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EPR lineshape study of the incommensurate phase in γ-irradiated K2SeO4
Solid State Communications, Vol. 37, pp. 123—125. Pergamon Press Ltd. 1981. Printed in Great Britain.
EPA LINESHAPE STUDY
OF THE INCOMMENSURATE PHASE A.S.
Chaves and R.
IN y-IRRADIATEO K
2SeO4
Gazzinelli
Departamento de Fisica Universidade Federal de Minas Gerais Caixa Postal 702 6810 Horizonte - Brazil R. Blinc 3. Stefan Institute, University of Ljubljana Ljubljana, Yugoslavia (Received September
4
1960 by R.C.C. Leite) 4 EPR frequencies and The the temperature asyrimetric dependence broadening of of the the SeQ ~PR lines in the incommensurate phase of K SeQ can be explained by an incommensurate spatial mo~ula~ionof the g tensors which corresponds to the ‘broad” phase soliton limit. A comparison between the experimental and calculated lineshape shows a ~l% volume fraction of commensurate regions in the middle of the incommensurate phase at 110 K.
(1,2) K SeO was found by Aiki to un2two4succesSive phase tran~3fformstions dergo at T 1 129.5K and T = 93K. The high temperature paraelect~ic (Pj5phase tains four (T > T1) is SeQ orthorhombic (02h) and con4 groups. The low tempera ture commensurate (C) phase IT < T ) is ferroelectric and monoclinic (C~~)cwith the spontaneous polarization appearing along the c-axis and the unit cell being tripled (p = 3) along the a-axis. The intermediate phase is f~ommensurate and was described as corresponding to a frozen-in displacement wave with a periodicity alp + ~) where 6 is an irrational number and p = 3. It has bee~4r~centlysuggested that phase solitons exist in the phase of K2SeO4 a relatively wide tempers ture range so that the incommensurate phase actually consists of commensura te low temperature regions separated by regions of phase slip. In order to check on this suggestion and on the local natu re of the I phase in general we decided to study the EPR line shapes in y-irradiated K SeQ in some detail. EPR as well as ~n ~ should be ideally suited to discrinata between the two limiting descriptions of the phase, modulation models ~ . the In “phase the “plane the “plane wave” soliton” wave” modulation model, the resonance lines are expected to broaden into a cont~guum limited by two edge singulari ties whereas one would expect to see sharp “commensurate” lines superimposed on a broad background in the “narrow soliton”(~mit. Aiki showed that the four room temperature Se04 13-band EPR lines split into twelve bro~d, partially overlapping
asymmetric lines cooling below T
below T one finds 1 .
-
view of Onin further
the domain spliting - twenty four rela tively narrow symmetric SeO~ 4magnetic lines for a general orientation of the field with respect to the crystal axes. Our results showed that such a splitting and asymmetric b~oadening occurs not only fo~ the Sea4 but also for some of the SeO3 radical lines. The te~perature dependence of th~ 13-bend SeO - EPR frequencies for correspo~ding to Fig. 12 of Aiki is shown in Fig. 1. In the hatched re gion which is limited by two edge singu larities there is a continuum distribution of resonance lines. The lowest and the highest field SoO~ spectrashow superimposed on the continuum - an additional line which is quite pronoun ced over most of the I phase. The sp~Qtra are qualitative~ simil,r to the Rb NMR lines observed in the I phasAs of Rb2ZnC1 and Rb ZnBr In view ofthe asymmetry ~f the 1~nesh~peand the appea rance of additional lines superimposed — on the continuum a quantitative lineshape analysis has to be performed before one can discriminate between the “plane wave” and the 4 “soliton” spectra of modulation spinless models. Se isotope ca~ be described by the The SeQ spin Hamiltonian
~
AK?
.4 =
~ H
—~ ,
s
(1)
where the g tensor is a function of the order parameter q and thus varies in space in the I phase in a way which reflects the spatial variation of the incommensurate modulation.
123
124
THE INCOMMENSURATE PHASE IN 7—IRRADIATED K
2SeOz~
I
in eq.
