Curie temperature

From Wikipedia, the free encyclopedia
  (Redirected from Curie point)
Jump to: navigation, search
Figure 1 Below the Curie temperature, neighbouring magnetic spins align in a ferromagnet in the absence of an applied magnetic field.
Figure 2 Above the Curie temperature, the magnetic spins are randomly aligned in a paramagnet unless a magnetic field is applied.

In physics and materials science, the Curie temperature (Tc), or Curie point, is the temperature where a material's permanent magnetism changes to induced magnetism. The force of magnetism is determined by magnetic moments.

The Curie temperature is the critical point where a material's intrinsic magnetic moments change direction. Magnetic moments are permanent dipole moments within the atom which originate from electrons' angular momentum and spin. Materials have different structures of intrinsic magnetic moments that depend on temperature. At a material's Curie Temperature those intrinsic magnetic moments change direction.

Permanent magnetism is caused by the alignment of magnetic moments and induced magnetism is created when disordered magnetic moments are forced to align in an applied magnetic field. For example, the ordered magnetic moments (ferromagnetic, figure 1) change and become disordered (paramagnetic, figure 2) at the Curie Temperature.

Higher temperatures make magnets weaker as spontaneous magnetism only occurs below the Curie Temperature. Magnetic susceptibility only occurs above the Curie Temperature and can be calculated from the Curie-Weiss Law which is derived from Curie's Law.

In analogy to ferromagnetic and paramagnetic materials, the Curie temperature can also be used to describe the temperature where a material's spontaneous electric polarisation changes to induced electric polarisation or the reverse upon reduction of the temperature below the Curie temperature.

The Curie temperature is named after Pierre Curie who showed that magnetism was lost at a critical temperature.[1]

Curie Temperature of materials [2][3][4]
Material Curie
temperature (K)
Iron (Fe) 1043
Cobalt (Co) 1400
Nickel (Ni) 627
Gadolinium (Gd) 292
Dysprosium (Dy) 88
MnBi 630
MnSb 587
CrO2 386
MnAs 318
EuO 69
Iron(III) oxide (Fe2O3) 948
Iron(II,III) oxide (FeOFe2O3) 858
NiOFe2O3 858
CuOFe2O3 728
MgOFe2O3 713
MnOFe2O3 573
Y3Fe5O12 560

Magnetic moments[edit]

Magnetic moments are permanent dipole moments within the atom which are made up from electrons angular momentum and spin.[5]

Electrons inside atoms contribute magnetic moments from their own angular momentum and from their orbital momentum around the nucleus. Magnetic moments from the nucleus are insignificant in contrast to magnetic moments from electrons.[6] Thermal contribution will result in higher energy electrons causing disruption to their order and alignment between dipoles to be destroyed.

Ferromagnetic, paramagnetic, ferrimagnetic and antiferromagnetic materials have different structures of intrinsic magnetic moments. It is at a material's specific Curie Temperature where they change properties. The transition from antiferromagnetic to paramagnetic (or vice versa) occurs at the Néel Temperature which is analogous to Curie Temperature.

Below Tc Above Tc
Ferromagnetic ↔ Paramagnetic
Ferrimagnetic ↔ Paramagnetic
Antiferromagnetic ↔ Paramagnetic

Materials with magnetic moments that change properties at the Curie temperature[edit]

Ferromagnetic, paramagnetic, ferrimagnetic and antiferromagnetic structures are made up of intrinsic magnetic moments. If all electrons within the structure are paired, these moments cancel out due to having opposite spins and angular momentum. Thus even with an applied magnetic field will have different properties and no Curie Temperature.[7][8]

Paramagnetic[edit]

Main article: Paramagnetism

A material is paramagnetic only above its Curie Temperature. Paramagnetic materials are non-magnetic when a magnetic field is absent and magnetic when a magnetic field is applied. When the magnetic field is absent the material has disordered magnetic moments; that is, the atoms are unsymmetrical and not aligned. When the magnetic field is present the magnetic moments are temporarily realigned parallel to the applied field;[9][10] the atoms are symmetrical and aligned.[11] The magnetic moment in the same direction is what causes an induced magnetic field.[11][12]

