**The Origin of Magnetism and Negative Temperature Characteristic in Finite Level Magnetic System**

The magnetism of matter is a deceptively simple problem that is actually quite subtle. In history, magnetism was discovered by humans very early. But it is strange that such “simple” magnetism cannot be described by classical physics, which seems so powerful! Specifically, applying the conclusions of classical electrodynamics within the framework of statistical mechanics fails to give any magnetism! This difficulty was not resolved until the emergence and development of quantum mechanics in the 20th century. Magnetism can only be given by the framework of statistical mechanics applied under the description of quantum mechanics! This paper attempts to explain why magnetism cannot be given by classical electrodynamics through rigorous arguments. Then why can magnetism and correct macroscopic magnetic laws be given by introducing quantum models. Finally, let’s discuss a very peculiar property in the (quantum) finite energy level magnetic system – negative temperature!

1 Statistical mechanics procedures applied to classical electrodynamics cannot give material magnetic properties

Consider a system of non-interacting N particles in an external field. When the external field has only magnetic field and no electric field, the system Hamiltonian given by classical electrodynamics is:

where the magnetic field is given by the curl of the magnetic vector potential :

So the canonical partition function of the system is:

[Note that and in the above formula are respectively the embodiment of the principle of quantum identity (identical particles are indistinguishable) and the principle of quantum uncertainty in the classical partition function (equivalent to these two retained factors). But since the magnetic field is not included in these two factors, they are only unimportant constants in the following calculations. 】

So the Helmholtz free energy of the system is:

Since the Helmholtz free energy has nothing to do with the magnetic field, the average magnetic moment of the system at any temperature is:

So classical electrodynamics cannot give magnetism within the framework of statistical mechanics! That is to say, there is no place for magnetism in classical physics!

2 Under the description of quantum mechanics, statistical mechanics procedures are applied to give the material magnetism 2.1 Semi-classical model — classical description of spin magnetic moment

Consider a system of N particles. Each particle is on a grid point. Under the framework of quantum mechanics, if the interaction energy between the magnetic moment and the coupling between the magnetic moment on the lattice point is not considered, the Hamiltonian of the magnetic system can be written as:

For the sake of simplicity, it is assumed that a classical spin is placed on each lattice point (that is, the magnetic moment has not been quantized into discrete values in the intrinsic space), and each spin can change continuously in the direction of , then the regularity of the system The partition function is:

So the Helmholtz free energy of the system is:

So the average magnetic moment of the system is:

When or (i.e. high temperature and low field), the average magnetic moment can be further simplified as:

So the magnetic susceptibility is:

So this model gives magnetism, and gives the magnetic (paramagnetic) equation of state. This equation of state is exactly the same as the form of Curie’s law satisfied by the magnetism measured by macroscopic experiments, that is, the inverse relationship between magnetic susceptibility and temperature! So we have successfully deduced macroscopic magnetism from this microscopic model!

2.2 Pure quantum model — quantum description of spin magnetic moment

Although the model of (2.1) gives the correct form of the magnetic equation of state, we notice that the spin in the Hamiltonian of this model is still classical (this is an unreasonable assumption we made earlier for the sake of simplicity )! So in order to get a more accurate conclusion, we must replace it with the real quantum spin!

Among them is the Bohr magneton. So the quantum canonical partition function of the system is:

where the parameter is a function of the magnetic field :

So the Helmholtz free energy of the system is:

So the average magnetic moment of the system is:

Note that in the limit case (the quantum effect of the discrete spatial orientation of the intrinsic spin disappears), the above formula for the average magnetic moment degenerates into the case where each lattice point is a classical spin:

When or (i.e. high temperature and low field), . Therefore, the average magnetic moment can be further simplified as:

For electrons without spin (S)-orbital (L) coupling, , . Substitute into the above formula to get:

And then get the magnetic susceptibility:

Therefore, after our model fully considers the quantized value of the spin in the intrinsic space, we still get the magnetic (paramagnetic) equation of state consistent with the previous form, which satisfies the macroscopic magnetic Curie’s law! But it is worth noting that since we have considered the quantum effect of the spin on each lattice point, the Curie coefficient at this time is 3 times the original! That is, after considering the quantum effect of the spin itself, the magnetism becomes larger!

Negative temperature characteristics in 3 (quantum) finite energy level magnetic systems

Finally, let’s discuss the singular characteristic of negative temperature unique to (quantum) finite-level magnetic systems like (2.2)! For the sake of simplicity, we only take the two-level electron system without spin (S)-orbital (L) coupling as an example, at this time, , . Substituting into the expressions of the quantum canonical partition function and the Helmholtz free energy in (2.2), we get:

From this, it is easy to find the dependence of internal energy and entropy on temperature:

Reverse the temperature as a function of internal energy:

So can be expressed as a function of :

It is easy to find that is an even function of . In order to intuitively see the relationship between and , [Figure 1] made a sketch of the functional relationship of – :

the Figure 1 – Sketch of entropy (S) – internal energy (U) function relationship. where the ordinate is entropy and the abscissa is internal energy

It can be seen that: when the internal energy is less than 0, the slope of the – function is positive; when the internal energy is greater than 0, the slope of the – function is negative. And the slope of the – function is just proportional to the inversion temperature, namely:

The above formula can be regarded as the definition formula of temperature. It is easy to see – the positive or negative of the slope of the function directly determines the positive or negative of the temperature T! So U less than 0 (that is, the left half of Figure 1) corresponds to a normal positive temperature (normal equilibrium thermodynamic temperature). But U greater than 0 (that is, the right half of Figure 1) corresponds to a negative temperature! ! At negative temperature, since the internal energy is greater than 0, the high-energy level occupies more than the low-energy level in the two-level electronic system, which realizes the so-called “particle population inversion”! According to the Boltzmann distribution, when the system reaches thermal equilibrium, the occupancy number of particles in the low energy level is always greater than that in the high energy level (the occupancy number of particles in the high and low energy levels is just equal at the temperature of positive infinity). So in order to pump more than half of the particles to a higher energy level, the negative temperature is actually a higher temperature than the positive infinity temperature! Since the number of particles is in an inverted distribution at negative temperatures, the negative temperature system is not a stable and balanced system. Most particles occupied by high energy levels will quickly jump down to low energy levels. Such negative temperature systems rarely occur in nature, but many useful negative temperature systems, such as lasers, can be produced by man-made.

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