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Local order parameter near the critical point: A Monte Carlo study
Solid State Communications,Vol. 15, pp. 1135—1139, 1974.
Pergamon Press.
Printed in Great Britain
LOCAL ORDER PARAMETER NEAR THE CRITICAL POINT: A MONTE CARLO STUDY H. Muller~Krumbhaar*and K.A. MUller IBM Zurich Research Laboratory, 8803 RUschlikon, Switzerland (Received 16 May 1974 by J.L. Olsen)
The probability distribution of the local magnetization in a Heisenberg ferromagnet has been investigated near the critical temperature. The distribution function depends strongly upon the number of atoms generating the local field. A cusp-like critical broadening is observed with different behavior above and below the critical temperature.
RECENT experiments with NMR and EPR techniques have aroused interest in the behavior of the resonance line width and line shape dependent on the choice of the material near critical points.1 EPR investigations2 of a structural phase transition in SrTiO 3 have shown a line broadening as T T~,changeover of the line broadening from fast-motion to slow-motion regime, and asymmetry of the width of the resonance lines with respect to T~.In principle, the same observation can also be made 3by NMR experiments near magnetic phase transitions. In systems with short-range interaction it may be assumed that these observations are caused by fluctuations in the local exchange and dipole field, which is produced by the neighboring magnetic atoms (i.e., the magnetic ~ on the resonant Br79, Br8’ in CrBr 3 19 in KMnF ferromagnets, or Mn~~ on F 3 or MnF2 antiferromagnets).’ Although the appearance of critical fluctuations near 7 is a well-known phenomenon,4 the detailed behavior of small groups of spins around the critical point has not been investigated so far.
results concerning the critical behavior are reported here. Though we do not claim to explain recent experi. ments in detail yet, the results should qualitively hold for all systems with short-range interaction.
The line shapes observed in resonance experiments may, in principle, assume two different forms. In the fast-motion regime, where the average frequency of thermal fluctuations is high compared with the character teristic of the experiment,ofthe shape isresonance essentiallyfrequency Lorentzian, a consequence the time integration over pair-correlation functions.6 Sufficiently close to the critical point, the thermal fluctuations have slowed down and the observation time is short compared with the relaxation time of the local fields. Therefore, in this slow-motion regime the
—~
Owing to the principal interest in this problem now, a Heisenberg ferromagnet was studied by means of the “selfconsistent” Monte Carlo method5 (SMC) above and below 7,. The probability distributions of various local quantities were investigated and a few principal *
IBM postdoctoral fellow from the Physik-Department E14 Technischen Universität, MUnchen, Deutschland. 1135
line shape reflects the statistical distribution of the 7 quasi-constant local fields. We thus restrict our investigation to the instantaneous probability distribution of the local order parameter of a classical Heisenberg ferromagnet, assuming that the relaxation times of the correlation functions are already much longer than the inverse frequency of the line width of the resonance experiment. A simple cubic model-system of N = 1 6~classical spins with nearest-neighbor interaction was treated by means of the selfconsistent Monte Carlo method.5 The probability distributions for the magnetization of one single spin (m 1), of two neighboring spins (m2) of
1136
LOCAL ORDER PARAMETER NEAR THE CRITICAL POINT
three spins in a line (m3) and of a cluster of one central spin and its six surrounding neighbors (m7) were calculated both above and below 7~,outside a region of lt I (T T~)/T~ I 0.02. Above T~the probability distributions P(mi), 1 = 1, 2, 3, 7, are symmetric around m1 = 0; in the ferromagnetic region below 1~the functions are asymmetric owing to the appearance of a spontaneous magnetization. —
Some results for the two-spin and seven-spin clusters above 1 are shown in Figs. 1(a) and (b). In Fig. 1(a) the seven-spin cluster probability distribution P(m7) shows a Gaussian form but with a flattened top as a consequence of the considerable correlation between particles as T approaches T~.Quite different is the two-spin distribution P(m2), exhibiting a nonsteady slope at m2 = 0, since P(m2) assumes a triangular form as T goes to infinity,
P(m
——
3
Vol. 15, No.6
where the probability distribution changes its slope unsteadily. This pronounced discontinuity occurs only if no additional contributions to the local magnetization from other spins through weak long-range forces become important, as for example magnetic dipole forces. The result would be a rounding of the discontinuity depending on the number of additional interacting spins as well as their effective interaction parameter. At temperatures slightly below T~,the probability density distributions F(m1) become asymmetric with respect to the position of their maximum [Fig. 2(a)]. Therefore the characteristic magnetization of the peak position is always larger than the expectation value m of the spdntaneous magnetization. This asymmetry is also clearly exhibited in the derivatives aP(m1)/am1 [Fig.2(b)]. However, the particular shape of these functions is due to the symmetry properties of the isotropic classical Heisenberg ferromagnet.
