thus magnetic effects are wiped out by thermal fluctuation for k B T > µ B B (7) at about 10 K! Thus experiments which measure the magnetism of non-interacting systems must be carried out at low temperatures. These experiments typically measure the susceptibility of the system with a Faraday balance or a magnetometer (SQUID). For a collection of isolated moments (spin 1/2), the susceptibility may be calculated from the moment s = 1 2 ( ) 1 e β 1 2 gµ BB e β 1 2 gµ BB m = gµ B ( ) (8) 2 e β 1 2 gµ BB + e β 1 2 gµ BB χ g µ B 2 tanh(βµ µ B B BB) µ B T = m B µ2 B k B T (g 2) (9) Once again, the energy of the moment-field interaction is roughly E µ2 B T B2 ( ev ) T e T ( ) 10 4 ev 2 B 2, (k = 1) (10) K When E T, thermal fluctuations destroy the orientation of the moments with the external field. If B 10T e E 100 K (11) T or E T at 10 K! However, we know that systems such as iron exist for which a small field can induce a relatively large moment at room temperature. This is surprising since for a metal, or a free electron gas, the susceptibility is much smaller than the free electron result, since only the spins near the Fermi surface can participate, χ = µ2 B EF. Note that this is even smaller than the free electron result by a factor of k B T E F 1! 5

6 E kt/e F D(E) Figure 2: In a metal, only the electrons near the Fermi surface, which are not paired into singlets, contribute to the bulk susceptibility χ 2 Coulombic Correlation Effects 2.1 Moment Formation kt E F µ 2 B k B T µ2 B D(E F ) Of course real materials are not composed of free isolated electrons. Nevertheless some insulators act almost as if they are composed of non-interacting atoms (ions) with moments given by Hunds rules: maximum S maximum L which leads to large atomic moments. Hunds rules reflect the atom s attempt to lower its Coulombic energy, see Fig. 3. By maximizing the total spin S, the spin part of the wavefunction Max S ψ(x) Max L x e- e- Figure 3: Hunds rules, Maximize S and L, both result from minimizing the Coulomb energy. becomes symmetric under electron exchange (i.e. for two electrons with s = 1 the 2 maximum value of total spin is S = 1 with a wavefunction + ). Then, since 6

7 the total wavefunction must be antisymmetric under exchange, the spatial part must be antisymmetric requiring it to have a node. The node keeps the electrons apart, minimizing their Coulomb energy. The second Hunds rule is also due to Coulombic interactions. Maximizing L tends to keep the electrons apart, much like a centrifuge. (alternatively, the radial Schroedinger equation obtains an angular momentum barrier L(L + 1)/r 2 ). µ B U ε Figure 4: A simple tight-binding model with a local Coulomb repulsion U. If U = 0, the rate that electrons hop on and off any site may be approximated using Fermi s golden rule π B 2 D(E F ) 1. Then by the uncertainty principal E t h so t each site energy acquires an uncertainty or width E h t π B 2 D(E F ) Γ. The sites will form moments (see Fig. 5) if Γ U, ɛ To illustrate how band formation modifies this scenerio, lets consider a simple tight binding model (See Fig. 4). By Fermi s goldon rule, each level acquires a width (uncertainty in its energy) Γ = πb 2 D(E F ) and each level can be in one of the four states shown in Fig. 5. Clearly, the states and do not have a moment, 2ε+U moment forms µ Γ ε Figure 5: A moment forms on an orbital provided that Γ U, ɛ. U e2 a e a r T F r T F is small for a metal, large for an insulator and the states and do. If these states mix equally a moment will not form. The mixing between the states with moments is only through one of the other two states ( or ) and may be suppressed, as can the occupancy of the 7

8 I L ξ e n m B Figure 6: Here ξ = 1 c φ t, m = µ BL and φ = Bπa 2 moment-less states, by increasing the energy of the states without moments. Ie., a moment will form on each site if ɛ Γ and ɛ + U Γ. In this limit U B, the system will act more like a system of free moments than a free electron gas. Thus, one might expect χ insulator χ metal (12) for noninteracting systems. However, this is not the case. The reason is that I have only told you half of the story. A real atom, or a system composed of such atoms, has a diamagnetic response due to the angular momentum L of the rotating electrons. This effect is due to Lenz Law. So that any introduced magnetic induction will induce an EMF and hence a current that opposes the electron current which reduces the moment. diamagnetism with χ a 2. In the free electron limit (see J.M. Ziman, ) [ χ = µ 2 BD(E F ) 1 1 ( m ) ] 2 3 m (13) E 10 8 ( ev T For insulators, often with m m < 1 the diamagnetism wins; whereas for metals, generally with m m ) 2 a10 1 ev 1 B ev B 2 (T ) (14) 1, the Pauli paramagnetism wins. 8

9 ξ a Figure 7: If the magnetic moments in a small volume are correlated, then the magnetic susceptibility is strongly enhanced. 2.2 Magnetism and Intersite Correlations From both Hund s rules and a simple tight binding picture, we argued that moment formation in solids results from local Coulomb correlations between electrons. We also saw that a collection of such isolated moments is rather boring since all magnetic behavior is washed out by thermal fluctuations at very low temperatures. Consider once again an isolated moment of magnitude mµ B in an external field. χ (mµ B) 2 k B T E (mµ BB) 2 k B T (15) (16) For magnetism to be significant at room temperature (300K) we must increase the energy of our system in a field. This may be accomplished by increasing the effective moment m by correlating adjacent moments. If the range of this correlation is ξ, so that roughly 4πξ3 3a 3 moments are correlated, then let ξ a = 3 so that 102 moments are correlated and m This increases E by about 10 4, so that E k b T at T 10 3 K. The observed (measured) susceptibility also then increases by about 10 4, all by only correlating moments in a range of 3 lattice spacings. Clearly correlations between adjacent spins can make magnetism in materials rele- 9

