- Email: [email protected]
Coercive field and Grüneisen's law
Accepted Manuscript Coercive field and Grüneisen's law M. Salazar, G.A. Pérez Alcazar PII:
S0921-4526(19)30048-1
DOI:
https://doi.org/10.1016/j.physb.2019.01.048
Reference:
PHYSB 311300
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 23 November 2018 Revised Date:
27 January 2019
Accepted Date: 29 January 2019
Please cite this article as: M. Salazar, G.A.Pé. Alcazar, Coercive field and Grüneisen's law, Physica B: Physics of Condensed Matter (2019), doi: https://doi.org/10.1016/j.physb.2019.01.048. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
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COERCIVE FIELD AND GRÜNEISEN’S LAW
30 31 32 33 34 35 36 37 38 39 40 41
M AN U
TE D
a
Pitt Community College, P.O. Drawer 7007, Greenville, NC 27835-7007, USA. b
Departamento de Física, Universidad del Valle, A.A. 25360, Cali, Colombia
EP
29
M. Salazara* and G.A. Pérez Alcazarb
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8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
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7
* Corresponding author: Manuel Salazar. E-mail address: [email protected]
1
ACCEPTED MANUSCRIPT
42
Abstract
43 The general expression of the blocking temperature for a superparamagnetic system depends on
45
the volume of the particle, but in this expression the volume is independent of temperature.
46
Some experimental curves of the coercive field as a function of temperature show a minimum at
47
relatively low temperatures. One way to obtain theoretical curves that qualitatively describe this
48
behavior is by considering that these systems have two different regimes of thermal dilation
49
when temperature increases: one for low temperatures and the other for high temperatures. In
50
the present work it is shown how these theoretical curves of the coercive field can be calculated
51
assuming these two different regimes and considering Grüneisen’s law for low temperatures.
52
Using the expression of the blocking temperature we have obtained the theoretical curve for the
53
coercive field against temperature. In order to adjust the theoretical curve with the experimental
54
one we have considered a volume model with respect to temperature taking into account
55
Grüneisen’s law for low temperatures.
56 57
Keywords: Coercive field, Grüneisen’s law, blocking temperature
59
61
1. INTRODUCTION
TE D
58
60
M AN U
SC
RI PT
44
In the development of the fundamental expressions of the theory of superparamagnetism it has
63
previously been considered that the volume of the nanoparticles does not depend on temperature
64
[1, 2]. This consideration allows to obtain an expression of the blocking temperature that
65
depends on the volume of the nanoparticles but not on the temperature. For the calculation of
66
the coercive field, which depends on temperature, the value of the blocking temperature is taken
67
into account. In the representation of the experimental data of the coercive field, it was found
68
that in some samples it presents a minimum at relatively low temperatures [3]. Some
69
experimental studies had reported this behavior, for example in systems such as Cu97Co3 and
70
Cu90 Co10 [4], in micrometric powders of Fe50Mn10Al40 [5], in magnetic materials like
71
Macroporus Cobalt [6], and in many others materials. A proposal to explain the occurrence of
72
this minimum presented by Nunes et al. [4] considered a distribution of nanoparticle size. Other
AC C
EP
62
2
ACCEPTED MANUSCRIPT
73
studies considered the effect of magnetic interactions on the expresion of the coercive field [7,
74
8]. However, an analytical prediction for the occurrence of this minimum has yet to be
75
developed.
76
In this study, we present an analytical method to explain this minimum considering in the expression of the blocking temperature a dependency of the volume of the sample with
78
temperature. Because Grüneisen’s law relates heat capacity with the thermal dilation coefficient,
79
and this law is valid at low temperatures, this law can be considered to explain the coercive field
80
at temperatures lower than the minimum. In contrast, for temperatures higher than that of the
81
minimum, the typical dependence of the thermal dilation coefficient with temperature (a linear
82
dependence) is considered.