For paramagnetism this response to an applied magnetic field is positive and known as magnetic susceptibility.[7] The magnetic susceptibility only applies above the Curie Temperature for disordered states.[13]

Sources of Paramagnetism (Materials which have Curie Temperatures);[14]

  • All atoms which have unpaired electrons;
  • Atoms where inner shells are incomplete in electrons;
  • Free radicals;
  • Metals

Above the Curie Temperature the atoms are excited, the spin orientation becomes randomised,[8] but can be realigned in an applied field and the material paramagnetic. Below the Curie Temperature the intrinsic structure has under gone a phase transition,[15] the atoms are ordered and the material is ferromagnetic.[11] The paramagnetic materials induced magnetic fields are very weak in comparison to ferromagnetic materials magnetic fields.[15]

Ferromagnetic[edit]

Main article: Ferromagnetism

Materials are only ferromagnetic below their corresponding Curie temperatures. Ferromagnetic materials are magnetic in the absence of an applied magnetic field.

When a magnetic field is absent the material has spontaneous magnetization which is a result of the ordered magnetic moments; that is, for ferromagnetism, the atoms are symmetrical and aligned in the same direction creating a permanent magnetic field.

The magnetic interactions are held together by exchange interactions; otherwise thermal disorder would overcome the weak interactions of magnetic moments. The exchange interaction has a zero probability of parallel electrons occupying the same point in time, implying a preferred parallel alignment in the material.[16] The Boltzmann factor contributes heavily as it prefers interacting particles to be aligned in the same direction.[17] This causes ferromagnets to have strong magnetic fields and high Curie temperatures of around 1000K.[18]

Below the Curie temperature, the atoms are aligned and parallel, causing spontaneous magnetism; the material is ferromagnetic. Above the Curie temperature the material is paramagnetic, as the atoms lose their ordered magnetic moments when the material undergoes a phase transition.[15]

Ferrimagnetic[edit]

Main article: Ferrimagnetism

Not to be confused with ferromagnetic.

Materials are only ferrimagnetic below their materials corresponding Curie Temperature. Ferrimagnetic materials are magnetic in the absence of an applied magnetic field and are made up of two different ions.[19]

When a magnetic field is absent the material has a spontaneous magnetism which is the result of ordered magnetic moments; that is, for ferrimagnetism one ion's magnetic moments are aligned facing in one direction with certain magnitude and the other ion's magnetic moments are aligned facing in the opposite direction with a different magnitude. As the magnetic moments are of different magnitudes in opposite directions there is still a spontaneous magnetism and a magnetic field is present.[19]

Similar to ferromagnetic materials the magnetic interactions are held together by exchange interactions. The orientations of moments however are anti-parallel which results in a net momentum by subtracting their momentum from one another.[19]

Below the Curie Temperature the atoms of each ion are aligned anti-parallel with different momentums causing a spontaneous magnetism; the material is ferrimagnetic. Above the Curie Temperature the material is paramagnetic as the atoms lose their ordered magnetic moments as the material undergoes a phase transition.[19]

Antiferromagnetic and the Néel temperature[edit]

Main article: Antiferromagnetism

Materials are only antiferromagetic below their corresponding Néel Temperature. This is similar to the Curie Temperature as above the Néel Temperature the material undergoes a phase transition and becomes paramagnetic.

The material has equal magnetic moments aligned in opposite directions resulting in a zero magnetic moment and a net magnetism of zero at all temperatures below the Néel Temperature. Antiferromagnetic materials are weakly magnetic in the absence or presence of an applied magnetic field.

Similar to ferromagnetic materials the magnetic interactions are held together by exchange interactions preventing thermal disorder from overcoming the weak interactions of magnetic moments.[16][20] When disorder occurs it is at the Néel Temperature.[20]

Curie-Weiss law[edit]

The Curie-Weiss law is an adapted version of Curie's law.