m
~-‘l.032 2
(a)
~ /
~
A 4 -IP(m,)
“
________
=
T —~1.O32 Tc 5~
ÔP(m,) ôm,
——
m~
~
(a)
_../
0
-1
+1
Q
me
,,~
(b) + —
‘V
\
/
~—_xJ
,_1
10
—
~
-
5
3’8 the first derivative .
.
0.975
~
ÔP(m,) öm,
m~
FIG. 1. (a) Probability distribution of the magnetization of a cluster of two spins (m2) and of seven spins (m7) above 7~.(b) Derivatives of the distribution functions (a), corresponding to resonance lines in the slowmotion regime. -
- —
/
-
In resonance experiments’ aP(m1)/am1 as shown in Fig. 1(b) would be measurea 2)/am instead. The aP(m 2 curve shows discontinuities
(b)
-
~ 10
-
15
-
20
-
— —
m®
—
FIG. 2. (a) Probability distribution of the magnetization
of a cluster of two spms (m
2) and of seven spins (m,) below 1~.(b) Derivatives oftthe distribution functions (a), corresponding to resonance lines in the slow-motion regime.
Vol. 15, No.6
LOCAL ORDER PARAMETER NEAR THE CRITICAL POINT
From the experimental point of view there is now more interest in antiferromagnetic materials,3 because of their low total magnetization in external fields. The situation is somewhat more complicated, as discussed in the following, The order parmeter of an antiferromgnet is not the total magnetization m but the sublattice magnetization rn.9 The local properties of this order parameter are the same as in the ferromagnet because of the equivalent Hamiltonian. The essential modification is due to the “staggered” orientation of magnetization in two sublattices. Below 1~,the local sublattice magnetization exhibits two peaks symmetrically about m 1 = 0, to be obtained from our ferromagnetic calculations by simply adding up the two sides of the distribution functions. Because of the linearity of this procedure the derivatives 3F(rn1)/am, analogous to Fig. 2(b) are obtained in the same way. The results are now quite different if the local quantity is composed of contributions from both sublattices simultaneously. The corresponding two-spin probability density distribution, for example, has the same triangular form for T oo as in the ferromagnetic case. At lower temperatures instead the appearing curvature is not concave as in the ferromagnet [Fig. 1(a)], but convex with a ~ which is even more pronounced at m = 0. This shape persists also below 1~without a deviation of the peak from its central position. Therefore, no double peak structure is to be observed below 1~,while the discontinuity in the denvative aP(m2)/a(m2) exists throughout the whole temperature range T ~ T~. -~
stress the factdepends that the strongly qualitative hehavior of theseWe local properties upon the number of particles involved in the specific averaging. Goin from two to about twenty particles considered at a time, the probability distribution will rapidly approach Gaussian form in accordance with the central limit theorem. The properties of the local order parameter (sublattice an antiferromagnet in somemagnetization) way similar to m1 theofcase observed by EPRare measurements of a structural phase transition, e.g., in SrTiO 2 There, the order parameter can be described 3. as “staggered” finite rotation of the Ti0 2 6-octahedron
1137
corresponding to some extent— to the staggered spins in an antiferromagnet. Since every resonant paramagnetic Fe~~ ion (substituting a Ti4~)“feels” the local field in its peculiar octahedron only, a doublepeak probability distribution below ~1,appears,8 like in the symmetricized ferromagnet. We mention briefly here that the situation can still be more complicated. If, for example, the effective molecular interaction between oxygen atoms in the antiferrodistortive case exhibits a double-well potential, two peaks in the probability distribution will then appear also above 7~,11 This would about correspond to an antiferromagnet with uniaxial anisotropy. —