SC
RI PT
77
84
M AN U
83
2. BLOCKING TEMPERATURE, COERCIVE FIELD, AND GRÜNEISEN’S LAW
85 86
Considering weak interactions between nanoparticles and for the particular case of the critical
87
volume V0 and a relaxation time equal to that of the measurement, τ = τm, it is possible to
88
demonstrate that for the current existing model the blocking temperature, TB, is given by [2]:
90
ܶ = (1)
92
EP
91
ܸܭ ߬ ݇ ݊ܮ ߬
TE D
89
where it is assumed that the volume does not depend on temperature.
AC C
93 94
Next, a system of non-interacting single domain particles whose energy is only due to the
95
uniaxial anisotropy energy is considered. For the range of temperatures from 0 to TB, that is,
96
when all the particles are blocked, the coercive field is given by [4]:
97
98
ଵ
2ܭ ܶ ଶ ܪ ሺܶሻ = ߙ 1 − ൬ ൰ ܯௌ ܶ (2)
3
ACCEPTED MANUSCRIPT
99 100
Here, α = 1 if the particles with easy axis are aligned, or α = 0.48 if they are randomly oriented [8].
101 For the current case, for identical weak interacting particles, the expression of TB that previous
103
studies have used is that of Eq. (1). In the present study, considering now that the volume V0 of
104
Eq. (1) depends on temperature, it can be written as V0 = V(T) in the corresponding blocking
105
temperature:
(3)
ܸܭሺܶሻ ߬ ݇ ݊ܮ ߬
SC
107
ܶ ሺܶሻ =
M AN U
106
RI PT
102
108 where
110
112
ܸ ሺܶ ሻ = ܸ ሺ1 + ߙ் ܶሻ
113
(4)
111
TE D
109
114 115
in which Vi is the initial volume and αT is the thermal dilation coefficient. Taking this into
116
account it is clear that the coercive field now takes the following form:
117
2ܭ ܶ ଶ 1 − ൬ ൰ ܪ ሺܶሻ = ߙ ܯௌ ܶ ሺܶሻ
118
(5)
119 120
However, it is necessary to study with detail the behavior of the blocking temperature because it
121
depends on V(T) and must be written in the following form:
122
ܶ ሺܶሻ =
123
(6)
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ଵ
ܸܭ ߬ ሾ1 + ߙ ் ܶ ሿ ݇ ݊ܮ ߬
4
ACCEPTED MANUSCRIPT
124
The values for TB(T) depend on the critical volume, and this volume is not constant due to the
125
existence of dilation. When temperatures are extremely low, it is necessary to consider
126
Grüneisen’s law.
127 Grüneisen’s law describes the relationship between the thermal dilation coefficient and the heat
129
capacity. This relationship is independent of temperature. It is known that at low temperatures
130
the contribution of the lattice to the heat capacity in a solid with cubic symmetry is proportional
131
to T3 (Debye’s formula). Therefore, it can be stated that at low temperatures the thermal dilation
132
coefficient will also be proportional to T3; then, at low temperatures it has the following form (α1
133
constant):
134
ߙ் ሺܶሻ = ߙଵ ܶ ଷ
136
SC
M AN U
135
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128
(7)
137
It is clear that for low temperatures this dependency of the thermal dilation coefficient is
139
necessary to correctly explain the dependency of the volume with temperature. However, a
140
“transition” temperature must exist that links this relationship with the known formula for the
141
dependency of the volume with respect to temperature where the thermal dilation coefficient is
142
no longer a function of temperature.
