Phase Transitions and Magnetic Correlations in TwoDimensional AntiferromagnetsR. J. Birgeneau, J. Skalyo Jr., and G. Shirane Citation: Journal of Applied Physics 41, 1303 (1970); doi: 10.1063/1.1658917 View online: http://dx.doi.org/10.1063/1.1658917 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/41/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic properties of two-dimensional nanodots: Ground state and phase transition AIP Advances 3, 122121 (2013); 10.1063/1.4858416 Correlation Effect on the TwoDimensional Peierls Phase AIP Conf. Proc. 850, 1323 (2006); 10.1063/1.2355195 Phase diagram of the two-dimensional quantum antiferromagnet in a magnetic field J. Appl. Phys. 99, 08H503 (2006); 10.1063/1.2172209 Phase transitions in twodimensional systems J. Appl. Phys. 49, 1315 (1978); 10.1063/1.325029 Phase Transition in TwoDimensional Liquid Crystals J. Chem. Phys. 55, 4678 (1971); 10.1063/1.1676823

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JOURNAL OF APPL1ED PHYSICS VOLUME 41, NUMBER 1 MARCH 1970

Phase Transitions D. W. HONE, Chairman

Phase Transitions and Magnetic Correlations in Two-Dimensional Antiferromagnets

R. J. BrRGENEAU

Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey 07974 and Brookhaven National Laboratory,* Upton, New York 11973

AND

J. SKALYO, JR., AND G. SHIRANE

Brookhaven National Laboratory,* Upton, Ne"dJ York 11973

Recent neutron scattering work on the planar antiferromagnet K2NiF. has indicated that this system undergoes a true two-dimensional antiferromagnetic phase transition. In this paper we review these studies together with measurements of the magnons at low temperatures and in the vicinity of TN. New high precision measurements of the sublattice magnetization around TN are presented. It is shown that the power law

M( T) / M(4.2) =0.973 (1- T/92.23)o.138±o.OO4

holds for 2XlO-l>1-T/TN >3X10-4• Preliminary measurements of the wavevector-dependent staggered susceptibility xafJ (q) in K 2NiF. show that only x", the component parallel to the anisotropy axis, has pronounced diffusive character, diverging at TN. This provides definitive proof that the phase transition is anisotropy-induced.

I. INTRODUCTION

The cooperative properties of two-dimensional and near-two-dimensional systems have been of some interest to the physics community, both experimen­talists and theoreticians, for a number of decades.! In recent years this has been stimulated by the renewed interest in the problem of phase transitions since, as is well known, two-dimensional systems playa unique role in this area.2- 4 The two-dimensional ([2J) S= t Ising model is the only system exhibiting a phase transition for which exact results are available.2,5 The [2J Heisen­berg model, on the other hand, can show no transition to true long-range order,6 although it has been suggested that the susceptibility may nevertheless diverge at some nonzero temperature.7 This prohibition of true long­range order in the Heisenberg model, however, can be removed by even very small amounts of anisotropy. Indeed, as early as 1952 it was shown by Kubo,s using conventional spin wave theory, that the sublattice magnetization, and hence TN in simple theory for the

indeed exhibit genuine [2J antiferromagnetic phase transitions.9•lO Recently H. J. Guggenheim of Bell Telephone Laboratories succeeded in growing large single crystals of a number of these, including K2NiF4, Rb2MnF4, Rb2FeF4• Consequently, a detailed investiga­tion of their magnetic properties using elastic and inelastic neutron scattering techniques was undertaken. In this paper, we review the work on K2NiF4,1l together with measurements of the magnon dispersion relations in K2NiF4 at 5°K and around TN.!2 In addition, we present more precise measurements of the sublattice magnetization of K2NiF4 with emphasis on the region in the immediate neighborhood of TN. We shall also give a discussion of some preliminary high-resolution measurements of the wavevector-dependent staggered susceptibility. As we shall see, K2NiF4, may be regarded as a near-Heisenberg [2J antiferromagnet exhibiting an anisotropy-induced [2J phase transition. I1 ·!2

II. PRELIMINARY INFORMATION

antiferromagnet, depends logarithmically on the anisot- In this section, we review such properties of these ropy. On the experimental side, however, there has been compounds as the crystal structures, reciprocal lattices, a paucity of information. Until recently no true [2J and spin Hamiltonians. The crystal structure and low­phase transitons had been found in nature, although the temperature magnetic structure of K2NiF4 are shown in bulk thermodynamic properties of a number of planar Fig. 1. The crystal is composed of successive simple pseudo-two-dimensional systems had been examined in square NiF2 planes separated by two KF planes.9 ,1l.!3 some detail,! In the past several years, however, a These NiF2 planes are stacked in such a way that in the number of investigations have been carried out on the Neel state within the molecular field approximation family of compounds with the K2NiF4 structure there is no net coupling between adjacent planes. (see Fig. 1) and it has been postulated that these do Indeed, Lines has shown that within the RPA Green's

1303

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1304 B I R G ENE AU, S K A L Y:O, JR., AND S H I RAN E

300 (031) 302 (033

.d I

I 1 fJ

!