10 1 2 r e 12 e r 1A r 1B r 2A r 2B e + A R AB e B + Figure 8: Geometry of two electrons, 1 and 2, bound to two ions A and B. vant. Such correlations are due to electronic effects and are hence usually short ranged due to electronic (Thomas-Fermi) screening. If we consider two s = 1 2 spins, 1 2, then the correlation is usually parameterized by the Heisenberg exchange Hamiltonian, or H = 2Jσ 1 σ 2 (17) where J is the exchange splitting between the singlet and triplet energies. + E t (18) The trick then is to calculate J! { } E s (19) E t E s = J (20) The Exchange Interaction Between Localized Spins Imagine that we have two hydrogen atoms A and B which localize two electrons 1 and 2. As these two electrons approach, their spins will become correlated. H = H 1 + H 2 + H 12 (21) H 1 = h2 2m 2 e2 r 1A e2 r 1B (22) 10

11 H 12 = e2 + e2 (23) r 12 R AB As we did in Chap. 1 to describe binding, we will use the atomic wave functions to approximate the molecular wavefunction ψ 12. ψ 12 = (φ A (1) + φ B (1)) (φ A (2) + φ B (2)) spin part = (φ A (1)φ A (2) + φ B (1)φ B (2) + φ A (1)φ B (2) + φ A (2)φ B (1)) spin part (24) If e2 r 12 is strong (it is) then the first two states with both electrons on the same ion are suppressed, especially if the ions are far apart. Thus we neglect them, and make the Heitler-London approximation; for example ψ 12 (φ A (1)φ B (2) + φ B (1)φ A (2)) spin singlet (25) The spatial wave function is symmetric, and thus appropriate for the spin singlet state since the total electronic wave function must be antisymmetric. For the symmetric spin triplet states, the electronic wave function is ψ 12 = (φ A (1)φ B (2) φ B (1)φ A (2)) spin triplet (26) or ψ 12 = φ A (1)φ B (2) ± φ B (1)φ A (2) spin part (27) The energy of these states may then be calculated by evaluating ψ 12 H ψ 12 ψ 12 ψ 12. E = ψ 12 H ψ 12 ψ 12 ψ 12 = 2E I + C ± A, + singlet, triplet (28) 1 ± S where E I = } d 2 r 1 φ A(1) { h2 2m 2 1 e2 φ A (1) < 0 (29) r 1A the Coulomb integral { 1 C = e 2 d 3 r 1 d 3 r } φ A (1) 2 φ B (2) 2 < 0 (30) R AB r 12 r 2A r 1B 11

12 the exchange integral { 1 A = e 2 d 3 r 1 d 3 r } φ R AB r 12 r 2A r A(1)φ A (2)φ B (1)φ B(2) (31) 1B and finally, the overlap integral is S = d 3 r 1 d 3 r 2 φ A(1)φ A (2)φ B (1)φ B(2) (0 < S < 1) (32) All E I, C, A, S R. So J = J = J = E t E s = 2E I + C A 1 S C A 1 S C + A 1 + S > 0 { 2E I + C + A } 1 + S 2 A SC 1 S 2 < 0 (33) where the inequality follows since the last two terms in the {} dominate the integral for A and in the Heitler-London approximation S 1. Or, for the effective Hamiltonian. H = 2J σ 1 σ 2, J < 0 (34) Clearly this favors an antiparallel or antiferromagnetic alignment of the spins (See Fig. 9) since then (classically) σ 1 σ 2 < 0 and E < 0, so minimizing the energy. This type of interaction is clearly appropriate for insulators which may be approximate as a collection of isolated atoms. Indeed antiferromagnets are generally insulators for this and other reasons. H = 2J σ i σ j, J < 0 (35) ij Exchange Interaction for Delocalized Spins Ferromagnetism, where adjacent spins tend to align forming a bulk magnetic moment, is most often seen in conducting metals such as Fe. As we will see in this section, the Pauli principle, the Coulomb interactions, and the itinerancy of free (metallic) electrons favors a ferromagnetic (J > 0) exchange interaction. 12

13 Figure 9: r i e V r j e Figure 10: triplet-symmetric Consider two like-spin (triplet) free electrons in a volume V (See Fig. 10). If we describe the spatial part of their wave function with plane waves, then ψ ij = 1 { e ik i r i e ik j r j e ik i r j e ik j r i } 2V = 1 2V e ik i r i e ik j r j { 1 e i(k i k j ) (r i r j ) } (36) The probability that the electrons are in volumes d 3 r i and d 3 r j is ψ ij 2 d 3 r i d 3 r j = 1 V 2 {1 cos [(k i k j ) (r i r j )]} d 3 r i d 3 r j (37) As required by the Pauli principle, this probability vanishes when r i = r j. This would not be the case for electrons in the singlet spin state (if the coulomb interaction continues to be ignored). Thus there is a hole, called the exchange hole, in the probability density for r i r j for triplet spin electrons, but not singlet spin ones. 13

14 Now consider the effects of the electron-ion and the electron-electron coulomb interactions (See Figure 11). If one electron comes near an ion, it will screen the b e a e Ze + + Figure 11: Electron a screens the potential seen by electron b, raising its energy. Anything which keeps pairs of electrons apart, but costs no energy like the exchange hole for the electronic triplet, will lower the energy of the system. Thus, triplet formation is favored thermodynamically. potential of that ion seen by other electrons; thereby raising their energy. Thus the effect of allowing electrons to approach each other, is to increase the electron-ion coulomb energy, and of course the electron-electron Coulomb energy. Thus, anything which keeps them apart without an energy cost, like the exchange hole for triplet spin electrons, will reduce their energy. As a result, like-spin electrons have lower energy Ze and are thermodynamically favored Ferromagnetism. i.e. r 0 i O e r= r j e Figure 12: Geometry to calculate the exchange interaction. To determine the range of this FM exchange interaction, we must average the effect over the Fermi sea. If one of the electrons is fixed at the origin (See Figure 12), 14