TE D
138
143
3. EXPLAINING THE MINIMUM OF THE COERCIVE FIELD USING GRÜNEISEN’S
145
LAW
AC C
146
EP
144
147
Considering the previous definitions and relationships, the dependency of the volume with
148
respect to temperature in the interval 0 < T ≤ T* is here defined in the following way, where T*
149 150 151 152
is the “transition” temperature, ߙ ் ሺܶሻ = ߙଵ ܶ ଷ :
ܸଵ ሺܶ ሻ = ܸ ሺ1 + ߙଵ ܶ ସ ሻ (8)
153
5
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154
For temperatures T > T*, the dependency of the volume with respect to temperature is now
155
defined in the usual way, where αT(T)=αT:
156
ܸଶ ሺܶ ሻ = ܸ ሺ1 + ߙ ் ܶ ሻ
158
(9)
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157
159
For these temperatures, αT is independent of temperature. The two volume functions are here
161
linked in the “transition” temperature T = T* by means of a continuous function that is here
162
defined as follows: ܸଶ ሺܶሻ = ܸଵ ሺܶ ∗ ሻ ቈ1 +
164 165
ܸଵᇱ ሺܶ ∗ ሻ ሺܶ − ܶ ∗ ሻ ܸଵ ሺܶ ∗ ሻ
(10)
166
M AN U
163
SC
160
In the expression of the blocking temperature of Eq. (3) it is now considered that:
168
169
ܸ ሺܶ ሻ, ܶ ≤ ܶ ∗ ܸ ሺܶ ሻ = ൜ ଵ ܸଶ ሺܶ ሻ, ܶ ≥ ܶ ∗
170
(11)
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167
171 172
The value of T* allows to control the fast drop of the experimental curve for values close to zero
174
Kelvin, and corresponds to that temperature where the experimental curve changes its tendency
175
and passes by a minimum. Therefore, the following expressions for the coercive field are
176
obtained:
177 178
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173
in the interval 0 < T ≤ T*
179 180
6
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181
ଶ ۓ ۍ ۗ ې 2ۖ ܭ ێ ܶ ۖ ۑ ܪ ሺܶሻ = ߙ 1−ێ ۑ ܸܭ ܯ௦ ۔ ሺ1 + ߙଵ ܶ ସ ሻۘ ۑ ێ ߬ ۖ ۖ ݇ ݊ܮ ۏ ۙ ے ߬ ە
(12)
182
185
ଵ
ଶ ۓ ۍ ۗ ې ۖ ێ ܶ ۖ ۑ 2ܭ ܪ ሺܶሻ = ߙ 1−ێ ۑ ܸܭ ܯ௦ ۔ ሺ1 + ߙ ் ܶ ሻۘ ۑ ێ ߬ ۖ ۖ ݇ ݊ܮ ۏ ے ߬ ە ۙ
(13)
186
SC
184
and in the region for T > T*
M AN U
183
187
In its general form, Eq. (12) can be written using 0 < x ≤ x*,
188
ݔ ଶ ൨ ൡ ݂ሺ ݔሻ = ܥଵ ൝1 − ସ ܥଶ ሺ1 + ܥଷ ݔሻ
191
(14)
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190
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189
ଵ
It is here proposed that xmin = x* , where xmin is the minimum of the coercive field in the
193
experimental curve, is the point at which the two expressions of the volume are connected. The
194
transition temperature corresponds to the same minimum value of the experimental curve and of
195
the function f(x). Once the function starts to decrease after having passed by the minimum, Eq.
196
(8) would indicate that the volume would increase in an abnormal way proportional to the fourth
197
power of T. This abnormal increase in volume that would take place at high T is the reason why
198
departing from this minimum a normal increase in the volume proportional to the first power of
199
T is here proposed and why at this point a conection with the function g(x) has to be made. This
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192
7
ACCEPTED MANUSCRIPT
200
means that the transition temperature after this minimum point separates the two behaviors of the
201
volume.
202
204
205
In its general form, Eq. (13) can be written using x > x*:
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203
ଵ
ݔ ଶ ൨ ൡ ݃ሺ ݔሻ = ܥଵ ൝1 − ܥଶ ሺ1 + ܥସ ݔሻ (15)
208
209
Taking into account the linked relationship between these general functions, we obtain: ݃ሺ ݔሻ = ݂ሺ ∗ ݔሻ ቈ1 + (16)
݂ ᇱ ሺ ∗ ݔሻ ሺ ݔ− ∗ ݔሻ ݂ሺ ∗ ݔሻ
211
This can be plotted using
212
݂ ሺ ݔሻ, 0 < ∗ ݔ ≤ ݔ ܨሺ ݔሻ = ൜ ݃ሺݔሻ, ∗ ݔ > ݔ
213
(17)
215
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214
Figure 1 shows the graph for this function, F(x), for which it was considered that:
216
C3~10-11 and C4~10-5
218
(18)
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217
219
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210
M AN U
207
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206
220
It can be noted that the function F(x) qualitatively describes the behavior of the coercive field
221
with temperature for this type of samples, which presents a minimum.
