Goal of the work: study of the second-order phase transition ferromagnet–paramagnet, determination of the dependence of spontaneous magnetization on temperature and verification of the Curie-Weiss law.
Introduction
In nature, there are various abrupt changes in the state of matter, called phase transformations. Such transformations include melting and solidification, evaporation and condensation, the transition of metals to a superconducting state and the reverse transition, and so on.
One of the phase transitions is the transformation from a ferromagnetic to a paramagnetic state in some substances, such as metals of the iron group, some lanthanides and others.
The ferromagnetic–paramagnetic transition is widely studied in our time, not only because of its importance in materials science, but also because a very simple model (the Ising model) can be used to study it, and, therefore, this transition can be studied in the most detail mathematically, which is important for creating the still missing general theory of phase transitions.
This work examines the ferromagnetic-paramagnetic transition in a two-dimensional crystal lattice, studies the dependence of spontaneous magnetization on temperature, and verifies the Curie–Weiss law.
Classification of magnetic materials
All substances, to one degree or another, have magnetic properties, that is, they are magnets. Magnets are divided into two large groups: highly magnetic and weakly magnetic substances. Strongly magnetic substances have magnetic properties even in the absence of an external magnetic field. These include ferromagnets, antiferromagnets and ferrimagnets. Weakly magnetic substances acquire magnetic properties only in the presence of an external magnetic field. They are divided into diamagnetic and paramagnetic.
Diamagnets include substances whose atoms or molecules do not have a magnetic moment in the absence of an external field. The atoms of these substances are arranged in such a way that the orbital and spin moments of the electrons entering them exactly compensate each other. An example of diamagnetic materials are inert gases, the atoms of which have only closed electron shells. When an external magnetic field appears due to the phenomenon of electromagnetic induction, the atoms of diamagnetic materials are magnetized, and they acquire a magnetic moment directed, according to Lenz’s rule, against the field.
Paramagnetic substances include substances whose atoms have non-zero magnetic moments. In the absence of an external field, these magnetic moments are randomly oriented due to chaotic thermal motion, and therefore the resulting magnetization of the paramagnetic is zero. When an external field appears, the magnetic moments of the atoms are oriented predominantly along the field, so a resulting magnetization appears, the direction of which coincides with the direction of the field. It should be noted that the paramagnetic atoms themselves in a magnetic field are magnetized in the same way as the diamagnetic atoms, but this effect is always weaker than the effect associated with the orientation of the moments.
The main feature of ferromagnets is the presence of spontaneous magnetization, which manifests itself in the fact that a ferromagnet can be magnetized even in the absence of an external magnetic field. This is due to the fact that the interaction energy of any pair of neighboring ferromagnetic atoms depends on the mutual orientation of their magnetic moments: if they are directed in one direction, then the interaction energy of the atoms is less, and if in opposite directions, then it is greater. In the language of forces, we can say that short-range forces act between magnetic moments, which try to force the neighboring atom to have the same direction of the magnetic moment as that of the given atom itself.
The spontaneous magnetization of a ferromagnet gradually decreases with increasing temperature, and at a certain critical temperature - the Curie point - it becomes equal to zero. At higher temperatures, a ferromagnet behaves in a magnetic field as a paramagnet. Thus, at the Curie point, a transition from the ferromagnetic to the paramagnetic state occurs, which is a second-order phase transition or a continuous phase transition.
Ising model
A simple Ising model was created to study magnetic and atomic ordering. In this model, it is assumed that atoms are located motionlessly, without oscillating, at the nodes of an ideal crystal lattice. The distances between lattice nodes are constant; they do not depend on temperature or magnetization, that is, this model does not take into account the thermal expansion of a solid.
The interaction between magnetic moments in the Ising model is taken into account, as a rule, only between nearest neighbors. It is believed that the magnitude of this interaction is also independent of temperature and magnetization. Interaction is usually (but not always) considered central and pairwise.
However, even in such a simple model, the study of the ferromagnetic–paramagnetic phase transition encounters enormous mathematical difficulties. Suffice it to say that an exact solution to the three-dimensional Ising problem in the general case has not yet been obtained, and the use of more or less accurate approximations in this problem leads to great computational difficulties and is on the verge of the capabilities of even modern computer technology.
Entropy
Let us consider a magnet in a two-dimensional Ising lattice (Fig. 1). Let the nodes form a square lattice. Magnetic moments directed upwards will be denoted by A, and down – B.