A rather general feature of the local order parameter is the behavior of the line width of P(m1) here defined as the half-width &n1 near the critical ternperature. In Fig. 3 the half-widths ~m2 and z~m7are plotted, showing a cusp-like increase near the critical temperature. Owing to the finiteness of the local fields this cannot be a divergence. Characteristic is also the asymmetry of the effective distribution width about T~,the width always being larger above 1~at the same IT— T~ I. Qualitatively corresponding results have been found recently in EPR measurements of SrTiO3 ~2 —
—
For an antiferromagnet this asymmetry should be observable if the local magnetization is composed by contributions from an unequal number of spins from both sublattices as in the anisotropic MnF2 and FeF2 If the contributions come from an equal number of spins as in KMnF3 or RbMnF3, no marked critical broadening of the probability density distribution can be expected. This is already indicated by the non-dh~rgence of the total uniform susceptibility of the the anti9 Nevertheless ferromagnet at alloftemperatures. strong influence the particle number involved upon the distribution function may be observable in the slow-motion regime by appropriate NMR investigations. Therefore we discuss briefly the question, how an NMR experiment may be set up to investigate the discontinuity in the aP(m 1)/am2 function above T~. .~
In the antiferromagnetic substances 9 nucleus (I = 1/2)mentioned couples to above, the F’ nearest Mnresonant or Fe neighbors. The hyperime interaction (AH) between the F’9 nucleus and the Mn~or Fe~ leads to hyperfine fields of HHF = AH sigN/IN 40kG. Magnetic dipole fields can probably be neglected
1138
LOCAL ORDER PARAMETER NEAR THE CRITICAL POINT
f ~m,
Vol. 15, No.6
-
~ T:;~’~~-
FIG. 3. Critical broadening of the half-width of the probability distributions Fig. 1(a), Fig. 2(b), plotted vs temperature. Both curves have a cusp at 7~. because of the antiferromagnetic exchange and the large hyperfine constant. In FeF2 a line width of 3 kG could still be detected at I (T T~)/T~I 0.02.~Therefore we require an approximately 100-fold increased sensitivity which can be achieved by using advanced integration techniques and an external field of 20 kG. This field may also be sufficient to scan ±10 per cent, i.e. ±4 (7”— kG ofT~)/T~0.Ol the aP/am anomaly nearbem2 0. An estimated = r would at = least neccessary to reach the slow-motion regime in FeF 3 2 for rfor >0such andr’s conventional scanning techniques. However, crystal inhomogeneities in T~ can become shape determining. On the other hand, it is possible that other magnetic substances with a more extended critical region become known, or special resonance techniques can be used to overcome this difficulty.’ —
Of course, a large external field changes the Néel temperature and also the behavior of the system near
2~.However, the selfconsistent Monte Carlo method might also be used for obtaining quantitative results in such antiferromagnets below the critical temperature, in order to explain or to stimulate appropriate experiments. Finally, we would like that,form fromofathe 12 to themention qualitative synergetic of view, distributionpoint function close to 7~(Figs. 1 and 2) and its effective width (Fig. 3) appear to be meaningful also in the field of social sciences.13 There, the local fluctuation amplitudes of small groups participating in a cooperative phenomenon of a large entity obviously lead to most important effects on a single individual.
Acknowledgements We have benefited from discussions and comments with P. Heller and 1. Schneider. —
REFERENCES 1.
See the various contributions in Proc. mt. School Phys. “Enrico Fermi Summer School on Local Properties at Phase Transitions, Varenna, Italy, July 1973, (edited by MULLER K.A. and RIGAMONTI A.), Academic Press (to be published).
2.
VON WALDKIRCH T.H., MUELLER K.A. and BERLINGER W., Phys. Rev. B7, 1052(1973); MUELLER K.A., BERLINGER W., WEST C. and HELLER P., Phys. Rev. Lett. 32, 160 (1974).
