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The critical behavior of a weak ferromagnet
Solid State Communications Vol. 4, pp. 147-151, 1966. Pergamon Press Ltd. Printed in Great Britain.
THE CRITICAL BEHAVIOR OF A WEAK FERROMAGNET* § G. Gorodetsky, S. Shtrikman and ID. Trevesli Department of Electronics, The Welzmann Institute of Science, Rehovoth, Israel (Received 7 February 1966 by E. Burstein) Measurements oil the temperature dependence of the magnetization in a weak ferromagnet YFeO3, along the ferromagnetic direction in the critical region 0.996 < T/Tc < 1.003, are reported. Analysis of the data gives the following results for the critical exponents: B = 0.55 * 0.04, ~‘= 1.33*0.04, y’ = 0.7±0.1, 1/6 = 0.36 *0.03. FOLLOWING THE RECENT attention paid to the magnetic phase transition, we have made measurements of the critical behavior of the weak ferromagnet YFeO3. The results are reported below. The following set of relations for: the reduced spontaneous magnetization (1), the initial susceptibility (2) and the magnetic moment vs. field critical isotherm (3), are believed to hold near the Curie point. 1 =
x (T)
D(1
-
T/T~)B
~ T
=
(-~-~
C =
H=O (T-T~)”
=
C’ (T -T)~’
(T
Tc)
(1)
(T n Tc)
(2)
S
temperature Tc 643~K 2 and the spontaneous magnetic moment parallel to the c axis. The measurements of the magnetic momTheonhysteresis loop was found be ent were3done a carefully chosen singletocrystal (10 mg).in the entire temperature region T < Tc. square This confIrme~1that the specimen was in a single domain state. ~ The magnetic moment was continuously measured using a vibrating sample magnetometer. The magnetic field was applied along the c axis. The criterion for the rate of change of the temperature the disappearance of the thermal hysteresis. was Temperature reproducibility was within * 0.02°C. The magnetic field dependence of the e. m.f., of the chromelalumel thermocouple that was used, was corrected for. The same results were also obtained using a copper-constantan thermocouple. The standard error of the magnetic moment measurement was less than * 3 per cent.
(T s T ) C
Figure 1 a shows the measured temperature dependence of the magnetic moment with the applied field as a parameter, the vicinity the Curie temperature.5 The In accuracy of theof
C
c~(H)= BH
“
(T
=
T C)
(3)
Curie point determination is extremely important in analysing these results. This,point was de-
We have attempted a measurement of these critical exponents in YFeO 3. This material is an orthohombic weak ferromagnet with a Curie
termined from (a) the isotherm o’ the = f(H/~) 6 and (b) peak that In the passes through the origin
*The research in this document has been sponsored in part by the Air Force Materials Laboratory, Research and Technology Division, AFSC through the European Office of Aerospace Research, U. S. Air Force. §This work was done in partial fulfillment of the Ph.D. requirement by one of the aurhots (0. G.). ¶Temporary address: Ampex Corporation, California, U. S. A. 147
148
THE CRITICAL BEHAVIOR OF A WEAK FERROMAGNET I
—
I
I
I
I
I
Vol. 4, No. 3 I
4.~lO2
-
b
~
2
A
-
~
o— b —
422 0. 922
0.
~
0
1.004
I
1.002
0.998
I
I
0.996
I/Ic
FIG. 1 (a) Magnetization vs. temperature of single Crystal YFeO3 with the applied magnetic field as a parameter. The broken line Is the magnetization extrapolated to H = 0. (b) The susceptibility In the direction of the spontaneous magnetic moment, calculated for different fields, from the results of Fig. la. 7 (see Fig. ib). ferromagnetic Both results agreed susceptibility to within the temperature reproducibility. By plotting log ~ 5(T)/c5(O) vs. log (1-T/Tc) it was found that B = 0. 55 *0.04 and that D = 3.9 (see Fig. 2a). Measurements on another YFeO3 Itislnterestlngtonotethat 8=0.54±0.05. crystal, weighIng 8 mg, gave ad analysis of the magnetostatic measurements reported by Smith8 on magnetite for T/T~< 0.998, gives 8~0. 53 and D 3. 5. The exponent
~‘
was calculated following
that in the 1<° T/T~< 1.001, (see y = Fig. 1.33 2b) * Kouvel and range Fisher. It was found 0.04 and that at higher temperatures, y decreases. Measurements on the 8 mg crystal gave y 1.3.