222 223 224
8
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225 226
48 47
227
46
228
45 43
230
42
231
40
232
39
41
0
50
100
150
233
200
250
300
350
400
x
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F(x) 44
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229
Figure 1. Graph showing the newly defined function, F(x), that describes the minimum of the
235
coercive field.
M AN U
234
236 237
4. CONCLUSIONS
238
Using the expression of the blocking temperature we have obtained the theoretical curve for the
240
coercive field against temperature. In order to adjust the theoretical curve with the experimental
241
one we have considered a volume model with respect to temperature that takes into account
242
Grüneisen’s law for low temperatures.
243
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239
Acknowledgements
245
The authors are thankful to A. Heimann for her assistance translating and editing the English
246
version of the manuscript.
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248 249 250 251 252 253
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247 REFERENCES
[1] C.P. Bean and J.D. Livingston, Superparamagnetism, J. Appl. Phys. Supplement to Vol. 30. No. 4, 120S-129S (1959). [2] E.F. Kneller and F.E. Luborsky, Particle size dependence of coercivity and remanence of single‐domain particles, J. Appl. Phys. 34 656 (1963).
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ACCEPTED MANUSCRIPT
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[3] M. Knobel, W.C. Nunes, L.M. Socolovsky, E. De Biasi, J.M. Vargas, and J.C. Denardin,
255
Superparamagnetism and other magnetic features in granular materials: a review on ideal
256
and real systems, J. Nanosc. Nanotech. 8, 2830 (2008).
260 261 262 263 264 265
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259
coercive field in single-domain particle systems, Phys. Rev. B 70, 014419 (2004).
[5] L.E. Zamora, G.A. Perez Alcazar, J.A. Tabares, J.J. Romero, A. Martinez, J.M. Gonzalez, F.J. Palomares, J.F. Marco, Physica B 407 2306-2312 (2012).
[6] I.N. Krivorotov, H. Yan, E.D. Dahlberg, and A. Stein, Exchange bias in macroporous Co/Co, J. Magn. Magn. Mate. 226-230, 2, 1800-1802 (2001).
SC
258
[4] W.C. Nunes, W.S.D. Folly, J.P. Sinnecker, and M.A. Novak, Temperature dependence of the
[7] D. Kechrakos and K.N. Trohidou, Magnetic properties of dipolar interacting single-domain particles, Phys. Rev. B 58, 12169 (1998).
M AN U
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[8] W.C. Nunes, F. Cebollada, M. Knobel, and D. Zanchet, Effects of dipolar interactions on the
266
magnetic properties of γ-Fe2O3 nanoparticles in the blocked state, J. Appl. Phys. 99, 08N705
267
(2006).
268 269
[9] L. Landau and E. Lifshitz, Statistical Physics, Part 1, Vol. 5, 3rd Ed., ButterworthHeinemann (2005).
[10] R.H. Carr, R.D. Mc Cammon and G.K. White, The thermal expansion of copper at low
271
temperatures, Proceedings of the Royal Society of London, Series A, Mathematical and
272
Physical Sciences 280, No. 1380, 72-84 (1964).
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S0921-4526(19)30048-1
DOI:
https://doi.org/10.1016/j.physb.2019.01.048
Reference:
PHYSB 311300
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 23 November 2018 Revised Date:
27 January 2019
Accepted Date: 29 January 2019
Please cite this article as: M. Salazar, G.A.Pé. Alcazar, Coercive field and Grüneisen's law, Physica B: Physics of Condensed Matter (2019), doi: https://doi.org/10.1016/j.physb.2019.01.048. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
1 2 3 4
RI PT
5 6
COERCIVE FIELD AND GRÜNEISEN’S LAW
30 31 32 33 34 35 36 37 38 39 40 41
M AN U
TE D
a
Pitt Community College, P.O. Drawer 7007, Greenville, NC 27835-7007, USA. b
Departamento de Física, Universidad del Valle, A.A. 25360, Cali, Colombia
EP
29
M. Salazara* and G.A. Pérez Alcazarb
AC C
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
SC
7
* Corresponding author: Manuel Salazar. E-mail address: [email protected]
1
ACCEPTED MANUSCRIPT
42
Abstract
43 The general expression of the blocking temperature for a superparamagnetic system depends on
45
the volume of the particle, but in this expression the volume is independent of temperature.
