If we proceed from the definition of temperature that was given at the beginning of this book, that is, that temperature is proportional to the average kinetic energy of the particles, then the title of this section seems to be devoid of meaning: after all, kinetic energy cannot be negative! And for those atomic systems in which the energy contains only the kinetic energy of particle motion, negative temperature does not really have physical meaning.
But remember that in addition to the molecular-kinetic definition of temperature, we in Ch. I also noted the role of temperature as a quantity that determines the energy distribution of particles (see p. 55). If we use this more general concept of temperature, then we come to the possibility of the existence (at least in principle) and negative temperatures.
It is easy to see that Boltzmann's formula (9.2)
formally "allows" the temperature to take not only positive, but also negative values.
Indeed, in this formula, this is the fraction of particles in a state with energy, and this is the number of particles in a state with a certain initial energy, from which the energy is counted. It is seen from the formula that the higher the lower the fraction of particles with this energy. So, for example, at times less than the base of natural logarithms). And the energy is already possessed by a much smaller fraction of particles: in this case, times less It is clear that in the equilibrium state, to which, as we know, Boltzmann's law applies, it is always less than
Taking the logarithm of equality (9.2), we get: whence
From this expression for it is seen that if then
If, however, it turned out that there is such an atomic system in which there can be more than that, this would mean that the temperature can also take negative values, since at becomes negative.
It will be easier for us to understand under what circumstances this is possible if we consider not a classical system (in which a negative temperature cannot be realized), but a quantum one, and use, in addition, the concept of entropy, which,
as we have just seen, is the quantity that determines the degree of disorder in the system.
Let the system be represented by a diagram of its energy levels (see, for example, Fig. 1, p. 17). At absolute zero temperature, all particles of our system are at their lowest energy levels, and all other levels are empty. Under such conditions, the system is maximally ordered and its entropy is zero (its heat capacity is also zero).
If we now increase the temperature of the system by supplying energy to it, then the particles will also move to higher energy levels, which, thus, also turn out to be partially populated, and the higher the temperature, the greater the “population” of higher energy levels. The distribution of particles over energy levels is determined by the Boltzmann formula. This means that it will be such that there will be fewer particles at higher levels than at lower ones. The "dispersal" of particles over many levels, of course, increases the disorder in the system and its entropy increases with increasing temperature. The greatest disorder, and hence the maximum entropy, would be achieved with such a distribution of particles by energy, at which they are uniformly distributed over all energy levels. Such a distribution would mean that in the formula means, Therefore, a uniform distribution of particles by energy corresponds to an infinitely high temperature and maximum entropy.
However, in the quantum system we are talking about here, such a distribution is impossible, because the number of levels is infinitely large, and the number of particles is finite. Therefore, the entropy in such a system does not pass through a maximum, but increases monotonically with temperature. At an infinitely high temperature, the entropy will also be infinitely high.
Let us now imagine such a system (quantum), which has an upper limit of its internal energy, and the number of energy levels is finite. This, of course, is possible only in such a system in which the energy does not include the kinetic energy of particle motion.
In such a system, at absolute zero temperature, the particles will also occupy only the lowest energy levels, and the entropy will be equal to zero. As the temperature rises, the particles "disperse" at higher levels, causing a corresponding increase in entropy. In fig. 99, and a system with two energy levels is presented. But, since the number of energy levels of the system, as well as the number of particles in it, is now finite, then in the end a state can be achieved in which the particles are evenly distributed over the energy levels. As we have just seen, this state corresponds to an infinitely high temperature and maximum entropy.
In this case, the energy of the system will also be some maximum, but not infinitely large, so that our old definition of temperature as the average energy of particles becomes inapplicable.
If now in some way to inform the system, already at an infinitely high temperature, additional energy, then the particles will continue to move to a higher energy level, and this will lead to the fact that the "population" of this high energy level will be greater than that of the lower (Fig. 99, b). It is clear that such a predominant accumulation of particles at high levels already signifies some ordering in comparison with the complete disorder that existed when the particles were uniformly distributed over energies. The entropy, which has reached a maximum at, begins, therefore, to decrease with a further supply of energy. But if with increasing energy the entropy does not grow, but falls, then this means that the temperature is not positive, but negative.
