**2.1 The liquid state**

The ability of the liquids to form a free surface differs from that of the gases, which occupy the entire volume available and have diffusion coefficients (<sup>∼</sup> 0, 5 cm2s−1) of several orders of magnitude higher than those of liquids (<sup>∼</sup> <sup>10</sup>−<sup>5</sup> cm2s−1) or solids (<sup>∼</sup> <sup>10</sup>−<sup>9</sup> cm2s−1). Moreover, if the dynamic viscosity of liquids (between 10−<sup>5</sup> Pa.s and 1 Pa.s) is so lower compared to that of solids, it is explained in terms of competition between *configurational* and *kinetic* processes. Indeed, in a solid, the displacements of atoms occur only after the breaking of the bonds that keep them in a stable configuration. At the opposite, in a gas, molecular transport is a purely kinetic process perfectly described in terms of exchanges of energy and momentum. In a liquid, the continuous rearrangement of particles and the molecular transport combine together in appropriate proportion, meaning that the liquid is an intermediate state between the gaseous and solid states.

of Simple Liquids 3

Thermodynamic Perturbation Theory of Simple Liquids 841

to the position of the nearest neighbors around an origin atom. It should be noted that the pair correlation function *g*(*r*) is accessible by a simple Fourier transform of the experimental

The pair correlation function is of crucial importance in the theory of liquids at equilibrium, because it depends strongly on the *pair potential u*(*r*) between the molecules. In fact, one of the goals of the theory of liquids at equilibrium is to predict the thermodynamic properties using the pair correlation function *g*(*r*) and the pair potential *u*(*r*) acting in the liquids. There are a large number of potential models (hard sphere, square well, Yukawa, Gaussian, Lennard-Jones...) more or less adapted to each type of liquids. These interaction potentials have considerable theoretical interest in statistical physics, because they allow the calculation of the properties of the liquids they are supposed to represent. But many approximations for

Note that there is a great advantage in comparing the results of the theory with those issued from the numerical simulation with the aim to test the models developed in the theory. Beside, the comparison of the theoretical results to the experimental results allows us to test the potential when the theory itself is validated. Nevertheless, comparison of simulation results with experimental results is the most efficient way to test the potential, because the simulation provides the exact solution without using a theoretical model. It is a matter of fact that simulation is generally identified to a numerical experience. Even if they are time consuming, the simulation computations currently available with thousands of interacting

In the theory of simple fluids, one of the major achievements has been the recognition of the quite distinct roles played by the repulsive and attractive parts of the pair potential in determining the microscopic properties of simple fluids. In recent years, much attention has been paid in developing analytically solvable models capable to represent the thermodynamic and structural properties of real fluids. The hard-sphere (HS) model - with its diameter *σ* - is the natural *reference system* for describing the general characteristics of liquids, i.e. the local atomic order due to the excluded volume effects and the *solidification* process of liquids into a solid ordered structure. In contrast, the HS model is not able to predict the *condensation* of a gas into a liquid, which is only made possible by the existence of dispersion forces represented

Another reference model that has proved very useful to stabilize the local structure in liquids is the hard-core potential with an attractive Yukawa tail (HCY), by varying the hard-sphere diameter *σ* and screening length *λ*. It is an advantage of this model for modeling real systems with widely different features (1), like rare gases with a screening length *λ* ∼ 2 or colloidal suspensions and protein solutions with a screening length *λ* ∼ 8. An additional reason that does the HCY model appealing is that analytical solutions are available. After the search of the original solution with the mean-spherical approximation (2), valuable simplifications have been progressively brought giving simple analytical expressions for the thermodynamic properties and the pair correlation function. For this purpose, the expression for the free energy has been used under an expanded form in powers of the inverse temperature, as

At this stage, it is perhaps salutary to claim that no attempt will be made, in this article, to discuss neither the respective advantages of the pair potentials nor the ability of various approximations to predict the structure, which are necessary to determine the thermodynamic properties of liquids. In other terms, nothing will be said on the theoretical aspect of correlation functions, except a brief summary of the experimental determination of the pair correlation function. In contrast, it will be useful to state some of the concepts

*structure factor S*(*q*) (intensity of scattered radiation).

calculating the pair correlation function *g*(*r*) exist too.

by an attractive long-ranged part in the potential.

derived by Henderson *et al*. (3).

particles gives a role increasingly important to the simulation methods.