200 202

T ,

CI -I

I

j I ~

10'

I ! 100 (011) 102 (013)

i fSCAN A I I I I I

}Q.O.9_*_ °002

+-SCAN B 1

1 1 f') 1

.d -,..

\' £

FIG. 1. Chemical and magnetic structure of K2NiF •. Inverting the central spin exchanges, the a and b axes. The reciprocal lat­tice displays both the [OlOJ and [l00J magnetic zones. The nu­clear Bragg peaks are indicated by double circles. The thick lines indicate the vicinity in which two-dimensional critical scattering is observed.

function approximation any coupling between the· planes tends to inhibit ordering. lO It is this cancellation effect which accounts for the two-dimensionality.

The [0, 1, OJ and [1, 0, OJ zones of the corresponding reciprocal lattice are also shown in Fig. 1. Both zones are always present as separate domains; the two domains are related by flipping the body center spin in the structure shown in Fig. 1. The single filled and open dots are reciprocal lattice points of the [3J magnetic struc­ture. Any purely [2J phenomena, either true long-range order or long-range [2J correlations should give rise to a scattering cross section which has the form of ridges along (1,0, l), (3,0, I), etc. Conceptually, therefore, an experiment to establish the [2J nature of the system is quite straightforward. It consists merely of searching for the existence of a ridge using scans of the sort portrayed in the figure (scan A, scan B). For a com­plete discussion of the neutron scattering cross section for [2J systems, see Refs. 11 and 14.

The spin Hamiltonian in K2NiF4 may be approxi­mated by

3C= L: L: JnnS;,Sj+ L: gJ.l.BH.-tSzi i i<i i

+X (distant neighbor), (1)

where the nearest-neighbor isotropic exchange term is dominant. As we shall see later, measurements of the magnons at low temperatures12 give Jnn= 77.9 cm-I, gfJ.BHA=0.59 cm-I . The coupling between planes is found to be at least 270 times smaller than the intra­planar interactions and, in addition, more distant in­plane exchange is smalL Thus K2NiF4 is quite close to being a pure two-dimensional nearest-neighbor Heisen­berg system, the anisotropy being only 1 part in 500.

III. EXPERIMENTAL RESULTS: GENERAL SURVEY

A general survey of the diffusive and Bragg scatter­ing, without employing precise resolution corrections, has been carried out by Birgeneau, Guggenheim, and Shiranell in K2NiF4, Rb2MnF4, R~FeF4. Their experi­mental results in K2NiF4 are shown in Figs. 2 and 3. We shall now discuss these results in some detaiL

Figure 2 shows the results of ridge scans of types A, B (see Fig. 1) at 99°, 95°, and 4.2°K. From the figure it may be seen that at 99°K in K2NiF4 the ridge does indeed exist. Scan B along (1, 0, l), that is along the top of the ridge, gives a constant value far above the back­ground. The decrease in intensity at large l is due to the tensorial nature of x"'''' plus the form factor, where x"'''' is the ath component of the wavevector-dependent susceptibility. We shall discuss this in detail in Sec. V. The important feature is that there is no peaking whatsoever about (1,0,0) and (0, 1, 1) as would occur in a normal [3J system. Scan A along (h, 0, 0.25), that is, perpendicular to the ridge, shows a sharp peak with a linewidth determined by the instrumental resolution. The lack of concavity in scan B together with the sharpness of scan A shows unambiguously that at 99°K, K2NiF4 behaves as a pure two-dimensional anti ferro-

(L..Q.,!)

1500

o 00 0 0 0

o 00

0 00 0 0000°0 0°00

o-99'K .. -95·K ._ 5°K

1000 0

<n ~ 500 :> z :i

'" .... Vl ... z 6 1500 u

1000

500

_~ _____________ ~~~@~D

-0.5· a

o

0.8 0.9

0.5 R ....

1.0 h-

1.1

1.0 1.5

1.2

FIG. 2. Ridge scans in K2NiF •. The upper set of curves corres­pond to scan B along the top of the ridge as shown in Fig. 1; the lower set of curves correspond to scan A across the ridge. Data taken from Ref. 11.