Rice. 1
Let the number of upward magnetic moments be equal to N A, and down – N B, the total number of moments is N. It's clear that
N A + N IN = N. (1)
Number of ways you can place N A moments of sorts A And N B moments of sorts IN By N nodes is equal to the number of permutations of all these nodes with each other, that is, equal to N!. However, out of this total, all permutations of identical magnetic moments with each other do not lead to a new state (they are called indistinguishable permutations). That is, to find out the number of ways to place moments, you need N! divided by the number of indistinguishable permutations. Thus, we obtain the value
. (2)
This quantity is the total number of microstates corresponding to a macrostate with a given magnetization, i.e., the statistical weight of the macrostate.
When calculating the statistical weight using formula (2), a fairly strong approximation was made, namely that the appearance of a specific magnetic moment at some lattice site does not depend on what magnetic moments atoms have at neighboring sites. In fact, atoms with moments of any orientation, due to the interaction of particles with each other, “try” to surround themselves with atoms with the same magnetic moments, but this is not taken into account in formula (2). It is said that in this case we do not take into account the correlation in the arrangement of moments. This approximation in the theory of magnetism is called the Bragg–Williams approximation. Let us note that the problem of taking into account correlation is one of the most difficult problems in any theory dealing with a group of particles interacting with each other.
If we apply the Stirling formula ln N! N (ln N – 1), fair for big ones N, then from formula (2) we can obtain an expression for the entropy associated with the location of the magnetic moments (it is called configuration entropy):
Let us introduce the probability of the appearance of a magnetic moment “up”: 
. Similarly, you can enter the probability of the appearance of a downward magnetic moment: 
. Then the expression for entropy will be written as follows:
From formula (1) it follows that the probabilities introduced above are related by the relation:
.
(3)
Let us introduce the so-called long-range order parameter:
(4)
Then from formulas (3) and (4) we can express all probabilities through the order parameter:
Substituting these relations into the expression for entropy, we obtain:
. (6)
Let us find out the physical meaning of the long-range order parameter . Magnetization of a magnet M is determined in our model by the excess of atoms with one of two possible orientations of the magnetic moment, and it is equal to:
where 
, Where M max =
N
– maximum magnetization achieved with parallel orientation of all magnetic moments ( – value of the magnetic moment of one atom). Thus, the order parameter is the relative magnetization, and it can vary from –1 to +1. Negative values of the order parameter only indicate the direction of the preferential orientation of the magnetic moments. In the absence of an external magnetic field, the order parameter values +
and – are physically equivalent.
Energy
Atoms interact with each other, and this interaction is observed only at fairly short distances. In a theoretical consideration, it is easiest to take into account the interaction of only the atoms closest to each other. Let there be no external field ( N = 0).
Let only neighboring atoms interact. Let the interaction energy of two atoms with identically directed magnetic moments (both “up” or both “down”) be equal to – V(attraction corresponds to negative energy), and with oppositely directed + V.
Let the crystal be such that each atom has z nearest neighbors (for example, in a simple cubic lattice z = 6, in body-centered cubic z = 8, square z = 4).
The energy of interaction of one atom, the magnetic moment of which is directed “upward,” with its immediate environment (i.e., with z p A moments “up” and with z p B moments “down”) in our model is equal to – V z (p A – p B). A similar value for an atom with a “downward” moment is equal to V z (p A – p B). At the same time, we again used the Bragg–Williams approximation, which was already used in deriving the formula for entropy, and does not take into account correlations in the arrangement of atoms, that is, we assumed that the probability of the appearance of a specific magnetic moment at some lattice site does not depend on what magnetic moments the atoms have on neighboring nodes.
In this approximation, the total energy of the magnet is:
where the factor ½ appeared so that the interaction of all neighboring atoms with each other would not be taken into account twice.
Expressing N A And N B through probabilities, we get:
.
(7)
Equilibrium equations
The energy of interaction reflects the tendency of the system to establish complete order in it, precisely with complete order (in our case with = 1) energy is minimal, which would correspond to stable equilibrium in the absence of thermal motion. The entropy of the system, on the contrary, reflects the tendency towards maximum molecular chaos and maximum thermal motion. The stronger the thermal motion, the greater the entropy, and if there were no interaction of molecules with each other, then the system would tend to maximum chaos with maximum entropy.
In a real system, both of these tendencies exist, and this is manifested in the fact that at constant volume and temperature in a state of thermodynamic equilibrium, it is not the energy or the entropy that reaches its extreme (minimum) value, but the Helmholtz free energy:
F = U – T S.
For our case, from formulas (6) and (7) we can obtain:
In a state of thermodynamic equilibrium, the degree of ordering must be such that the free energy is minimal, so we must examine function (8) for an extremum, taking its derivative with respect to and equating it to zero. Thus, the equilibrium condition will take the form:
. (9)
In this equation 
– dimensionless temperature.