46
Some experimental curves of the coercive field as a function of temperature show a minimum at
47
relatively low temperatures. One way to obtain theoretical curves that qualitatively describe this
48
behavior is by considering that these systems have two different regimes of thermal dilation
49
when temperature increases: one for low temperatures and the other for high temperatures. In
50
the present work it is shown how these theoretical curves of the coercive field can be calculated
51
assuming these two different regimes and considering Grüneisen’s law for low temperatures.
52
Using the expression of the blocking temperature we have obtained the theoretical curve for the
53
coercive field against temperature. In order to adjust the theoretical curve with the experimental
54
one we have considered a volume model with respect to temperature taking into account
55
Grüneisen’s law for low temperatures.
56 57
Keywords: Coercive field, Grüneisen’s law, blocking temperature
59
61
1. INTRODUCTION
TE D
58
60
M AN U
SC
RI PT
44
In the development of the fundamental expressions of the theory of superparamagnetism it has
63
previously been considered that the volume of the nanoparticles does not depend on temperature
64
[1, 2]. This consideration allows to obtain an expression of the blocking temperature that
65
depends on the volume of the nanoparticles but not on the temperature. For the calculation of
66
the coercive field, which depends on temperature, the value of the blocking temperature is taken
67
into account. In the representation of the experimental data of the coercive field, it was found
68
that in some samples it presents a minimum at relatively low temperatures [3]. Some
69
experimental studies had reported this behavior, for example in systems such as Cu97Co3 and
70
Cu90 Co10 [4], in micrometric powders of Fe50Mn10Al40 [5], in magnetic materials like
71
Macroporus Cobalt [6], and in many others materials. A proposal to explain the occurrence of
72
this minimum presented by Nunes et al. [4] considered a distribution of nanoparticle size. Other
AC C
EP
62
2
ACCEPTED MANUSCRIPT
73
studies considered the effect of magnetic interactions on the expresion of the coercive field [7,
74
8]. However, an analytical prediction for the occurrence of this minimum has yet to be
75
developed.
76
In this study, we present an analytical method to explain this minimum considering in the expression of the blocking temperature a dependency of the volume of the sample with
78
temperature. Because Grüneisen’s law relates heat capacity with the thermal dilation coefficient,
79
and this law is valid at low temperatures, this law can be considered to explain the coercive field
80
at temperatures lower than the minimum. In contrast, for temperatures higher than that of the
81
minimum, the typical dependence of the thermal dilation coefficient with temperature (a linear
82
dependence) is considered.
SC
RI PT
77
84
M AN U
83
2. BLOCKING TEMPERATURE, COERCIVE FIELD, AND GRÜNEISEN’S LAW
85 86
Considering weak interactions between nanoparticles and for the particular case of the critical
87
volume V0 and a relaxation time equal to that of the measurement, τ = τm, it is possible to
88
demonstrate that for the current existing model the blocking temperature, TB, is given by [2]:
90
ܶ = (1)
92
EP
91
ܸܭ ߬ ݇ ݊ܮ ߬
TE D
89
where it is assumed that the volume does not depend on temperature.
AC C
93 94
Next, a system of non-interacting single domain particles whose energy is only due to the
95
uniaxial anisotropy energy is considered. For the range of temperatures from 0 to TB, that is,
96
when all the particles are blocked, the coercive field is given by [4]:
97
98
ଵ
2ܭ ܶ ଶ ܪ ሺܶሻ = ߙ 1 − ൬ ൰ ܯௌ ܶ (2)
3
ACCEPTED MANUSCRIPT
99 100
Here, α = 1 if the particles with easy axis are aligned, or α = 0.48 if they are randomly oriented [8].