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TWO-DIMENSIONAL ANTIFERROMAGNETS 1305

magnet with very long-range correlations within the planes (> 1000 A) and no measurable correlations between the planes. The scattering remains essentially identical in form down to 97.2°K. However, in cooling from 97.2° to 97.0oK a rather unusual phase transition is observed. Sharp Bragg peaks appear on top of the ridge at the magnetic reciprocal lattice points (1,0,0), (0,1,1), etc. Their linewidth is just the mosaic spread of the crystal, indicating that true LRO in all 3 dimen­sions has been established.

In order to ascertain the nature of the [2J region above 97.1 OK it is necessary to study the evolution of the rod with temperature. A series of scans of types A, B show that the ridge remains well defined up to at least 2oooK. The (1,0,0.25) peak intensities together with the (1,0, l) ridge linewidths as a function of tem­perature are shown in Fig. 3. From the figure it may be seen that the (1, 0, I) ridge reaches its limiting intensity and linewidth at TN. The behavior in the range from 97.2° to 2000 K thus seems to be exactly analogous to the critical region in a [3J system except that the temperature scale is greatly expanded.15

Let us now consider the behavior for T < TN. First, as shown in Fig. 3, the ridge decreases extremely rapidly in intensity with decreasing temperature. Concomi­tantly the (1, 0, 0) peak intensity increases equally rapidly with decreasing temperature. 0,20 scans

100 xlO' r-------------~

'" w I­::> z i '" .... '" I-Z ::> o

50xlO'

u fOOO

500

•

PEAK INTENSITY

(1,0,0.25)

) • •

50

PEAK INTENSITY {t ,0,0)

'" '::: z

0.22 :I: I-0

i 0.1 w

Z ..J

FIG. 3. Scattering intensity in K2NiF. at peak (1, 0, 0) top and. (1, 0,. 0.25) bot~om together with the linewidth in reciprocal lattice umts (full Width at half-maximum) for scan A in Fig. 1 as a function of temperature. Data taken from Ref. 11.

TABLE I. K2NiF. Sublattice magnetization parameters.

~10-3 ~10-3

<1-T/TN<0.9 <1-T/TN<6XlO-2

1.02±0.01 97. 05±0. 009 0.14±0.01

0.98±0.03 97.04±0.07 0.14±0.01

through (1,0,0) at 95°K show no evidence at all for [3J critical scattering. Thus the diffuse scattering below TN retains the form of a ridge. In the immediate neighborhood of TN, at least, this diffuse scattering may be thought of as the T < TN counterpart of the [2J critical scattering observed above TN.

The magnetic Bragg scattering intensity in an anti­ferromagnet is proportional to the square of the sub­lattice magnetization. In the critical region it is expected that the sublattice magnetization will vary like

M(T)/M(O) =B(1- T/TN)fJ, (2)

where both /3 and B may be calculated from theory in certain cases.2 Thus, in the critical region one expects

I(Bragg) (T)/I(Bragg) (0) =B2(1- T/TN)2fJ, (3)

and therefore the critical exponent (3 and the factor B may easily be determined simply by measuring the intensity of a Bragg superlattice point as a function of temperature.

The variation of the (1,0,0) Bragg peak intensity as a function of temperature is shown in Fig. 3. The data analysis is quite straightforward. The background is first subtracted off and the data is then reduced to the form M(T)/M(4.2°K).16 This may then be fitted to a power law for various ranges of reduced temperature. Quite remarkably, these calculations show that for K2NiF4, the data can be fitted satisfactorily at all temperatures to a simple power law of the formll

M(T)/M(4.2°K) = 1.02[1- (T/97.05)]0.14. (4)

This fit holds over the range "-'1Q-3<1-(T/97.05) < 0.9. Birgeneau et al.!1 then show that if one instead limits the data to the more usual "critical region" a somewhat improved fit is found with

M(T) /M( 4.2°K) =0.98[1- (T /97.04) JO.14. (5)

This holds for "-'1Q-3<1-(T/97.04) <6X1Q-2. The parameters are tabulated in Table I together with the associated errors. The fact that the fit holds at all tem­peratures at first seems very anomalous. However, more detailed consideration of the sublattice magnetization at lower temperatures shows that the data up to ,,-,0. 7 TN is at least as well explained using conven­tional spin wave theoryll,17; thus the apparent agreement with the power law at all temperatures is probably a coincidence.