The more energy is supplied to the system, the more particles will be at the highest energy levels. In the limit, one can imagine a state in which all particles will collect at the highest levels. This state, obviously, is also quite orderly. It is in no way "worse" than the state when all particles occupy the lowest levels: in both cases, complete order prevails in the system, and the entropy is equal to zero. We can therefore denote the temperature at which this second well-ordered state is established by -0, in contrast to the "usual" absolute zero. The difference between these two "zeros" is that we arrive at the first of them from the negative side, and to the second - from the side of positive temperatures.
Thus, the conceivable temperatures of the system are not limited to the interval from absolute zero to infinity, but extend from through to, and coincide with each other. In fig. 100 shows the curve of the dependence of the entropy on the energy of the system. The part of the curve to the left of the maximum corresponds to positive temperatures, to the right of it - to negative temperatures. At the maximum point, the temperature value is
From the point of view of orderliness, and hence entropy, the following three extreme states are possible, therefore:
1. Complete ordering - particles are concentrated at the lowest energy levels. This state corresponds to "normal" absolute zero
2. Complete disorder - particles are evenly distributed over all energy levels. This state corresponds to the temperature
3. Complete ordering again - particles occupy only the highest energy levels. The temperature corresponding to this condition is assigned a value of -0.
We are dealing here, therefore, with a paradoxical situation: in order to reach negative temperatures, we had to not cool the system below absolute zero, which is impossible, but, on the contrary, to increase its energy; negative temperature turns out to be higher than infinitely high temperature!
There is a very important difference between the two well-ordered states that we just mentioned - states with temperatures.
The state of "ordinary" absolute zero, if it could be created in the system, would persist in it for an arbitrarily long time, provided that it is reliably isolated from the environment, isolated in the sense that no energy is supplied from this environment to the system. This state is a state of stable equilibrium, from which the system itself, without outside interference, cannot get out. This is due to the fact that the energy of the system in this state has a minimum value.
On the other hand, the state of negative absolute zero is an extremely nonequilibrium state, since. the energy of the system is maximum. If it were possible to bring the system to this state, and then leave it to itself, then it would immediately come out of this nonequilibrium, unstable state. It could be preserved only with a continuous supply of energy to the system. Without this, particles located at higher energy levels will certainly "fall" to lower levels.
The common property of both "zeros" is their unattainability: their achievement requires the expenditure of infinitely large energy.
However, not only the state corresponding to a temperature of -0 is unstable, nonequilibrium, but also all states with negative temperatures. All of them correspond to the values of and for equilibrium, the inverse relationship is necessary
We have already noted that negative temperatures are higher temperatures than positive ones. Therefore, if you bring
a body heated (one cannot say: cooled) to negative temperatures, in contact with a body whose temperature is positive, then the energy will transfer from the first to the second, and not vice versa, and this means that its temperature is higher, although it is negative. When two bodies with a negative temperature come into contact, the energy will transfer from a body with a lower absolute value of temperature to a body with a higher numerical value of temperature.
Being in an extremely disequilibrium state, a body heated to a negative temperature very willingly gives up energy. Therefore, in order for such a state to be created, the system must be reliably isolated from other bodies (at least from systems that are not similar to it, that is, they do not have a finite number of energy levels).
However, a state with a negative temperature is so non-equilibrium that even if a system in this state is isolated and there is no one to transfer energy to it, it can still give off energy in the form of radiation until it goes into a state (equilibrium) with a positive temperature ...
It remains to add that atomic systems with a limited set of energy levels, in which, as we have seen, a state with a negative temperature can be realized is not only a conceivable theoretical construction. Such systems actually exist, and in fact, negative temperatures can be obtained in them. The radiation arising from the transition from a negative state to a state with an ordinary temperature is practically used in special devices: molecular generators and amplifiers - masers and lasers. But we cannot dwell on this issue in more detail here.