The characterization of the three states of matter can be done in an advantageous manner by comparing the kinetic energy and potential energy as it is done in figure (1). The nature and intensity of forces acting between particles are such that the particles tend to attract each other at great distances, while they repel at the short distances. The particles are in equilibrium when the attraction and repulsion forces balance each other. In gases, the kinetic energy of particles, whose the distribution is given by the Maxwell velocity distribution, is located in the region of unbound states. The particles move freely on trajectories suddenly modified by binary collisions; thus the movement of particles in the gases is essentially an *individual movement*. In solids, the energy distribution is confined within the potential well. It follows that the particles are in tight bound states and describe harmonic motions around their equilibrium positions; therefore the movement of particles in the solids is essentially a *collective movement*. When the temperature increases, the energy distribution moves towards high energies and the particles are subjected to anharmonic movements that intensify progressively. In liquids, the energy distribution is almost entirely located in the region of bound states, and the movements of the particles are strongly anharmonic. On approaching the critical point, the energy distribution shifts towards the region of unbound states. This results in important fluctuations in concentrations, accompanied by the destruction and formation of aggregates of particles. Therefore, the movement of particles in liquids is thus the result of a combination of individual and collective movements.

Fig. 1. Comparison of kinetic and potential energies in solids, liquids and gases.

When a crystalline solid melts, the long-range order of the crystal is destroyed, but a residual local order persits on distances greater than several molecular diameters. This local order into liquid state is described in terms of the *pair correlation function*, *g*(*r*) = *<sup>ρ</sup>*(*r*) *ρ*∞ , which is defined as the ratio of the mean molecular density *ρ*(*r*), at a distance *r* from an arbitrary molecule, to the bulk density *ρ*∞. If *g*(*r*) is equal to unity everywhere, the fluid is completely disordered, like in diluted gases. The deviation of *g*(*r*) from unity is a measure of the local order in the arrangement of near-neighbors. The representative curve of *g*(*r*) for a liquid is formed of maxima and minima rapidly damped around unity, where the first maximum corresponds 2 Thermodynamics book 1

The characterization of the three states of matter can be done in an advantageous manner by comparing the kinetic energy and potential energy as it is done in figure (1). The nature and intensity of forces acting between particles are such that the particles tend to attract each other at great distances, while they repel at the short distances. The particles are in equilibrium when the attraction and repulsion forces balance each other. In gases, the kinetic energy of particles, whose the distribution is given by the Maxwell velocity distribution, is located in the region of unbound states. The particles move freely on trajectories suddenly modified by binary collisions; thus the movement of particles in the gases is essentially an *individual movement*. In solids, the energy distribution is confined within the potential well. It follows that the particles are in tight bound states and describe harmonic motions around their equilibrium positions; therefore the movement of particles in the solids is essentially a *collective movement*. When the temperature increases, the energy distribution moves towards high energies and the particles are subjected to anharmonic movements that intensify progressively. In liquids, the energy distribution is almost entirely located in the region of bound states, and the movements of the particles are strongly anharmonic. On approaching the critical point, the energy distribution shifts towards the region of unbound states. This results in important fluctuations in concentrations, accompanied by the destruction and formation of aggregates of particles. Therefore, the movement of particles in liquids is thus

the result of a combination of individual and collective movements.

Fig. 1. Comparison of kinetic and potential energies in solids, liquids and gases.

liquid state is described in terms of the *pair correlation function*, *g*(*r*) = *<sup>ρ</sup>*(*r*)

When a crystalline solid melts, the long-range order of the crystal is destroyed, but a residual local order persits on distances greater than several molecular diameters. This local order into

as the ratio of the mean molecular density *ρ*(*r*), at a distance *r* from an arbitrary molecule, to the bulk density *ρ*∞. If *g*(*r*) is equal to unity everywhere, the fluid is completely disordered, like in diluted gases. The deviation of *g*(*r*) from unity is a measure of the local order in the arrangement of near-neighbors. The representative curve of *g*(*r*) for a liquid is formed of maxima and minima rapidly damped around unity, where the first maximum corresponds

*ρ*∞

, which is defined

to the position of the nearest neighbors around an origin atom. It should be noted that the pair correlation function *g*(*r*) is accessible by a simple Fourier transform of the experimental *structure factor S*(*q*) (intensity of scattered radiation).