Rice. 2
Equation (9) is transcendental and can be solved by numerical methods. However, its solution can be explored graphically. To do this, you need to build graphs of the functions on the left and right sides of the equation for different values of the parameter . Let us denote these functions accordingly F 1 and F 2
(Fig. 2).
Function F 1 does not depend on the parameter , it is a curve with two vertical asymptotes at values of the variable equal to +1 and –1. This function increases monotonically, it is odd, its derivative at the origin is equal to 
. Function F 2
is depicted as a straight line passing through the origin of coordinates, its slope depends on the parameter : the smaller , the greater the tangent of the angle of inclination, which is equal to 
.
If 1, then 
, then the curves intersect only at the origin, that is, in this case, equation (9) has only one solution =
0. At 1, the curves intersect at three points, that is, equation (9) has 3 solutions. One of them is still zero, the other two differ only in sign.
It turns out that the zero solution for A and IN(i.e. “up” and “down” moments).
Substituting the value = 1, we obtain the value of the temperature separating two types of solutions to equation (9):
.
This temperature is called the ferromagnetic-paramagnetic transition temperature or Curie point, or simply the critical temperature.
At lower temperatures, the magnet exists in an ordered ferromagnetic state, and at higher temperatures, there is no long-range order in the arrangement of the magnetic moments of the atoms, and the substance is paramagnetic. Note that this transition is a second-order phase transition; the order parameter gradually decreases with increasing temperature and becomes equal to zero at the critical point.
The dependence of the order parameter on the reduced temperature , obtained from solving equation (9), is shown in
rice. 3.
Free energy (8) for a ferromagnet in an external field will be written:
Rice. 3
where is the magnetic moment of the atom. In this formula, the second term represents the energy of interaction of the magnetic moments of atoms with an external magnetic field, equal to 
. The general case of a ferromagnet in a magnetic field is quite difficult to study mathematically; we will limit ourselves to only considering a ferromagnet at temperatures above the Curie point. Then the equilibrium equation, similar to (9), will take the form:
.
Let us restrict ourselves to the case of weak magnetization, which is observed at temperatures significantly above the Curie point
(T ≫ T C) and weak magnetic fields. For ≪ 1, the left side of this equation can be expanded into a series, limited to linear terms, i.e.
ln (1+) . Then 2 kT = Н +2 kT C, and magnetization 
, i.e. paramagnetic susceptibility 
. Thus, the susceptibility of a ferromagnet at temperatures above the Curie point in weak magnetic fields is inversely proportional to ( T – T C), i.e., there is agreement between the theory and the experimental Curie–Weiss law.
Description of work
A frame from a computer laboratory work is shown in Fig. 4. The ferromagnet is modeled by a fragment of a simple square lattice of 100 nodes, on which the “up” and “down” magnetic moments are located, depicted by respectively directed arrows. The temperature of the magnet is set in reduced units 
and external magnetic field strength.
You need to do two exercises. In the first of them, it is necessary to determine the dependence of magnetization on temperature in the absence of an external magnetic field. In the second exercise, you need to investigate the magnetization of a magnet by an external field at a temperature above the Curie point and check the Curie-Weiss law.
Progress
1. Press the "RESET" button, and the "START" button will appear.
2. Set the required field strength values N and reduced temperature 
.
3. Press the “START” button, and an image of a ferromagnet will appear, in which the number of magnetic moments “up” and “down” are determined by the specified parameters. The number of magnetic moments “up” will appear in the corresponding window.
4. Calculate the value of the order parameter. It should be borne in mind that the total number of magnetic moments is 100.
5. Carry out the experiment described above for other values of field strength and temperature, calculating the order parameter each time.
6. It is recommended to select field strength values in the range from 2 to 10 units (4–5 values), and the given temperature – in the range from 4 to 15–20 (4–5 values).
7. For each temperature, plot the dependence of magnetization on the field strength and determine the magnetic susceptibility at a given temperature as the slope of the corresponding graph.
8. Assess the implementation of the Curie-Weiss law, for which construct a graph of the dependence of susceptibility on the ratio 
. According to the Curie-Weiss law, this dependence should be linear.
9. Plot the dependence of magnetization on the reduced temperature at field strength N = 0 at temperatures below the Curie point (the given temperature values should be taken in the range from 0.5 to 1).
Control questions
What substances are called highly magnetic?
What is spontaneous magnetization?
What is the reason that a ferromagnet has spontaneous magnetization?
What is a ferromagnet at temperatures above the Curie point?
Why does a paramagnetic material not have spontaneous magnetization?
What are the main features of the Ising model?
What is the physical meaning of the degree of long-range order?
What is the nature of the interaction between magnetic moments?