15 then the probability that a second is located a distance r away, in a volume element d 3 r is P (r)d 3 r = n d 3 r (1 cos [(k i k j ) r]) }{{} Fermi sea average n = 1 2 n = 1 # electrons 2 volume In terms of an electronic charge density, this is (38) (39) ρ ex (r) = en 2 (1 cos [(k i k j ) r]) = en { 1 1 kf 2 ( 4 d 3 k i d 3 1 ( k j e ı(k i k j ) r + e )} ı(k i k j ) r 3 k3 F )2 o 2 = en { 1 ( 4 kf kf } 2 3 k3 F ) 2 d 3 k i e ık i r d 3 k j e ık j r 0 0 = en {1 9 (sin k } F r k F r cos k F r) 2 2 (k F r) 6 Note that both of the exponential terms in the second line are the same, since we integrate over all k i k i & k j k j. Since we have only been considering Pauliprinciple effects, the electronic density of spin down electrons remains unchanged. Thus, the total charge density around the up spin electron fixed at the origin is { ρ eff (r) = en 1 9 } (sin k F r k F r cos k F r) 2 (41) 2 (k F r) 6 The size of the exchange hole, and the range of the corresponding ferromagnetic exchange potential, is 1 k F a which is rather short. (40) 3 Band Model of Ferromagnetism Due to the short range of this potential, its Fourier transform is essentially flat in k. This fact may be used to construct a band theory of FM where the mean effect of a spin-up electron is to lower the energy of all other band states of spin up electrons by a small amount, independent of k. E (k) = E(k) IN N ; I < 1eV (42) 15

16 ρ eff en 1 1/2 2 4 k r F Figure 13: Electron density near an electron fixed at the origin. Coulomb effects would reduce the density for small r further, but would not significantly effect the size of the exchange hole or the range of the corresponding potential, both 1/k F. Likewise for spin down E (k) = E(k) IN N. (43) Where I, the stoner parameter, quantifies the exchange hole energy. The relative spin occupation R is related to the bulk moment Then R = (N ( N ) N, M = µ B N V E σ (k) = E(k) I(N +N ) 2N ) R (44) σir, (σ = ±) (45) 2 Ẽ(k) σir 2. (46) If R is finite and real, then we have ferromagnetism. R = 1 1 { } N exp (Ẽ(k) IR/2 E F )/k B T + 1 k 1 { } exp (Ẽ(k) + IR/2 E F )/k B T + 1 (47) For small R, we may expand around Ẽ(k) = E F. f(x a) f(x + a) = 2af 2 3! a3 f (48) 16

17 f f E F E F E E f f E E Figure 14: All derivatives will be evaluated at Ẽ(k) = E F, so f < 0 and f > 0. Thus, R = 2 IR 1 f 2 N Ẽ(k) 2 ( ) 3 IR 1 3 f (49) 6 2 N k E F k Ẽ3 (k) E F This is a quadratic equation in R 1 I f = 1 N E(k) 24 k E I3 R f (50) N E 3 (k) F k E F which has a real solution iff 1 I f > 0 (51) N E(k) k E F Or, the derivative of the Fermi function summed over the BZ must be enough to overcome the -1 and produce a positive result. Clearly this is most likely to happen f at T = 0, where δ(ẽ E F ) EF T = 0, 1 N E(k) k f E k = dẽ V 2N D(Ẽ)δ(Ẽ E F ) = V 2N D(E F ) = D(E F ) (52) 17

18 So, the condition for FM at T = 0 is I D(E F ) > 1. This is known as the Stoner criterion. I is essentially flat as a function of the atomic number, thus materials such I (ev) 1.0 D(E ) (ev-1) F Fe Ni Co 1.0 Li Na Z 50 Z Figure 15: as Fe, Co, & Ni with a large D(E F ) are favored to be FM. 3.1 Enhancement of χ Even those systems without a FM ground state have their susceptibility strongly enhanced by this mechanism. Let us reconsider the effect of an external field (gs = 1) on the band energies. Then R E σ (k) = E(k) In σ N µ BσB (53) = 1 N k f Ẽk (IR + 2µ B B) = D(E F )(IR + 2µ B B) (54) N or as M = µ B R, we get V or M = 2µ 2 N D(E F ) B V 1 I D(E F ) B (55) χ = M B = χ 0 1 I D(E F ) (56) χ 0 = 2µ 2 B N V D(E F ) (57) = µ 2 B D(E F ) (58) 18

19 Thus, when I D(E F ) < 1, the susceptibility can be considerably enhanced over the non-interacting result χ 0. However, this approximation usually overestimates χ since it neglects diamagnetic contributions, and spin fluctuations (at T 0). As we will see, the latter especially are important for estimating T c. ξ a Figure 16: Spin fluctuations can reduce the total moment within the correlated region, and even reduce ξ itself. Both effects lead to a reduction in the bulk susceptibility χ moment Finite T Behavior of a Band Ferromagnet In principle, one could start from an ab-initio calculation of the electronic band structure of E(k) and I, such as Ni, and calculate the temperature dependence of R (and hence the magnetization) using R = 1 f(ẽk IR N 2 µ BB 0 E F ) f(ẽk + IR 2 + µ BB 0 E F ) (59) k with f(x) = 1. However, this would be pointless since all of the approximations e βx +1 made to this point have destroyed the quantitative validity of the calculation. How- 19

20 D(E) s electrons (no moments) not correlated d electrons (with moments) correlated Figure 17: In metallic Ni, the d-orbitals are compact and hybridize weakly due to low overlap with the s-orbitals (due to symmetry) and with each other (due to low overlap). Thus, moments tend to form on the d-orbitals and they contribute narrow features in the electronic density of states. The s-orbitals hybridize strongly and form a broad metallic band. ever, it still retains a qualitative use. For Ni, we can do this by approximating the very narrow d-electron feature in D(E) as a δ function and performing the integral. However, only the d-electrons have a strong exchange splitting I and hence only they will tend to contribute to the magnetization. Thus our D(E) should reflect only the d-electron contribution, we will accommodate this by setting D(E) Cδ(E E F ), (C < 1) (60) C, an unknown constant, will be determined by the T = 0 behavior. Then { R = C f( IR 2 µ BB 0 ) f( IR } 2 + µ BB 0 ) (61) Let R C R and T c = IC 4k B, then if B 0 = 0 R = ( exp 1 2 RT c T ) + 1 ( exp 1 2 RT c T ) + 1 = tanh RT c T (62) n n. For Ni, the measured ground state magnetiza- N If T = 0, then R = 1 = R = 1 C C tion per Ni atom is µeff µ B = 0.54 = n n N For small x, tanh x x 1 3 x3, and for large x. Therefore, C = 0.54 = µ eff µ B. tanh x = sinh x cosh x = ex e x 1 e 2x = e x + e x 1 + e 2x = (1 e 2x )(1 e 2x ) 1 2e 2x (63) 20