The pair correlation function is of crucial importance in the theory of liquids at equilibrium, because it depends strongly on the *pair potential u*(*r*) between the molecules. In fact, one of the goals of the theory of liquids at equilibrium is to predict the thermodynamic properties using the pair correlation function *g*(*r*) and the pair potential *u*(*r*) acting in the liquids. There are a large number of potential models (hard sphere, square well, Yukawa, Gaussian, Lennard-Jones...) more or less adapted to each type of liquids. These interaction potentials have considerable theoretical interest in statistical physics, because they allow the calculation of the properties of the liquids they are supposed to represent. But many approximations for calculating the pair correlation function *g*(*r*) exist too.

Note that there is a great advantage in comparing the results of the theory with those issued from the numerical simulation with the aim to test the models developed in the theory. Beside, the comparison of the theoretical results to the experimental results allows us to test the potential when the theory itself is validated. Nevertheless, comparison of simulation results with experimental results is the most efficient way to test the potential, because the simulation provides the exact solution without using a theoretical model. It is a matter of fact that simulation is generally identified to a numerical experience. Even if they are time consuming, the simulation computations currently available with thousands of interacting particles gives a role increasingly important to the simulation methods.

In the theory of simple fluids, one of the major achievements has been the recognition of the quite distinct roles played by the repulsive and attractive parts of the pair potential in determining the microscopic properties of simple fluids. In recent years, much attention has been paid in developing analytically solvable models capable to represent the thermodynamic and structural properties of real fluids. The hard-sphere (HS) model - with its diameter *σ* - is the natural *reference system* for describing the general characteristics of liquids, i.e. the local atomic order due to the excluded volume effects and the *solidification* process of liquids into a solid ordered structure. In contrast, the HS model is not able to predict the *condensation* of a gas into a liquid, which is only made possible by the existence of dispersion forces represented by an attractive long-ranged part in the potential.

Another reference model that has proved very useful to stabilize the local structure in liquids is the hard-core potential with an attractive Yukawa tail (HCY), by varying the hard-sphere diameter *σ* and screening length *λ*. It is an advantage of this model for modeling real systems with widely different features (1), like rare gases with a screening length *λ* ∼ 2 or colloidal suspensions and protein solutions with a screening length *λ* ∼ 8. An additional reason that does the HCY model appealing is that analytical solutions are available. After the search of the original solution with the mean-spherical approximation (2), valuable simplifications have been progressively brought giving simple analytical expressions for the thermodynamic properties and the pair correlation function. For this purpose, the expression for the free energy has been used under an expanded form in powers of the inverse temperature, as derived by Henderson *et al*. (3).

At this stage, it is perhaps salutary to claim that no attempt will be made, in this article, to discuss neither the respective advantages of the pair potentials nor the ability of various approximations to predict the structure, which are necessary to determine the thermodynamic properties of liquids. In other terms, nothing will be said on the theoretical aspect of correlation functions, except a brief summary of the experimental determination of the pair correlation function. In contrast, it will be useful to state some of the concepts

of Simple Liquids 5

Thermodynamic Perturbation Theory of Simple Liquids 843

Fig. 2. Schematic representations of the Lennard-Jones potential (*m* − *n*) and the diagram

*dT* <sup>=</sup> *Lvap*

*dT* of the branch TC at ambient pressure, we can estimate the ratio *Lvap*

well depth *ε*, the slope of the liquid-vapor coexistence curve decreases as *n* decreases. For liquid metals, it should be mentioned that the repulsive part of the potential is softer than for liquid rare gases. Moreover, even if *ε* is slightly lower for metals than for rare gases, the

TC curve compared to that of rare gases. It is worth also to indicate that some *flat-bottomed* potentials (6) are likely to give a good description of the physical properties of substances that

*TC* � 0, 56, or for organic and inorganic liquids, for which 0, 25 <sup>&</sup>lt; *TT*

return, it might be useful as empirical potential for metals with low melting point such as

The value of the slope of the branch TC also depends on the attractive part of the potential as

where *Lvap* is the latent heat of vaporization at the corresponding temperature *Tvap* and (*Vvap* − *Vliq*) is the difference of specific volumes between vapor and liquid. To evaluate the

*Tvap*(*Vvap* − *Vliq*)

*Tvap* � 85 J.K−1.mol−1), and the difference in volume (*Vvap* <sup>−</sup> *Vliq*) in terms of width of the potential well. Indeed, in noting that the quantity (*Vvap* − *Vliq*) is an increasing function of the width of potential well, which itself increases when *n* decreases, we see that, for a given

*<sup>ε</sup>* is much higher (between 2 and 4), which explains the elongation of the

*TC* . Such a potential is obviously not suitable for liquid rare gases,

*TC* < 0, 1.