The Curie-Weiss law is a simple model derived from a mean-field approximation, this means it works well for the materials temperature,T, much greater than their corresponding Curie Temperature,Tc, i.e. T >> Tc; however fails to describe the magnetic susceptibility, χ, in the immediate vicinity of the Curie point because of local fluctuations between atoms.[21]

Both Curie's law and the Curie-Weiss law do not hold for T< Tc.

Curie's law for a paramagnetic material;[22]

\chi = \frac{M}{H} =\frac{M \mu_0}{B} =\frac{C}{T}
Definition
χ the magnetic susceptibility; the influence of an applied magnetic field on a material
M the magnetic moments per unit volume
H the macroscopic magnetic field
B the magnetic field
C the material-specific Curie constant
C = \frac{\mu_0 \mu_B^2}{3 k_B}N g^2 J(J+1)[23]
µ0 the permeability of free space. Note - in CGS units is taken to equal one.[24]
g the Landé g-factor
J(J+1) the eigenvalue for eigenstate J2 for the stationary states within the incomplete atoms shells (electrons unpaired)
µB the Bohr Magneton
kB Boltzmann's constant
total magnetism is N number of magnetic moments per unit volume

The Curie-Weiss law is then derived from Curie's law to be

\chi = \frac{C}{T-T_c}

where

T_C = \frac{C \lambda }{\mu_0}

λ is the Weiss molecular field constant.[23][25]

For full derivation see Curie-Weiss law

Physics of Curie temperature[edit]

Approaching Curie temperature from above[edit]

As the Curie-Weiss Law is an approximation a more accurate model is needed when the temperature,T, approaches the materials Curie Temperature,TC.

Magnetic susceptibility occurs above the Curie Temperature.

An accurate model of critical behaviour for magnetic susceptibility with critical exponent γ;

\chi \sim \frac{1}{(T - T_{c})^\gamma}

The critical exponent differs between materials and for the mean-field model is taken as γ=1.[26]

As temperature is inversely proportional to magnetic susceptibility when T approaches TC the denominator tends to zero and the magnetic susceptibility approaches infinity allowing magnetism to occur. This is a spontaneous magnetism which is a property of ferromagnetic and ferrimagnetic materials.[27][28]

Approaching Curie temperature from below[edit]

Magnetism depends on temperature and spontaneous magnetism occurs below the Curie Temperature. An accurate model of critical behaviour for spontaneous magnetism with critical exponent β;

M \sim (T - T_C)^\beta

The critical exponent differs between materials and for the mean-field model as taken as β=0.5 where T<<TC.[26]

The spontaneous magnetism approaches zero as the temperature increases towards the materials Curie Temperature.

Approaching absolute zero (0 Kelvin)[edit]

The spontaneous magnetism, occurring in ferromagnetic, ferrimagnetic and antiferromagnetic materials, approaches zero as the temperature increases towards the material's Curie Temperature. Spontaneous magnetism is at its maximum as the temperature approaches 0K.[29] That is, the magnetic moments are completely aligned and at their strongest magnitude of magnetism due to no thermal disturbance.

In paramagnetic materials temperature is sufficient to overcome the ordered alignments. As the temperature approaches 0K the entropy decreases to zero, that is, the disorder decreases and becomes ordered. This occurs without the presence of an applied magnetic field and obeys the third law of thermodynamics.[16]

Both Curie's Law and the Curie-Weiss law fail as the temperature approaches 0K. This is because they depend on the magnetic susceptibility which only applies when the state is disordered.[30]

Gadolinium Sulphate continues to satisfy Curie's law at 1K. Between 0-1K the law fails to hold and a sudden change in the intrinsic structure occurs at the Curie Temperature.[31]

Ising model of phase transitions[edit]

The Ising model is mathematically based and can analyse the critical points of phase transitions in ferromagnetic order due to spins of electrons having magnitudes of either +/- ½. The spins interact with their neighbouring dipole electrons in the structure and here the Ising model can predict their behaviour with each other.[32][33]

This model is important for solving and understanding the concepts of phase transitions and hence solving the Curie Temperature. As a result many different dependencies that effect the Curie Temperature can be analysed.