The value of y’ in the region 1 > T/TC > 0. ~ was obtained by (a) applying the method of Kouvel and Fisher to ~(T) ~5(T) and (b) from the ratio (c1H)/(~M)obtained from the magnetization vs. field isotherms, as H — 0. 2ckIt was We found that, y’ =1tr3andC’~1.6. findC~3.5 0. 7 *0. 1 (see Fig. 10 (emu (0K)Y/Oe. mole). -
The magnitude of the exponent 1/a was derived directly from the experimental results and was found t 9 6/mole). pe 0. 36 * 0.03 (Fig. 2d), D 1.6 (emu. (Oe)’/ A comparison of our results with theory and the available experimental data is given in
THE CRITICAL BEHAVIOR OF A WEAK FERROMAGNET
Vol. 4, No. 3
I1T/TcI
11T1’TdI 2.10
u.n.,,,.
10’
—
O~(T)/Ui(0)•O(11ITc
b
149
P.0.55*0.04
-——
-a.-
1
IT’ Tc)
...t_ X’.C’(T~~T)~’
~
Tc)
0.7* 0.1 (7’ — k..vSI S Fis~r
—‘
—
0 —
£
—
2.I0~~
...
J 14—0
2,0?
~oI
b I 02
102 3 E
ft
-
/
$/$0.36t0.05(TTc(
— U
——— __.1_
I— — — —— ,
.~
-—-s
(‘~Tc)
1~
101
e
.—
—c~—~e7
7.1.33*0.04
/
50
.4
x;’
* O —
d
~(T_T~)_~$hI$
,~
‘
— —
I
io2
2.101
IO~
ii[o.]
It.~T/TCI
FIG.2
Logarithmic plots of (a) reduced spontaneous magnetic moment vs. reduced temperature, (b) initial reciprocal susceptibility above the Curie point vs. reduced temperature, (c) initial reciprocal susceptibility below the Curie point vs. reduced temperature and (d) magnetic moment vs. field at the critical point Tc. Table 1. It is seen that our results for B and a are in agreement with the classical theory.9 The exponent B changes from 1/2 very close to T~to 1/3 as one goes further away from Tc. A1J This change is in accordance with the theoretical work of Callen and Callen” and with Mössbauer experiments on Fe57 In nickel. 12 -~
The value obtained for the exponent immediately above T~,agrees with that derived from the Heinsberg model by the high temperatore series expansion)3 Similar values for y have been reported recently in several materlals. 6, 14-li Thus the magnitude of ~ may follow a general property of second order phase transitions.
Finally a word ~hou1d be said about the specific heat anomaly. The zero fie1c~specific heat is assumed to vary as (Tc T)~ (& ~ 0), where T approaches T~,from below)~ The interrelation of a’ and other critical exponents is determined by the following thermodynamic inequalities18~19 2$
‘~‘,
+
‘?~ 2
-
a’
and (1
+
6) B ~ 2
-
a.’.
It follows from the first one that a’
150
THE CRITICAL BEHAVIOR OF A WEAK FERROMAGNET
Vol. 4, No. 3
TABLE 1 Theoretical and Experimental Values for the Critical Exponents Exponent B y
i/a
Theory (three dimensional Heisenberg model)
Magnitude
Molecular field, Landau theory, 9 ~wo spin cluster. GreenfunctionR.P.A. and C.D.1 Molecular field, Landau theory. High temperature series expansion, Padé approximant. 13 Green function. 20
1 4/3 2
Molecular field, Landau theory.