What is the Bragg–Williams approximation and what does it mean that this approximation does not take into account correlations in the arrangement of magnetic moments?
How is the entropy of a ferromagnet determined?
What are the conditions for thermodynamic equilibrium of a ferromagnet?
Graphic solution of the equilibrium equation.
What does the Curie temperature depend on?
What is the Curie-Weiss law?
How can one study the dependence of the magnetization of a ferromagnet on temperature?
How to determine the magnetic susceptibility of a ferromagnet above the Curie point?
Paramagnetic substances include substances in which the magnetic moment of atoms or molecules is non-zero in the absence of an external magnetic field:
Therefore, paramagnets, when introduced into an external magnetic field, are magnetized in the direction of the field. In the absence of an external magnetic field, the paramagnet is not magnetized, since due to thermal motion all the magnetic moments of the atoms are randomly oriented, and therefore the magnetization is zero (Fig. 2.7 a). When a paramagnetic substance is introduced into an external magnetic field, a preferential orientation of the magnetic moments of atoms along the field is established (Fig. 2.7 b). Complete orientation is prevented by the thermal motion of atoms, which tends to scatter the moments. As a result of this preferential orientation, the paramagnet is magnetized, creating its own magnetic field, which, superimposed on the external one, strengthens it. This effect is called the paramagnetic effect or paramagnetism.
Fig.2.7. Paramagnetic in
absence of field(s) and in
external magnetic field (b)
Paramagnetic materials also exhibit Larmor precession and the diamagnetic effect, as in all substances. But the diamagnetic effect is weaker than the paramagnetic one and is suppressed by it, remaining invisible. For paramagnets, χ is also small, but positive, on the order of ~10 -7 –10 -4 , which means μ is slightly greater than one.
Just as for diamagnetic materials, the dependence of the magnetic susceptibility of paramagnetic materials on the external field is linear ( Fig.5.8). 
The preferential orientation of magnetic moments along the field depends on temperature. As the temperature increases, the thermal movement of atoms increases, therefore, orientation in one direction becomes difficult and magnetization decreases. The French physicist P. Curie established the following pattern: where C is the Curie constant, depending on the type of substance. The classical theory of paramagnetism was developed in 1905 by P. Langevin.
2.10 Ferromagnetism. Ferromagnets. Domain structure of ferromagnets.
.7. Ferromagnetism. Ferromagnets. @
Ferromagnets are solid crystalline substances that have spontaneous magnetization in the absence of an external magnetic field. .Atoms (molecules) of such substances have a non-zero magnetic moment. In the absence of an external field, magnetic moments within large regions are oriented in the same way (more on this later). Unlike weakly magnetic dia- and paramagnets, ferromagnets are highly magnetic substances. Their internal magnetic field can be hundreds and thousands of times greater than the external one. For ferromagnets, χ and μ are positive and can reach very large values, on the order of ~10 3 . Only ferromagnets can be permanent magnets.
Why do ferromagnetic bodies exhibit such strong magnetization? Why does thermal motion in them not interfere with the establishment of order in the arrangement of magnetic moments? To answer this question, let's look at some important properties of ferromagnets.
If we depict the main magnetization curve in coordinates (B, H) (Fig. 2.10, curve 0-1), we get a slightly different picture: since , then when the value J us is reached, the magnetic induction continues to grow along with the growth linearly: 
= μ 0 + const, const = μ 0 J us.
Ferromagnets are characterized by the phenomenon hysteresis(from the Greek hysteresis – lag, delay).
We will bring the magnetization of the body to saturation, increasing the external field strength (Fig. 2.10, point 1), and then we will decrease H. In this case, the dependence B(H) follows not the original curve 0-1, but the new curve 1-2. When the voltage decreases to zero, the magnetization of the substance and magnetic induction will disappear. At Н=0, the magnetic induction has a non-zero value V ost, which is called residual induction. The magnetization J ost, corresponding to B ost, is called residual magnetization, and the ferromagnet acquires the properties of a permanent magnet. V ost and J ost become zero only under the influence of a field opposite in direction to the original one. The value of the field strength H c at which the residual magnetization and induction vanish is called coercive force(from Latin coercitio - retention). Continuing to act on the ferromagnet with an alternating magnetic field, we obtain the curve 1-2-3-4-1, called hysteresis loop. In this case, the body’s reaction (B or J) seems to lag behind the causes that cause it (H).