First, we note that the concept of states with negative absolute temperature does not contradict Nerst's theorem on the impossibility of reaching absolute zero.
Consider a system with negative absolute temperature and only two energy levels. At absolute zero temperatures, all particles are at the lowest level. As the temperature rises, some of the particles begin to move from the lower level to the upper one. The ratio between the number of particles at the first and second levels at different temperatures will satisfy the energy distribution in the form:
As the temperature rises, the number of particles at the second level will approach the number of particles at the first level. In the limiting case of infinitely high temperatures, there will be the same number of particles at both levels.
Thus, for any ratio of the number of particles in the interval
our system can be assigned a certain statistical temperature in the interval determined by equality (12. 44). However, under special conditions, it is possible to achieve that in the system under consideration the number of particles at the second level is greater than the number of particles at the first level. A state with such a ratio of the number of particles can, by analogy with the first considered case, also be assigned a certain statistical temperature or modulus of distribution. But, as follows from (12. 44), this modulus of the statistical distribution must be negative. Thus, the considered state can be attributed to a negative absolute temperature.
From the considered example, it is clear that the negative absolute temperature introduced in this way is in no way a temperature below absolute zero. Indeed, if at absolute zero the system has a minimum internal energy, then with increasing temperature the internal energy of the system increases. However, if we consider a system of particles with only two energy levels, then its internal energy will change as follows. When all the particles are at a lower level with energy, therefore, the internal energy At an infinitely high temperature, the particles are evenly distributed between the levels (Fig. 71) and the internal energy:
that is, it has a finite value.
If we now calculate the energy of the system in the state to which we assigned a negative temperature, it turns out that the internal energy in this state will be greater than the energy in the case of an infinitely large positive temperature. Really,
Thus, negative temperatures correspond to higher internal energies than positive ones. With thermal contact of bodies with negative and positive temperatures, the energy will transfer from bodies with negative absolute temperatures to bodies with positive temperatures. Therefore, bodies at negative temperatures can be considered "hotter" than at positive ones.
Rice. 71. To an explanation of the concept of negative absolute temperatures
The above considerations about the internal energy with a negative modulus of distribution allow us to consider the negative absolute temperature as if it were higher than the infinitely large positive temperature. It turns out that on the temperature scale the region of negative absolute temperatures is not “below absolute zero”, but “above infinite temperature”. In this case, an infinitely large positive temperature "is next to" an infinitely large negative temperature, that is,
A decrease in the negative temperature in absolute value will lead to a further increase in the internal energy of the system. At, the energy of the system will be maximum, since all particles will collect at the second level:
The entropy of the system turns out to be symmetric with respect to the sign of the absolute temperature at equilibrium states.
The physical meaning of negative absolute temperature is reduced to the concept of a negative modulus of statistical distribution.
Whenever the state of the system is described using a statistical distribution with a negative modulus, the concept of negative temperature can be introduced.
It turns out that similar states for some systems can be realized under different physical conditions. The simplest of them is the finiteness of the energy of the system with weak interaction with surrounding systems with positive temperatures and the ability to maintain this state by external forces.
Indeed, if you create a state with a negative temperature, that is, do more, then thanks to spontaneous transitions, particles will be able to move from a state with a state with a lower energy. Thus, a state with a negative temperature will be unstable. To maintain it for a long time, it is necessary to replenish the number of particles at the level by decreasing the number of particles at the level
It turned out that systems of nuclear magnetic moments satisfy the requirement that the energy be finite. Indeed, spin magnetic moments have a certain number of orientations and hence energy levels in a magnetic field. On the other side; in a system of nuclear spins, using nuclear magnetic resonance, it is possible to transfer most of the spins to the state with the highest energy, i.e., to the highest level. For the reverse transition to the lower level, the nuclear spins will have to exchange energy with the crystal lattice, which will take quite a long time. During time intervals less than the spin-lattice relaxation time, the system can be in states with a negative temperature.