(5)
Vol. 37, No. 2
is given by
2 dv = - ~ CO5~) + \J3 coe 4~+.~ (6) dx dx peaked wherever dV/dx becomes small. If a The frequency distribution will be
I2580-~ -S
single line has the lineshape L (~-~ ),the incommensurate line shape will be gi~/en by FN) IL Iv -~ C ) f Iv) dv (7) C
I
—
o
C
I
P
I2~40
In the “plane wave” limit the phase ~ (x) is a linear function of x, whereas it is a non-linear function in the “soliton” limit. Bo~~cases are covered if ~ Ix) is determined as a solution of th~ sine— -Gordon equation 2 d ~ = nT. sin (p~) dx
2530
12460
for different values of ct(T). The sineGordon equation can be reduced to a pen dultim-like equation which admits both plane wave-like solutions as wa~I1 as multi soliton lattice-like solutions (for pendulum making complete revolutions 8)) The equation is sufficiently simple to allow direct numerical integration on a computer and a synthesis of EPR line-sha pee given by eq. (7). Fig. 2 shows the spatial variation of the
____
I
80
Tc 00
120 Ti
40
60
Temperature (K)
FIGURE 1 4 EPR frequencies Temperature dependence of of y-irradiated the 13-bend KSe0 S~O for an orientation corresponding t~ Fi~. 12 of Alki (2)• The magnetic field is 65° from the c-axis in the (120) plane. In the hatched region in the p phase there is a continuum distribution of resonance frequencies.
In vie~i 4 sites both and second or7~f the first C1 symmetry at the der4terms may appear in the expansion of Se0 the EPR frequencies in powers of the order parameter r~ V
=
V
+
0
1
a 1q
+
~-
(8)
phase2~ for several values of the parame tar A which is proportional to the diffe dulum the energy necessary — rence end between the total energy to of go the topen
2 TT flS
‘2 tan5u2~(1+A2)h/2
3
2 a2q
4 +
.
.
.
(2)
6~
1 2
Since order parameter determine~ by its the amplitude A and its isphase ~ =
A cos
1
2
Ix)
~‘2 ~
~ 4
3
~ ~2.
0.3
~ ~2= 0.003
one finds V
i
=
2
V~ V~ coo ~(x)+ ~ V
5 roe ~(x)÷...
de
TI
e 10
(4)
In the C phase below T cosc~ takes on site. In~n the phase, a discrete value4 depending the I positiontheof other on the Se0 ha~d, cos~(x) may take on nearly continuously all values between +1 and -1. The density of spectf~} lines at the frequency V will be given by N fly) = dV/dx s~here N is the number of Se0~ sites per unit length. The derivative appearing
E~+cos28/2J’/2 I
0
0.25
0.5
0.75
1.0
FIGURE 2 Spatial variation = ~(x) as determined from ofeq.the(8)phase for 4several values of the parameter A2,showing the continuous variation from the multisoli ton lattice to the plane wave limit. —
Vol. 37, No. 2
THE INCOMMENSURATE PHASE IN i—IRRADIATED K
2SeO~
the ~op 10 = p41 = 11). The particular cc se A 0 cor~esponos to a single soliton whereas fo~ A > 0 one has a multisoliton wave limit. lattice. A..-~ corresponds to the plane Fig. 3 shows the Comparison between the experimental and the theoretical ~-band EPR lineshapes for one of the SeQ lines in the middle of the p phase at ll~K. The sharp symmetric paraelectric line becomes double peaked and asymmetric in the
ACKNOWLEDGEMENTS: The authors want to thank C.B. Pinheiros and R.A. Nogueira for their assistance on the computations. This work was partially supported by CNPq. FINEP, IAEA and CNEN:
220) Experiment Broad SoIiton(4~ Two Lore ntzian Lines
————
I phase. The experimental lineshape cannot be explained as a sum of two Lorentzians (Fig.2). The fit with two Gaussians - not shown in the figure — is even ~rse. The broad soliton modulation model yields a reasonably good fit with V 1 = 3.9 Gauss, V2 2.1 Gauss and 2 the 20.=pareelectric what corresL Iv - v I taken from c phase, to ponds as a well 1% volume as 4A fraction of commensu rate regions Ir~ I at this temperature. This is equival~nt to a phase solitonThe density of 99% = 1-q = 0.99). middle of thaj phase of K~SeO 4 thus corresponds to the “broad” and not to the “narrow” soliton limit. The plane wave modulation model is thus a very good approximation. The results seem to agree with the theory wh~’~fluctuations are taken into account . A more detailed treatment of this problem will be publish ed elsewhere.