The existence of residual magnetization makes it possible to manufacture permanent magnets, because ferromagnets with Bres ≠ 0 have a constant magnetic moment and create a constant magnetic field in the space surrounding them. Such a magnet retains its properties better, the greater the coercive force of the material from which it is made. Magnetic materials are usually divided according to the value of Hc into magnetically soft(i.e. with low H of the order of 10 -2 A/m and, accordingly, with a narrow hysteresis loop) and magnetically hard(H with ~10 5 A/m and a wide hysteresis loop). Soft magnetic materials are required for the manufacture of transformers, the cores of which are constantly remagnetized by alternating current. If the transformer core has a large hysteresis, it will heat up during magnetization reversal, which will waste energy. Transformers therefore require materials that are as hysteresis-free as possible. Ferromagnets with a narrow hysteresis loop include alloys of iron with nickel or iron with nickel and molybdenum (permalloy and supermalloy).
Magnetically hard materials (including carbon, tungsten, chromium and aluminum-nickel steels) are used to make permanent magnets.
Residual permanent magnetization will exist indefinitely if the ferromagnet is not exposed to strong magnetic fields, high temperatures and deformation. All information recorded on magnetic tapes - from music to video programs - is stored thanks to this physical phenomenon.
An essential feature of ferromagnets is the enormous values of magnetic permeability and magnetic susceptibility. For example, for iron μ max ≈ 5000, for permalloy – 100000, for supermalloy – 900000. For ferromagnets, the values of magnetic susceptibility and magnetic permeability are functions of the magnetic field strength H (Fig. 2.11). With increasing field strength, the value of μ first quickly increases to μ max, and then decreases, approaching the value μ=1 in very strong fields. Therefore, although the formula B = μμ 0 H remains valid for ferromagnetic substances, the linear relationship between B and H is violated.
The second magnetomechanical effect is Villari effect– change and even disappearance of the residual magnetization of a body when it is shaken or deformed (discovered by E. Villari in 1865). It is because of this that permanent magnets should be protected from shock.
Heating acts on ferromagnets in a similar way to deformation. With increasing temperature, the residual magnetization begins to decrease, weakly at first, and then, upon reaching a certain sufficiently high temperature characteristic of each ferromagnet, a sharp decrease in magnetization to zero occurs. The body then becomes paramagnetic. The temperature at which such a change in properties occurs is called Curie point, in honor of P. Curie who discovered it. For iron, the Curie point is 770ºC, for cobalt - 1130ºC, for nickel - 358ºC, for gadolinium - 16ºC. This transition is not accompanied by the release or absorption of heat and is a second-order phase transition. All these phenomena find their explanation when considering the structure of ferromagnets.
According to their magnetic properties, all substances are divided into weakly magnetic and strongly magnetic. In addition, magnets are classified depending on the magnetization mechanism.
Diamagnets
Diamagnets are classified as weakly magnetic substances. In the absence of a magnetic field, they are not magnetized. In such substances, when they are introduced into an external magnetic field, the movement of electrons in molecules and atoms changes so that an oriented circular current is formed. The current is characterized by a magnetic moment ($p_m$):
where $S$ is the area of the coil with current.
The magnetic induction created by this circular current, additional to the external field, is directed against the external field. The value of the additional field can be found as:
Any substance has diamagnetism.
The magnetic permeability of diamagnetic materials differs very slightly from unity. For solids and liquids, the diamagnetic susceptibility is of the order of approximately $(10)^(-5),\ $for gases it is significantly less. The magnetic susceptibility of diamagnetic materials does not depend on temperature, which was discovered experimentally by P. Curie.
Diamagnets are divided into “classical”, “anomalous” and superconductors. Classical diamagnetic materials have a magnetic susceptibility $\varkappa
In weak magnetic fields, the magnetization of diamagnetic materials is proportional to the magnetic field strength ($\overrightarrow(H)$):
where $\varkappa$ is the magnetic susceptibility of the medium (magnet). Figure 1 shows the dependence of the magnetization of a “classical” diamagnetic on the magnetic field strength in weak fields.
Paramagnets
Paramagnetic substances are also classified as weakly magnetic substances. Paramagnetic molecules have a permanent magnetic moment ($\overrightarrow(p_m)$). The energy of the magnetic moment in an external magnetic field is calculated by the formula:
The minimum energy value is achieved when the direction of $\overrightarrow(p_m)$ coincides with $\overrightarrow(B)$. When a paramagnetic substance is introduced into an external magnetic field in accordance with the Boltzmann distribution, a preferential orientation of the magnetic moments of its molecules appears in the direction of the field. Magnetization of the substance appears. The induction of the additional field coincides with the external field and accordingly enhances it. The angle between the direction $\overrightarrow(p_m)$ and $\overrightarrow(B)$ does not change. The reorientation of magnetic moments in accordance with the Boltzmann distribution occurs due to collisions and interactions of atoms with each other. Paramagnetic susceptibility ($\varkappa $) depends on temperature according to Curie’s law:
or the Curie-Weiss law:
where C and C" are the Curie constants, $\triangle $ is a constant that can be greater or less than zero.