For example the surface and bulk properties depend on the alignment and magnitude of spins and the Ising model can determine the effects of magnetism in this system.

Weiss domains and surface and bulk Curie temperatures[edit]

Figure 3 The Weiss domains in a ferromagnetic material; the magnetic moments are aligned in domains.

Materials structures consist of intrinsic magnetic moments which are separated into domains called Weiss domains.[34] This can result in ferromagnetic materials having no spontaneous magnetism as domains could potentially balance each other out.[34] The position of particles can therefore have different orientations around the surface than the main part (bulk) of the material. This property directly affects the Curie Temperature as there can be a bulk Curie Temperature TB and a different surface Curie Temperature TS for a material.[35]

This allows for the surface Curie Temperature to be ferromagnetic above the bulk Curie Temperature when the main state is disordered, i.e. Ordered and disordered states occur simultaneously.[32]

The surface and bulk properties can be predicted by the Ising model and electron capture spectroscopy can be used to detect the electron spins and hence the magnetic moments on the surface of the material. An average total magnetism is taken from the bulk and surface temperatures to calculate the Curie Temperature from the material, noting the bulk contributes more.[32][36]

The angular momentum of an electron is either +ħ/2 or - ħ/2 due to it having a spin of ½, which gives a specific size of magnetic moment to the electron; the Bohr Magneton.[37] Electrons orbiting around the nucleus in a current loop create a magnetic field which depends on the Bohr Magneton and magnetic quantum number.[37] Therefore the magnetic moments are related between angular and orbital momentum and affect each other. Angular momentum contributes twice as much to magnetic moments than orbital.[38]

For terbium which is a rare earth metal and has a high orbital angular momentum the magnetic moment is strong enough to affect the order above its bulk temperatures. It is said to have a high anisotropy on the surface, that is it is highly directed in one orientation. It remains ferromagnetic on its surface above its Curie Temperature while its bulk becomes ferrimagnetic and then at higher temperatures its surface remains ferrimagnetic above its bulk Néel Temperature before becoming completely disordered and paramagnetic with increasing temperature. The anisotropy in the bulk is different from its surface anisotropy just above these phase changes as the magnetic moments will be ordered differently or ordered in paramagnetic materials.[35]

Changing a material's Curie temperature[edit]

Composite materials[edit]

Composite materials, that is, materials composed from other materials with different properties, can change the Curie Temperature. For example a composite which has silver in can create spaces for oxygen molecules in bonding which decreases the Curie Temperature[39] as the crystal lattice will not be as compact.

The alignment of magnetic moments in the composite material affects the Curie Temperature. If the materials moments are parallel with each other the Curie Temperature will increase and if perpendicular the Curie Temperature will decrease[39] as either more or less thermal energy will be needed to destroy the alignments.

Preparing composite materials through different temperatures can result in different final compositions which will have different Curie Temperatures.[40] Doping a material can also affect its Curie Temperature.[40]

The density of nanocomposite materials changes the Curie Temperature. Nanocomposites are compact structures on a nano-scale. The structure is built up of high and low bulk Curie Temperatures, however will only have one mean-field Curie Temperature. A higher density of lower bulk temperatures results in a lower mean-field Curie Temperature and a higher density of higher bulk temperature significantly increases the mean-field Curie Temperature. In more than one dimension the Curie Temperature begins to increase as the magnetic moments will need more thermal energy to overcome the ordered structure.[36]

Particle size[edit]

The size of particles in a material's crystal lattice changes the Curie Temperature. Due to the small size of particles (nanoparticles) the fluctuations of electron spins become more prominent, this results in the Curie Temperature drastically decreasing when the size of particles decrease as the fluctuations cause disorder. The size of a particle also affects the anisotropy causing alignment to become less stable and thus lead to disorder in magnetic moments.[32][41]

The extreme of this is superparamagnetism which only occurs in small ferromagnetic particles and is where fluctuations are very influential causing magnetic moments to change direction randomly and thus create disorder.

