Magnetic Properties of Materials: Paramagnetism and Ferromagnetism
PARAMAGNETISM
The atoms of paramagnetic substances possess permanent magnetic dipoles.
In the absence of external magnetic field, the atomic dipoles are distributed randomly throughout the paramagnetic material as shown in fig. (5.6a). The external field of individual atoms cancel each other’s effect and hence do not exhibit any magnetic properties in the absence of external magnetic field.
Now, when the substance is subjected to an external field B, each of the atomic dipole experiences a torque. Due to this torque, the atomic dipoles rotate and try to align themselves along the direction of magnetic field B as shown in fig. 5.6(b). The alignment of individual dipoles gives rise to the magnetic properties in the substance. However, the alignment of atomic dipoles is restricted by the thermal agitation of the atoms. The alignment predominates at low temperature and gives stronger magnetic fields.
Langevin’s Theory of Paramagnetism
It is a known fact that in paramagnetic substances, every atom or molecule possesses a net unbalanced magnetic moment. The contribution towards the magnetic moment due to electron spins and orbital motion of electrons do not cancel each other and the atom thus possesses a net magnetic moment as shown in fig 5.6(a) and 5.6(b).
Langevin considered the paramagnetic solids as paramagnetic gas in which each particle is assumed to have a net magnetic moment, p. Let the number of atoms/molecules per unit volume be n and when magnetic field B is applied, magnetic dipoles try to align themselves along the direction of B but thermal agitation tries to disorient the orderly state and hence magnetic dipoles align themselves at a certain angle with magnetic field direction as shown in fig. 5.7
Now, the energy of interaction, known as potential energy of magnetic dipole moment is given bj
->
U = -Pm• B=-Pm B cos 0
…(5.43)
This energy is least, when Pm and B are parallel i.E., magnetic dipole get aligned in the field direction but due to thermal agitation as discussed earlier, they are randomly oriented.
Now, according to Maxwell-Boltzmann’s statistics for an assembly of particles (atoms/molecules), the number of particles
(atoms/molecules), the number of particles (dn) having energy state U is
(1) proportional to e-U/KT, where K is Boltzmann constant and T is absolute temperature. (i) proportional to solid angle (d2) between the two cones with semi-vertical angles and O+de i.E., d2=2m sine de i.E.,
dn proportional e^U/KT × d2.
dn proportional 2л sinthita dthita.E^U/KT dn =С.2ле KT*Pm costhita/kt *sin O de
where C is a constant of proportionality.
The integration of dn over all the values of thita from thita = 0 to thita = л must be equal to n i.E., nintegration of dn from 0 to pi
Thus, total number (n) of magnetic dipoles (atoms/molecules) per unit volume of the specimen or number density of dipoles are obtained by integrating eqn. On both sides i.E.,
dn proportional e^U/KT x d2.
dn proportional 2л sinthita dthita.E^U/KT dn =С.2ле KT*Pm costhita/kt *sin O do
where C is a constant of proportionality.
The integration of dn over all the values of thita from thita = 0 to thita = must be equal to n i…,
nintegration of dn from 0 to pi
Thus, total number (n) of magnetic dipoles (atoms/molecules) per unit volume of the specimen or number density of dipoles are obtained by integrating eqn. On both sides
i.E.,
i.E.,
n = 2C integration of e^Pm B cos®/KT, sino de from 0 to pi
=2C integration of eucosO.SinO dO, where u = pmB/kT
C=n/2m integration of eucosO.SinO.De from Oto Tr
Putting the value in C in euation and taking pmB/KT=u, we get dn=(ne^ucosO.SinOd®)/integration of e^ucosO.Sine.De from 0 to pi
Here, each of the dipole (atom/ molecule) will contribute a magnetic moment myu1=pmcos© and this magnetic moment is parallel to the magnetic field. Therefore, total magnetic moment pe volume (I) in the direction of magnetising field due to all the dipoles is given by I=integration of dn.Pmcos© from 0 to pi
Inpm. Integration of eucos@.Cos.Sin@.De/integration of eucos®.Sin@de from 0 to pi
Here, perpendicular component contribute zero magnetizsation. Therefore, we consider the effect due to horizontal components only. To solve eqn., let us put ucos®=x. Differentiating both sides, we get -usin©d®=dx Now, when ©=0,x=u and when O=pi, x=-u putting these values in eqn, we get Inpm|cothu-1/u]
this eqn is called langevin’s ean.