21 Thus, R R = 1 2e 2Tc T, for T T c (64) = 3(1 T T C ) 1 2, for small R or T < T c (65) However, neither of the formulas is verified by experiment. The critical exponent Eq (64) R Eq(65) T/T c Figure 18: β = 1 in Eq. 65 is found to be reduced to 1, and Eq. 64 loses its exponential 2 3 form, in real systems. Using more realistic D(E) or values of I will not correct these problems. Clearly something fundamental is missing from this model (spin waves). Elementary Excitations spin flip B 0 Included spin wave Not Included Figure 19: Local spin-flip excitations, left, due to thermal fluctuations are properly treated by mean-field like theories such as the one discussed in Secs. 3 and 4. However, non-local spin fluctuations due to intersite correlations between the spins are neglected in mean-field theories. These low-energy excitations can fundamentally change the nature of the transition. 21

22 3.2.1 Effect of B If there is an external field B 0 0, then R = tanh { RTc + µ B B 0 /2k B T } (66) Or for small R and B 0, (or rather, large T T c.) R = µ B 2kT B 0 + T c T R R = µ B 1 B 0 (67) 2k T T c Thus since M = µ BN R = Cµ BN V V R χ = M B 0 = Cµ2 B 2kV N T T c. This form for χ χ = Const T T c (68) is called the Curie-Weiss form which is qualitatively satisfied for T T c ; however, the values of Const and T c predicted by band structure are inaccurate. Again, this is due to the neglect of low-energy excitations. 4 Mean-Field Theory of Magnetism 4.1 Ferromagnetism for localized electrons (MFT) Some of the rare earth metals or ionic materials with valence d or f electrons are both ferromagnetic and have largely localized electrons for which the band theory of FM is inappropriate (A good example is CeSi 2 x, with x > 0.2). As we have seen, systems with localized spins are described by the Heisenberg Hamiltonian. H = J iδ S i S iδ gµ B B 0 S i (69) iδ i In general, this Hamiltonian has no solution, and we must resort to (further) approximation. In this case, we will approximate the field (exchange plus external magnetic) felt by each spin as the average field due to the neighbors of that spin and the external field. (See Fig. 21.) Then 22

23 δ = 2 δ = 3 i δ = 1 δ = 4 Figure 20: Terms in the Heisenberg Hamiltonian H = iδ J iδs i S iδ gµ B B 0 i S i Here i refers to the sites and δ refers to the neighbors of site i. S i J Each site υ nearest neighbors with exchange interaction J Figure 21: The mean or average field felt by a spin S i at site i, due to both its neighbors and the external magnetic field, is 1 gµ B δ J iδs iδ + B 0 = B ieff. Where δ J iδs iδ is the internal field, due to the neighbors of site i. H i gµ B B ieff S i = i { } S i J iδ S iδ + gµ B B 0 (70) δ If J iδ = J is a constant (independent of i and δ) describing the exchange between the spin at site i and its ν nearest neighbors, then B ieff = J iδ S iδ + gµ B B 0 gµ B = Jν gµ B S + B 0 (71) M = gµ B N V S ; ν = #nn. (72) 23

24 For a homogeneous, ordered system, and B eff = V νjm + B Ng 2 µ 2 0 = B MF + B 0 (73) B H gµ B B eff S i (74) ie., a system of independent spins in a field B eff. The probability that a particular spin is up, is then i P e β( gµ 1 BB eff 2) (75) and P e β(+gµ 1 BB eff 2) (76) so, on average and, since N + N = N M = 1 2 gµ N N B V N N = e βgµ BB eff (77) = 1 ( ) 2 gµ N β B V tanh 2 gµ BB eff (78) Since tanh is odd and B eff M, this will only have nontrivial solutions if J > 0 (if B 0 = 0). If we identify M s = N gµ B V 2 ; T c = 1 4 ν J k ( ) M Tc M = tanh M s T M s (79) (80) and again for T = 0, M(T = 0) = M s, and for T < T c M ) 1 3 (1 TTc 2 M s (81) Again, we get the same (wrong) exponent β = 1 2. When is this approximation good? When each spin really feels an average field. Suppose we have an ordered solid, so that B MF = Jν 2gµ B = B real (82) 24

25 a = tanh (ba) y = a y = tanh (ba) initial slope = b Figure 22: Equations of the form a = tanh(ba), i.e. Eq. 80, have nontrivial solutions (a 0) solutions for all b > 1. Now, consider one spin flip excitation adjacent to site i only, Fig. 23. If there are an infinite # of spins then B MF remains unchanged but for ν < B real = Jν ν 2 2gµ B ν B MF = Jν 2gµ B. (83) Clearly, for this approximation to remain valid, we need B real = B MF, which will only happen if ν 2 = 1 or ν 2. The more nearest neighbors to each spin, the better ν MFT is! (This remains true even when we consider other lower energy excitations, other than a local spin flip, such as spin waves). 4.2 Mean-Field Theory of Antiferromagnets Oxides of Fe Co Ni and of course Cu often display antiferromagnetic coupling between the transition-metal d orbitals. Lets assume we have such a magnetic system on a bipartite lattice composed of two inter-penetrating sublattices, like bcc. We consider the magnetization of each lattice separately: For example, the central site shown in Fig. 24 feels a mean field from the ν = 8 near-neighbor spins on the down sublattice. so 25