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1306 BIRGENEAU, SKALYO, JR., AND SHIRANE

, (---)

30

:; .. -S w 20

10

- - - - - - -x- - -- - ---x

<6.0

FIG. 4. Magnon dispersion curve in K2NiF. in the [OlOJ zone. The solid line is for [r, 0, OJ while the dashed lines are for [0, 0, rJ at 2.3 meV and [0.45, 0, rJ at 38.1 meV. Data taken from Ref. 12.

Detailed consideration of the above results, particu­larly the apparent absence of any [3J critical scattering, immediately leads one to the conclusion that the phase transition is a genuine [2J phase transition.!! By this one means that the system achieves long-range order solely because of the [2J properties; loosely speaking, one may envisage the system as achieving LRO in two dimen­sions and then by necessity ordering in three dimensions, since even a microscopic interaction between nnn planes is then amplified by N, the number of spins in a plane. A more precise description of the phase transition probably requires one to envisage the system as having two critical regions. The first, which is purely [2J in form, and which is the only one Birgeneau et al. have been able to monitor experimentally, corresponds to the growth of the correlations within the planes. At some point, the correlation length within the planes must become sufficiently long (perhaps macroscopic) that these spin "globules" in the different nnn planes be­come aware of each other's existence. At this tempera­ture, the critical behavior must go over to being [3J in character. However, in terms of absolute temperature this [3J region may be extremely small, and indeed

they were unable to obtain any experimental evidence for its existence. Most importantly, the [3J aspects of the system do not seem to alter the pure [2J nature of the phase transition for values of /1- T / TN / down to at least 10-3• It should be noted that Birgeneau et al.l1

also observe behavior essentially identical to that discussed above for K2NiF4 in the isostructural com­pound Rb2MnF4 and, to a lesser extent, in Rb2FeF4•

IV. MAGNONS AT LOW AND HIGH TEMPERATURES IN K2NiF4

Measurements of anti ferromagnetic resonance in K2NiF4 as a function of temperature have been carried out by Birgeneau, DeRosa, and Guggenheim.ls Studies of the magnons over the entire [0, 1, OJ zone have been made by Skalyo, Shirane, Birgeneau and Guggenheiml2 using inelastic neutron scattering techniques. For the pure q = ° mode AFMR has the advantage that no resolution corrections are required, whereas, of course, only neutrons are satisfactory for q~O. Since K2NiF4 corresponds to a conventional two-sublattice antiferro­magnet the AFMR frequency may be writtenl9

w= g/kBI [HA (2HE+HA) JI/2±H.1 , (6)

where HA and HE are effective anisotropy and exchange fields and Hz is a static externally applied field in the preferred direction. At 4.2°K a single magnetic dipole active mode is observed at 19.1 cm-I • Applications of a field along the c axis causes the mode to split in accord­ance with Eq. (1). The mode splitting is linear with applied field with a g value of 2.22±0.06. As the tem­perature increases above 4.2°K the mode decreases in energy and begins to broaden. In fact, the mode energy is found to be exactly proportional to the sublattice magnetization. By 8s oK the mode is barely discernible. Finally at lOsoK, approximately 80 above TN, no resonant absorption is detected.

The neutron measurements of Skalyo et al.!2 are shown in Figs. 4 and 5. Due to the existence of two domains, measurements were by necessity made simultaneously in both the [1, 0, OJ and [0, 1, OJ zones. The spin-wave spectra along [.1,0, OJ and [0, .I, OJ are the same if only intraplanar exchange exists.

The dispersion curve in the [.\,0, OJ direction is shown in Fig. 4. All measurements were taken from magnetic Bragg points of the [0, 1, OJ zone. Even though the two zones give rise to two magnon surfaces, at no point was there any more than one magnon observed. This was particularly evident at [0.45, 0, OJ near the zone boundary, where measurements were made with the constant Q technique with the spectrometer in a well-focussed condition. Because of this sharp focussing around [0.45,0, OJ it was advantageous to study the dispersion in the c direction along [0.45, 0, r]. The results are shown in Fig. 4. Quite remarkably, there is no measurable dispersion at all. All points gave values

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TWO-DIMENSIONAL ANTIFERROMAGNETS 1307

of 38.1±0.1 meV. A similar lack of dispersion was also found along the [0,0, rJ direction, although with higher relative uncertainty.