1
Molecular field, Landau theory.
1/3
1/2
Experiment MnF 21 72 Fe in Ni
8
0.33*0.01 2
YFeO 3, Fe304
NI 6, 14, Fe 15, 16
~
17
0. 5
YFeO3
YFeO3 1/~
Ni 6
—
1. 33
0.7*0.1 17
-~0.25
YFeO3
0.36*0.03
is necessarily ~ 0. 1 and from the second, that a’ ~ 0. Thus the present results are at least
not in contradiction with the general belief namely ~‘ 0. 1
References 1. 2.
FISHER M.E., Lectures in Theoretical Physics, Vol. VUc, p. 1. University of Colorado Press, Boulder, Colorado (1965). EIBSCHtITZ M., GORODETSKY G., SHTRIKMAN S. and TREVES D., J. Appi. Phys. Suppl. 35, 1071 (1964).
3.
The crystals were kindly supplied by J. P. Remeika, Bell Telephone Labs.
4.
REICH S., SHTRIKMAN S. and TREVES D., J. Appl. Phys. 36, 140 (1965).
5.
As these materials are very weakly ferromagnetic, demagnetization effects can be neglected.
6.
KOUVEL J. S. and FISHER M. E., Phys. Rev. 136, A1626 (1964).
7.
MORIYA T., Magnetism, Vol. 1, p. 86. Academic Press, Inc., New York (1963).
8.
SMITH D. 0., Phys. Rev. 102, 959 (1956).
—
Vol. 4, No. 3 9.
THE CRITICAL BEHAVIOR OF A WEAK FERROMAGNET
BELOV K. P., Magnetic Transition. Inc., New York (1961).
151
Translation from Russian. Consultants Bureau Enterprises,
10.
EIBSCHUTZ M., SHTRIKMAN S. and TREVES D., Solid State Communications 4, 141 (1966). The same results hold also for the ferromagnetic moment, as It was found that the canting angle is independent of T for T/TC < 0.99. TREVES D., J. Appl. Phys. Suppl. 36, 1033 (1965).
11.
CALLEN E. and CALLEN H., J. Appl. Phys. Suppl. 36, 1140 (1965).
12.
HOWABI) D. G., DUNLAP B.D. and DASH J. G., Phys. Rev. Letters 15, 628 (1965).
13.
DOMB C. and SYKES M. F., Phys. Rev. 128, 168 (1962); BAKER G. A., Phys. Rev. 129, 99 (1963); GAMMEL J., MARSHALL W. and MORGAN L., Proc. Roy. Soc. (London) A2’T~257 (1963).
14.
ARAJS S., J. Appl. Phys. Suppl. 36, 1136 (1965).
15.
ARAJS S. and COLVIN R.V., J. Appi. Phys. 35, 2424 (1964).
16.
NOAKES J. E. and ARROTT A., J. Appl. Phys. 35, 931 (1964).
17.
GRAHAM C.D.,Jr., J. Appl. Phys. Suppl. 36, 1135 (1965).
18.
RUSHBROOKE G.S., J. Chem. Phys. 39, 842 (1963).
19.
GRIFFITHS R. B., Phys. Rev. Letters 14, 623 (1965).
20.
TAHIR-KBELI R., Phys. Rev. 132, 689 (1963).
21.
HELLER P. and BENEDEIC G.E., Phys. Rev. Letters 8, 428 (1962).
,YFeO3 0,996 !n’c~-tpi
“V’~tI
D3~D~I15”~V
T/T~ C 1..003
ri~Tnn~i~
‘D’lpfl
~ ofti~
O’K~n a’~i~n ~XZ~3
0,7 + 0.1.