26 i Figure 23: The flip of a single spin adjacent to site i makes a significant change in the effective exchange field, felt by spin S i, if the site has few nearest neighbors. J < 0 "down" sublattice "up" sublattice Figure 24: Antiferromagnetism (the Neel state) on a bcc lattice is composed of two interpenetrating sc sublattices lattices. M + = 1 { } 2 gµ N + B V tanh gµb V νjm (84) 2kT N g 2 µ 2 B M = (+ )... (85) where M + is the magnetization of the up sublattice. These equations have the same form as that for the ferromagnetic case! We can make a closer analogy by realizing that N + = N and M + = M, so that M + = 1 2 gµ N + B { V tanh } +, 2kT N + gµ B J < 0 (86) M = M + (87) 26

27 Figure 25: 2 2 B = V/( N g µ )νjm MF B Again, these equations will saturate at M s + = Ms = 1 2 gµ N + B V (88) so M + M + s { TN = tanh T } M + M s where T N = 1 νj 4 k B Now consider the effect of a small external field B 0. This will yield a small increase or decrease in each sublattice s magnetization M ±. M + + M + = 1 { [ 2 gµ N + B V tanh gµb B 0 + V νj ( M + M )]} 2kT N g 2 µ 2 B M + M = (+ )... (90) Or, since d dx tanh x = 1 cosh 2 x, then { M = M B eff B eff }. M = M + + M = 1 [ 2 gµ N + gµ B 1 B V 2kT cosh 2 B 0 + x V νj 2N g 2 µ 2 B ] M (89) (91) where x = T N M + T. For T > T M s + N, M + = 0 and so x = 0, and M = 2 g2 µ 2 B N 8V k B T [ B 0 4k ] BT N V M Ng 2 µ 2 B (92) T M = g2 µ 2 B N 4V k B B 0 MT N (93) 27

28 M = χ = g 2 µ 2 B N 4V k B (T + T N ) B 0 (94) g 2 µ 2 B N 4V k B (T + T N ) (95) χ T N 0 T N T Figure 26: Sketch of χ = Const/(T + T N ). Unlike the ferromagnetic case, the bulk susceptibility χ does not diverge at the transition. However, as we will see, this equation only applies for the paramagnetic state (T > T N ), and even here, there are important corrections. Below the transition, T < T N, the susceptibility displays different behaviors depending upon the orientation of the applied field. For T T N and a small B 0 parallel to the axis of the sublattice magnetization, we can approximate M + (T ) M s + and x T N T in Eq. 91 B 0 Figure 27: When T T N, a weak field applied parallel to the sublattice magnetization axis only weakly perturbs the spins. Here M + (T ) M + s and x T N T 28

29 χ g2 µ 2 B N 4V k B 1 T cosh 2 ( T NT ) + TN (96) χ g2 µ 2 B N e 2 T NT (97) 4V k B Now consider the case where B 0 is perpendicular to the magnetic axis. The α B 0 B 0 B MF Figure 28: When T T N, a weak field B 0 applied perpendicular to the sublattice magnetization, can still cause a rotation of each spin by an angle proportional to B 0 /B MF. external field will cause each spin to rotate a small angle α. (See Fig. 28) The energy of each spin in this external field and the mean field ν J gµ B to the first order in B 0 is E = 1 2 gµ BB 0 sin + 1 νj cos α (98) 2 Equilibrium is obtained when E = 0. Since B α 0 is taken as small, α 1. E 1 2 gµ BB 0 α + 1 ( 2 νj 1 1 ) 2 α2 or E α = 0 = 1 2 gµ BB να α = gµ BB 0 νj The induced magnetization is then M = 1 2 gµ B B 0 α = g2 µ 2 B NB 0 V 2νJV (99) (100) (101) so χ = g2 µ 2 B N = constant (102) 2ν J V Of course, in general, in a powdered sample, the susceptibility will reflect an average of the two forms, see for example Fig

30 χ χ χ χ χ powdered sample T N T T Figure 29: Below the Neel transition, the lattice responds very differently to a field applied parallel or perpendicular to the sublattice magnetization. However, in a powdered sample, or for a field applied in an arbitrary direction, the susceptibility looks something like the sketch on the right. 5 Spin Waves We have discussed the failings of our mean-field approaches to magnetism in terms of their inability to account for low-energy processes, such as the flipping of spins. (S α S α ) However, we have yet to discuss the lowest energy spin flip processes which are spin waves. We will approach spin-waves two ways. First following Ibach and Luth we will determine a spin wave in a ferromagnet. Second, we will argue that they should be quantized and then introduce a (canonical) transformation to a Boson representation. Consider a ferromagnetic Heisenberg Model H = J iδ S i S i+δ (103) where S i = ˆxS i x + ŷs y i + ẑsz i. If we define α = 1 (i.e. ), β = 0 (i.e. ) 0 1 so that S z 0 = 1 0 (104) and S x = 1 0 i S y = S z = (105) 2 i

31 1.2e-5 1.1e-5 1.0e-5 9.0e-6 8.0e-6 7.0e-6 6.0e-6 5.0e-6 4.0e-6 3.0e-6 2.0e-6 1.0e-6 0.0e T (K) Figure 30: High temperature superconductor Y123 with 25% Fe substituted on Cu, courtesy W. Joiner, data from a SQUID magnetometer. [ S α, S β] = iɛ αβγ S γ (106) It is often convenient to introduce spin lowering and raising operators S and S +. S + = S x + is y = 0 1 [ S z, S ±] = ±S +/ (107) 0 0 S = S x is y = 0 0 [ S 2, S ±] = 0 (108) 1 0 S + 0 = 1, S 0 = 0... (109) They allow us to rewrite H as H = J iδ S z i S z i+δ ( S + i S i+δ + S ) S + i+δ (110) Since J > 0, the ground state is composed of all spins oriented, for example 0 = Π i α i (i.e. all up) (111) 31

32 This is an eigenstate of H, since S + i S i+δ 0 = 0 (112) and Si z Si+δ z 0 = 1 0 (113) 4 so H 0 = 1 4 JνN 0 E 0 0 (114) where N is the number of spins each with ν nearest neighbors. Now consider a spin-flip excitation. (See Fig. 31) j Figure 31: A single local spin-flip excitation of a ferromagnetic system. The resulting state is not an eigenstate of the Heisenberg Hamiltonian. j = S j Π n α n (115) This is not an eigenstate since the Hamiltonian operator S + j S j+δ will move the flipped spin to an adjacent site, and hence create another state. However, if we delocalize this spin-flip excitation, then we can create a lower energy excitation (due to the non-linear nature of the inter-spin potential) which is an eigenstate. Consider the state j Figure 32: If we spread out the spin-flip over a wider region then we can create a lower energy excitation. A spin-wave is the completely delocalized analog of this with one net spin flip. 32