Because of this complete lack of dispersion in the c direction, the magnon energies are given simply by

E(qx) =4 1 I 1 sin(q",a/2)1[1-(J2+2J3)/11J+"', (7)

assuming 1 2+213«11• This then gives

1 1[1- (J2+213)/ iIJ= -9.68±0.03 meV,

with the proviso that 1 2+213«11 • Utilizing the accurate antiferromagnetic resonance of 2.37 meV for the q=O magnon energy,18 the anisotropy energy was determined to be g/J.BH A = 0.073 me V. It should also be noted that a full least-squares fit to the exact expression allowing II, 1 2+213, g/J.]JHA to vary freely gave 1 2+213 =0 within the errors, but with II and 1 2+213

very strongly correlated. The authors have also estimated the strength of

interplanar exchange consistent with their observations. Measurements along [0.45,0, rJ are of high relative accuracy; the magnon energy is 38.1 meV, while the dispersion is most certainly less than ±0.14 meV. If it is assumed that the nearest neighbor interplanar exchange II' is much less than II, one obtains away from the zone center for [qx, 0, qzJ magnons

E(qx, qz) = 41 II sin (q,ra/2) 111-[ (J2+21a) /11JI

X[1+CN/l1) cos(qzc/2)J+···. (8)

E(q1J, qz), on the other hand, has no first-order de­pendence on II'. As there is no dispersion to within 1 part in 270 for the mean energy of E(qx, qz) and E(qy, qz), then 1 1>270iI'.

Let us now consider the temperature evolution of the magnon energies and lifetimes as measured and dis­cussed by Skalyo et al. 12 Previous experiments in three­dimensional antiferromagnets have shown that as TN is approached the spin wave energies are considerably renormalized and the half-widths become comparable to the energy itself.20 Nevertheless, at short wavelengths, magnon-like propagating modes are still in evidence above TN. In K2NiF4 , because of the anomalously wide temperature region over which there are long-range two-dimensional correlations above TN, one might expect more dramatic behavior.

In Fig. 5 the results of constant energy scans at 7 meV in the vicinity of q = (0.05, 0, 0) are shown. This wavevector is equivalent to a wavelength of ~11O A. From the figure it may be seen that there is no measur­able change in q or the width up to 105°K, that is, to ~1.1TN. Thus not only do these relatively long wave­length magnons remain well-defined above the phase transition, they are, in fact, indistinguishable from their counterparts at T= OaK. Finally, by 146°K the spin wave mode has merged with the longitudinal diffusive mode and is no longer resolvable.

K2 Ni F4 TN = 97 OK

CONSTANT ENERGY SCANS - 7 meV [100] MAG NON

0: 0: 0

77°K ::' 600 146°K '= 600 z Z 0 0 ::;; ::;; ~

0400 • 8 400 0 0 0 co co "- "-I- I-

~ 200 ~ 200 0 0 <.) <.)

01.0 01.0

• 0: 60 9soK a: 60 • • 105°K 0 ::' I- Z • Z • 0 0 • ::;; ::;;

40 40 "" ~ • 0 0 0 • 0 co

~ • "-I- 20 I- 20 z z ::> ::> • 0 0 <.) <.)

°1.0 °1.0

FIG. S. Temperature dependence of [0.05, 0, OJ magnon in K2NiF4• Note the change in counting statistics in going from 77° to 98°K. Data taken from Ref. 12.

In summary, results of AFMR and neutron scatter­ing show that at 5°K the observed magnon dispersion relations can be accurately described by simple [2J spin-wave theory employing nearest-neighbor Heisen­berg interactions together with an anisotropy field 500 times smaller than the exchange field. The lack of dispersion in the c direction necessitates that the inter­planar exchange be at least a factor of 270 smaller than the intraplanar interactions. Thus the dynamics show that K2NiF4 is very close to being a pure [2J Heisenberg system although the small anisotropy does play a crucial role, as we shall see in the next section. As the temperature is increased to TN the pure q = 0 mode decreases in energy and broadens as expected. However, away from q = 0, at q",,0.05 A-I, no appreciable thermal effects are seen up to 1.1 TN.

V. NEW RESULTS

All of the experiments which we have described so far were carried out in the spirit of a general survey. Two obvious extensions of the static measurements are required. First, a more detailed study of the Bragg scattering around TN with better temperature control is desirable both to determine a more precise value of (3 and to see if any [3J critical scattering exists. Secondly, detailed measurements of the ridge around TN should give values for the critical exponents v, v' (correlation lengths above and below TN), ,)" ')" (q = 0 staggered susceptibilities) I .", .,,' (deviation from Orstein-Zernike).