The magnetic susceptibility ($\varkappa $) of a paramagnetic is greater than zero, but, like that of a diamagnetic, it is very small.
Paramagnets are divided into normal paramagnets, paramagnetic metals, and antiferromagnets.
For paramagnetic metals, magnetic susceptibility does not depend on temperature. These metals are weakly magnetic $\varkappa \approx (10)^(-6).$
In paramagnetic materials there is a phenomenon called paramagnetic resonance. Let us assume that in a paramagnetic material that is in an external magnetic field, an additional periodic magnetic field is created, the induction vector of this field is perpendicular to the induction vector of a constant field. As a result of the interaction of the magnetic moment of an atom with an additional field, a moment of force ($\overrightarrow(M)$) is created, which tends to change the angle between $\overrightarrow(p_m)$ and $\overrightarrow(B).$ If the frequency of the alternating magnetic field and the frequency the precession of the atomic motion coincides, then the torque created by the alternating magnetic field either constantly increases the angle between $\overrightarrow(p_m)$ and $\overrightarrow(B)$, or decreases. This phenomenon is called paramagnetic resonance.
In weak magnetic fields, magnetization in paramagnetic materials is proportional to the field strength and is expressed by formula (3) (Fig. 2).
Ferromagnets
Ferromagnets are classified as highly magnetic substances. Magnets whose magnetic permeability reaches large values and depends on the external magnetic field and previous history are called ferromagnets. Ferromagnets can have residual magnetization.
The magnetic susceptibility of ferromagnets is a function of the strength of the external magnetic field. The J(H) dependence is shown in Fig. 3. Magnetization has a saturation limit ($J_(nas)$).
The existence of a magnetization saturation limit indicates that the magnetization of ferromagnets is caused by the reorientation of some elementary magnetic moments. In ferromagnets, the phenomenon of hysteresis is observed (Fig. 4).
Ferromagnets, in turn, are divided into:
- Soft magnetically. Substances with high magnetic permeability, easily magnetized and demagnetized. They are used in electrical engineering, where they work with alternating fields, for example in transformers.
- Magnetically hard. Substances with relatively low magnetic permeability, difficult to magnetize and demagnetize. These substances are used to create permanent magnets.
Example 1
Assignment: The dependence of magnetization for a ferromagnet is shown in Fig. 3. J(H). Draw the B(H) curve. Is there saturation for magnetic induction, why?
Since the magnetic induction vector is related to the magnetization vector by the relation:
\[(\overrightarrow(B)=\overrightarrow(J\ )+\mu )_0\overrightarrow(H)\ \left(1.1\right),\]
then the curve B(H) does not reach saturation. A graph of the dependence of magnetic field induction on the strength of the external magnetic field can be presented as shown in Fig. 5. Such a curve is called a magnetization curve.
Answer: There is no saturation for the induction curve.
Example 2
Assignment: Obtain the formula for paramagnetic susceptibility $(\varkappa)$, knowing that the mechanism of magnetization of a paramagnet is similar to the mechanism of electrification of polar dielectrics. For the average value of the magnetic moment of a molecule in projection onto the Z axis, we can write the formula:
\[\left\langle p_(mz)\right\rangle =p_mL\left(\beta \right)\left(2.1\right),\]
where $L\left(\beta \right)=cth\left(\beta \right)-\frac(1)(\beta )$ is the Langevin function with $\beta =\frac(p_mB)(kT).$
At high temperatures and small fields, we get that:
Therefore, for $\beta \ll 1$ $cth\left(\beta \right)=\frac(1)(\beta )+\frac(\beta )(3)-\frac((\beta )^3 )(45)+\dots $ , restricting the function by a linear term in $\beta $ we obtain:
Substituting the result (2.3) into (2.1), we obtain:
\[\left\langle p_(mz)\right\rangle =p_m\frac(p_mB)(3kT)=\frac((p_m)^2B)(3kT)\ \left(2.4\right).\]
Using the relationship between magnetic field strength and magnetic induction ($\overrightarrow(B)=\mu (\mu )_0\overrightarrow(H)$), taking into account that the magnetic permeability of paramagnetic materials differs little from unity, we can write:
\[\left\langle p_(mz)\right\rangle =\frac((p_m)^2(\mu )_0H)(3kT)\left(2.5\right).\]
Then the magnetization will look like:
Knowing that the relationship between the magnetization modulus and the voltage vector modulus has the form:
For paramagnetic susceptibility we have:
\[\varkappa =\frac((p_m)^2m_0n)(3kT)\ .\]
Answer: $\varkappa =\frac((p_m)^2(\mu )_0n)(3kT)\ .$
Goal of the work : determine the Neel temperature for a ferrimagnet (ferrite rod)
Brief theoretical information
Every substance is magnetic, i.e. is capable of acquiring a magnetic moment under the influence of a magnetic field. Thus, the substance creates a magnetic field, which is superimposed on the external field. Both fields add up to the resulting field:
The magnetization of a magnet is characterized by the magnetic moment per unit volume. This quantity is called the magnetization vector
where is the magnetic moment of an individual molecule.