The Curie Temperature of nanoparticles are also affected by the crystal lattice structure, body-centred cubic (bcc), face-centred cubic (fcc) and a hexagonal structure (hcp) all have different Curie Temperatures due to magnetic moments reacting to their neighbouring electron spins. fcc and hcp have tighter structures and as a results have higher Curie Temperatures than bcc as the magnetic moments have stronger effects when closer together.[32] This is known as the coordination number which is the number of nearest neighbouring particles in a structure. This indicates a lower coordination number at the surface of a material than the bulk which leads to the surface becoming less significant when the temperature is approaching the Curie Temperature. In smaller systems the coordination number for the surface is more significant and the magnetic moments have a stronger affect on the system.[32]

Although fluctuations in particles can be minuscule, they are heavily dependent on the structure of crystal lattices as they react with their nearest neighbouring particles. Fluctuations are also affected by the exchange interaction[41] as parallel facing magnetic moments are favoured and therefore have less disturbance and disorder, therefore a tighter structure influences a stronger magnetism and therefore a higher Curie Temperature.

Pressure[edit]

Pressure changes a material's Curie Temperature. Increasing pressure on the crystal lattice decreases the volume of the system. Pressure directly affects the kinetic energy in particles as movement increases causing the vibrations to disrupt the order of magnetic moments. This is similar to temperature as it also increases the kinetic energy of particles and destroys the order of magnetic moments and magnetism.[42]

Pressure also affects the density of states (DOS).[42] Here the DOS decreases causing the number of electrons available to the system to decrease. This leads to the number of magnetic moments decreasing as they depend on electron spins. It would be expected because of this that the Curie Temperature would decrease however it increases. This is the result of the exchange interaction. The exchange interaction favours the aligned parallel magnetic moments due to electrons being unable to occupy the same space in time[16] and as this is increased due to the volume decreasing the Curie Temperature increases with pressure. The Curie Temperature is made up of a combination of dependencies on kinetic energy and the DOS.[42]

It is interesting to note that the concentration of particles also affects the Curie Temperature when pressure is being applied and can result in a decrease in Curie Temperature when the concentration is above a certain percent.[42]

Orbital ordering[edit]

Orbital ordering changes the Curie Temperature of a material. Orbital ordering can be controlled through applied strains.[43] This is a function that determines the wave of a single electron or paired electrons inside the material. Having control over the probability of where the electron will be allows the Curie Temperature to be altered. For example the delocalised electrons can be moved onto the same plane by applied strains within the crystal lattice.[43]

The Curie Temperature is seen to increase greatly due to electrons being packed together in the same plane, they are forced to align due to the exchange interaction and thus increases the strength of the magnetic moments which prevents thermal disorder at lower temperatures.

Curie temperature in ferroelectric and piezoelectric materials[edit]

In analogy to ferromagnetic and paramagnetic materials, the Curie Temperature can also used to describe the temperature where a material's spontaneous electric polarisation changes to induced electric polarisation, or vice versa.[44]

Electric polarisation is a result of aligned electric dipoles. Aligned electric dipoles are composites of positive and negative charges where all the dipoles are facing in one direction. The charges are separated from their stable placement in the particles and can occur spontaneously, from pressure or an applied electric field.[45]

Ferroelectric, dielectric (paraelectric) and piezoelectric materials have electric polarisation. In ferroelectric materials there is a spontaneous electric polarisation in the absence of an applied electric field.[44] In dielectric materials there is electric polarisation aligned only when an electric field is applied.[45] Piezoelectric materials have electric polarisation due to applied mechanical stress distorting the structure from pressure.[46]

T0 is the temperature where ferroelectric materials lose their spontaneous polarisation as a first or second order phase change occurs, that is the internal structure changes or the internal symmetry changes.[44] In certain cases T0 is equal to the Curie Temperature however the Curie Temperature can be 10 kelvin lower than T0.[47]