3.
HELLER P., in reference I and GOTTLIEB A.M. and HELLER P., Phys. Rev. B3, 3615 (1971).
‘~
Vol. 15, No.6
LOCAL ORDER PARAMETER NEAR THE CRITICAL POINT
1139
4.
DOMB C. and GREEN M.S., Phase Transitions and critical Phenomena, Academic Press, New York (1972); STANLEY H.E., Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford (1971); FISCHER M.E., Rep. Prog. Phys. 30,615 (1967).
5.
MUELLER-KRUMBHAAR H. and BINDER K., Z. Phys. 254, 269 (1972); BINDER K. and MUELLER-KRUMBHAAR H.,Phys. Rev. B7, 3297 (1973).
6.
KUBO R. and TOMITAK.,J. Phys. Soc. Japan 9, 888 (1954); MORIYA T.,Progr. Theoret. Phys. (Kyoto) 28, 371 (1962).
7.
KUBO R., in Fluctuation, Resonance and Relaxation in Magnetic Systems, Scottish Universities Summer School (edited by TER HAAR D.) (1961).
8.
MUELLERK.A. and BERLINGERW.,Phys. Rev. Lett. 29, 715 (1972).
9.
HALPERIN B.I. and HOHENBERG P.C.,Phys. Rev. 177, 952 (1969).
10.
MUELLER-KRUMBHAAR H., (to be published).
11.
SCHNEIDER T. and STOLL E., in reference (1), also private communication.
12.
HAKEN H., Synergetics, Teubner Verlag, Stuttgart (1972).
13.
WEIDLICH W., in reference (12), p. 269.
Die Wahrscheinlichkeitsverteilung der lokalen Magnetisierung eines Heisenberg-Ferromagneten wurde in der Umgebung der kritischen Temperatur untersucht. Die Verteilungsfunktion hängt stark von der Anzahl von Atomen ab, welche das lokale Feld erzeugen. Es tritt kritische Verbreiterung auf mit einer Spitze bei der Curie-Temperatur und mit unterschiedlichem Verhalten oberhalb und unterhalb T~.
Pergamon Press.
Printed in Great Britain
LOCAL ORDER PARAMETER NEAR THE CRITICAL POINT: A MONTE CARLO STUDY H. Muller~Krumbhaar*and K.A. MUller IBM Zurich Research Laboratory, 8803 RUschlikon, Switzerland (Received 16 May 1974 by J.L. Olsen)
The probability distribution of the local magnetization in a Heisenberg ferromagnet has been investigated near the critical temperature. The distribution function depends strongly upon the number of atoms generating the local field. A cusp-like critical broadening is observed with different behavior above and below the critical temperature.
RECENT experiments with NMR and EPR techniques have aroused interest in the behavior of the resonance line width and line shape dependent on the choice of the material near critical points.1 EPR investigations2 of a structural phase transition in SrTiO 3 have shown a line broadening as T T~,changeover of the line broadening from fast-motion to slow-motion regime, and asymmetry of the width of the resonance lines with respect to T~.In principle, the same observation can also be made 3by NMR experiments near magnetic phase transitions. In systems with short-range interaction it may be assumed that these observations are caused by fluctuations in the local exchange and dipole field, which is produced by the neighboring magnetic atoms (i.e., the magnetic ~ on the resonant Br79, Br8’ in CrBr 3 19 in KMnF ferromagnets, or Mn~~ on F 3 or MnF2 antiferromagnets).’ Although the appearance of critical fluctuations near 7 is a well-known phenomenon,4 the detailed behavior of small groups of spins around the critical point has not been investigated so far.
results concerning the critical behavior are reported here. Though we do not claim to explain recent experi. ments in detail yet, the results should qualitively hold for all systems with short-range interaction.