The magnetization vector is related to the magnetic field strength by the following relationship:
Where c- a characteristic value for a given substance, called magnetic susceptibility.
The magnetic induction vector is related to the magnetic field strength:
The dimensionless quantity is called relative magnetic permeability.
All substances according to their magnetic properties can be divided into three classes:
1) paramagnetic materials m> 1 in which magnetization increases the total field
2) diamagnetic materials m < 1 в которых намагниченность вещества уменьшает суммарное поле
3) ferromagnets m>> 1 magnetization increases the total magnetic field.
A substance is ferromagnetic if it has a spontaneous magnetic moment even in the absence of an external magnetic field. Saturation magnetization of a ferromagnet I S is defined as the spontaneous magnetic moment per unit volume of a substance.
Ferromagnetism is observed in 3 d-metals ( Fe , Ni , Co ) and 4 f metals ( Gd , Tb , Er , Dy , Ho , Tm ) In addition, there are a huge number of ferromagnetic alloys. It is interesting to note that only the 9 pure metals listed above have ferromagnetism. They all have unfinished d - or f - shells.
The ferromagnetic properties of a substance are explained by the fact that there is a special interaction between the atoms of this substance, which does not take place in dia- and paramagnets, leading to the fact that the ionic or atomic magnetic moments of neighboring atoms are oriented in the same direction. The physical nature of this special interaction, called exchange, was established by Ya.I. Frenkel and W. Heisenberg in the 30s of the 20th century on the basis of quantum mechanics. The study of the interaction of two atoms from the point of view of quantum mechanics shows that the energy of interaction of atoms i And j, having spin moments S i And S j , contains a term due to the exchange interaction:
Where J– exchange integral, the presence of which is associated with the overlap of the electron shells of atoms i And j. The value of the exchange integral strongly depends on the interatomic distance in the crystal (the period of the crystal lattice). In ferromagnets J>0, if J<0 вещество является антиферромагнетиком, а при J=0 – paramagnetic. Metabolic energy has no classical analogue, although it is of electrostatic origin. It characterizes the difference in the energy of the Coulomb interaction of the system in the cases when the spins are parallel and when they are antiparallel. This is a consequence of the Pauli principle. In a quantum mechanical system, a change in the relative orientation of the two spins must be accompanied by a change in the spatial distribution of charge in the overlap region. At a temperature T=0 K, the spins of all atoms must be oriented in the same way; with increasing temperature, the order in the orientation of the spins decreases. There is a critical temperature called the Curie temperature T S, at which the correlation in the orientations of individual spins disappears, the substance changes from a ferromagnet to a paramagnet. Three conditions can be identified that favor the emergence of ferromagnetism:
1) the presence of significant intrinsic magnetic moments in atoms of matter (this is only possible in atoms with unfinished d - or f - shells);
2) the exchange integral for a given crystal must be positive;
3) density of states in d - And f - zones should be large.
The magnetic susceptibility of a ferromagnet obeys Curie-Weiss law :
, WITH– Curie constant.
Ferromagnetism of bodies consisting of a large number of atoms is due to the presence of macroscopic volumes of matter (domains), in which the magnetic moments of atoms or ions are parallel and identically directed. These domains exhibit spontaneous spontaneous magnetization even in the absence of an external magnetizing field.
Model of the atomic magnetic structure of a ferromagnet with a face-centered cubic lattice. Arrows indicate the magnetic moments of atoms.
In the absence of an external magnetic field, a generally unmagnetized ferromagnet consists of a larger number of domains, in each of which all spins are oriented in the same way, but the direction of their orientation differs from the directions of spins in neighboring domains. On average, in a sample of a non-magnetized ferromagnet, all directions are equally represented, so a macroscopic magnetic field is not obtained. Even in a single crystal there are domains. The separation of matter into domains occurs because it requires less energy than an arrangement with identically oriented spins.
