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Biquadratic exchange in the diluted Ising ferromagnet
Journal of Magnetism and Magnetic Materials 67 (1987) 33-36 North-Holland, Amsterdam
33
BIQUADRATIC EXCHANGE IN T H E DILUTED ISING F E R R O M A G N E T M. TIWARI, N. C H A T U R V E D I * and R.N. SHRIVASTAVA Department of Electrical Engineering, Institute of Technology, Banaras Hindu University, Varanasi-221005, India Received 7 October 1986
Effects of biquadratic exchange on Curie temperature and specific heat have been studied in the light of a new effective field theory for the lsing Spin System. The nearest neighbour interaction is taken into account.
1. Introduction The magnetism of amorphous and disordered materials has achieved much attention among scientists from theoretical as well as experimental point of view. Here we have concentrated on the site diluted magnets that is, systems in which magnetic atoms are replaced by nonmagnetic atoms in increasing amounts. Authors [1,2] have discussed the diluted magnetic systems based on effective field theory and Coherent Potential Approximation (CPA). Recently Kaneyoshi et al. [3] developed a method based on the Callen identity [1] for the spin S = 1 / 2 system and applied it to different problems. In our paper we have made an attempt to study the effect of biquadratic exchange and percolation concentration together on Curie temperature following a new effective field theory developed by Kaneyoshi [3]. The expression for internal energy and specific heat has been derived and discussed in detail.
Jr=
- • E
-
(2)
The exact correlation function is given by ( # i ) = (tanh flEi),
(3)
where fl = 1/kT,
E, = ~,Jifl~jaj + a ' / 2 Z J , f l ~ j , J
J
where t~7 = Oj, n is an integer. Introducing the differential operator into (3), we have (t~,) = (e °r''''"A''' +~')) tanh x l~=0 = 0
2. Curie temperature Following the notation of ref. [3], Hamiltonian of such a system is given by •,~= - ½~ J , j0,0~,t5 -- 50l ~ , EJij(t~iOj]li#j) /j
where #, and #j are the sins on i and j sites, the parameter 0, is random variable which takes on the value 0 or 1. Jq is the usual exchange interaction, a' is the biquadratic exchange which varies between 0 and 1. For the spin S = 1 / 2 system, there is simple a identity to reduce biquadratic to bilinear at once. Hence the Hamiltonian (1) is reduced to
2
(1)
(4)
with a" = a ' / 2
tij = J,j/kT,
D = 3/3x.
For a diluted ferromagnet
/j
• School of Materials Science and Technology, Institute of Technology, Banaras Hindu University, Varanasi-221005, India.
0304-8853/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
= m,=
mj=
= [cosh(Pt') + m sinh(Dt')] " X tanh xlx_ 0,
(5)
M. Tiwari et aL / Biquadratic exchange in the diluted Ising ferromagnet
34
Taking only linear terms in m, we have
where
t' =J, J k r ( 1
pz ~.
+ a " ) = t(1 + a " )
and (Oj), = p, which is concentration of magnetic atoms. Applying the Laplace transform to the above equation, we get 1 c+i~ 2,rri E g p z ( a ) t a n h ( a t ' )
m-
1
1 = ~-~ - - ( f " - ) t a n h
da,
, t~(pz-
v)
v=O
+ t a n h t'(2v + 2 - p z ) .
(11)
This is the required expression for the Curie temperature.
(6)
c-Joe
3. Specific heat
where
gpz(a)=
The internal energy U of the system is given by
e -~y [cosh y" + m sinh y']Pz d y
ff
'
U = - ½Y'~J, jO,(#, tanh tiEs)O s q
= f0 ~° e--,,y'
- l a " E J , sO,2(# , tanh tiEs)O ~, e y" _ e y ' ] pz
e.V' + e - y '
x
2
+ m ---------~ l
dy.
where (7)
Now applying the binomial expansion
Es= EJ,..,O,, i
(gs) = tanh tiE s, (a+b)"=
(12)
q
~ (7)a"-'b"
(#}) = tanh tiEs',
with
r=O
Ej = a"Y'.4jm0,.