125
I
.
/
‘
I
I,
11
‘
. ‘..._
I
—
I
I
—
/
Ii I
/
I
.
/ I
.•
_. ~.—.
______________________________________
2495
12490
12485 (G)
FIGURE 3 Comparison between the experimental end calculedEPR for of onetheof incomthe ~~4lines inlineshape the middle mens ura
te phase
at I
=
110K.
REFERENCES
1. 2. 3. 4.
5.
6. 7. 8. 9,
K.Aiki, K. Hukuda and 0. Matumura, 3. Phys. Soc. Japan, 26, 1064 (1969) K. Aiki, 3. Phys. Soc. Japan, 29, 379 (1970) M. Lizumi, 3.0. Axe, G. Shirane and K. Shimaoka, Phys. Rev. 615, 4392 (1977) a) W.L. McMillan, Phys. Rev. Bl2, 1187 (1975)1 ibid. 614, l4~ (1976) b) W. Rehwald, A. Vonlanthen,]~ Krbger, R. Wal1erius~~d H.G. Unruh, paper presented at the Meco Meeting, Budapest (1980) A. Blinc, S. Juznic, V. Rutar. 3. Seliger and S. Zumer, Phys. Rev. Lett. 44 609 (1980)1 A. Blinc, V. Rutar, 3. Seliger, S. Zumer, Th. Raising and I.PT Alekeandrova, Solid State Comm., 34, 895 (1980) A.Q. Bruce and R.A. Cowley, 3. Phys. Cil, 3609 (1978) and references therein A.K. Thin, Sol. State Comm. 31, 237 (1979) J.K. Perring and T.H.R. Skyrme, Nuci. Phys. 31, 550 (1962) T. Natterman, 3. Phys. C13, L265 (1960).
EPA LINESHAPE STUDY
OF THE INCOMMENSURATE PHASE A.S.
Chaves and R.
IN y-IRRADIATEO K
2SeO4
Gazzinelli
Departamento de Fisica Universidade Federal de Minas Gerais Caixa Postal 702 6810 Horizonte - Brazil R. Blinc 3. Stefan Institute, University of Ljubljana Ljubljana, Yugoslavia (Received September
4
1960 by R.C.C. Leite) 4 EPR frequencies and The the temperature asyrimetric dependence broadening of of the the SeQ ~PR lines in the incommensurate phase of K SeQ can be explained by an incommensurate spatial mo~ula~ionof the g tensors which corresponds to the ‘broad” phase soliton limit. A comparison between the experimental and calculated lineshape shows a ~l% volume fraction of commensurate regions in the middle of the incommensurate phase at 110 K.
(1,2) K SeO was found by Aiki to un2two4succesSive phase tran~3fformstions dergo at T 1 129.5K and T = 93K. The high temperature paraelect~ic (Pj5phase tains four (T > T1) is SeQ orthorhombic (02h) and con4 groups. The low tempera ture commensurate (C) phase IT < T ) is ferroelectric and monoclinic (C~~)cwith the spontaneous polarization appearing along the c-axis and the unit cell being tripled (p = 3) along the a-axis. The intermediate phase is f~ommensurate and was described as corresponding to a frozen-in displacement wave with a periodicity alp + ~) where 6 is an irrational number and p = 3. It has bee~4r~centlysuggested that phase solitons exist in the phase of K2SeO4 a relatively wide tempers ture range so that the incommensurate phase actually consists of commensura te low temperature regions separated by regions of phase slip. In order to check on this suggestion and on the local natu re of the I phase in general we decided to study the EPR line shapes in y-irradiated K SeQ in some detail. EPR as well as ~n ~ should be ideally suited to discrinata between the two limiting descriptions of the phase, modulation models ~ . the In “phase the “plane the “plane wave” soliton” wave” modulation model, the resonance lines are expected to broaden into a cont~guum limited by two edge singulari ties whereas one would expect to see sharp “commensurate” lines superimposed on a broad background in the “narrow soliton”(~mit. Aiki showed that the four room temperature Se04 13-band EPR lines split into twelve bro~d, partially overlapping
asymmetric lines cooling below T
below T one finds 1 .