,y
1.33
4
,51315fl
,‘1’5~IbD~. I1t’~3~3~I~TD3 II’~I~fl1~i231J~1flJ~’3~
0,04
,
.i./6
8
0.55 • 0.04 0,36
+
0.03
THE CRITICAL BEHAVIOR OF A WEAK FERROMAGNET* § G. Gorodetsky, S. Shtrikman and ID. Trevesli Department of Electronics, The Welzmann Institute of Science, Rehovoth, Israel (Received 7 February 1966 by E. Burstein) Measurements oil the temperature dependence of the magnetization in a weak ferromagnet YFeO3, along the ferromagnetic direction in the critical region 0.996 < T/Tc < 1.003, are reported. Analysis of the data gives the following results for the critical exponents: B = 0.55 * 0.04, ~‘= 1.33*0.04, y’ = 0.7±0.1, 1/6 = 0.36 *0.03. FOLLOWING THE RECENT attention paid to the magnetic phase transition, we have made measurements of the critical behavior of the weak ferromagnet YFeO3. The results are reported below. The following set of relations for: the reduced spontaneous magnetization (1), the initial susceptibility (2) and the magnetic moment vs. field critical isotherm (3), are believed to hold near the Curie point. 1 =
x (T)
D(1
-
T/T~)B
~ T
=
(-~-~
C =
H=O (T-T~)”
=
C’ (T -T)~’
(T
Tc)
(1)
(T n Tc)
(2)
S
temperature Tc 643~K 2 and the spontaneous magnetic moment parallel to the c axis. The measurements of the magnetic momTheonhysteresis loop was found be ent were3done a carefully chosen singletocrystal (10 mg).in the entire temperature region T < Tc. square This confIrme~1that the specimen was in a single domain state. ~ The magnetic moment was continuously measured using a vibrating sample magnetometer. The magnetic field was applied along the c axis. The criterion for the rate of change of the temperature the disappearance of the thermal hysteresis. was Temperature reproducibility was within * 0.02°C. The magnetic field dependence of the e. m.f., of the chromelalumel thermocouple that was used, was corrected for. The same results were also obtained using a copper-constantan thermocouple. The standard error of the magnetic moment measurement was less than * 3 per cent.
(T s T ) C
Figure 1 a shows the measured temperature dependence of the magnetic moment with the applied field as a parameter, the vicinity the Curie temperature.5 The In accuracy of theof
C
c~(H)= BH
“
(T
=
T C)
(3)
Curie point determination is extremely important in analysing these results. This,point was de-
We have attempted a measurement of these critical exponents in YFeO 3. This material is an orthohombic weak ferromagnet with a Curie
termined from (a) the isotherm o’ the = f(H/~) 6 and (b) peak that In the passes through the origin
*The research in this document has been sponsored in part by the Air Force Materials Laboratory, Research and Technology Division, AFSC through the European Office of Aerospace Research, U. S. Air Force. §This work was done in partial fulfillment of the Ph.D. requirement by one of the aurhots (0. G.). ¶Temporary address: Ampex Corporation, California, U. S. A. 147
148
THE CRITICAL BEHAVIOR OF A WEAK FERROMAGNET I
—
I
I
I
I
I
Vol. 4, No. 3 I
4.~lO2
-
b
~
2
A
-
~
o— b —
422 0. 922
0.
~
0
1.004
I
1.002
0.998
I
I
0.996
I/Ic
FIG. 1 (a) Magnetization vs. temperature of single Crystal YFeO3 with the applied magnetic field as a parameter. The broken line Is the magnetization extrapolated to H = 0. (b) The susceptibility In the direction of the spontaneous magnetic moment, calculated for different fields, from the results of Fig. la. 7 (see Fig. ib). ferromagnetic Both results agreed susceptibility to within the temperature reproducibility. By plotting log ~ 5(T)/c5(O) vs. log (1-T/Tc) it was found that B = 0. 55 *0.04 and that D = 3.9 (see Fig. 2a). Measurements on another YFeO3 Itislnterestlngtonotethat 8=0.54±0.05. crystal, weighIng 8 mg, gave ad analysis of the magnetostatic measurements reported by Smith8 on magnetite for T/T~< 0.998, gives 8~0. 53 and D 3. 5. The exponent
~‘
was calculated following
that in the 1<° T/T~< 1.001, (see y = Fig. 1.33 2b) * Kouvel and range Fisher. It was found 0.04 and that at higher temperatures, y decreases. Measurements on the 8 mg crystal gave y 1.3.