Figure 4 (Below T0) Ferroelectric polarisation P in an applied electric field E.
Figure 5 (Above T0) Dielectric polarisation P in an applied electric field E.
Below T0 Above T0[48]
Ferroelectric ↔ Dielectric

All ferroelectric materials are piezoelectric.[44]

Piezoelectric

An external force applies pressure on particles inside the material which affects the structure of the crystal lattice. Particles in a unit cell become unsymmetrical which allows a net polarisation from each particle. Symmetry would cancel the opposing charges out and there would be no net polarisation.[49] Below the transition temperature T0 displacement of electric charges causes polarisation. Above the transition temperature T0 the structure is cubic and symmetric, causing the material to become dielectric. Electric charges are also agitated and disordered causing the material to have no electric polarisation in the absence of an applied electric field.

Ferroelectric and Dielectric

Materials are only ferroelectric below their corresponding transition temperature T0.[44] Ferroelectric materials are all piezoelectric and therefore have a spontaneous electric polarisation as the structures are unsymmetrical.

Materials are only dielectric above their corresponding transition temperature T0.[50] Dielectric materials have no electric polarisation in the absence of an applied electric field. The electric dipoles are unaligned and have no net polarisation. In analogy to magnetic susceptibility, electric susceptibility only occurs above T0.

Ferroelectric materials when polarised are influenced under hysteresis (Figure 4); that is they are dependent on their past state as well as their current state. As an electric field is applied the dipoles are forced to align and polarisation is created, when the electric field is removed polarisation remains. The hysteresis loop depends on temperature and as a result as the temperature is increased and reaches T0 the two curves become one curve as shown in the dielectric polarisation (Figure 5).[51]

Relative Permittivity

A modified version of the Curie Weiss law applies to the dielectric constant, also known as the relative permittivity:[47][52]

\epsilon = \epsilon_0 + \frac{C}{T-T_c}.

Applications[edit]

A heat-induced ferromagnetic-paramagnetic transition is used in magneto-optical storage media, for erasing and writing of new data. Famous examples include the Sony Minidisc format, as well as the now-obsolete CD-MO format. Other uses include temperature control in soldering irons, and stabilizing the magnetic field of tachometer generators against temperature variation.[53]

See also[edit]

Notes[edit]

  1. ^ "Pierre Curie - Biography". Nobelprize.org. The Nobel Foundation 1903. Retrieved 2013-03-14. 
  2. ^ Buschow 2001, p5021, table 1
  3. ^ Jullien 1989, p. 155
  4. ^ Kittel 1986
  5. ^ Hall 1994, p. 200
  6. ^ Jullien 1989, pp. 136–138
  7. ^ a b Lüth, Harald Ibach, Hans (2009). Solid-state physics : an introduction to principles of materials science (4th extensively updated and enlarged ed. ed.). Berlin: Springer. ISBN 978-3-540-93803-3. 
  8. ^ a b Levy 1968, pp. 236–239
  9. ^ Dekker 1958, pp. 217–220
  10. ^ Levy 1968
  11. ^ a b c Fan 1987, pp. 164–165
  12. ^ Dekker 1958, pp. 454–455
  13. ^ Mendelssohn 1977, p. 162
  14. ^ Levy 1968, pp. 198–202
  15. ^ a b c Cusack 1958, p. 269
  16. ^ a b c d Hall 1994, pp. 220–221
  17. ^ Palmer 2007
  18. ^ Hall 1994, p. 220
  19. ^ a b c d Jullien 1989, pp. 158–159
  20. ^ a b Jullien 1989, pp. 156–157
  21. ^ Jullien 1989, pp. 153
  22. ^ Hall 1994, pp. 205–206
  23. ^ a b Levy 1968, pp. 201–202
  24. ^ Kittel 1996, pp. 444
  25. ^ Myers 1997, pp. 334–345
  26. ^ a b Hall 1994, pp. 227–228
  27. ^ Kittel 1986, pp. 424–426
  28. ^ Spaldin 2010, pp. 52–54
  29. ^ Hall 1994, pp. 225
  30. ^ Mendelssohn 1977, pp. 180–181
  31. ^ Mendelssohn 1977, p. 167
  32. ^ a b c d e f Bertoldi 2012
  33. ^ Brout 1965, pp. 6–7
  34. ^ a b Jullien 1989, p. 161
  35. ^ a b Rau 1988
  36. ^ a b Skomski 2000
  37. ^ a b Jullien 1989, pp. 138
  38. ^ Hall 1994
  39. ^ a b Hwang 1998
  40. ^ a b Jones 2003
  41. ^ a b Lopez-Dominguez 2012
  42. ^ a b c d Bose 2011
  43. ^ a b Sadoc 2010
  44. ^ a b c d e Myers 1997, pp. 404–405
  45. ^ a b Jullien 1989, pp. 56–59
  46. ^ Hall 1994, p. 275
  47. ^ a b Webster 1999
  48. ^ Kovetz 1990, p. 116
  49. ^ Pascoe 1973, pp. 186–187
  50. ^ Hummel 2001, pp. 189
  51. ^ Pascoe 1973, pp. 190–191
  52. ^ Webster, John G. (1999). The measurement, instrumentation, and sensors handbook ([Online-Ausg.] ed.). Boca Raton, Fla.: CRC Press published in cooperation with IEEE Press. pp. 6.55–6.56. ISBN 9780849383472. 
  53. ^ Pallàs-Areny & Webster 2001, pp. 262–263