The considered example is not the only way to obtain systems with negative temperatures.
Systems with negative temperatures have one interesting feature. If radiation with a frequency corresponding to the energy level difference is passed through such a system, then the transmitted radiation
will stimulate the transitions of particles to the lower level, accompanied by additional radiation. This effect is used in the operation of quantum generators and quantum amplifiers (masers and lasers).
negative absolute temperature, a quantity introduced to describe nonequilibrium states of a quantum system, in which higher energy levels are more populated than lower ones. In equilibrium, the probability of having energy E n is defined by the formula:
Here E i - system energy levels, k- Boltzmann constant, T is the absolute temperature characterizing the average energy of the equilibrium system U = Σ (W n E n), It is seen from (1) that for T> 0, the lower energy levels are more populated with particles than the upper ones. If the system under the influence of external influences passes into a nonequilibrium state, characterized by a greater population of the upper levels in comparison with the lower ones, then formally you can use formula (1), putting in it T < 0. Однако понятие О. т. применимо только к квантовым системам, обладающим конечным числом уровней, так как для создания О. т. для пары уровней необходимо затратить определённую энергию.
In thermodynamics, absolute temperature T is determined through the reciprocal of 1 / T equal to the derivative of entropy (See Entropy) S by the average energy of the system with the constancy of other parameters NS:
From (2) it follows that O. t. Means a decrease in entropy with an increase in the average energy. However, O. t. Is introduced to describe nonequilibrium states to which the application of the laws of equilibrium thermodynamics is conditional.
An example of a system with a crystal lattice is a system of nuclear Spins in a crystal in a magnetic field, interacting very weakly with thermal vibrations of the crystal lattice (see Vibrations of a crystal lattice), that is, practically isolated from thermal motion. The time required to establish thermal equilibrium of the spins with the lattice is measured in tens of minutes. During this time, the system of nuclear spins can be in a state with O. t., Into which it has passed under external influence.
In a narrower sense, O. T. is a characteristic of the degree of population inversion of two selected energy levels of a quantum system. In the case of thermodynamic equilibrium of the population N 1 and N 2 levels E 1 and E 2 (E 1 < E 2), i.e., the average numbers of particles in these states are related by the Boltzmann formula:
where T - absolute temperature of the substance. It follows from (3) that N 2 < N 1... If the equilibrium of the system is disturbed, for example, by acting on the system with monochromatic electromagnetic radiation, the frequency of which is close to the frequency of the transition between levels: ω 21 = ( E 2 - E 1)/ħ and differs from the frequencies of other transitions, then it is possible to obtain a state in which the population of the upper level is higher than the lower N 2 > N 1... If we conditionally apply the Boltzmann formula to the case of such a nonequilibrium state, then with respect to a pair of energy levels E 1 and E 2 you can enter O. t. according to the formula:
Thermodynamic systems in which the probability of finding a system in a microstate with a higher energy is higher than in a microstate with a lower one.
In quantum statistics, this means that there is a greater likelihood of finding a system at one higher energy level than at a lower energy level. An n-fold degenerate level is counted as n levels.
In classical statistics, this corresponds to a higher probability density for points in the phase space with a higher energy compared to points with a lower energy. At a positive temperature, the ratio of probabilities or their densities is the opposite.
For the existence of equilibrium states with a negative temperature, convergence of the partition function is required at this temperature. Sufficient conditions for this are: in quantum statistics - the finiteness of the number of energy levels of the system, in classical statistical physics - the fact that the phase space available to the system has a limited volume, and all points in this available space correspond to energies from a certain finite interval.
In these cases, there is the possibility that the energy of the system will be higher than the energy of the same system in equilibrium distribution with any positive or infinite temperature. Uniform distribution will correspond to infinite temperature and the final energy will be lower than the maximum possible. If such a system has an energy higher than the energy at infinite temperature, then the equilibrium state at such an energy can be described only in terms of negative absolute temperature.