The value of y’ in the region 1 > T/TC > 0. ~ was obtained by (a) applying the method of Kouvel and Fisher to ~(T) ~5(T) and (b) from the ratio (c1H)/(~M)obtained from the magnetization vs. field isotherms, as H — 0. 2ckIt was We found that, y’ =1tr3andC’~1.6. findC~3.5 0. 7 *0. 1 (see Fig. 10 (emu (0K)Y/Oe. mole). -
The magnitude of the exponent 1/a was derived directly from the experimental results and was found t 9 6/mole). pe 0. 36 * 0.03 (Fig. 2d), D 1.6 (emu. (Oe)’/ A comparison of our results with theory and the available experimental data is given in
THE CRITICAL BEHAVIOR OF A WEAK FERROMAGNET
Vol. 4, No. 3
I1T/TcI
11T1’TdI 2.10
u.n.,,,.
10’
—
O~(T)/Ui(0)•O(11ITc
b
149
P.0.55*0.04
-——
-a.-
1
IT’ Tc)
...t_ X’.C’(T~~T)~’
~
Tc)
0.7* 0.1 (7’ — k..vSI S Fis~r
—‘
—
0 —
£
—
2.I0~~
...
J 14—0
2,0?
~oI
b I 02
102 3 E
ft
-
/
$/$0.36t0.05(TTc(
— U
——— __.1_
I— — — —— ,
.~
-—-s
(‘~Tc)
1~
101
e
.—
—c~—~e7
7.1.33*0.04
/
50
.4
x;’
* O —
d
~(T_T~)_~$hI$
,~
‘
— —
I
io2
2.101
IO~
ii[o.]
It.~T/TCI
FIG.2
Logarithmic plots of (a) reduced spontaneous magnetic moment vs. reduced temperature, (b) initial reciprocal susceptibility above the Curie point vs. reduced temperature, (c) initial reciprocal susceptibility below the Curie point vs. reduced temperature and (d) magnetic moment vs. field at the critical point Tc. Table 1. It is seen that our results for B and a are in agreement with the classical theory.9 The exponent B changes from 1/2 very close to T~to 1/3 as one goes further away from Tc. A1J This change is in accordance with the theoretical work of Callen and Callen” and with Mössbauer experiments on Fe57 In nickel. 12 -~
The value obtained for the exponent immediately above T~,agrees with that derived from the Heinsberg model by the high temperatore series expansion)3 Similar values for y have been reported recently in several materlals. 6, 14-li Thus the magnitude of ~ may follow a general property of second order phase transitions.
Finally a word ~hou1d be said about the specific heat anomaly. The zero fie1c~specific heat is assumed to vary as (Tc T)~ (& ~ 0), where T approaches T~,from below)~ The interrelation of a’ and other critical exponents is determined by the following thermodynamic inequalities18~19 2$
‘~‘,
+
‘?~ 2
-
a’
and (1
+
6) B ~ 2
-
a.’.
It follows from the first one that a’
150
THE CRITICAL BEHAVIOR OF A WEAK FERROMAGNET
Vol. 4, No. 3
TABLE 1 Theoretical and Experimental Values for the Critical Exponents Exponent B y
i/a
Theory (three dimensional Heisenberg model)
Magnitude
Molecular field, Landau theory, 9 ~wo spin cluster. GreenfunctionR.P.A. and C.D.1 Molecular field, Landau theory. High temperature series expansion, Padé approximant. 13 Green function. 20
1 4/3 2
Molecular field, Landau theory.