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1308 B I R G ENE AU, S K A L YO, J'R., AND S H I RAN E

TABLE II. K2NiF4 (100) integrated intensity.

77.0 81.10 87.04 91.21 92.17 93.28 95.00 95.60 95.81 96.04 96.27 96.28 96.39 96.50 96.74 96.85 96.97 97.080 97.101 97.147 97.170 97.196 97.198 97.240 97.286 97.38

[(Bragg) (100) (arbitrary

units)

21.53 18.60 17.77 15.93 14.08 13.15 11.08 10.44 9.93 9.53 9.07 8.69 8.58 8.41 7.41 7.06 6.29 5.484 5.231 4.527 4.143 3.541 3.475 1.928 0.614 o

" Normalized to data given in Ref. 11.

M(T)jM (4.2)8

0.795 0.738 0.722 0.683 0.643 0.621 0.570 0.553 0.540 0.529 0.515 0.505 0.501 0.497 0.466 0.455 0.429 0.401 0.392 0.364 0.349 0.322 0.319 0.238 0.134 o

In this section we report new experimental results on the former together with some preliminary measure­ments on the latter.

A. Accurate Measurement of {3 in K2NiF4

These experiments were carried out on a two-axis crystal spectrometer at the Brookhaven High Flux Beam Reactor using neutrons of wavelength 1.029 A. The crystal was the same one used for previous studiesll •12 ; 10' collimation before and after the scatter­ingwas employed. The monochromator was a germanium crystal reflecting from (3, 1, 1) in transmission geom­etry. The crystal was mounted in a Cryogenics Associate temperature control Dewar similar to that employed previously but with a control system which gave relative temperatures reproducibly to better than 20 mdeg K over the entire temperature range studied.

In order to study the sublattice magnetization, a series of scans along (1,0,1) through the Bragg peak at (1,0,0) were carried out. At all temperatures from 77°K through the phase transition at 97.2°K, the scattering was given accurately by the sum of a ridge and a [3J peak centered at (1,0,0), where the [3J part always had the line shape appropriate to the crystal

mosaic. Values for the integrated peak intensity with the background removed (including the ridge) are tabulated in Table II. The absolute value of the reduced magnetization was found by normalizing to the previous data in Ref. 11. In general, the agreement between these results and the previous ones is well within the experi­mental error. From Table II it may be seen that the [3J scattering persists up to ""'97.29°K, although there seems to be a definite rounding off of the intensity above 97.200K. This rounding off was not observed previously due to the poorer temperature control.

We now wish to fit the data in: Table II to a function of the form of Eq. (2) where each of B, TN, {3 must be determined from the fit. It is clear that the apparent rounding observed above 97.2°K may present some difficulties. In order to analyze the data the following procedure was adopted. A series of least-squares fits including successively higher temperature points were performed and the mean square % error was examined. The results of these fits are given in Table III. Up to to 97.198°K the quality of the fit actually seems to improve with increasing temperature. Also up to this temperature B, {3, and to a lesser extent TN, are quite insensitive to T max. However, when the data point at 97.240oK is included the fit becomes much worse, and finally the effect of the point at T=97.286°K is catastrophic.

We may conclude, therefore, that over the tem­perature range from 77° to 97.198°K, that is, for

TABLE III. Statistical analysis of K2NiF4 sublattice magnetization."

Final Mean square temperature % error B

96.970 0.164X10--a 0.980 97.101 0.148X 10--a 0.979 97.147 0.151X10--a 0.975 97.170 0.145X 1000a 0.974 97.198 0.133X10-3 0.973

97.240 0.231X10--a 0.996 97.286 0.330X10--2 1.094

Correlation matrix B TN {1

B (1.00 0.65 0.95)

TN 0.65 1.00 0.78

{1 0.95 0.78 1.00

97.252 97.248 97.233 97.230 97.229

97.245 97.287

68% support plane

confidence limits

B=±0.02

{1=±0.004

0.141 0.141 0.139 0.138 0.138

0.145 0.173

a The function minimized was always the sum of the SQ uares of the percentage deviations.