101 For the current case, for identical weak interacting particles, the expression of TB that previous
103
studies have used is that of Eq. (1). In the present study, considering now that the volume V0 of
104
Eq. (1) depends on temperature, it can be written as V0 = V(T) in the corresponding blocking
105
temperature:
(3)
ܸܭሺܶሻ ߬ ݇ ݊ܮ ߬
SC
107
ܶ ሺܶሻ =
M AN U
106
RI PT
102
108 where
110
112
ܸ ሺܶ ሻ = ܸ ሺ1 + ߙ் ܶሻ
113
(4)
111
TE D
109
114 115
in which Vi is the initial volume and αT is the thermal dilation coefficient. Taking this into
116
account it is clear that the coercive field now takes the following form:
117
2ܭ ܶ ଶ 1 − ൬ ൰ ܪ ሺܶሻ = ߙ ܯௌ ܶ ሺܶሻ
118
(5)
119 120
However, it is necessary to study with detail the behavior of the blocking temperature because it
121
depends on V(T) and must be written in the following form:
122
ܶ ሺܶሻ =
123
(6)
AC C
EP
ଵ
ܸܭ ߬ ሾ1 + ߙ ் ܶ ሿ ݇ ݊ܮ ߬
4
ACCEPTED MANUSCRIPT
124
The values for TB(T) depend on the critical volume, and this volume is not constant due to the
125
existence of dilation. When temperatures are extremely low, it is necessary to consider
126
Grüneisen’s law.
127 Grüneisen’s law describes the relationship between the thermal dilation coefficient and the heat
129
capacity. This relationship is independent of temperature. It is known that at low temperatures
130
the contribution of the lattice to the heat capacity in a solid with cubic symmetry is proportional
131
to T3 (Debye’s formula). Therefore, it can be stated that at low temperatures the thermal dilation
132
coefficient will also be proportional to T3; then, at low temperatures it has the following form (α1
133
constant):
134
ߙ் ሺܶሻ = ߙଵ ܶ ଷ
136
SC
M AN U
135
RI PT
128
(7)
137
It is clear that for low temperatures this dependency of the thermal dilation coefficient is
139
necessary to correctly explain the dependency of the volume with temperature. However, a
140
“transition” temperature must exist that links this relationship with the known formula for the
141
dependency of the volume with respect to temperature where the thermal dilation coefficient is
142
no longer a function of temperature.
TE D
138
143
3. EXPLAINING THE MINIMUM OF THE COERCIVE FIELD USING GRÜNEISEN’S
145
LAW
AC C
146
EP
144
147
Considering the previous definitions and relationships, the dependency of the volume with
148
respect to temperature in the interval 0 < T ≤ T* is here defined in the following way, where T*
149 150 151 152
is the “transition” temperature, ߙ ் ሺܶሻ = ߙଵ ܶ ଷ :
ܸଵ ሺܶ ሻ = ܸ ሺ1 + ߙଵ ܶ ସ ሻ (8)
153
5
ACCEPTED MANUSCRIPT
154
For temperatures T > T*, the dependency of the volume with respect to temperature is now
155
defined in the usual way, where αT(T)=αT:
156
ܸଶ ሺܶ ሻ = ܸ ሺ1 + ߙ ் ܶ ሻ
158
(9)
RI PT
157
159
For these temperatures, αT is independent of temperature. The two volume functions are here
161
linked in the “transition” temperature T = T* by means of a continuous function that is here
162
defined as follows: ܸଶ ሺܶሻ = ܸଵ ሺܶ ∗ ሻ ቈ1 +
164 165
ܸଵᇱ ሺܶ ∗ ሻ ሺܶ − ܶ ∗ ሻ ܸଵ ሺܶ ∗ ሻ
(10)
166
M AN U
163
SC
160
In the expression of the blocking temperature of Eq. (3) it is now considered that:
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ܸ ሺܶ ሻ, ܶ ≤ ܶ ∗ ܸ ሺܶ ሻ = ൜ ଵ ܸଶ ሺܶ ሻ, ܶ ≥ ܶ ∗
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(11)
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The value of T* allows to control the fast drop of the experimental curve for values close to zero
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Kelvin, and corresponds to that temperature where the experimental curve changes its tendency
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and passes by a minimum. Therefore, the following expressions for the coercive field are
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obtained:
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in the interval 0 < T ≤ T*
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ଶ ۓ ۍ ۗ ې 2ۖ ܭ ێ ܶ ۖ ۑ ܪ ሺܶሻ = ߙ 1−ێ ۑ ܸܭ ܯ௦ ۔ ሺ1 + ߙଵ ܶ ସ ሻۘ ۑ ێ ߬ ۖ ۖ ݇ ݊ܮ ۏ ۙ ے ߬ ە
(12)
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ଶ ۓ ۍ ۗ ې ۖ ێ ܶ ۖ ۑ 2ܭ ܪ ሺܶሻ = ߙ 1−ێ ۑ ܸܭ ܯ௦ ۔ ሺ1 + ߙ ் ܶ ሻۘ ۑ ێ ߬ ۖ ۖ ݇ ݊ܮ ۏ ے ߬ ە ۙ
(13)
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and in the region for T > T*