-
view of Onin further
the domain spliting - twenty four rela tively narrow symmetric SeO~ 4magnetic lines for a general orientation of the field with respect to the crystal axes. Our results showed that such a splitting and asymmetric b~oadening occurs not only fo~ the Sea4 but also for some of the SeO3 radical lines. The te~perature dependence of th~ 13-bend SeO - EPR frequencies for correspo~ding to Fig. 12 of Aiki is shown in Fig. 1. In the hatched re gion which is limited by two edge singu larities there is a continuum distribution of resonance lines. The lowest and the highest field SoO~ spectrashow superimposed on the continuum - an additional line which is quite pronoun ced over most of the I phase. The sp~Qtra are qualitative~ simil,r to the Rb NMR lines observed in the I phasAs of Rb2ZnC1 and Rb ZnBr In view ofthe asymmetry ~f the 1~nesh~peand the appea rance of additional lines superimposed — on the continuum a quantitative lineshape analysis has to be performed before one can discriminate between the “plane wave” and the 4 “soliton” spectra of modulation spinless models. Se isotope ca~ be described by the The SeQ spin Hamiltonian
~
AK?
.4 =
~ H
—~ ,
s
(1)
where the g tensor is a function of the order parameter q and thus varies in space in the I phase in a way which reflects the spatial variation of the incommensurate modulation.
123
124
THE INCOMMENSURATE PHASE IN 7—IRRADIATED K
2SeOz~
I
in eq.
(5)
Vol. 37, No. 2
is given by
2 dv = - ~ CO5~) + \J3 coe 4~+.~ (6) dx dx peaked wherever dV/dx becomes small. If a The frequency distribution will be
I2580-~ -S
single line has the lineshape L (~-~ ),the incommensurate line shape will be gi~/en by FN) IL Iv -~ C ) f Iv) dv (7) C
I
—
o
C
I
P
I2~40
In the “plane wave” limit the phase ~ (x) is a linear function of x, whereas it is a non-linear function in the “soliton” limit. Bo~~cases are covered if ~ Ix) is determined as a solution of th~ sine— -Gordon equation 2 d ~ = nT. sin (p~) dx
2530
12460
for different values of ct(T). The sineGordon equation can be reduced to a pen dultim-like equation which admits both plane wave-like solutions as wa~I1 as multi soliton lattice-like solutions (for pendulum making complete revolutions 8)) The equation is sufficiently simple to allow direct numerical integration on a computer and a synthesis of EPR line-sha pee given by eq. (7). Fig. 2 shows the spatial variation of the
____
I
80
Tc 00
120 Ti
40
60
Temperature (K)
FIGURE 1 4 EPR frequencies Temperature dependence of of y-irradiated the 13-bend KSe0 S~O for an orientation corresponding t~ Fi~. 12 of Alki (2)• The magnetic field is 65° from the c-axis in the (120) plane. In the hatched region in the p phase there is a continuum distribution of resonance frequencies.
In vie~i 4 sites both and second or7~f the first C1 symmetry at the der4terms may appear in the expansion of Se0 the EPR frequencies in powers of the order parameter r~ V
=
V
+
0
1
a 1q
+
~-
(8)
phase2~ for several values of the parame tar A which is proportional to the diffe dulum the energy necessary — rence end between the total energy to of go the topen
2 TT flS
‘2 tan5u2~(1+A2)h/2
3
2 a2q
4 +
.
.
.
(2)
6~
1 2
Since order parameter determine~ by its the amplitude A and its isphase ~ =
A cos
1
2
Ix)
~‘2 ~
~ 4
3
~ ~2.
0.3
~ ~2= 0.003
one finds V
i
=
2
V~ V~ coo ~(x)+ ~ V
5 roe ~(x)÷...
de
TI
e 10
(4)
In the C phase below T cosc~ takes on site. In~n the phase, a discrete value4 depending the I positiontheof other on the Se0 ha~d, cos~(x) may take on nearly continuously all values between +1 and -1. The density of spectf~} lines at the frequency V will be given by N fly) = dV/dx s~here N is the number of Se0~ sites per unit length. The derivative appearing
E~+cos28/2J’/2 I
0
0.25
0.5
0.75
1.0
FIGURE 2 Spatial variation = ~(x) as determined from ofeq.the(8)phase for 4several values of the parameter A2,showing the continuous variation from the multisoli ton lattice to the plane wave limit. —
Vol. 37, No. 2
THE INCOMMENSURATE PHASE IN i—IRRADIATED K
2SeO~
the ~op 10 = p41 = 11). The particular cc se A 0 cor~esponos to a single soliton whereas fo~ A > 0 one has a multisoliton wave limit. lattice. A..-~ corresponds to the plane Fig. 3 shows the Comparison between the experimental and the theoretical ~-band EPR lineshapes for one of the SeQ lines in the middle of the p phase at ll~K. The sharp symmetric paraelectric line becomes double peaked and asymmetric in the
ACKNOWLEDGEMENTS: The authors want to thank C.B. Pinheiros and R.A. Nogueira for their assistance on the computations. This work was partially supported by CNPq. FINEP, IAEA and CNEN:
220) Experiment Broad SoIiton(4~ Two Lore ntzian Lines
————
I phase. The experimental lineshape cannot be explained as a sum of two Lorentzians (Fig.2). The fit with two Gaussians - not shown in the figure — is even ~rse. The broad soliton modulation model yields a reasonably good fit with V 1 = 3.9 Gauss, V2 2.1 Gauss and 2 the 20.=pareelectric what corresL Iv - v I taken from c phase, to ponds as a well 1% volume as 4A fraction of commensu rate regions Ir~ I at this temperature. This is equival~nt to a phase solitonThe density of 99% = 1-q = 0.99). middle of thaj phase of K~SeO 4 thus corresponds to the “broad” and not to the “narrow” soliton limit. The plane wave modulation model is thus a very good approximation. The results seem to agree with the theory wh~’~fluctuations are taken into account . A more detailed treatment of this problem will be publish ed elsewhere.
125
I
.
/
‘
I
I,
11
‘
. ‘..._
I
—
I
I
—
/
Ii I
/
I
.
/ I
.•
_. ~.—.
______________________________________
2495
12490
12485 (G)
FIGURE 3 Comparison between the experimental end calculedEPR for of onetheof incomthe ~~4lines inlineshape the middle mens ura
te phase
at I
=
110K.
REFERENCES
1. 2. 3. 4.
5.
6. 7. 8. 9,
K.Aiki, K. Hukuda and 0. Matumura, 3. Phys. Soc. Japan, 26, 1064 (1969) K. Aiki, 3. Phys. Soc. Japan, 29, 379 (1970) M. Lizumi, 3.0. Axe, G. Shirane and K. Shimaoka, Phys. Rev. 615, 4392 (1977) a) W.L. McMillan, Phys. Rev. Bl2, 1187 (1975)1 ibid. 614, l4~ (1976) b) W. Rehwald, A. Vonlanthen,]~ Krbger, R. Wal1erius~~d H.G. Unruh, paper presented at the Meco Meeting, Budapest (1980) A. Blinc, S. Juznic, V. Rutar. 3. Seliger and S. Zumer, Phys. Rev. Lett. 44 609 (1980)1 A. Blinc, V. Rutar, 3. Seliger, S. Zumer, Th. Raising and I.PT Alekeandrova, Solid State Comm., 34, 895 (1980) A.Q. Bruce and R.A. Cowley, 3. Phys. Cil, 3609 (1978) and references therein A.K. Thin, Sol. State Comm. 31, 237 (1979) J.K. Perring and T.H.R. Skyrme, Nuci. Phys. 31, 550 (1962) T. Natterman, 3. Phys. C13, L265 (1960).