to eq. (7), we have
i
1
gpz( ) =
pz
When we apply the differential operator to the above equation, we get,
E (;z) v--O
1
X a _ p z + 2v (1
+
m)P:-~(1 - m) ~
U = - ½E Oj( E s e °OL; ) t a n h x , = 0
(8)
J
- ½ Y ' . O 2 ( E , ' e n#e;) tanh xlx= 0, From eqs. (6) and (8), we have 1 /c+i~ m=~j.¢_io~
We again apply another differential operator
1 P~ 2 p~ y" ( l + m ) P ~ - ' ( 1 - m ) ' ( ~ ) v--O
G / ( y , D) = (e°#E,"),
tanh a t ' d a X a - ( p z - 2v) "
(9)
From contour integration
G/,(y, D) = (eD#E; y)
to the above equations and get the reduced equation for energy:
U = - ~ - - ~ [ ~°sr° -fyG1(y, -
~. - ~1- ( 1
+ mS'Z-'(1 - m
), (y~)
OJz[O---G (y,
oti t Oy /'
v--O
X tanh t 2 ( pz - v).
(10)
D ) ] tanh
x x--0 y~0
X -- X 0
m=
(13)
S
D)] tanh x ~=o" ,=0
(14)
M. Tiwari et a L / Biquadratic exchange in the diluted lsing ferromagnet
35
0'25r ¢': 1.0
0"20 ¢' sO.2 o~' = 0'6
¢',
-
0'8-
0'15
~ ' = I"0 ~
¢t' ~0"4
0'10 ~i= 0
0"05
O 0
~ 2
1
3
4
5
0'0V 0
6
pz
A
I 1
,
I i 2 t ----~
I 3
t
I t,
Fig. 1. Variation of the reduced Curie temperature kTc/J with percolation concentration pz for different values of the biquadratic exchange parameter i.e. a ' = 0, 0.2, 0.6, 0.8 and 1, respectively.
Fig. 2. Variation of the specific heat C / C o with the reduced Curie temperature t = J/kT¢ for different a " s e.g. a ' = 0, 0.4, 0.6, 0.8 and 1.
For y = 1
or,
( U ) , = - ½NpgZ[cosh Dt + m sinh Dt] p`-'
C/C o = ] ( t 2 sech2t + t '2 sech2t'),
- ½NpaYZ[cosh Dt" + m sinh Dt'] p~-I
(15) Forpz=l,
m=O
[NpJ
(U)r= - T
NpJ
sinhOt---ff-
sinhDt'
] ,
where we have used the relation
e~°f(x) = f ( x + a). Specific heat is given by C= b
( U ) , = -Npk - ~ , (t 2 sech2t + t, 2 sech2t ' )
(16)
(17)
where Co= ~Npk. Eqs. (11) and (17) are solved numerically by iteration process for different values of biquadratic exchange parameter a'. It is clear from fig. 1 that the Curie temperature increases with percolation concentration for each value of the biquadratic exchange and at the same time the Curie temperature decreases with the increase of a'. Fig. 2 shows that the specific heat at the transition temperature have broad maxima for smaller a " s but the sharp maximum appears for higher a ' 's. The numerical value of specific heat increases with a ' for fixed temperature. For higher values of a', the specific heat decreases rapidly. In this way, the biquadratic exchange influences specific heat data very much. The effect
36
M. Tiwari et al. / Biquadratic exchange in the diluted
of biquadratic exchange a' on magnetization m for higher p z values will be studied in the future.
Acknowledgements We are thankful to Prof. M. Bhattacharya and Dr. R.N. Mukherjee for their valuable suggestions.
lsing ferromagnet
References [1] T. Kaneyoshi, J. Phys. C8 (1975) 3415. [2] T. Tahir-Kheli, Phys. Rev. B6 (1972) 2808. [3] T. Kaneyoshi, I.P. Fittipaldi and H. Beyer, Phys. Stat. Sol. (b) 102 (1980) 393. [4] H.B. Callen, Phys. Lett. 4 (1963) 161.