References[edit]

  • Buschow, K. H. J. (2001). Encyclopedia of materials : science and technology. Elsevier. ISBN 0-08-043152-6. 
  • Kittel, Charles (1986). Introduction to Solid State Physics (sixth ed.). John Wiley & Sons. ISBN 0-471-87474-4. 
  • Pallàs-Areny, Ramon; Webster, John G (2001). Sensors and Signal Conditioning (2nd ed.). John Wiley & Sons. pp. 262–263. ISBN 978-0-471-33232-9. 
  • Spaldin, Nicola A. (2010). Magnetic materials : fundamentals and applications (2nd ed.). Cambridge: Cambridge University Press. ISBN 9780521886697. 
  • Ibach, Harald; Lüth, Hans (2009). Solid-state physics : an introduction to principles of materials science (4th extensively updated and enlarged ed.). Berlin: Springer. ISBN 9783540938033. 
  • Levy, Robert A (1968). Principles of Solid State Physics. Academic Press. ISBN 978-0124457508. 
  • Fan, H.Y (1987). Elements of Solid State Physics. Wiley-Interscience. ISBN 9780471859871. 
  • Dekker, Adrianus J (1958). Solid State Physics. Macmillan. ISBN 9780333106235. 
  • Cusack, N (1958). The Electrical and Magnetic Properties of Solids. Longmans, Green. 
  • Hall, J.R. Hook, H.E. (1994). Solid state physics (2nd ed.). Chichester: Wiley. ISBN 0471928054. 
  • Jullien, André Guinier ; Rémi (1989). The solid state from superconductors to superalloys (Pbk. ed.). Oxford: Oxford Univ. Press. ISBN 0198555547. 
  • Mendelssohn, K. (1977). The quest for absolute zero : the meaning of low temperature physics. with S.I. units. (2nd ed.). London: Taylor and Francis. ISBN 0850661196. 
  • Myers, H.P. (1997). Introductory solid state physics. (2nd ed. ed.). London: Taylor & Francis. ISBN 0748406603. 
  • Kittel, Charles (1996). Introduction to solid state physics (7. ed. ed.). New York [u.a.]: Wiley. ISBN 0471111813. 
  • Palmer, John (2007). Planar Ising correlations ([Online-Ausg.]. ed.). Boston: Birkhäuser. ISBN 9780817646202. 
  • Bertoldi, Dalía S; Bringa, Eduardo M; Miranda, E N (6 June 2012). "Analytical solution of the mean field Ising model for finite systems". Journal of Physics: Condensed Matter 24 (22): 226004. Bibcode:2012JPCM...24v6004B. doi:10.1088/0953-8984/24/22/226004. Retrieved 12/02/2013.  Check date values in: |accessdate= (help)
  • Brout, Robert (1965). Phase Transitions. New York, Amsterdam: W.A.Benjamin.INC. 
  • Rau, C.; Jin, C.; Robert, M. (1 January 1988). "Ferromagnetic order at Tb surfaces above the bulk Curie temperature". Journal of Applied Physics 63 (8): 3667. Bibcode:1988JAP....63.3667R. doi:10.1063/1.340679. 