The line shapes observed in resonance experiments may, in principle, assume two different forms. In the fast-motion regime, where the average frequency of thermal fluctuations is high compared with the character teristic of the experiment,ofthe shape isresonance essentiallyfrequency Lorentzian, a consequence the time integration over pair-correlation functions.6 Sufficiently close to the critical point, the thermal fluctuations have slowed down and the observation time is short compared with the relaxation time of the local fields. Therefore, in this slow-motion regime the
—~
Owing to the principal interest in this problem now, a Heisenberg ferromagnet was studied by means of the “selfconsistent” Monte Carlo method5 (SMC) above and below 7,. The probability distributions of various local quantities were investigated and a few principal *
IBM postdoctoral fellow from the Physik-Department E14 Technischen Universität, MUnchen, Deutschland. 1135
line shape reflects the statistical distribution of the 7 quasi-constant local fields. We thus restrict our investigation to the instantaneous probability distribution of the local order parameter of a classical Heisenberg ferromagnet, assuming that the relaxation times of the correlation functions are already much longer than the inverse frequency of the line width of the resonance experiment. A simple cubic model-system of N = 1 6~classical spins with nearest-neighbor interaction was treated by means of the selfconsistent Monte Carlo method.5 The probability distributions for the magnetization of one single spin (m 1), of two neighboring spins (m2) of
1136
LOCAL ORDER PARAMETER NEAR THE CRITICAL POINT
three spins in a line (m3) and of a cluster of one central spin and its six surrounding neighbors (m7) were calculated both above and below 7~,outside a region of lt I (T T~)/T~ I 0.02. Above T~the probability distributions P(mi), 1 = 1, 2, 3, 7, are symmetric around m1 = 0; in the ferromagnetic region below 1~the functions are asymmetric owing to the appearance of a spontaneous magnetization. —
Some results for the two-spin and seven-spin clusters above 1 are shown in Figs. 1(a) and (b). In Fig. 1(a) the seven-spin cluster probability distribution P(m7) shows a Gaussian form but with a flattened top as a consequence of the considerable correlation between particles as T approaches T~.Quite different is the two-spin distribution P(m2), exhibiting a nonsteady slope at m2 = 0, since P(m2) assumes a triangular form as T goes to infinity,
P(m
——
3
Vol. 15, No.6
where the probability distribution changes its slope unsteadily. This pronounced discontinuity occurs only if no additional contributions to the local magnetization from other spins through weak long-range forces become important, as for example magnetic dipole forces. The result would be a rounding of the discontinuity depending on the number of additional interacting spins as well as their effective interaction parameter. At temperatures slightly below T~,the probability density distributions F(m1) become asymmetric with respect to the position of their maximum [Fig. 2(a)]. Therefore the characteristic magnetization of the peak position is always larger than the expectation value m of the spdntaneous magnetization. This asymmetry is also clearly exhibited in the derivatives aP(m1)/am1 [Fig.2(b)]. However, the particular shape of these functions is due to the symmetry properties of the isotropic classical Heisenberg ferromagnet.
m
~-‘l.032 2
(a)
~ /
~
A 4 -IP(m,)
“
________
=
T —~1.O32 Tc 5~
ÔP(m,) ôm,
——
m~
~
(a)
_../
0
-1
+1
Q
me
,,~
(b) + —
‘V
\
/
~—_xJ
,_1
10
—
~
-
5
3’8 the first derivative .
.
0.975
~
ÔP(m,) öm,
m~
FIG. 1. (a) Probability distribution of the magnetization of a cluster of two spins (m2) and of seven spins (m7) above 7~.(b) Derivatives of the distribution functions (a), corresponding to resonance lines in the slowmotion regime. -
- —
/
-
In resonance experiments’ aP(m1)/am1 as shown in Fig. 1(b) would be measurea 2)/am instead. The aP(m 2 curve shows discontinuities
(b)
-
~ 10
-
15
-
20
-
— —
m®
—
FIG. 2. (a) Probability distribution of the magnetization
of a cluster of two spms (m
2) and of seven spins (m,) below 1~.(b) Derivatives oftthe distribution functions (a), corresponding to resonance lines in the slow-motion regime.