, (2)

*Tvap* with Trouton's

*TC* < 0, 45. In

*p*(*T*), as a function of the values of the parameters *m* and *n*.

mercury, gallium, indium, tin, etc., the ratio of which being *TT*

*dp*

shown by the Clausius-Clapeyron equation:

slope *dp*

rule ( *Lvap*

quantity (*TC*−*TT* )*kB*

whose ratio *TT*

have a low value of the ratio *TT*

of statistical thermodynamics providing a link between the microscopic description of liquids and classical thermodynamic functions. Then, it will be given an account of the thermodynamic perturbation theory with the analytical expressions required for calculating the thermodynamic properties. Finally, the HCY model, which is founded on the perturbation theory, will be presented in greater detail for investigating the thermodynamics of liquids. Thus, a review of the thermodynamic perturbation theory will be set up, with a special effort towards the pedagogical aspect. We hope that this paper will help readers to develop their inductive and synthetic capacities, and to enhance their scientific ability in the field of thermodynamic of liquids. It goes without saying that the intention of the present paper is just to initiate the readers to that matter, which is developed in many standard textbooks (4).

#### **2.2 Phase stability limits versus pair potential**

One success of the numerical simulation was to establish a relationship between the shape of the pair potential and the phase stability limits, thus clarifying the circumstances of the liquid-solid and liquid-vapor phase transitions. It has been shown, in particular, that the hard-sphere (HS) potential is able to correctly describe the atomic structure of liquids and predict the liquid-solid phase transition (5). By contrast, the HS potential is unable to describe the liquid-vapor phase transition, which is essentially due to the presence of attractive forces of dispersion. More specifically, the simulation results have shown that the liquid-solid phase transition depends on the steric hindrance of the atoms and that the coexistence curve of liquid-solid phases is governed by the details of the repulsive part of potential. In fact, this was already contained in the phenomenological theories of melting, like the Lindemann theory that predicts the melting of a solid when the mean displacement of atoms from their equilibrium positions on the network exceeds the atomic diameter of 10%. In other words, a substance melts when its volume exceeds the volume at 0 K of 30%.

In restricting the discussion to simple centrosymmetric interactions from the outset, it is necessary to consider a realistic pair potential adequate for testing the phase stability limits. The most natural prototype potential is the Lennard-Jones (LJ) potential given by

$$
\mu\_{Lf}(r) = 4\varepsilon\_{Lf} \left[ (\frac{\sigma\_{Lf}}{r})^m - (\frac{\sigma\_{Lf}}{r})^n \right],\tag{1}
$$

where the parameters *m* and *n* are usually taken to be equal to 12 and 6, respectively. Such a functional form gives a reasonable representation of the interactions operating in real fluids, where the well depth *εL J* and the collision diameter *σL J* are independent of density and temperature. Figure (2a) displays the general shape of the Lennard-Jones potential (*m* − *n*) corresponding to equation (1). Each substance has its own values of *εL J* and *σL J* so that, in reduced form, the LJ potentials have not only the same shape for all simple fluids, but superimpose each other rigorously. This is the condition for substances to conform to the *law of corresponding states*.

Figure (2b) represents the diagram *p*(*T*) of a pure substance. We can see how the slope of the coexistence curve of solid-liquid phases varies with the repulsive part of potential: the higher the value of *m*, the steeper the repulsive part of the potential (Fig. 2a) and, consequently, the more the coexistence curve of solid-liquid phases is tilted (Fig. 2b).

We can also remark that the LJ potential predicts the liquid-vapor coexistence curve, which begins at the triple point T and ends at the critical point C. A detailed analysis shows that the length of the branch TC is proportional to the depth *ε* of the potential well. As an example, for rare gases, it is verified that (*TC* − *TT*)*kB* � 0, 55 *ε*. It follows immediately from this condition that the liquid-vapor coexistence curve disappears when the potential well is absent (*ε* = 0). 4 Thermodynamics book 1

of statistical thermodynamics providing a link between the microscopic description of liquids and classical thermodynamic functions. Then, it will be given an account of the thermodynamic perturbation theory with the analytical expressions required for calculating the thermodynamic properties. Finally, the HCY model, which is founded on the perturbation theory, will be presented in greater detail for investigating the thermodynamics of liquids. Thus, a review of the thermodynamic perturbation theory will be set up, with a special effort towards the pedagogical aspect. We hope that this paper will help readers to develop their inductive and synthetic capacities, and to enhance their scientific ability in the field of thermodynamic of liquids. It goes without saying that the intention of the present paper is just to initiate the readers to that matter, which is developed in many standard textbooks (4).