33 k = 1 e ik r j j. (116) N j It is an eigenstate. Consider: H k = 1 N j e ik r j νj j 1 2 J δ { 1 4 νj(n 2) j ( j+δ + j δ ) } (117) where the sum in the last two terms on the right is over the near-neighbors δ to site j. The last two terms may be rewritten: 1 e ik r j j+δ = 1 N N m j e ik (rm r δ) m (118) where r j+δ = r j + r δ = r m (119) so H k = { 1 4 νjn + νj 1 2 J δ ( e ik r δ + e δ) } ik r 1 e ik r j j (120) N j Thus k is an eigenstate with an eigenvalue { } E = E 0 + Jν 1 1 cos k r δ ν δ (121) Apparently the energy of the excitation described by k vanishes as k 0. What is k? First, consider S z k = Si z k = S z 1 i e ik r j j i i N j = 1 e ik r j Si z j = (SN 1) k (122) N j i I.e., it is an excitation of the ground state with one spin flipped. Apparently, since E k=0 = E 0, the energy to flip a spin in this way vanishes as k 0. 33

34 S z n Table 1: The correspondence between S z and the number of spin-wave excitations on a site with S = 3/ Second Quantization of Ferromagnetic Spin Waves In the ground state all of the spins are up. If we flip a spin, using a spin-wave excitation, then S z k = (SN 1) k, S z 0 = SN 0 (123) If we add another spin wave, then S z = (SN 2). For spin 1 2, Sz = N 2 n, where n is the number of the spin waves. Since S z is quantized, so must be the number of spin waves in each mode. Thus, we may describe spin waves using Boson creation and annihilation operators a and a. By specifying the number n k excitations in each mode k, the corresponding excited spin state can be described by a Boson state vector n 1 n 2... n N We can introduce creation and anhialation operators to describe the spin excitations on each site. Suppose, in the ground state, the spin is saturated in the state S z = S, then n = 0. If S z = S 1, then n = 1, and so on. Apparently S z i = S a i a i S + i a i S i a i (124) If these excitations are Boselike, then [ ] a i, a i = 1 (125) 34

35 a i n = n i n 1 (126) a i n = n i + 1 n + 1 (127) This transformation is faithful (canonical) and will maintain the dynamical properties of the system (given by θ = i h [H, θ]) if it preserves the commutator algebra t [ S + i, S i ] = 2S z i, [ S i, ] Sz i = 2S i, [ S + i, ] Sz i = 2S + i (128) consider { S + S S S +} n = 2S z n = 2(S n) n (129) If S + = a, and S = a, then the left-hand side of the above equation would be {(n + 1) n} n = n 2(S n) n. In order to maintain the commutators, we need Then S + = 2S n a S = a 2S n (130) [S +, S ] n = S + S n S S + n = 2S a aaa 2S a a n a (2S a a)a n = (2S n)(n + 1) n n(2s (n 1)) n = (2Sn + 2S n 2 n 2Sn + n 2 n) n = 2(S n) n (131) You can check that this transformation preserves the other commutators. Of course, we need one other constraint, since S S z S, we also must have n 2S (132) This transformation S + i = 2S a i a ia i S i = a i 2S a i a i Si z = S a i a i (133) is called the Holstein Primakoff transformation. 35

36 If we Fourier transform these operators, a i = 1 e ik R i a k ; a i = 1 e ik R i a k (134) N N k then (since the Fourier transform is unitary) these new operators satisfy the same commutation relations [a k, a k ] = δ kk [a k, a k ] k = [a k, a k ] = 0 (135) To convert the Hamiltonian into this form, assume the number of magnons in each mode is small and expand S + i = 2S n i a i 2S(1 n i 4S )a i (136) { } 1 e ik R i a k 1 e i(p+q k) R i a N 4SN 3 k a pa q 2 Of course this is only exact for n i excitations, and large S (the classical spin limit). In this limit the Hamiltonian may be approximated as S + i k kpq 2S, i.e. for low T where there are few spin 2S e ik R i a k (137) N k 2S e ik R i a k (138) N k Si z = S 1 e i(k k ) Ri a k N a k, (139) kk S i H = J iδ {S zi S zi+δ + 12 (S+i S i+δ + S i S+i+δ ) } (140) H NJνS 2 + 2JνS a k a k k 2JνS ( ) 1 e ik R δ a k ν a k + O(a 4 k) (141) k δ 36

37 H E 0 + k 2JνS(1 γ k )a k a k + O(a 4 k) (142) where γ k = 1 ν δ eikr δ. This is the Hamiltonian of a collection of harmonic oscillators k-q k +q k k k k Figure 33: The fourth order correction to Eq. 142 corresponds to interactions between the spin waves, giving them a finite lifetime plus some other term of order O(a 4 k ) which corresponds to interactions between the spin waves. These interactions are a result of our definition of a spin-wave as an itinerant spin flip in an otherwise perfect ferromagnet. Once we have one magnon, another cannot be created in a perfect ferromagnetic background. Clearly if the number of such excitations is small (T small) and S is large, then our approximation should be valid. Furthermore, since these are the lowest energy excitations of our spin system, they should dominate the low-t thermodynamic properties of the system such as the specific heat and the magnetization. Consider E = k hω k e β hω k 1. (143) For small k, hω k = 2JνS(1 γ k ) = 2JνSk 2 on a cubic lattice. Then let s assume that the k-space is isotropic, so that d 3 k k 2 dk, then γ k = 2 ν (cos k x + cos k y + ) = 1 k2 ν (144) and E k 2JνSk 2 e β2jνsk2 1 0 k 4 dk e βαk2 1 (145) x = βαk 2 k = ( ) 1 x 2 βα dk = 1 2 ( ) x 1 2 dx (146) βα 37

