EntrancePhysics: Determination of the temperature of the ferromagnetic-paramagnetic phase transition, Laboratory work. Under what conditions does a ferromagnet turn into a paramagnetic? Phase transition ferromagnetic paramagnetic
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, what is important for creating the still missing general theory 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 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, in the absence of an external field, do not have magnetic moment. 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 sides, then more. 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 the atoms are located motionless, 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's find out physical meaning 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 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?
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 spontaneous magnetization is greater, the lower absolute temperature. 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. (number blue arrows 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 physical properties complex systems near their phase transition points.
Pages:
Ufr>=
Based on the obtained relationships, calculations were carried out,
y(\)
characterizing the order of the power singularityy =1 - - at the top
composite wedge atu = i/2, a2 = i(Table 1). For occasionssch - sch= 2zh/3,p1= 0.5 ,
0L-
, X -3 and L - 0.01, isothermal lines are plotted (Fig. 2 and Fig. 3, respectively).
SUMMARY
Different questions mechanics of composite materials, heat conductivity, electrostatics, magnetostatics, mathematical biology result in boundary problems of elliptic type for piecewise ■ homogeneous mediums. When the border of area has angular points for correct determinationO/physical fields it is necessary to have the information about fields singularities In an angular point- Itisconsidereduproblem of the potential theory for compound wedge . Green's function Is built for situation when the concentrated source works in one of phases .
BIBLIOGRAPHY
1. ArcesionV.Ya., Mental physics. Basic equations and special functions.-SCHScience, 1966.
UDC 537.624
PARAMAGNETIC-FERROMAGNETIC PHASE TRANSITION IN A SYSTEM OF SINGLE-DOMAIN FERROMAGNETIC PARTICLES
S.I. Denisov, prof.; V.F.Iefedchenko, smallpox
It is well known that the reason for the appearance of long-range magnetic order in most currently known magnetic materials is.-.^:..-. exchange interaction. At the same time, still in1946 year- _^ g:g Tissa theoreticallySHJVMLYaih gi mpgnptidiolcasinteraction can also serve this role. Since the latter exchange-element is, as a rule, much weaker than the exchange one, the transition temperaturefromordered state of atomic theory
moment, interactingMaychitolnpol^nsh oOrl.chig,:,
is caused by very small and amounts to a fraction of a degree Kelvin. This
Goodness, as well as the absence of substances in which the hierarchicalrilmagnetic interactions begin with the magnetic-dipole, longschzhldid not allow experimental verification of this
->s.And only recently, a corresponding test, based on the net conclusion of Luttinger and Tissa, was carried out on crystals of salts of the KOREANS of the earth, having the chemical formulaCs^Naii(N02)e.
"Kvase systems in which magnetic dipole interaction
structural elements plays a major role, also includes systems
"domain ferromagnetic particles randomly distributed in
in a magnetic solid matrix. The study of such systems is extremely
from a practical point of view, a lot of literature is devoted to this.
Oivako, the study of cooperative effects in them began only in
last years. The main result obtained both numerically,
and both analytical and direct experimental data,
is that, just as in systems of atomic magnetic
moments, in systems of single-domain ferromagnetic particles can
„■walk (one-time transition ferromagnetic state. Although
Some features of this transition have been studied in,
remained
many important issues remain unresolved. Among them, in particular,
urgent question about the impact on phase transition anisotropy
raster for reading particles in space. The point is that analytical
methods developed in,predict the existence of a phase
transition and for isotropic particle distribution. However, this conclusion
contradicts one of the results, according to which in the system
h. ;. :-.b.x dipoles located at nodesdowntimecoupon
lattice, a phase transition to a ferromagnetic state does not occur.
The question of the influence of finite size was also not considered.
Shh§amagkite particles by the value of the average magnetic field,
action on any particle on the part of the others. Meanwhile
its solution is necessary, in particular, for constructing a quantitative
-- cooperative effects in YISTAMAYA PDOTNvuIaYaYaYiH particles.