Vol. 15, No.6
LOCAL ORDER PARAMETER NEAR THE CRITICAL POINT
From the experimental point of view there is now more interest in antiferromagnetic materials,3 because of their low total magnetization in external fields. The situation is somewhat more complicated, as discussed in the following, The order parmeter of an antiferromgnet is not the total magnetization m but the sublattice magnetization rn.9 The local properties of this order parameter are the same as in the ferromagnet because of the equivalent Hamiltonian. The essential modification is due to the “staggered” orientation of magnetization in two sublattices. Below 1~,the local sublattice magnetization exhibits two peaks symmetrically about m 1 = 0, to be obtained from our ferromagnetic calculations by simply adding up the two sides of the distribution functions. Because of the linearity of this procedure the derivatives 3F(rn1)/am, analogous to Fig. 2(b) are obtained in the same way. The results are now quite different if the local quantity is composed of contributions from both sublattices simultaneously. The corresponding two-spin probability density distribution, for example, has the same triangular form for T oo as in the ferromagnetic case. At lower temperatures instead the appearing curvature is not concave as in the ferromagnet [Fig. 1(a)], but convex with a ~ which is even more pronounced at m = 0. This shape persists also below 1~without a deviation of the peak from its central position. Therefore, no double peak structure is to be observed below 1~,while the discontinuity in the denvative aP(m2)/a(m2) exists throughout the whole temperature range T ~ T~. -~
stress the factdepends that the strongly qualitative hehavior of theseWe local properties upon the number of particles involved in the specific averaging. Goin from two to about twenty particles considered at a time, the probability distribution will rapidly approach Gaussian form in accordance with the central limit theorem. The properties of the local order parameter (sublattice an antiferromagnet in somemagnetization) way similar to m1 theofcase observed by EPRare measurements of a structural phase transition, e.g., in SrTiO 2 There, the order parameter can be described 3. as “staggered” finite rotation of the Ti0 2 6-octahedron
1137
corresponding to some extent— to the staggered spins in an antiferromagnet. Since every resonant paramagnetic Fe~~ ion (substituting a Ti4~)“feels” the local field in its peculiar octahedron only, a doublepeak probability distribution below ~1,appears,8 like in the symmetricized ferromagnet. We mention briefly here that the situation can still be more complicated. If, for example, the effective molecular interaction between oxygen atoms in the antiferrodistortive case exhibits a double-well potential, two peaks in the probability distribution will then appear also above 7~,11 This would about correspond to an antiferromagnet with uniaxial anisotropy. —
A rather general feature of the local order parameter is the behavior of the line width of P(m1) here defined as the half-width &n1 near the critical ternperature. In Fig. 3 the half-widths ~m2 and z~m7are plotted, showing a cusp-like increase near the critical temperature. Owing to the finiteness of the local fields this cannot be a divergence. Characteristic is also the asymmetry of the effective distribution width about T~,the width always being larger above 1~at the same IT— T~ I. Qualitatively corresponding results have been found recently in EPR measurements of SrTiO3 ~2 —
—
For an antiferromagnet this asymmetry should be observable if the local magnetization is composed by contributions from an unequal number of spins from both sublattices as in the anisotropic MnF2 and FeF2 If the contributions come from an equal number of spins as in KMnF3 or RbMnF3, no marked critical broadening of the probability density distribution can be expected. This is already indicated by the non-dh~rgence of the total uniform susceptibility of the the anti9 Nevertheless ferromagnet at alloftemperatures. strong influence the particle number involved upon the distribution function may be observable in the slow-motion regime by appropriate NMR investigations. Therefore we discuss briefly the question, how an NMR experiment may be set up to investigate the discontinuity in the aP(m 1)/am2 function above T~. .~