38 so that E β 2 β 1/2 Thus, the specific heat at constant volume C V experiment. 0 dx x3/2 e x 1 T 5/2 (147) T 3/2, which is in agreement with M/M(0) 1 - T 3/2 T Figure 34: The magnetization in a ferromagnet versus temperature. At low temperatures, the spin waves reduce the magnetization by a factor proportional to T 3/2, which dominates the reduction due to local spin fluctuations, derived from our mean-field theory. This result is also consistent with experiment. If we increase the temperature from zero, then the change in the magnetization is proportional to the number of magnons generated M(0) M(T ) = n k k gµ B V (148) since each magnon corresponds to spin flip. Thus k 2 dk M(T ) M(0) e βαk2 1 T 3 2 (149) which clearly dominates the exponential form found in MFT (1 2e 2Tc T ). This is also consistent with experiment! 38

39 5.2 Antiferromagnetic Spin Waves Since the antiferromagnetic ground state is unknown, the spin wave theory will perturb around the Neel mean-field state in which there are both a spin up and down sublattices. Spin operators can then be written in terms of the Boson creation and down sublattice up sublattice Figure 35: To formulate an antiferromagnetic spin-wave theory, we once again consider a bipartite lattice, which may be decomposed into interpenetrating spin up and spin down sublattices. annihilation operators as before up sublattice down sublattice S + i = ( S i S z i = S n i S z i = S + n i (150) ) + = 2Sfi (S)a i S + i = ( S i ) + = 2Sa i f i(s) where f i (S) = 1 n i 2S and n i = a i a i (151) Again this transformation is exact (canonical) within the manifold of allowed states 0 n i 2S S S z S. (152) The Hamiltonian H = J iδ S z i S z i+δ = 1 2 ( S + i S i+δ + ) S i S+ i+δ (153) 39

40 may be rewritten in terms of Boson operators as H = +JS 2 Nν + J a i a ia i+δ a i+δ iδ JS { a i a i + a i+δ a i+δ iδ } + f i (S)a i f i+δ (S)a i+δ + a i f i(s)a i+δ f i(s) (154) Once again, we will expand f i (S) = 1 n i 2S = 1 n i 4S n2 i 32S (155) 2 and include terms in H only to O(a 2 ) H JS 2 Nν JS iδ { } a i a i + a i+δ a i+δ + a i a i+δ + a i a i+δ (156) This Hamiltonian may be diagonalized using a Fourier transform and the Bogoliubov transform Here the α k a i = 1 e ik R i a k (157) N k a k = α k cosh u k α k sinh u k (158) a k = α k cosh u k α k sinh u k (159) are also Boson operators tanh 2u k = γ k (160) γ k = 1 e ik R δ ν (161) δ [ ] α k, α k canonical, we must ensure that the commutators are preserved. = 1. To see if this transform is 1 = [a k, a k ] = [α k C k α k S k, α k C k α k S k ] { } = Ck[α 2 k, α k ] + S2 k[α k, α k] δ kk (162) = { C 2 k S 2 k} δkk = δ kk 40

41 where C k (S k ) is shorthand for cosh u k (sinh u k ). You should check that the other relations, [a k, a k ] = [a k, a k ] = 0, are preserved. After this transformation, H JNνS(S + 1) + k ( hω k α k α k + 1 ) + O(a 4 ) (163) 2 where hω k = 2JSν 1 γ 2 k. Notice that for small k, hω k 2JSνk Ck (C = 2JSν is the spin-wave velocity). The ground state energy of this system (no magnons), is E 0 = JNνS(S + 1) JSν k 1 γ 2 k (164) If γ k = 0, then each spin decouples from the fluctuations of its neighbors and E 0 = JNνS 2 (J < 0) which is the energy of the Neel state. However, since γ k 0, the 2 E N = JNνS Figure 36: The Neel state of an antiferromagnetic lattice. Due to zero point motion, this is not the ground state of the Heisenberg Hamiltonian when J < 0 and S is finite. ground state energy E 0 < E N. Thus the ground state is not the Neel state, and is thus not composed of perfectly antiparallel aligned spins. Each sublattice has a small amount of disorder n i + 1/2 in its spin alignment. The linear dispersion of the antiferromagnet means that its bulk thermodynamic 41

42 properties will emulate those of a phonon lattice. For example hω k E e β hω k k 1 k αk 3 dk 0 e βαk 1 E T 4 x 3 dx e x 1 0 αk e βαk 1 (165) C = E T T 3 like phonons! (166) Which means that a calorimeter experiment cannot distinguish phonon and magnon excitations of an antiferromagnet. n E i thermal spinpolarized neutrons sample E f θ dθ dωdω S(k, ω) I {F(-i [a(t),a (0)] )} 2θ k = k i k j h ω = E E i f n Figure 37: Polarized neutrons are used for two reasons. First if we look at only spin flip events, then we can discriminate between phonon and magnon contributions to S(k, ω). Second the dispersion may be anisotropic, so excitations with orthogonal polarizations may disperse differently. Therefore, perhaps the most distinctive experiment one may perform on an antiferromagnet is inelastic neutron scattering. If spin-polarized neutrons are scattered from a sample, then only those with flipped spins have created a magnon. If the neutron creates a phonon, then its spin remains unchanged. The time of flight of the neutron allows us to determine the energy loss or gain of the neutron. Thus, if we plot the differential cross section of neutrons with flipped spins, we learn about magnon dispersion and lifetime. Notice that the peak in S(k, ω) has a width. This is not just due to the instrumental resolution of the experiment; rather it also reflects the fact 42

43 S(k,ω) n magnon γ k ω k n phonon junk subtracted off ω k ω 2θ Figure 38: Sketch of neutron structure factor from scattering off of a magnetic system. The spin-wave peak is centered on the magnon dispersion. It has a width due to the finite lifetime of magnon excitations. that magnons have a finite life time δt,which broadens their neutron signature by γ k. γ k δt h δt 1 (167) γ k However in the quadratic spin wave approximation the lifetime of the modes hω k is infinite. It is the neglected terms in H, of order O(a 4 ) and higher which give the magnons a finite lifetime. a ia ia a i+δ i+δ Figure 39: 43