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TWO-DIMENSIONAL ANTIFERROMAGNETS 1309

2X1o-1>1-T/TN >3XIQ-4 the sublattice magnetiza­tion is accurately described by

M(T)/M( 4.2) =0.973(1- T/97.23)o.138. (9)

The fit to this function is shown in Fig. 6. Table III also includes the correlation matrix and the 68% support plane confidence limits.21-23 The latter are obtained by projecting the error ellipse in parameter space onto the parameter axes.23 TN in Eq. (21) is nearly 0.2°K higher than that in Eq. (13). This is due to the dif­ferent temperature control systems used for the two experiments; both give relative temperatures around TN to the accuracies quoted, but no attempt, par­ticularly in the previous experiments, was made to ascertain the absolute temperatures to more than ±0.2°K precision. The value of (3 we obtain for K2NiF4, 0.138±0.OO4, should be compared with 0.125 for the [2J Ising model and 0.33-0.40 for typical [3J systems.2,3

From Table II it may be seen that some [3J scattering occurs above TN(fitted) =97.23°K. That observed at 97.240oK can be explained on the basis of the uncer­tainty in the temperature of ±20 mdeg K but the point at 97.286°K lies outside of these limits. Possible explanations are (a) there is a spread in TN of this order, (b) we are seeing real [3J critical scattering effects. We cannot decide unambiguously between these two with our present data, although the former seems more likely.

B. Wavevector-Dependent Susceptibility in K2NiF4

The diffusive part of the scattering cross section in the quasi elastic approximation may be written2

(du/dn)D = A (k, k') (kT/ g2JJ.B2)

X L: [o"~-Q,,Q~Jx"~(Q), (to) ,,~

where x"~(Q) is the wavevector-dependent susceptibil-

1.0

.8

.S

~ i .5

"-;:: .4

i .3

10-2

I-T/TN

FIG. 6. Reduced sublattice magnetization vs temperature in K2NiF4•

z ~

1500

~ 1000

"-~ z :::> o u 500

0.8 0.9 1.0 h

o (h,O,.2)

• (h,O,-S.8)

1.1 1.2

FIG. 7. Two-dimensional critical scattering scans in K2NiF4•

ity. For a [2J system a simple Ornstein-Zernike form for x"" cannot be correct since it implies a pair correla­tion function which varies as lnr for large r. The simplest modification of the Lorentzian is

X"~(Q) 0: 1/(K,,~2+qx2+ql)I-(~/2), (11)

where, in the [2J S=t Ising model at K=O, l1=i. Experiments to measure x"~(Q) in K2NiF4 as a function of temperature are now under way. Neither the data analysis nor the experiments are complete at the time of the writing of this paper, so it is not possible to quote final values for the critical parameters. However, there are a number of qualitative results which are of interest.

The measurements were carried out using the experimental configuration discussed in Pt. A of this section. A series of scans were made across the ridge at (1,0,0.2) and (1,0, -6.8). If <J> is the angle the momentum transfer vector Q makes with the c axis, then

(du/dn) 0:.f(Q)[Sin2<llx11+(1-i- cos2<J>)x.1J, (12)

so that at (1,0,0.2),

(13)

whereas at (1,0, -6.8),

where f(Q) is the form factor, XII is the component of xa~ parallel to the anisotropy axis (c axis) and hence perpendicular to the planes, and X.1 is the component of xa~ within the planes. From Eqs. (13) and (14) we see that the two scans should enable us to separate out XII, X.1· Experimental results at tosOK are shown in Fig. 7.

The scan at (h, 0, 0.2) gives a well-defined Lorentzian­shaped peak with a linewidth which is approximately twice the resolution width; this corresponds to a correlation length 1/ K of approximately 400 A. At

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1310 BIRGENEAU, SKALYO, JR., AND SHIRANE

(h,O, -6.8), however, the scattering is barely above background. Since the ratios of the squares of the form factors for the two scans is only 0.5, this difference must arise from the different linear combination of XII, XJ.. Consideration of Eqs. (13), (14), and Fig. 7 leads one to conclude immediately that the intense diffusive peak at (1,0, 0.2) is associated almost entirely with XII whereas XJ. must be nearly flat. This same general pattern is observed at all temperatures both above and below TN.

This marked difference in character between XII and XJ. is at first quite surprising, since we know that the "exchange field" which is isotropic in space has a strength of ,...,450oK whereas the "anisotropy field" which distinguishes XII from XJ. is only O.85°K. If the phase transition in K2NiF4 were of the Stanley-Kaplan type1 as has been proposed by a number of authors,24,25 then since it arises from the isotropic part of the Hamiltonian alone we would not expect such a drastic difference between XII and XJ.. Thus the anisotropy must be playing a crucial role in the spin dynamics. We mav conclude therefore that at 97.23°K K2NiF4 undergoes an anisotropy-induced [2J phase transition.