This is precisely what the this work. Let us consider an ensemble of spherical single-domain ferromagnetic
RadiusG,randomly distributed l non-magnetic solid
hgtrice. We will simulate the distribution of particles in the matrix,
Whattheir centers with probabilityRoccupy idle nodes
tetragonal lattice having periodsdx(>2r)(along the axesXAndat) AndLg(>2g\(along the axis2 - fourth order axes). We will also^re.glio.tag,that the particles are uniaxial, their easy axes of magnetizationz±:-=:;-;:cular planeshu,particle interaction, _-- ;-. ;,:gilyuee, and the dynamics of the magnetic momentt=chp|i|OrRvavoA¬ ..th particle is described by the stochastic Lanlau equation
...
m - -utax(H+h) - (Hujm)mTomxH (m(0) = e,m). (1)
4vka ,4>0)- gyromagnetic ratio;I -dissipation parameter;m=|m|;e.- unit vector along the axisG;N --rfVfcia- effective,= S-.lZUi. 1999. X>2(13)
13 a magnetic field;W- magnetic energy of the particle;h- thermal magnetic field, determined by the relations:
to w= O.+?) = pcs%0Ш$0д,(2)
WhereT- absolute temperature; $ts# - Kronener simiol;a,fi=x,y.zSht)-(i-function,and the bar denotes averaging over implementationsh.
According to the selected modelVapproaching mean zero we have
W -(Haj2m)ml - H(t)m, , (3)
WhereN/,- magnetic anisotropy field;H(t)
~
the average magnetic field acting on a selected particle from the rest. In (3) we took into account that, in accordance with symmetry considerations, in the case under consideration the average field has only2
-component. By placing the origin of coordinates at the lattice node occupied by the selected particle, and numbering the rest with an index і,
expression forH(tjLet's represent it in the form
(7) Finally, identifying in (7) the expression in brackets withtg(i), taking into account the relation ШПу^м - Р and defining the function1 v2-li-4
G2 2 r2 2"i.™s,"a ["і + 1d +WITH,"P§
(8) (g= d2/dl),for the average magnetic field we obtain the following expression:
Shy^ShShchtM,(9)
gayl =pfd-fd?- particle concentration.
A characteristic feature of the functionS(^),conditioning
features of the magnetic properties of three-dimensional
ensemble of single-domain particles, anisotropic
distributed in space is
the inconstancy of its sign:S(
£)>0
atlj
S(g)<0
cri£>1(see Fig. 1). According to (9) this
means that whenf directions of averages
magnetic moments of particles and average
magnetic field coincide, and at£>1have
opposite directions.
^-Hence, ferromagnetic ordering
in systems of single-domain particles occurs
~only with Particularly, but completely
complianceWithLuttinger's prediction and
Tissy to case |- 3, corresponding to prime
Drawing і
cubic lattice, ferromagnetic
There is no such thing. We also note that there is no ferromagnetic order in the limiting case of a two-dimensional distribution of particles, whenf= ", aS(*>)*>-1,129.
According to (2), (3) and (9), the stochastic equation (1), interpreted according to Stratonovich, corresponds to the Fokker-Planck equation
- = - - j |a(ain29 + 2b(t)sinV) -cot antfjP + - J(10)
(AND)
gzeC(a,2ab)
(12) VisnikSIDDU".iS°S,№2(13)
15 (b=b(fj)).Let us determine the order parameter of the system under consideration
single-domain particles as/l- t,g(co)/t. Then, using the relation
(13)
And expressions (11) and (12), for/.і we get equation 2e°
C(a,ZT0c/g)
Sinn
T;G
(AND)where Г0 -onm2 ZS(£)/3k.