33
BIQUADRATIC EXCHANGE IN T H E DILUTED ISING F E R R O M A G N E T M. TIWARI, N. C H A T U R V E D I * and R.N. SHRIVASTAVA Department of Electrical Engineering, Institute of Technology, Banaras Hindu University, Varanasi-221005, India Received 7 October 1986
Effects of biquadratic exchange on Curie temperature and specific heat have been studied in the light of a new effective field theory for the lsing Spin System. The nearest neighbour interaction is taken into account.
1. Introduction The magnetism of amorphous and disordered materials has achieved much attention among scientists from theoretical as well as experimental point of view. Here we have concentrated on the site diluted magnets that is, systems in which magnetic atoms are replaced by nonmagnetic atoms in increasing amounts. Authors [1,2] have discussed the diluted magnetic systems based on effective field theory and Coherent Potential Approximation (CPA). Recently Kaneyoshi et al. [3] developed a method based on the Callen identity [1] for the spin S = 1 / 2 system and applied it to different problems. In our paper we have made an attempt to study the effect of biquadratic exchange and percolation concentration together on Curie temperature following a new effective field theory developed by Kaneyoshi [3]. The expression for internal energy and specific heat has been derived and discussed in detail.
Jr=
- • E
-
(2)
The exact correlation function is given by ( # i ) = (tanh flEi),
(3)
where fl = 1/kT,
E, = ~,Jifl~jaj + a ' / 2 Z J , f l ~ j , J
J
where t~7 = Oj, n is an integer. Introducing the differential operator into (3), we have (t~,) = (e °r''''"A''' +~')) tanh x l~=0 = 0
2. Curie temperature Following the notation of ref. [3], Hamiltonian of such a system is given by •,~= - ½~ J , j0,0~,t5 -- 50l ~ , EJij(t~iOj]li#j) /j
where #, and #j are the sins on i and j sites, the parameter 0, is random variable which takes on the value 0 or 1. Jq is the usual exchange interaction, a' is the biquadratic exchange which varies between 0 and 1. For the spin S = 1 / 2 system, there is simple a identity to reduce biquadratic to bilinear at once. Hence the Hamiltonian (1) is reduced to
2
(1)
(4)
with a" = a ' / 2
tij = J,j/kT,
D = 3/3x.
For a diluted ferromagnet
/j
• School of Materials Science and Technology, Institute of Technology, Banaras Hindu University, Varanasi-221005, India.
0304-8853/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
= m,=
mj=
= [cosh(Pt') + m sinh(Dt')] " X tanh xlx_ 0,
(5)
M. Tiwari et aL / Biquadratic exchange in the diluted Ising ferromagnet
34
Taking only linear terms in m, we have
where
t' =J, J k r ( 1
pz ~.
+ a " ) = t(1 + a " )
and (Oj), = p, which is concentration of magnetic atoms. Applying the Laplace transform to the above equation, we get 1 c+i~ 2,rri E g p z ( a ) t a n h ( a t ' )
m-
1
1 = ~-~ - - ( f " - ) t a n h
da,
, t~(pz-
v)
v=O
+ t a n h t'(2v + 2 - p z ) .
(11)
This is the required expression for the Curie temperature.
(6)
c-Joe
3. Specific heat
where
gpz(a)=
The internal energy U of the system is given by
e -~y [cosh y" + m sinh y']Pz d y
ff
'
U = - ½Y'~J, jO,(#, tanh tiEs)O s q
= f0 ~° e--,,y'
- l a " E J , sO,2(# , tanh tiEs)O ~, e y" _ e y ' ] pz
e.V' + e - y '
x
2
+ m ---------~ l
dy.
where (7)
Now applying the binomial expansion
Es= EJ,..,O,, i
(gs) = tanh tiE s, (a+b)"=
(12)
q
~ (7)a"-'b"
(#}) = tanh tiEs',
with
r=O
Ej = a"Y'.4jm0,.