The negative temperature of the system remains long enough if this system is sufficiently well insulated from bodies with a positive temperature. In practice, a negative temperature can be realized, for example, in a system of nuclear spins.
Equilibrium processes are possible with negative temperatures. With thermal contact of two systems with different signs of temperature, a system with a positive temperature begins to heat up, with a negative one - it cools. For the temperatures to become equal, one of the systems must pass through an infinite temperature (in the particular case, the equilibrium temperature of the combined system will remain infinite).
Absolute temperature + ∞ (\ displaystyle + \ infty) and - ∞ (\ displaystyle - \ infty)- this is the same temperature (corresponding to a uniform distribution), but the temperatures T = + 0 and T = -0 are different. Thus, a quantum system with a finite number of levels will be concentrated at the lowest level at T = + 0, and at the highest at T = -0. Passing through a number of equilibrium states, the system can enter the temperature range with a different sign only through an infinite temperature.
In a system of levels with population inversion, the absolute temperature is negative if it is determined, that is, if the system is close enough to equilibrium.
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Absolute temperature ➽ Physics grade 10 ➽ Video tutorial
The absolute temperature in the molecular kinetic theory is defined as a value proportional to the average kinetic energy of particles (see Section 2.3). Since the kinetic energy is always positive, the absolute temperature cannot be negative either. The situation will be different if we use a more general definition of the absolute temperature as a quantity characterizing the equilibrium distribution of the particles of the system over the energy values (see Section 3.2). Then, using Boltzmann's formula (3.9), we have
where N 1 - the number of particles with energy 𝜀 1 , N 2 - the number of particles with energy 𝜀 2 .
Taking the logarithm of this formula, we obtain
In the equilibrium state of the system N 2 is always less N 1 if 𝜀 2 > 𝜀 1 . This means that the number of particles with a higher energy value is less than the number of particles with a lower energy value. In this case, always T > 0.
If we apply this formula to such a nonequilibrium state when N 2 > N 1 at 𝜀 2 > 𝜀 1, then T < 0, т.е. состоянию с таким соотношением числа частиц можно формально по аналогии с предыдущим случаем приписать определенную отрицательную абсолютную температуру. Поскольку при этом формула Больцмана применена к неравновесному распределению частиц системы по энергии, то отрицательная температура является величиной, характеризующей неравновесные системы. Поэтому отрицательная температура имеет иной физический смысл, чем понятие обычной температуры, определение которой неразрывно связано с равновесием.
A negative temperature is attainable only in systems with a finite maximum energy value, or in systems with a finite number of discrete energy values that particles can take, i.e. with a finite number of energy levels. Since the existence of such systems is associated with the quantization of energy states, in this sense, the possibility of the existence of systems with negative absolute temperature is a quantum effect.
Consider a system with negative absolute temperature, having, for example, only two energy levels (Fig. 6.5). At absolute zero temperature, all particles are at the lowest energy level, and N 2 = 0. If the temperature of the system is increased by supplying energy to it, then the particles will begin to move from the lower level to the upper one. In the limiting case, one can imagine a state in which the number of particles is the same at both levels. Applying formula (6.27) to this state, we obtain that T = for N 1 = N 2, i.e. the uniform energy distribution of the particles of the system corresponds to an infinitely high temperature. If in some way additional energy is given to the system, then the transition of particles from the lower level to the upper one will continue, and N 2 becomes larger than N 1 . Obviously, in this case the temperature in accordance with the formula (6.27) will take a negative value. The more energy is supplied to the system, the more particles will be at the upper level and the greater the negative temperature will be. In the extreme case, one can imagine a state in which all the particles are collected at the upper level; wherein N 1 = 0. Therefore, this state will correspond to the temperature - 0K or, as they say, the temperature of negative absolute zero. However, the energy of the system in this case will already be infinitely large.
