So the phonon’s total energy:
U = 9N𝑘𝐵�T
When the temperature was low, we can make limit extended to infinity.
� 𝑑𝑥 𝑥3 𝑚𝑥− 1
∞
0 = 𝜋4
15 T>>θ, Heat capacity was closed to classical 3NKB. T<<θ,U ≅ 3𝜋4𝑁𝑘𝐵𝑇4/5𝜋𝜃3，and
Paramagnetic susceptibility χ is a small positive number, typically between 10-3 and 10-6, the weak magnetic diamagnetic, but whose magnetization direction is the same as the applied field. Constituting because paramagnetic atoms or molecules with an unpaired electron appears inside, the paramagnetic net magnetic moment of each atom is not zero, but because the magnetic moments between atoms cancel each other in different directions so overall net moment paramagnetic substance is zero, not magnetic. When a magnetic field is applied, the unpaired electron of this magnetic field susceptible to the effects of rotation of the magnetic moment of these atoms are arranged along the direction of the applied magnetic field, thereby generating a magnetic field in the same direction with the net magnetic moment. This net magnetic
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moment will heat disturbance, and makes the arrangement of atomic magnetic moment is damaged, so the magnetic susceptibility and temperature.
Figure.2.5.1-1 Paramagnetism
2.5.2 Ferromagnetism
Ferromagnetic materials include Fe, Co, Ni and their compounds and alloys and other materials, the main features is a large magnetic susceptibility, χ value is one that is large positive number. Ferromagnetic material and paramagnetic substances, atom itself has an unpaired electron atom resulting net magnetic moment is not zero.
However, ferromagnetic and paramagnetic materials have a very different place, is ferromagnetic material of the magnetic moments of the two atoms adjacent to each other, there is interaction between, the interaction of the magnetic moment that is adjacent to the same direction, which is a quantum effect.
The magnetic behavior of ferromagnetic materials the magnetic field can be used (magnetic domain) to make the concept overview. Each magnetic field ranges between a few microns to 1 mm, containing approximately 1015 ~ 1016 atoms. Even without an external magnetic field, still has a ferromagnetic material arranged in neat rows from spintronics and magnetic moment. With a magnetic domain magnetic moment, due to the magnetic field between the atoms in a strong magnetic dipole coupling effect, making the magnetic moments are arranged parallel to each other and
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pointing in the same direction, that is, each magnetic field itself will have a saturation magnetization value, called self magnetized. These magnetic domains wall between domains (domain wall) apart, each magnetic domain contains several millions of small moments. The applied magnetic field, the different moments of the magnetic field is not the same point, so the total magnetic moment of ferromagnetic substance to zero.
Figure.2.5.2-1 Ferromagnetic
Curie temperature
The heat causes ferromagnetic material deviation from perfect magnetic dipole moment parallel’s arrangement. When the temperature gradually increased, if the temperature is increased to heat the ferromagnetic material is greater than the magnetic dipole moment, the exchange arranged in parallel, because of the
disturbance of the magnetic moment by thermal excitation, so that the order became messy arrangement, when the ferromagnetic material disappear into the
ferromagnetic-paramagnetic. This change the magnetic properties of the temperature are called the Curie temperature TC. When a ferromagnetic material from a high temperature cooled to below the Curie temperature, the ferromagnetic Jiangzai magnetic field is formed back into a ferromagnetic material. General common bulk
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ferromagnetic materials, Fe, the Curie temperature is about 770 ℃, Co Curie temperature of about 1123 ℃, Ni Curie temperature of about 358 ℃
2.5.3 Diamagnetic
Diamagnetic susceptibility χ is a small negative value, usually at about -10-5.
Diamagnetic effect occurs is the result of the electron orbital motion. As the
anti-magnetic material each electron orbit electrons are paired, which each track has two electrons with opposite spins, the individual electronic bypass magnetic moments of nuclei generated offset each other, thus making diamagnetic substance itself is not magnetic, and its net magnetic moment is zero. When a magnetic field applied to this material, the orbit of the electrons out of balance, the electron orbital angular
momentum change to produce a net magnetic moment m, the cooling times (Lentz) law, the net magnetic moment in the direction of this magnetic field should be in the opposite direction. Diamagnetic effect in all material occurs, albeit to varying degrees.
The properties also have other magnetic materials that (eg: ferromagnetic and paramagnetic), diamagnetic almost negligible, it is called diamagnetic material is generally treated with non-magnetic substance. The atomic magnetic moment due to induction regardless of the direction, so does not affect the thermal disturbance magnetic permeability diamagnetic sense, so the susceptibility is independent of temperature.
2.5.4 Antiferromagnetism
Also a weak antiferromagnetic magnetic susceptibility χ is a small positive number, the same as the magnetization direction of the applied magnetic field.
Magnetic susceptibility and temperature dependent, but varies with temperature compared with paramagnetic special circumstances. When the temperature is lower
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than the Fresnel temperature TN (Neel temperature), with the temperature, the magnetic susceptibility increases, and when the temperature is higher than TN, the magnetic susceptibility, along with the increase in temperature becomes small, paramagnetic. When the temperature is below TN, the magnetic moments tended to antiparallel arrangement.
Neel temperature
In the anti-ferromagnetic material, there is also a similar characteristic temperature Curie temperature, when the temperature rises due to heat effect of a disturbance, so that the arrangement of the magnetic moment became chaotic, is no longer anti-parallel arrangement, the antiferromagnetic magnetic begins
antiferromagnetic into paramagnetic, which is a characteristic temperature of the Fresnel transformation temperature TN. When the temperature continued to drop, the magnetic moments of antiferromagnetic material will again form antiparallel.
Antiferromagnetic Fresnel bulk temperature of the CoO 290 K, therefore make the antiferromagnetic CoO showing the operation to be cooled.
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