One success of the numerical simulation was to establish a relationship between the shape of the pair potential and the phase stability limits, thus clarifying the circumstances of the liquid-solid and liquid-vapor phase transitions. It has been shown, in particular, that the hard-sphere (HS) potential is able to correctly describe the atomic structure of liquids and predict the liquid-solid phase transition (5). By contrast, the HS potential is unable to describe the liquid-vapor phase transition, which is essentially due to the presence of attractive forces of dispersion. More specifically, the simulation results have shown that the liquid-solid phase transition depends on the steric hindrance of the atoms and that the coexistence curve of liquid-solid phases is governed by the details of the repulsive part of potential. In fact, this was already contained in the phenomenological theories of melting, like the Lindemann theory that predicts the melting of a solid when the mean displacement of atoms from their equilibrium positions on the network exceeds the atomic diameter of 10%. In other words, a

In restricting the discussion to simple centrosymmetric interactions from the outset, it is necessary to consider a realistic pair potential adequate for testing the phase stability limits.

where the parameters *m* and *n* are usually taken to be equal to 12 and 6, respectively. Such a functional form gives a reasonable representation of the interactions operating in real fluids, where the well depth *εL J* and the collision diameter *σL J* are independent of density and temperature. Figure (2a) displays the general shape of the Lennard-Jones potential (*m* − *n*) corresponding to equation (1). Each substance has its own values of *εL J* and *σL J* so that, in reduced form, the LJ potentials have not only the same shape for all simple fluids, but superimpose each other rigorously. This is the condition for substances to conform to the *law*

Figure (2b) represents the diagram *p*(*T*) of a pure substance. We can see how the slope of the coexistence curve of solid-liquid phases varies with the repulsive part of potential: the higher the value of *m*, the steeper the repulsive part of the potential (Fig. 2a) and, consequently, the

We can also remark that the LJ potential predicts the liquid-vapor coexistence curve, which begins at the triple point T and ends at the critical point C. A detailed analysis shows that the length of the branch TC is proportional to the depth *ε* of the potential well. As an example, for rare gases, it is verified that (*TC* − *TT*)*kB* � 0, 55 *ε*. It follows immediately from this condition that the liquid-vapor coexistence curve disappears when the potential well is absent (*ε* = 0).

*σL J r* )*n* 

, (1)

The most natural prototype potential is the Lennard-Jones (LJ) potential given by

 ( *σL J <sup>r</sup>* )*<sup>m</sup>* <sup>−</sup> (

**2.2 Phase stability limits versus pair potential**

*of corresponding states*.

substance melts when its volume exceeds the volume at 0 K of 30%.

*uL J*(*r*) = 4*εL J*

more the coexistence curve of solid-liquid phases is tilted (Fig. 2b).

Fig. 2. Schematic representations of the Lennard-Jones potential (*m* − *n*) and the diagram *p*(*T*), as a function of the values of the parameters *m* and *n*.

The value of the slope of the branch TC also depends on the attractive part of the potential as shown by the Clausius-Clapeyron equation:

$$\frac{dp}{dT} = \frac{L\_{vap}}{T\_{vap}(V\_{vap} - V\_{liq})} \,\text{}\tag{2}$$

where *Lvap* is the latent heat of vaporization at the corresponding temperature *Tvap* and (*Vvap* − *Vliq*) is the difference of specific volumes between vapor and liquid. To evaluate the slope *dp dT* of the branch TC at ambient pressure, we can estimate the ratio *Lvap Tvap* with Trouton's rule ( *Lvap Tvap* � 85 J.K−1.mol−1), and the difference in volume (*Vvap* <sup>−</sup> *Vliq*) in terms of width of the potential well. Indeed, in noting that the quantity (*Vvap* − *Vliq*) is an increasing function of the width of potential well, which itself increases when *n* decreases, we see that, for a given well depth *ε*, the slope of the liquid-vapor coexistence curve decreases as *n* decreases. For liquid metals, it should be mentioned that the repulsive part of the potential is softer than for liquid rare gases. Moreover, even if *ε* is slightly lower for metals than for rare gases, the quantity (*TC*−*TT* )*kB <sup>ε</sup>* is much higher (between 2 and 4), which explains the elongation of the TC curve compared to that of rare gases. It is worth also to indicate that some *flat-bottomed* potentials (6) are likely to give a good description of the physical properties of substances that have a low value of the ratio *TT TC* . Such a potential is obviously not suitable for liquid rare gases, whose ratio *TT TC* � 0, 56, or for organic and inorganic liquids, for which 0, 25 <sup>&</sup>lt; *TT TC* < 0, 45. In return, it might be useful as empirical potential for metals with low melting point such as mercury, gallium, indium, tin, etc., the ratio of which being *TT TC* < 0, 1.