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In its general form, Eq. (12) can be written using 0 < x ≤ x*,
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ݔ ଶ ൨ ൡ ݂ሺ ݔሻ = ܥଵ ൝1 − ସ ܥଶ ሺ1 + ܥଷ ݔሻ
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(14)
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ଵ
It is here proposed that xmin = x* , where xmin is the minimum of the coercive field in the
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experimental curve, is the point at which the two expressions of the volume are connected. The
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transition temperature corresponds to the same minimum value of the experimental curve and of
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the function f(x). Once the function starts to decrease after having passed by the minimum, Eq.
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(8) would indicate that the volume would increase in an abnormal way proportional to the fourth
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power of T. This abnormal increase in volume that would take place at high T is the reason why
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departing from this minimum a normal increase in the volume proportional to the first power of
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T is here proposed and why at this point a conection with the function g(x) has to be made. This
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means that the transition temperature after this minimum point separates the two behaviors of the
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volume.
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In its general form, Eq. (13) can be written using x > x*:
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ݔ ଶ ൨ ൡ ݃ሺ ݔሻ = ܥଵ ൝1 − ܥଶ ሺ1 + ܥସ ݔሻ (15)
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Taking into account the linked relationship between these general functions, we obtain: ݃ሺ ݔሻ = ݂ሺ ∗ ݔሻ ቈ1 + (16)
݂ ᇱ ሺ ∗ ݔሻ ሺ ݔ− ∗ ݔሻ ݂ሺ ∗ ݔሻ
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This can be plotted using
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݂ ሺ ݔሻ, 0 < ∗ ݔ ≤ ݔ ܨሺ ݔሻ = ൜ ݃ሺݔሻ, ∗ ݔ > ݔ
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(17)
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Figure 1 shows the graph for this function, F(x), for which it was considered that:
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C3~10-11 and C4~10-5
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(18)
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It can be noted that the function F(x) qualitatively describes the behavior of the coercive field
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with temperature for this type of samples, which presents a minimum.
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Figure 1. Graph showing the newly defined function, F(x), that describes the minimum of the
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coercive field.
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4. CONCLUSIONS
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Using the expression of the blocking temperature we have obtained the theoretical curve for the
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coercive field against temperature. In order to adjust the theoretical curve with the experimental
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one we have considered a volume model with respect to temperature that takes into account
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Grüneisen’s law for low temperatures.
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Acknowledgements
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The authors are thankful to A. Heimann for her assistance translating and editing the English
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version of the manuscript.
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[3] M. Knobel, W.C. Nunes, L.M. Socolovsky, E. De Biasi, J.M. Vargas, and J.C. Denardin,
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Superparamagnetism and other magnetic features in granular materials: a review on ideal
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