L(u)=[cothu-1/u]
put the value of u and get langevin’s function.
Now, for the value of T approaching zero and very high value of B, the term u (= pmB/KT) becomes very large in comparison to unity.
With this limit we have, coth u= 1 and Therefore, equation reduces to #npm=|not(say)
This shows that the maximum value of magnetisation, called saturation magnetisation, takes place when all the molecular dipoles get aligned parallel to the given magnetic field B.
The qualitative variation of the magnetization (I) with the term u (= pmB/kT) is shown in fig
At room temperature T = 300 K and putting
Pm=9.3 x 10-24 J/T, k = 1.38 × 10-23 J/mole. K and B = 1 Tesla, we get u =PmB/KT
=9.3 × 10-24 × 1/1.38 × 10-23 × 300
=2.24 × 10-3
which shows that the value of u is very small.
For low value of B and at high temperature u pmB/KT<<1
Also,
(coth u-1/u)=e^u + e1-u/e^u-e^-u -1/u congurant to 1+u^2/2!+…./u+u^3/3!+…. – 1/u (coth u-1/u)=u/3=pmB/3KT
Therefore, for small value of u, the value of intensity of magnetisation from eqn. Is given by I Enpm^2 B/3KT
Enpm[Pm B /3KT]
=myu not прт^2H/3kT
Further,for small value of u, value of l is quite small than that of B, Under these conditions, the susceptibility of paramagnetic substances is given by
Xm=l/H=I.Myu not/B
Emyu not.Np^2m/3KT
=1/T(np^2m Myunot/3K)=C’/T
Xm proportional to 1/T
where, C’ is a cure constant. It is clear from equation that the susceptibility of paramagnetic substances under normal conditions is inversely proportional to the absolute temp. This law is called curie Law.
FERROMAGNETISM
Ferromagnetic materials are those materials which are strongly attracted by magnetic field and are able to retain their magnetic behaviour even after the removal of external field. These materials show very high degree of alignment of atomic dipoles as compared to paramagnetic substances. These materials show all the properties as that of paramagnetic substances but to a greater extent. The ferromagnetism is explained on the basis of domain theory suggested by Weiss. According to this theory, some of the atoms get grouped together in a region, called domain. These domains are formed in whole of the substance and there is a strong interaction between the atoms of each domain which give rise to alignment of atomic dipoles in that particular domain. Such interaction in each domain is called exchange coupling and it results into magnetic moment associated with each domain as shown in fig.
In the absence of any external magnetic field, the dipole moments of individual domains are distributed randomly in such a way that the net dipole moment of the substance becomes zero. But when the magnetic field (external) is applied, the domains start aligning along the direction of magnetic field. With an increase in the strength of external magnetic field, the degree of alignment gets enhanced till all the domains are aligned in the direction of applied field.
With further increase in external field, there appears no further increase in alignment of domains and thus, a saturation stage of magnetism has reached.
There is a strong competition between the alignment of domains due to external field and the randomness in domains produced by thermal agitation. With an increase in temperature, the latter factor dominates and beyond a particular temperature, called Curie point, the ferromagnetic substance turns into a paramagnetic substance. The curie point is different for different substances. For example, the curie point for iron is 750° C, for nickel is 358° C, for cobalt is 1127° C etc.