  • Skomski, R.; Sellmyer, D. J. (1 January 2000). "Curie temperature of multiphase nanostructures". Journal of Applied Physics 87 (9): 4756. Bibcode:2000JAP....87.4756S. doi:10.1063/1.373149. 
  • Lopez-Dominguez, Victor; Hernàndez, Joan Manel; Tejada, Javier; Ziolo, Ronald F. (8 January 2013). "Colossal Reduction in Curie Temperature Due to Finite-Size Effects in CoFe O Nanoparticles". Chemistry of Materials 25 (1): 6–11. doi:10.1021/cm301927z. 
  • Bose, S. K.; Kudrnovský, J.; Drchal, V.; Turek, I. (1 November 2011). "Pressure dependence of Curie temperature and resistivity in complex Heusler alloys". Physical Review B 84 (17). arXiv:1010.3025. Bibcode:2011PhRvB..84q4422B. doi:10.1103/PhysRevB.84.174422. 
  • Webster, John G. (1999). The measurement, instrumentation, and sensors handbook ([Online-Ausg.] ed.). Boca Raton, Fla.: CRC Press published in cooperation with IEEE Press. ISBN 0849383471. 
  • Kovetz, Attay (1990). The principles of electromagnetic theory. (1st published. ed.). Cambridge [England]: Cambridge University Press. ISBN 0-521-39997-1. 
  • Hummel, Rolf E. (2001). Electronic properties of materials (3. ed. ed.). New York [u.a.]: Springer. ISBN 0-387-95144-X. 
  • Pascoe, K.J. (1973). Properties of materials for electrical engineers. New York, N.Y.: J. Wiley and Sons. ISBN 0471669113. 
  • Jones, Paulsen, Jason A. Lo, Chester C H; Snyder, John E.; Ring, A. P.; Jones, L. L.; Jiles, David C. (Sep 2003). "Study of the Curie temperature of cobalt ferrite based composites for stress sensor applications". 39 , Issue: 5. pp. 3316–3318. 
  • Hwang, Hae Jin; Nagai, Toru; Ohji, Tatsuki; Sando, Mutsuo; Toriyama, Motohiro; Niihara, Koichi (21 January 2005). "Curie Temperature Anomaly in Lead Zirconate Titanate/Silver Composites". Journal of the American Ceramic Society 81 (3): 709–712. doi:10.1111/j.1151-2916.1998.tb02394.x. 
  • Sadoc, Aymeric; Mercey, Bernard; Simon, Charles; Grebille, Dominique; Prellier, Wilfrid; Lepetit, Marie-Bernadette (1 January 2010). "Large Increase of the Curie Temperature by Orbital Ordering Control". Physical Review Letters 104 (4). arXiv:0910.3393. Bibcode:2010PhRvL.104d6804S. doi:10.1103/PhysRevLett.104.046804. 
  • "Pierre Curie - Biography". Nobelprize.org, From Nobel Lectures, Physics 1901-1921, Elsevier Publishing Company, Amsterdam, 1967. The Nobel Foundation 1903. Retrieved 2013-03-14. 
  • Martin Kochmański, Tadeusz Paszkiewicz, Sławomir Wolski (2013). "Curie-Weiss magnet: a simple model of phase transition". European Journal of Physics 34: 1555–1573. arXiv:1301.2141. Bibcode:2013EJPh...34.1555K. doi:10.1088/0143-0807/34/6/1555. 

External links[edit]