Analysis of equation (14) shows that, in accordance with the physical considerations stated above, when££J(WhenTd<0) it has a unique solution /(=0 at any temperature, i.e., long-range order does not arise in this case. A nonzero solution can exist only at£<1. As with the Langevin equation,p=co\&nh(3Tnp./T)-T/3T0fi,to which the equation reduces(14) at Н„-*0, it exists if at/t~»0 the tangent of the angle of inclination of the tangent to the graph of the function defined by the right-hand side of (14) exceeds 1. It is easy to check that this condition is satisfied whenT<Т^Г, WhereTcr ~ temperature of the paramagnetic-ferromagnetic phase transition, which is defined as the solution to the equationT=3T0f(a) ( f(a)= is equal to zero. Diamagnets include many metals (for example, Bi, Ag, Au, Cu), most organic compounds, resins, carbon, etc.
Since the diamagnetic effect is caused by the action of an external magnetic field on the electrons of the atoms of a substance, diamagnetism is characteristic of all substances. However, along with diamagnetic materials, there are also paramagnets - substances that are magnetized in an external magnetic field in the direction of the field.
In paramagnetic substances, in the absence of an external magnetic field, the magnetic moments of the electrons do not compensate each other, and the atoms (molecules) of paramagnetic materials always have a magnetic moment. However, due to the thermal motion of molecules, their magnetic moments are randomly oriented, therefore paramagnetic substances do not have magnetic properties. When a paramagnetic substance is introduced into an external magnetic field, preferential orientation of magnetic moments of atoms on the field(full orientation is prevented by the thermal movement of atoms). Thus, the paramagnetic material is magnetized, creating its own magnetic field, which coincides in direction with the external field and enhances it. This Effect called paramagnetic.
When the external magnetic field is weakened to zero, the orientation of the magnetic moments due to thermal motion is disrupted and the paramagnet is demagnetized. Paramagnetic materials include rare earth elements, Pt, A1, etc. The diamagnetic effect is also observed in paramagnetic materials, but it is much weaker than the paramagnetic effect and therefore remains unnoticeable.
In addition to the two classes of substances considered - dia- and paramagnets, called weakly magnetic substances there are still highly magnetic substances - ferromagnets - substances with spontaneous magnetization, i.e. they are magnetized even in the absence of an external magnetic field. In addition to their main representative - iron (from which the name “ferromagnetism” comes) - ferromagnets include, for example, cobalt, nickel, gadolinium, their alloys and compounds.
Ferromagnets, in addition to the ability to be strongly magnetized, also have other properties that significantly distinguish them from dia- and paramagnets. If for weakly magnetic substances the dependence on is linear, then for ferromagnetic substances this dependence is quite complex. As you increase H magnetization J first grows quickly, then more slowly, and finally the so-called magnetic saturation J to c, no longer dependent on field strength.
Rice. 2
Similar nature of dependence J from N can be explained by the fact that as the magnetizing field increases, the degree of orientation of molecular magnetic moments along the field increases. However, this process will begin to slow down when there are fewer and fewer unoriented moments remaining, and finally, when all moments are oriented along the field, a further increase N stops and magnetic saturation occurs.
Rice. 3 
Magnetic induction B = μ 0 (N+ J) in weak fields increases rapidly with increasing N due to increasing J, and in strong fields, since the second term is constant ( J=JHac), IN increases with increasing N according to a linear law.
An essential feature of ferromagnets is not only large values μ
(for example, for iron - 5000, for supermalloy alloy - 800,000!), but also the dependence μ
from N(Fig. 3). At the beginning μ
grows with increasing N, then, reaching a maximum, begins to decrease, tending in the case of strong fields to 1 ( 
, so when J=JHac= const with increasing N relation , and μ → 1).