As for the entropy, which is known to be a measure of the disorder of the system, depending on the energy in ordinary systems, it will increase monotonically (curve 1, Fig. 6.6), so
| Rice. 6.6 |
as in conventional systems, there is no upper limit for the energy value.
In contrast to conventional systems, in systems with a finite number of energy levels, the dependence of entropy on energy has the form shown by curve 2. The area shown by the dotted line corresponds to negative values of the absolute temperature.
For a more visual explanation of this behavior of entropy, let us turn again to the example of a two-level system considered above. At absolute zero temperature (+ 0K), when N 2 = 0, i.e. all particles are at the lower level, the maximum ordering of the system takes place and its entropy is zero. As the temperature rises, the particles will begin to move to the upper level, causing a corresponding increase in entropy. At N 1 = N 2 particles will be evenly distributed across the energy levels. Since such a state of the system can be represented in the greatest number of ways, it will correspond to the maximum value of entropy. A further transition of particles to the upper level leads to a certain ordering of the system in comparison with what took place with a non-uniform distribution of particles over energies. Consequently, despite the increase in the energy of the system, its entropy will begin to decrease. At N 1 = 0, when all the particles are collected at the upper level, there will again be the maximum ordering of the system and therefore its entropy will become equal to zero. The temperature at which this happens will be the temperature of negative absolute zero (–0K).
Thus, it turns out that the point T= - 0K corresponds to the state farthest from the usual absolute zero (+ 0K). This is due to the fact that on the temperature scale the region of negative absolute temperatures is above the infinitely large positive temperature. Moreover, the point corresponding to an infinitely large positive temperature coincides with the point corresponding to an infinitely large negative temperature. In other words, the sequence of temperatures in ascending order (from left to right) should be as follows:
0, +1, +2, … , +
It should be noted that a negative temperature state cannot be achieved by heating a conventional system in a positive temperature state.
The state of negative absolute zero is unattainable for the same reason that the state of positive absolute zero of temperature is unattainable.
Despite the fact that the states with temperatures + 0K and –0K have the same entropy equal to zero and correspond to the maximum ordering of the system, they are two completely different states. At + 0K, the system has a maximum energy value and if it could be achieved, then it would be a state of stable equilibrium of the system. An isolated system could not leave such a state by itself. At –0K, the system has a maximum energy value, and if it could be reached, then it would be a metastable state, i.e. a state of unstable equilibrium. It could be preserved only with a continuous supply of energy to the system, since otherwise the system, left to itself, would immediately come out of this state. All states with negative temperatures are equally unstable.
If a body with a negative temperature is brought into contact with a body with a positive temperature, then the energy will pass from the first body to the second, and not vice versa (as in bodies with a normal positive absolute temperature). Therefore, we can assume that a body with any finite negative temperature is "warmer" than a body with any positive temperature. In this case, the inequality expressing the second law of thermodynamics (the second particular formulation)
can be written as
where is the amount by which the heat of a body with a positive temperature changes over a short period of time, is the amount by which the amount of heat of a body with a negative temperature changes over the same time.
Obviously, this inequality can be fulfilled for and only if the value = - is negative.
Since the states of a system with a negative temperature are unstable, then in real cases such states can be obtained only with good isolation of the system from surrounding bodies with a positive temperature and provided that such states are maintained by external influences. One of the first methods for obtaining negative temperatures was the method of sorting ammonia molecules in a molecular generator created by Russian physicists N.G. Basov and A.M. Prokhorov. Negative temperatures can be obtained using a gas discharge in semiconductors exposed to a pulsed electric field, and in a number of other cases.
It is interesting to note that since systems with negative temperatures are unstable, when radiation of a certain frequency passes through them, as a result of the transition of particles to lower energy levels, additional radiation will appear, and the intensity of the radiation passing through them will increase, i.e. systems have negative absorption. This effect is used in the operation of quantum generators and quantum amplifiers (in masers and lasers).
Note that the difference between the usual absolute zero of temperature and negative is that we approach the first from the side of negative temperatures, and the second - from the side of positive ones.