to eq. (7), we have
i
1
gpz( ) =
pz
When we apply the differential operator to the above equation, we get,
E (;z) v--O
1
X a _ p z + 2v (1
+
m)P:-~(1 - m) ~
U = - ½E Oj( E s e °OL; ) t a n h x , = 0
(8)
J
- ½ Y ' . O 2 ( E , ' e n#e;) tanh xlx= 0, From eqs. (6) and (8), we have 1 /c+i~ m=~j.¢_io~
We again apply another differential operator
1 P~ 2 p~ y" ( l + m ) P ~ - ' ( 1 - m ) ' ( ~ ) v--O
G / ( y , D) = (e°#E,"),
tanh a t ' d a X a - ( p z - 2v) "
(9)
From contour integration
G/,(y, D) = (eD#E; y)
to the above equations and get the reduced equation for energy:
U = - ~ - - ~ [ ~°sr° -fyG1(y, -
~. - ~1- ( 1
+ mS'Z-'(1 - m
), (y~)
OJz[O---G (y,
oti t Oy /'
v--O
X tanh t 2 ( pz - v).
(10)
D ) ] tanh
x x--0 y~0
X -- X 0
m=
(13)
S
D)] tanh x ~=o" ,=0
(14)
M. Tiwari et a L / Biquadratic exchange in the diluted lsing ferromagnet
35
0'25r ¢': 1.0
0"20 ¢' sO.2 o~' = 0'6
¢',
-
0'8-
0'15
~ ' = I"0 ~
¢t' ~0"4
0'10 ~i= 0
0"05
O 0
~ 2
1
3
4
5
0'0V 0
6
pz
A
I 1
,
I i 2 t ----~
I 3
t
I t,
Fig. 1. Variation of the reduced Curie temperature kTc/J with percolation concentration pz for different values of the biquadratic exchange parameter i.e. a ' = 0, 0.2, 0.6, 0.8 and 1, respectively.
Fig. 2. Variation of the specific heat C / C o with the reduced Curie temperature t = J/kT¢ for different a " s e.g. a ' = 0, 0.4, 0.6, 0.8 and 1.
For y = 1
or,
( U ) , = - ½NpgZ[cosh Dt + m sinh Dt] p`-'
C/C o = ] ( t 2 sech2t + t '2 sech2t'),
- ½NpaYZ[cosh Dt" + m sinh Dt'] p~-I
(15) Forpz=l,
m=O
[NpJ
(U)r= - T
NpJ
sinhOt---ff-
sinhDt'
] ,
where we have used the relation
e~°f(x) = f ( x + a). Specific heat is given by C= b
( U ) , = -Npk - ~ , (t 2 sech2t + t, 2 sech2t ' )
(16)
(17)
where Co= ~Npk. Eqs. (11) and (17) are solved numerically by iteration process for different values of biquadratic exchange parameter a'. It is clear from fig. 1 that the Curie temperature increases with percolation concentration for each value of the biquadratic exchange and at the same time the Curie temperature decreases with the increase of a'. Fig. 2 shows that the specific heat at the transition temperature have broad maxima for smaller a " s but the sharp maximum appears for higher a ' 's. The numerical value of specific heat increases with a ' for fixed temperature. For higher values of a', the specific heat decreases rapidly. In this way, the biquadratic exchange influences specific heat data very much. The effect
36
M. Tiwari et al. / Biquadratic exchange in the diluted
of biquadratic exchange a' on magnetization m for higher p z values will be studied in the future.
Acknowledgements We are thankful to Prof. M. Bhattacharya and Dr. R.N. Mukherjee for their valuable suggestions.
lsing ferromagnet
References [1] T. Kaneyoshi, J. Phys. C8 (1975) 3415. [2] T. Tahir-Kheli, Phys. Rev. B6 (1972) 2808. [3] T. Kaneyoshi, I.P. Fittipaldi and H. Beyer, Phys. Stat. Sol. (b) 102 (1980) 393. [4] H.B. Callen, Phys. Lett. 4 (1963) 161.