44 2. At the fixed point K = K K = R( K ) 6 Criticality and Exponents Before reading this section, please read Ken Wilson s [ Scientific American, 241, 158 (Aug, 1979)] paper on renormalization group (RG). From the this paper, you have learned a number of new ideas that serve as the basic paradigms of Fisher scaling which leads to a better understanding of criticality in terms of critical exponents. 1. Renormalization Group (RG) operates in the space of all possible Hamiltonian parameters K = (K 1, K 2, K 3 ) which renormalize such that K = R( K) where R depends on the system, the RG transform employed (since RG is not unique) and the renormalization scale b. 3. Transitions are indicated by an unstable fixed point. 4. At or near a transition, the parameters K = (K 1, K 2, K 3 ) change very slowly under repeated RG transformations (critical slowing down). If R is analytic at K, then for small K K we may linearlize, so that K α K α = β T αβ (K β K β) very close to K where the T-matrix T αβ = dk α dk β has real eigenvalues (symmetric, real). eigenvectors of T α K=K Let λ (i) and φ (i) α be the left eigenvalues and φ (i) α T αβ = λ (i) φ (i) β. 44

45 We define the scaling variables u i = α φ(i) α (K α Kα) which transform multiplicatively, so that u i = α = α = β φ i α (K α K α) φ (i) α ( ) T αβ Kβ Kβ β λ (i) φ (i) β ( Kβ K β) = λ (i) u i This means that under repeated RG transforms, u i grows or shrinks depending on the size of λ (i). If λ (i) > 1, then the variable u i is relevant since repeated RG transforms will make it grow movig the system away from the fixed point. λ (i) < 1, then it is irrelevant since RG transforms leave it at the fixed point. λ (i) = 1, then it is marginal. We can use this to describe the scaling behavior. The RG transformation is constructed to keep the partition function Z the same, and hence to also keep the free energy the same. So, if f = F/N is the free energy per site, then f(k) = f 0 (K) + b D f s (K ) where f 0 is the analytic part (no transitions and boring) and f s is the non-analytic part which describes the phase transformation. So, the singular part scales like f s ( K) = b D f s ( K ). Or, in terms of the scaling variables (fields) f s (u 1, u 2, ) = b D f s (λ 1 u 1, λ 2 u 2, ) = b nd f s (λ n 1u 1, λ n 2u 2, ) where the second line is after n RG steps and includes relevant variables only so that λ 1 > 1, λ 2 > 1, etc. so the arguments grow with n. Since n is arbitrary, we may 45

46 eliminate it by defining A, so that λ n 1u 1 = A 1 (where the inequality ensures that the linear approximation still holds). Then n ln(λ 1 ) + ln(u 1 ) = ln(a), which we use to eliminate n, so that ) ln A D ln λ f s (u 1, u 2 ) = b 1 b D ln u ln A 1 ln λ ln λ 1 f s (A, ln u 1, λ 1 ln λ 1 2, Now, to make better contact with conventional notation, we define y 1 = ln λ 1 / ln b, y 2 = ln λ 2 / ln b, etc. so that λ 1 = b y 1 etc. These y are called the scaling exponents. Generally, we define the relevant variables such that u 1 = t = (T T c )/T c, u 2 = h, or u 3 = 1/L, which are the reduced temperature, field and inverse system size, which are all obviously relevant. The corresponding set {y j } are related to the critical exponents {ν, γ, β }. One of the most important applications of these ideas is in a finite-sized scaling analysis. Suppose we have a simulation (e.g., Monte Carlo or Molecular Dynamics) of a system of size L and spatial dimension D. Then 1/L must be a relevant field u j since if L is finite there can be no transitions, since they require that the correlation length ξ diverge. Suppose, for the sake of an example, that the other relevant fields are t and h. Under a scale transformation of size b, f s must be scale invariant so that f s ( u1, u 2, L 1) = b D f s ( b y 1 u 1, b y 2 u 2, b 1 L 1) where L 1 is not only relevant, it must scale with exponent y L = 1 as shown. Let s identify, the other relevant variables as discussed, so that f s ( t, h, L 1 ) = b D f s ( b y 1 t, b y 2 h, bl L) This form must be true for any b, L 1, h or t close to the transition (in the scaling region). So, let s choose b = L to eliminate a parameter, so that f s ( t, h, L 1 ) = b D f s (b y 1 t, b y 2 h, 1) 46

47 Figure 40: Scaled susceptibility Suppose we want to calculate the susceptibility χ to the applied field h. It involves two derivatives with respect to h at h = 0) χ = L D+2y 2 g (L y 1 t, 0, 1) where g is the second derivative of f s function. Now, some of these exponents have well known identities that we may as well use, such as ξ = versus t = T T c /T c. By ξ 0 t 1/y 1 so that ν = 1/y 1, χ t γ with the scaling the susceptibility with an appropriate power of the cluster size, the curves for different cluster sizes L may be made to cross at the transition, identifying T c and γ/ν. A similar finite size scaling hyperscaling relation γ = (2y 2 D)/y 1, so that χ L γ/ν g(l 1/ν t, 0, 1) We may apply a similar analysis to the specific heat, which is proportional to a second derivative of f s with respect to t, again for h = 0, so that ansatz exists for the specific heat and yields another estimate for T c C L 2/ν D g (tl 1/ν, 0, 1) and Now ν suppose we calculate χ an C for several different clusters of size L. L must be large enough so that the system is in linear scaling region of the phase transition. Then we notice that since χ L γ/ν g(l 1/ν t, 0, 1), so that if we plot L γ/ν χ versus t, then the curves for different L must all have the same value and cross at the transition where T = T c so that t = 0, and g(0, 0, 1) is a constant (Fig. 6). This allows us to obtain an estimate for T c and γ/ν. Likewise, if I plot L D 2/ν C versus t, then again the curves for different L must cross at the transition where g (0, 0, 1) is constant. This produces another estimate for T c and for ν, which when combined with the scaling for χ gives γ. 47