1
Molecular field, Landau theory.
1/3
1/2
Experiment MnF 21 72 Fe in Ni
8
0.33*0.01 2
YFeO 3, Fe304
NI 6, 14, Fe 15, 16
~
17
0. 5
YFeO3
YFeO3 1/~
Ni 6
—
1. 33
0.7*0.1 17
-~0.25
YFeO3
0.36*0.03
is necessarily ~ 0. 1 and from the second, that a’ ~ 0. Thus the present results are at least
not in contradiction with the general belief namely ~‘ 0. 1
References 1. 2.
FISHER M.E., Lectures in Theoretical Physics, Vol. VUc, p. 1. University of Colorado Press, Boulder, Colorado (1965). EIBSCHtITZ M., GORODETSKY G., SHTRIKMAN S. and TREVES D., J. Appi. Phys. Suppl. 35, 1071 (1964).
3.
The crystals were kindly supplied by J. P. Remeika, Bell Telephone Labs.
4.
REICH S., SHTRIKMAN S. and TREVES D., J. Appl. Phys. 36, 140 (1965).
5.
As these materials are very weakly ferromagnetic, demagnetization effects can be neglected.
6.
KOUVEL J. S. and FISHER M. E., Phys. Rev. 136, A1626 (1964).
7.
MORIYA T., Magnetism, Vol. 1, p. 86. Academic Press, Inc., New York (1963).
8.
SMITH D. 0., Phys. Rev. 102, 959 (1956).
—
Vol. 4, No. 3 9.
THE CRITICAL BEHAVIOR OF A WEAK FERROMAGNET
BELOV K. P., Magnetic Transition. Inc., New York (1961).
151
Translation from Russian. Consultants Bureau Enterprises,
10.
EIBSCHUTZ M., SHTRIKMAN S. and TREVES D., Solid State Communications 4, 141 (1966). The same results hold also for the ferromagnetic moment, as It was found that the canting angle is independent of T for T/TC < 0.99. TREVES D., J. Appl. Phys. Suppl. 36, 1033 (1965).
11.
CALLEN E. and CALLEN H., J. Appl. Phys. Suppl. 36, 1140 (1965).
12.
HOWABI) D. G., DUNLAP B.D. and DASH J. G., Phys. Rev. Letters 15, 628 (1965).
13.
DOMB C. and SYKES M. F., Phys. Rev. 128, 168 (1962); BAKER G. A., Phys. Rev. 129, 99 (1963); GAMMEL J., MARSHALL W. and MORGAN L., Proc. Roy. Soc. (London) A2’T~257 (1963).
14.
ARAJS S., J. Appl. Phys. Suppl. 36, 1136 (1965).
15.
ARAJS S. and COLVIN R.V., J. Appi. Phys. 35, 2424 (1964).
16.
NOAKES J. E. and ARROTT A., J. Appl. Phys. 35, 931 (1964).
17.
GRAHAM C.D.,Jr., J. Appl. Phys. Suppl. 36, 1135 (1965).
18.
RUSHBROOKE G.S., J. Chem. Phys. 39, 842 (1963).
19.
GRIFFITHS R. B., Phys. Rev. Letters 14, 623 (1965).
20.
TAHIR-KBELI R., Phys. Rev. 132, 689 (1963).
21.
HELLER P. and BENEDEIC G.E., Phys. Rev. Letters 8, 428 (1962).
,YFeO3 0,996 !n’c~-tpi
“V’~tI
D3~D~I15”~V
T/T~ C 1..003
ri~Tnn~i~
‘D’lpfl
~ ofti~
O’K~n a’~i~n ~XZ~3
0,7 + 0.1.
,y
1.33
4
,51315fl
,‘1’5~IbD~. I1t’~3~3~I~TD3 II’~I~fl1~i231J~1flJ~’3~
0,04
,
.i./6
8
0.55 • 0.04 0,36
+
0.03