When a ferromagnet is placed in an external field, magnetic moments parallel to the field will have less energy than moments antiparallel to the field or directed in any other way. This gives an advantage to some domains that seek to increase in volume at the expense of others if possible. A rotation of magnetic moments within one domain can also occur. Thus a weak external field can cause a large change in magnetization.
When ferromagnets are heated to the Curie point, thermal motion destroys the regions of spontaneous magnetization, the substance loses its special magnetic properties and behaves like an ordinary paramagnet. The Curie temperatures for some ferromagnetic metals are given in the table.
| Substance | |
Fe | |
| Ni | |
| Co | |
| Gd |
In addition to ferromagnets, there is a large group of magnetically ordered substances in which the spin magnetic moments of atoms with unfinished shells are oriented antiparallel. As shown above, this situation arises when the exchange integral is negative. Just like in ferromagnets, magnetic ordering takes place here in the temperature range from 0 K to a certain critical QN, called the Néel temperature. If, with antiparallel orientation of localized magnetic moments, the resulting magnetization of the crystal is zero, then antiferromagnetism. If in this case there is no complete compensation of the magnetic moment, then they talk about ferrimagnetism. The most typical ferrimagnets are ferrites– double oxides of metals. A typical representative of ferrites is magnetite (Fe 3 O 4). Most ferrimagnets are ionic crystals and therefore have low electrical conductivity. In combination with good magnetic properties (high magnetic permeability, high saturation magnetization, etc.) this is an important advantage compared to conventional ferromagnets. It is this quality that has made it possible to use ferrites in ultrahigh frequency technology. Conventional ferromagnetic materials with high conductivity cannot be used here due to very high losses due to the formation of eddy currents. At the same time, many ferrites have a very low Néel point (100 – 300 °C) compared to the Curie temperature for ferromagnetic metals. In this work, to determine the temperature of the ferrimagnetic-paramagnetic transition, a rod made specifically of ferrite is used.
Phase transitions of the second kind are phase transformations in which the density of matter, entropy and thermodynamic potentials do not experience abrupt changes, but the heat capacity, compressibility, and thermal expansion coefficient of the phases change abruptly. Examples: transition of He to a superfluid state, Fe from a ferromagnetic state to a paramagnetic state (at the Curie point).
Paramagnetic-ferromagnetic phase transition
Magnetic systems are important due to the fact that all the terminology used in the theory of phase transitions is based on these systems. Consider a small sample made of iron placed in a magnetic field (). Let be the magnetization of this sample, depending on the magnetic field. Obviously, a decrease in the magnetic field leads to a decrease in magnetization. Two situations may occur. If the temperature is high, the magnetic moment becomes zero as the magnetic field approaches zero. The dependence of the magnetic moment on the magnetic field for this case is presented in Figure 3 a. .
Figure 3. Graph of magnetization versus magnetic field: a - at high; b - at low temperatures.
However, another situation is also possible, which occurs at low temperatures: the magnetization of the sample, which arose under the influence of an external magnetic field, is retained even when this field is reduced to zero. (Figure 3b). This residual magnetization is called spontaneous magnetization (). There is a very specific temperature at which spontaneous magnetization first appears. This temperature is called the Curie temperature. In the temperature range below the Curie temperature, the lower the absolute temperature, the greater the spontaneous magnetization. Magnetization is called the order parameter. A magnetic field, which is a variable thermodynamically conjugate to magnetization, is called an ordering field. Such pairs of conjugate variables will be very important for further theory. There is a very useful model of the paramagnetic-ferromagnetic phase transition. This model is called the Ising model. Let us consider an incompressible lattice, in each node of which there are magnetic needles. These arrows can be directed either up or down. Adjacent arrows interact in such a way that the forces acting between these arrows tend to position them parallel to each other.
Figure 4. Explanation of the Ising model.
It is assumed that the interaction energy of the arrows is positive. In this case, from an energy point of view, it is advantageous for the arrows to be parallel, i.e. so that all arrows point either up or all down. The energy of the system in this case is minimal. From an energy point of view, this state is the most favorable. However, there are only two such completely ordered states (all arrows are up and all arrows are down). In this sense, such ordered states are completely unfavorable from the point of view of entropy. Entropy “tends” to completely disorder the system
At high temperatures, entropy wins. There is disorder in the system and the average magnetization is zero. (the number of blue arrows is equal to the number of red arrows). At low temperatures, energy wins and spontaneous magnetization occurs in the system (the number of blue arrows is ten; and the number of red arrows is sixteen).
This means that in the system under consideration there is a temperature at which spontaneous magnetization appears in the system.
The behavior of all systems near phase transition points is completely universal. It is very comfortable. By studying the simplest system (such as the Ising model) around its critical point, we can predict the physical properties of complex systems around their phase transition points.