Fig.4 
A characteristic feature of ferromagnets is also that for them the dependence J from N(and therefore IN from N) is determined by the history of magnetization of the ferromagnet. This phenomenon is called magnetic hysteresis. If you magnetize a ferromagnet until saturation (Fig. 4, point 1), and then begin to reduce tension N magnetizing field, then, as experience shows, the decrease is described by the curve 1 - 2, above the curve 1 - 0. At N = 0 , J differs from zero, i.e. in a ferromagnet it is observed residual magnetization J oc .
The presence of residual magnetization is associated with the existence permanent magnets. Magnetization becomes zero under the influence of the field N s, having a direction opposite to the field that caused the magnetization. Tension N s is called coercive force.
With a further increase in the opposite field, the ferromagnet is remagnetized (curve 3 - 4), and at N = - N we reach saturation (point 4 ). Then the ferromagnet can be demagnetized again (curve 4 - 5-6) and remagnetized again until saturation (curve 6- 1 ).
Thus, when a ferromagnet is exposed to an alternating magnetic field, the magnetization J changes according to the curve 1-2-3-4- 5-6-1, which is called hysteresis loop (from the Greek “delay”). Hysteresis leads to the fact that the magnetization of a ferromagnet is not an unambiguous function N, i.e. to the same value N matches multiple values J.
Ferromagnets have another significant feature: for each ferromagnet there is a certain temperature, called Curie point, at which it loses its magnetic properties. When a sample is heated above the Curie point, the ferromagnet turns into an ordinary paramagnet. The transition of a substance from a ferromagnetic state to a paramagnetic state, which occurs at the Curie point, is not accompanied by the absorption or release of heat, i.e. At the Curie point, a second-order phase transition occurs.
Finally, the process of magnetization of ferromagnets is accompanied by a change in its linear dimensions and volume. This phenomenon is called magnetostriction . The magnitude and sign of the effect depend on the tension H magnetizing field, on the nature of the ferromagnet and the orientation of the crystallographic axes relative to the field.
Related information.
Determination of phase transition temperature
ferrimagnetic-paramagnetic
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 Sj , 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 | 769 |
| Ni | 364 |
| Co | 1121 |
| Gd | 18 |
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.
Completing of the work
Scheme of the experimental setup.
Experiment idea
The main part of this installation is a transformer with an open core made of ferrite. The primary winding, made of nichrome, also serves to heat the core. Voltage to the primary winding is supplied from the LATR to avoid overheating. The induced current is recorded using a voltmeter connected to the secondary winding. A single thermocouple, thermo-emf, is used to measure the core temperature. which is proportional to the temperature difference between the ambient air and the thermocouple junction. The core temperature can be calculated using the following formula: T=T 0 +23.5×e, where e is thermal emf. (in millivolts), T 0 – air temperature in the laboratory.
The idea of the experiment is as follows: induced emf in the secondary winding, where I i - current in the primary winding, L- inductance of the primary winding; it is known that where is the inductance of the secondary winding without a core, and m- magnetic permeability of the core.
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Magnetic permeability decreases with increasing temperature, and upon reaching the Néel point it drops sharply. Consequently, both the induced emf and the induced current drop sharply when .
Conducting an experiment
1. Assemble the installation according to the diagram shown in Fig. 2.
2. Set the LATR control knobs (there are two of them) to the extreme left position.
3. Turn on the LATR network and the millivoltmeter power supply.
4. Set the voltage at the output of the first LATR - 220V, at the output of the second - no more than 30V.
5. Take readings from the millivoltmeter every 1-2 divisions while simultaneously taking readings from the milliammeter.
6. After the Néel point is reached, turn off the LATR and allow the core to cool. Then repeat the measurements at least 3 times.
7. Construct graphs based on the table data. Determine from the graphs the temperature at which the value of the induced emf in the secondary winding begins to decrease sharply (see figure), we will take this temperature value equal to the Neel temperature in this experiment. Determine this way for each series of measurements. Calculate the average.
8. Determine the random error in measurements of the phase transition temperature.
Sample table for a report.
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Control questions
1. What are magnetic susceptibility and magnetic permeability?