Domain Theory of Ferromagnetism
The ferromagnetic substances get very high value of magnetisation even when they are subjected to weak fields. The magnetisation approaches a saturation value for ordinary values of H. Due to saturation of the magnetisation, such substances in a strong magnetising field experience a force which is just proportional to curlH/which is in contrast to that of diamagnetic and paramagnetic materials where force is proportional to HcurlH/Z thermal energy of molecules at ordinary temperatures is much larger (>100 times) than their magnetic potential energy even at quite high magnetic fields.
This leads to the fact that the saturation of the magnetisation in ferromagnetic materials can never be caused because of forces which are magnetic in origin.
Every ferromagnetic material becomes paramagnetic when its temperature is raised above a certain temperature Tc, called Curie point. In this range of temperature, it can no more gain intense magnetisation. Below the Curie temperatures, there exists what may be called the spontaneous alignment of molecular magnets parallel to each other. By spontaneous it is meant that the alignment of the molecular magnets is brought about without an external magnetising field. In fact, the individual molecular atoms of a ferromagnetic material divide into a large number of small regions called domains. In each domain, all the constituent dipoles (millions of atoms) align in same direction.
In an unmagnetised piece of iron, the dipoles in each of the domains are all parallel while the dipoles in neighbouring domains are also parallel but having different orientations as shown in fig.. On an average, all the directions are represented equally so that the specimen has no bulk magnetisation.
The division of bulk material into domains come into existence because the net potential energy for such division is much smaller than in the case when all the dipoles
are pointed parallel to each other. When magnetic field is applied to a material in direction parallel to which the atomic dipoles align, they possess a minimum energy. Thus, those domain in which the alignments of dipoles are parallel to the applied field as shown in fig. 5.10(b), become more favoured. At the boundary of these domains, the dipoles belonging to unfavoured domain tend to shift in favoured domain. The domains having axes parallel to the magnetising field start growing at the cost of other domains, thereby causing a shift in the domain boundary. In single crystals, even for small magnetising field, a large change in magnetisation results.
Mathematical Explanation:
Weiss assumed that the overlapping of fields of molecular magnets due to their alignment in the presence of magnetising field H is equivalent to a uniform magnetic field which have intensity equal to the intensity of magnetisation (1) and parallel to it. Therefore, if there is a molecular magnetic field given by
B= no gamma l
then the effective magnetic field on the given dipole will be
Beff=B+Bm=uH+uo gamma l
=uo(H+gamma l)
where B=uH and gamma is Weiss constant.
For very high temp., the intensity of magnetisation(l) calculated on the basis of Langvein theory of paramagnetisation is given by
I=npm^2B/3КТ
But in case of fermagnetic materials, we replace B by Beff as it is the actual magnetic field for any given molecular dipole. LEnpm^2B/3КТ
lEnpm^2 uo(H+gamma I)/3KT
3KT|=npm^2 woH+npm^2 uo gamma l
(3KT-npm^2 o gamma)l=npm^2 MoH
Inpm^2 MOH/(3КТ-npm^2 uo gamma)
As the susceptibility for feromagnetic materials is given by Xm=1/H
Enpm^2 o/(3KT-npm^2 uo gamma)
Enpm^2 H/3К(T-npm^2 wo gamma/3K)
=(npm^2 моН/3К)/3К(Т-(пр^2 мо/3K) gamma)
=C/t-thita
where C is a curie’s constant. And thita=gammaC represents the paramagnetic Curie temperature.
Eqn. Represents the well known Curie-Weiss law for ferromagnetic substances.
Special Case :
(1) If T >> 0, then the susceptibility (x) for ferromagnetic substances is similar in behaviour to that of paramagnetic substances i.E. The ferromagnetic substances behave as paramagnetic substances. (ii) If T < O, then some finite value of the intensity of magnetisation may exist even in the absence of external magnetising field (H). This magnetism is known as the spontaneous magnetisation which is the cause of residual magnetism in the absence of magnetising field i.E., at H=0.