A review of the spin wave results, which were carried out simultaneously with these measurements, shows that the transverse generalized susceptibility retains a spin wave character over most of the zone above TN; thus no appreciable Lorentzian diffuse scattering is ex­pected, as indeed none is observed. The longitudinal sus­ceptibility, on the other hand, is diffusive in form and xzz(O) is observed to diverge at TN as expected for an anisotropic antiferromagnet.

Another qualitative result which is of note is the behavior of XII(O) around TN. We find that XII(O) is extremely asymmetric in 11- T I TN 1 about TN, that is, XII(+)>>XII (_) at a given 11-TITN I. In mean field theory X" (+) Ix" ( - )':~d, whereas this ratio is ~5 for more realistic [3J .models, and ~37 for the [2J s=! Ising mode1.2 The gross asymmetry which we see, therefore, is again consistent with the [2J nature of the phase transition.

ACKNOWLEDGMENTS

We have benefited immensely from discussions of this work with M. Blume, B. I. Halperin, P. C. Hohenberg, M. E. Lines, and L. R. Walker.

* Work performed under the auspices of the U.S. Atomic Energy Commission.

1 For a review, see M. E. Lines, J. Appl. Phys. 40, 1352 (1969). 2 M. E. Fisher, Rep. Progr. Phys. 30, 615 (1967), Pt. I. 3 P. Heller, Rep. Progr. Phys. 30, 731 (1967), Pt. I. 4 L. P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E. Lewis,

V. Palcianskos, M. Rayl, J. Swift, D. Aspnes, and J. Kane, Rev. Mod. Phys. 39, 395 (1967).

5L. Onsager, Phys. Rev. 65, 117 (1944), Nuovo Cimento (Suppl.) 6, 261 (1949).

"N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).

7 H. E. Stanley and T. A. Kaplan, Phys. Rev. Lett. 17, 913 (1966) .

8 R. Kubo, Phys. Rev. 87, 568 (1952). 9 R. Plumier, J. Appl. Phys. 35, 950 (1964); J. Phys. Radium

24, 741 (1963). 10 M. E. Lines, Phys. Rev. 164, 736 (1967). 11 R. J. Birgeneau, H. J. Guggenheim, and G. Shirane, Phys.

Rev. Lett. 22, 720 (1969); Phys. Rev. (to be published). 12 J. Skalyo, Jr., G. Shirane, R. J. Birgeneau, and H. J. Guggen­

heim, Phys. Rev. Lett. 23, 1394, 1969. 13 D. Balz and K. Pleith, Z. Elektrochem. 59, 545 (1955). 14 W. Marshall and R. Lowde, Rep. Progr. Phys. 31,705 (1968),

p. II. 15 For purposes of comparison, see M. J. Cooper and R. Nathans,

J. Appl. Phys. 37, 1041 (1966). 16 Strictly speaking, the function of interest is M (T) / M (0) .

However, Birgeneau et at., (Ref. 11) 'point out that for all cases considered here this will be identical to M (T) / M (4.2) to within the errors quoted.

17 M. E. Lines (private communication). 18 R. J. Birgeneau, F. DeRosa, and H. J. Guggenheim, Solid

State Commun. (to be published). 19 For a review, see F. Keffer, Handbuch der Physik, S. Flugge,

Ed. (Springer-Verlag, Berlin, 1962), Vol. XVIII/2, p. 1. 20 K. C. Turberfield, A. Okazaki, and R. W. H. Stevenson,

Proc. Phys. Soc. (London) 85, 743 (1965). R. Nathans, F. Menzinger, and S. J. Pickart, J. Appl. Phys. 39, 1237 (1969).

21 D. W. Marquardt. J. Soc. Ind. Appl. Math. 11, 431 (1963). See also, D. W. Marquardt, "Leasts Squares Estimation of Non­linear Parameters," Share Program Library SDA 3094.

22 W. A. Burnette and C. S. Roberts, Bell Telephone Labora­tories Internal Rep. 1967 (unpublished).

23 H. Stone. discussion on paper by E. M. L. Beale, J. Roy. Statist. Soc., Ser. B, 22, 41 (1960).

24 D. Breed, Physica 37, 35 (1967). 2. E. P. Maarschall, A. C. Batterman, S. Vega, and A. R.

Midiema, Physica 41, 473 (1969).

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