of Simple Liquids 7

Thermodynamic Perturbation Theory of Simple Liquids 845

Now, if we consider an assembly of *N* identical atoms forming the liquid sample, the intensity

In a crystalline solid, the arrangement of atoms is known once and for all, and the representation of the scattered intensity *I* is given by spots forming the Laue or Debye-Scherrer patterns. But in a liquid, the atoms are in continous motion, and the diffraction experiment gives only the mean value of successive configurations during the experiment. Given the absence of translational symmetry in liquids, this mean value provides no information on long-range order. By contrast, it is a good measure of short-range order around each atom chosen as origin. Thus, in a liquid, the scattered intensity must be expressed

The first mean value, for *l* = *j*, is worth *N* because it represents the sum of *N* terms, each of them being equal to unity. To evaluate the second mean value, one should be able to calculate the sum of exponentials by considering all pairs of atoms (*j*, *l*) in all configurations counted during the experiment, then carry out the average of all configurations. However, this calculation can be achieved only by numerical simulation of a system made of a few particles. In a real system, the method adopted is to determine the mean contribution brought in by each pair of atoms (*j*, *l*), using the probability of finding the atoms *j* and *l* in the positions **r**� and **r**, respectively. To this end, we rewrite the double sum using the Dirac delta function in order to calculate the statistical average in terms of the *density of probability PN*(**r***N*, **p***N*) of the *canonical ensemble*1. Therefore, the statistical average can be written by using the *distribution*

<sup>1</sup> It seems useful to remember that the *probability density* function in the canonical ensemble is:

Λ = 

and *ZN*(*V*, *<sup>T</sup>*) =

The reader is advised to consult statistical-physics textbooks for further details.

*<sup>N</sup>*!*h*3*<sup>N</sup> QN*(*V*, *<sup>T</sup>*) exp

*QN*(*V*, *<sup>T</sup>*) = *ZN*(*V*, *<sup>T</sup>*)

with the *thermal wavelength* Λ, which is a measure of the thermodynamic uncertainty in the localization of a particle of mass *m*, and the *configuration integral ZN*(*V*, *T*), which is expressed in terms of the total

<sup>2</sup>*<sup>m</sup>* <sup>+</sup> *<sup>U</sup>*(**r***N*) is the *Hamiltonian* of the system, *<sup>β</sup>* <sup>=</sup> <sup>1</sup>

*h*2 <sup>2</sup>*πmkBT* ,

> *N* exp

Besides, the partition function *QN*(*V*, *T*) allows us to determine the free energy *F* according to the

*F* = *E* − *TS* = −*kBT* ln *QN*(*V*, *T*).

*<sup>N</sup>*!Λ3*<sup>N</sup>* ,

<sup>−</sup>*βU*(**r***N*)

 *d***r***N*.

*PN*(**r***N*, **<sup>p</sup>***N*) = <sup>1</sup>

+ *I*<sup>0</sup>

 *N* ∑ *j*=1

*N* ∑ *l*�=*j*

exp *i***q r***<sup>j</sup>* − **r***<sup>l</sup>*

<sup>−</sup>*βHN*(**r***N*, **<sup>p</sup>***N*)

 , . (4)

*kBT* and *QN*(*V*, *<sup>T</sup>*) the

*N* ∑ *l*=1

exp *i***q r***<sup>j</sup>* − **r***<sup>l</sup>*

.

scattered by the atoms in the direction *θ* (or **q**, according to Bragg's law) is given by:

*<sup>N</sup>* <sup>=</sup> *<sup>A</sup>*0*<sup>A</sup>* 0 *N* ∑ *j*=1

*I*(*q*) = *AN A*

as a function of *q* by the statistical average:

 *N* ∑ *l*=*j*=1

exp *i***q r***<sup>j</sup>* − **r***<sup>l</sup>*

*I*(*q*) = *I*<sup>0</sup>

where *HN*(**r***N*, **<sup>p</sup>***N*) = <sup>∑</sup> *<sup>p</sup>*<sup>2</sup>

*partition function* defined as:

relation:

*potential energy U*(**r***N*). They read:
