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Phase transitions in an ising ferromagnet with transverse single-ion anisotropy (spin 32 )
I Phvr Chn
Widr.
197l. Vol 38. pp 1411-1412.
Pergamon Pnss
Pmcd
in Great Bntsm
TECHNICAL NOTE PHASE TRANSITIONS IN AN ISINGFERROMAGNET WITH TRANSVERSE SINGLE-IONANISOTROPY (SPIN:) L.N. Physics
Department.
Physics
Th.D.S.B.
Department,
(Received II February
DW~VEDI
College.
Kumaun
C.
BHANDARI
M.
Allahabad
X = 41
C S,Z- DC (S;‘)‘- C
J(ij)S,‘S,“. 1.1
University,
1977; occepfed
The role of crystal field anisotropy has been widely studied in king model in recent years. Models describing these systems contain adjustable anisotropy parameters which can cause second-order phase transitions to become first-order one. A number of papers have been published recently on these effects in uniaxial crystals[ld]. In the present note we have extended Holas n al.1 lo] calculations for three level system to four level system to study the effect of transverse anisotropy on the phase transition. However utilizing Landau’s[ 1I] theory of secondorder phase transition the stability condition has been also derived. We consider the following Hamiltonian for a spin-4 (i = I) king ferromagnet with crystal anisotropy.
2d(R’
2Y + u2) cash Ir
= 3 ezd* sinh 2 + sinh 2 Ir p
Allahabad,
India
and the system
free energy
F /=NZIqJ2-Pln
1
per ion can be written
For a proper thermodynamic minimum. Therefore,
states
free
(5)
t
The first of above condition contained in eqn (2)
simply gives the information
already
ll= cash ? _ cash ?+ P P
(20-
e”” x 2x p
sinh ?! + (‘O ’ R, sinh ?! Y p cc
2Y 2 cash P
(6)
(2)
_ as
+
- (20 - R)*
I
+
‘=Bu The solution Solution for not physical. for various lower values while for R
(3) 1411
That
is for
(20 t I- 2p)y*t (2~ t R)* cash 2~ c
lrY2
4x”*(u
- 1)(2u - R) - 3pR*
and
1
curve.
e2h’ cos,, ?? P
2/J”
In order to obtain the stability condition the free energy of the system must be obtained within MFA. The free energy is related to the system Hamiltonian via the relationship
be
$0.
y=\/(R2tu2tuR)
-fiF = In Tr emaH
must
and
PI2
introduced
energy
-_ z-0
(2~ - 1 t Zp)x’
have ken
II (4)
The second condition gives so called stability finite magnetization, we obtain from eqn (5)
quantities
as
e5R/2r 2 em!’ cos,, 9 + 2 e-lolrl cash? [ cc fi
2 e’O” cash-t
where,
and reduced
263 002, India
in recked form 22 April 1977)
em* + u2) ezolr cash c+ p
Nainital
(1)
Here h represents an external field conjugate to S,‘. D is the magnitude of transverse single-ion anisotropy. and I(ij) > 0 is the effective exchange interaction between magnetic ions at sites i and j. Within Molecular Field Approximation (MFA) Hamiltonian (1) can be solved to give expression for magnetization as
2\l(R’
University.
4y”‘(u
+ 1x20 t R) - 3aR2 2py3’2
3
p
1 1
cc
eMr sinh 1_x
sinh 2ya P
0.
(7)
of eqn (7) has been represented in Fig. 1 for R = 0. the magnetization below this curve (for R = 0) are Fig. 2 represents the numerical solution of eqn (2) values of R. From the curve it is seen that for of R, u is single valued function of the temperature = 1, u becomes double valued function of tem-
1412
Technical Note
1.4
1.0 CT
05
0.5
0
0 05
IO
15
20
2.5
X
05
IO
1.5
2.0
2.5 X
P
P
Fig. I. Stability curve: (I vs p for
R - 0.
perature. Hence for this crystal field anisotropy transjtion becomes first-order one. For three tevel system this transition occurs for R =0.49 (Holas et a/.[lO]). Larger value of R which changes the phase from second-order to first-order clearly indicates the stronger co-operative phase present in four-level system. Thus addition of a single-ion transversal anisotropy term to magnetic exchange Hamiltonian suppresses the co-operative phase at certain range of the crystal field parameters and hence appearance of first-order phase transition. REFEBENCEs 1. Collen E. R.. Phys. Rev. 124, 1373 (I%!). 2. Blume M.. Phys. Rev. 141. 517 (1%6).
Fig. 2. Magnetization curve: Magnetization (u) as a function of temperature for various values of R. First order transition occurs for R = I. 3. Cape1 H. W.. Physica 32, 966 (1966);33, 295 (1%7f. 4. Taggart G. B. and Tahir Kheli R. A., Physica 44,321 (1969). 5. J. F. Devlin, Phys. Rev. B4, 136(1971). 6. Duda A. and Puszkarski H.. Phys. Stafus Solidi. (6) 58, 543 (1973). 7.
Blume M., Emery V. J. and Griffiths R. B., Phys. Rev. 84, 1071(1971). 8. Brankav 1. G., Przystawa J. and Praveczki E., J. Phys. C.; Solid State Phys. 5, 3387 (1972). 9. Sivardiere J. and Blume M., Phys. Rev. BS, 1126(1972). IO. Holas A.. Przystawa J. and Strycharski Z., J. Phys. Chem. Solids 37, I I9 (1976). I I. Landau L. D. and Lifshitz E. M., StatisticalPhysics. 2nd Edn., p. 424. Pergamon Press. London (1969).
Widr.
197l. Vol 38. pp 1411-1412.
Pergamon Pnss
Pmcd
in Great Bntsm
TECHNICAL NOTE PHASE TRANSITIONS IN AN ISINGFERROMAGNET WITH TRANSVERSE SINGLE-IONANISOTROPY (SPIN:) L.N. Physics
Department.
Physics
Th.D.S.B.
Department,
(Received II February
DW~VEDI
College.
Kumaun
C.
BHANDARI
M.
Allahabad
X = 41
C S,Z- DC (S;‘)‘- C
J(ij)S,‘S,“. 1.1
University,
1977; occepfed
The role of crystal field anisotropy has been widely studied in king model in recent years. Models describing these systems contain adjustable anisotropy parameters which can cause second-order phase transitions to become first-order one. A number of papers have been published recently on these effects in uniaxial crystals[ld]. In the present note we have extended Holas n al.1 lo] calculations for three level system to four level system to study the effect of transverse anisotropy on the phase transition. However utilizing Landau’s[ 1I] theory of secondorder phase transition the stability condition has been also derived. We consider the following Hamiltonian for a spin-4 (i = I) king ferromagnet with crystal anisotropy.
2d(R’
2Y + u2) cash Ir
= 3 ezd* sinh 2 + sinh 2 Ir p
Allahabad,
India
and the system
free energy
F /=NZIqJ2-Pln
1
per ion can be written
For a proper thermodynamic minimum. Therefore,
states
free
(5)
t
The first of above condition contained in eqn (2)
simply gives the information
already
ll= cash ? _ cash ?+ P P
(20-
e”” x 2x p
sinh ?! + (‘O ’ R, sinh ?! Y p cc
2Y 2 cash P
(6)
(2)
_ as
+
- (20 - R)*
I
+
‘=Bu The solution Solution for not physical. for various lower values while for R
(3) 1411
That
is for
(20 t I- 2p)y*t (2~ t R)* cash 2~ c
lrY2
4x”*(u
- 1)(2u - R) - 3pR*
and
1
curve.
e2h’ cos,, ?? P
2/J”
In order to obtain the stability condition the free energy of the system must be obtained within MFA. The free energy is related to the system Hamiltonian via the relationship
be
$0.
y=\/(R2tu2tuR)
-fiF = In Tr emaH
must
and
PI2
introduced
energy
-_ z-0
(2~ - 1 t Zp)x’
have ken
II (4)
The second condition gives so called stability finite magnetization, we obtain from eqn (5)
quantities
as
e5R/2r 2 em!’ cos,, 9 + 2 e-lolrl cash? [ cc fi
2 e’O” cash-t
where,
and reduced
263 002, India
in recked form 22 April 1977)
em* + u2) ezolr cash c+ p
Nainital
(1)
Here h represents an external field conjugate to S,‘. D is the magnitude of transverse single-ion anisotropy. and I(ij) > 0 is the effective exchange interaction between magnetic ions at sites i and j. Within Molecular Field Approximation (MFA) Hamiltonian (1) can be solved to give expression for magnetization as
2\l(R’
University.
4y”‘(u
+ 1x20 t R) - 3aR2 2py3’2
3
p
1 1
cc
eMr sinh 1_x
sinh 2ya P
0.
(7)
of eqn (7) has been represented in Fig. 1 for R = 0. the magnetization below this curve (for R = 0) are Fig. 2 represents the numerical solution of eqn (2) values of R. From the curve it is seen that for of R, u is single valued function of the temperature = 1, u becomes double valued function of tem-
1412
Technical Note
1.4
1.0 CT
05
0.5
0
0 05
IO
15
20
2.5
X
05
IO
1.5
2.0
2.5 X
P
P
Fig. I. Stability curve: (I vs p for
R - 0.
perature. Hence for this crystal field anisotropy transjtion becomes first-order one. For three tevel system this transition occurs for R =0.49 (Holas et a/.[lO]). Larger value of R which changes the phase from second-order to first-order clearly indicates the stronger co-operative phase present in four-level system. Thus addition of a single-ion transversal anisotropy term to magnetic exchange Hamiltonian suppresses the co-operative phase at certain range of the crystal field parameters and hence appearance of first-order phase transition. REFEBENCEs 1. Collen E. R.. Phys. Rev. 124, 1373 (I%!). 2. Blume M.. Phys. Rev. 141. 517 (1%6).
Fig. 2. Magnetization curve: Magnetization (u) as a function of temperature for various values of R. First order transition occurs for R = I. 3. Cape1 H. W.. Physica 32, 966 (1966);33, 295 (1%7f. 4. Taggart G. B. and Tahir Kheli R. A., Physica 44,321 (1969). 5. J. F. Devlin, Phys. Rev. B4, 136(1971). 6. Duda A. and Puszkarski H.. Phys. Stafus Solidi. (6) 58, 543 (1973). 7.
Blume M., Emery V. J. and Griffiths R. B., Phys. Rev. 84, 1071(1971). 8. Brankav 1. G., Przystawa J. and Praveczki E., J. Phys. C.; Solid State Phys. 5, 3387 (1972). 9. Sivardiere J. and Blume M., Phys. Rev. BS, 1126(1972). IO. Holas A.. Przystawa J. and Strycharski Z., J. Phys. Chem. Solids 37, I I9 (1976). I I. Landau L. D. and Lifshitz E. M., StatisticalPhysics. 2nd Edn., p. 424. Pergamon Press. London (1969).