In the antiferromagnetic substances 9 nucleus (I = 1/2)mentioned couples to above, the F’ nearest Mnresonant or Fe neighbors. The hyperime interaction (AH) between the F’9 nucleus and the Mn~or Fe~ leads to hyperfine fields of HHF = AH sigN/IN 40kG. Magnetic dipole fields can probably be neglected
1138
LOCAL ORDER PARAMETER NEAR THE CRITICAL POINT
f ~m,
Vol. 15, No.6
-
~ T:;~’~~-
FIG. 3. Critical broadening of the half-width of the probability distributions Fig. 1(a), Fig. 2(b), plotted vs temperature. Both curves have a cusp at 7~. because of the antiferromagnetic exchange and the large hyperfine constant. In FeF2 a line width of 3 kG could still be detected at I (T T~)/T~I 0.02.~Therefore we require an approximately 100-fold increased sensitivity which can be achieved by using advanced integration techniques and an external field of 20 kG. This field may also be sufficient to scan ±10 per cent, i.e. ±4 (7”— kG ofT~)/T~0.Ol the aP/am anomaly nearbem2 0. An estimated = r would at = least neccessary to reach the slow-motion regime in FeF 3 2 for rfor >0such andr’s conventional scanning techniques. However, crystal inhomogeneities in T~ can become shape determining. On the other hand, it is possible that other magnetic substances with a more extended critical region become known, or special resonance techniques can be used to overcome this difficulty.’ —
Of course, a large external field changes the Néel temperature and also the behavior of the system near
2~.However, the selfconsistent Monte Carlo method might also be used for obtaining quantitative results in such antiferromagnets below the critical temperature, in order to explain or to stimulate appropriate experiments. Finally, we would like that,form fromofathe 12 to themention qualitative synergetic of view, distributionpoint function close to 7~(Figs. 1 and 2) and its effective width (Fig. 3) appear to be meaningful also in the field of social sciences.13 There, the local fluctuation amplitudes of small groups participating in a cooperative phenomenon of a large entity obviously lead to most important effects on a single individual.
Acknowledgements We have benefited from discussions and comments with P. Heller and 1. Schneider. —
REFERENCES 1.
See the various contributions in Proc. mt. School Phys. “Enrico Fermi Summer School on Local Properties at Phase Transitions, Varenna, Italy, July 1973, (edited by MULLER K.A. and RIGAMONTI A.), Academic Press (to be published).
2.
VON WALDKIRCH T.H., MUELLER K.A. and BERLINGER W., Phys. Rev. B7, 1052(1973); MUELLER K.A., BERLINGER W., WEST C. and HELLER P., Phys. Rev. Lett. 32, 160 (1974).
3.
HELLER P., in reference I and GOTTLIEB A.M. and HELLER P., Phys. Rev. B3, 3615 (1971).
‘~
Vol. 15, No.6
LOCAL ORDER PARAMETER NEAR THE CRITICAL POINT
1139
4.
DOMB C. and GREEN M.S., Phase Transitions and critical Phenomena, Academic Press, New York (1972); STANLEY H.E., Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford (1971); FISCHER M.E., Rep. Prog. Phys. 30,615 (1967).
5.
MUELLER-KRUMBHAAR H. and BINDER K., Z. Phys. 254, 269 (1972); BINDER K. and MUELLER-KRUMBHAAR H.,Phys. Rev. B7, 3297 (1973).
6.
KUBO R. and TOMITAK.,J. Phys. Soc. Japan 9, 888 (1954); MORIYA T.,Progr. Theoret. Phys. (Kyoto) 28, 371 (1962).
7.
KUBO R., in Fluctuation, Resonance and Relaxation in Magnetic Systems, Scottish Universities Summer School (edited by TER HAAR D.) (1961).
8.
MUELLERK.A. and BERLINGERW.,Phys. Rev. Lett. 29, 715 (1972).
9.
HALPERIN B.I. and HOHENBERG P.C.,Phys. Rev. 177, 952 (1969).
10.
MUELLER-KRUMBHAAR H., (to be published).
11.
SCHNEIDER T. and STOLL E., in reference (1), also private communication.
12.
HAKEN H., Synergetics, Teubner Verlag, Stuttgart (1972).
13.
WEIDLICH W., in reference (12), p. 269.
Die Wahrscheinlichkeitsverteilung der lokalen Magnetisierung eines Heisenberg-Ferromagneten wurde in der Umgebung der kritischen Temperatur untersucht. Die Verteilungsfunktion hängt stark von der Anzahl von Atomen ab, welche das lokale Feld erzeugen. Es tritt kritische Verbreiterung auf mit einer Spitze bei der Curie-Temperatur und mit unterschiedlichem Verhalten oberhalb und unterhalb T~.