Because no direct interatomic interaction was considered in the above modeling work, the built models should subject to further atomic-coordinates optimization [

26] by molecular dynamics or first-principles simulations before applying them to the study of various physical properties of the alloys. Applications of the built models to some typical MPE alloys, recently reported by experimental researches, are given in the follows. These alloys are the quaternary FCC FeCoCrNi [

9,

27], quinary FCC CoCrFeMnNi [

28], quinary BCC AlCoCrFeNi [

29], and senary BCC AlCoCrCuFeNi [

30]. Firstly, the original bulk models of these MPE phases are created from the corresponding MaxEnt models by appropriate elemental substitution. Then, the atomic structures in these models are optimized by Molecular Mechanics simulation by the similar procedure as described in [

26]. Chen's lattice inversion pair-function potentials [

31] are used to calculate the interatomic interaction in the simulation. The optimized 8 × 8 × 8 bulk structures of these phases are illustrated in

Figure 5. The optimized lattice parameters together with the relevant experimental data are listed in

Table 3. It was demonstrated that the atomic structure of equimolar multi-component alloys have the 3D paracrystalline lattice configuration [

26]. This paracrystalline feature is also reflected in the optimized structure models in

Figure 5. The lattice distortion in paracrystals is measured by the lattice distortion parameter

g [

32], which is defined by:

where,

d is a lattice geometrical parameter of the structure, it can be the inter-planar spacing or local lattice parameter;

$\overline{d}$ and

$\overline{{d}^{2}}$ are the mean and square mean values of the lattice geometrical parameter, respectively. The local lattice constant

a_{l} is used in

g calculation in this study. The local lattice constant varies due to the lattice distortion in MPE alloys. The calculation method of local lattice constant is illustrated in

Figure 6. Parameter

g measures the statistical correlation of local lattice constant. The paracrystalline lattice distortion parameters

g of these alloys are given in the last column of

Table 3. The lattice constant given in this table is the averaged local lattice parameter

a_{l} for all lattice sites in the model. A relatively larger error in the theoretical lattice constant compared to the experimental one of these optimized MPE phases is seen in

Table 3. The reason for the problem could be mainly attributed to the empirical interatomic pair-potentials used in the model structure optimization. This problem will be further explored in the discussion section later. The

g parameter provides a quantitative evaluation of the lattice distortion in MPE phases, which can be directly measured through x-ray diffraction experiment [

32]. There are generally much more serious lattice distortions in BCC MPE alloys than in FCC ones as reflected in

Figure 5 and the

g parameters in

Table 3. The g parameter of BCC MPE alloys is about two to three times higher than that of the FCC ones. The reason for the phenomenon is largely because the atomic radii of the componential elements vary widely in the BCC phases. The degree of lattice distortion in MPE alloy is sensitively depended on its composition as reflected from the calculated

g-parameter. There is a close correlation between lattice distortion and the metallic radii [

33] of the componential elements. For examples, the FCC phases of FeCoCrNi and CoCrFeMnNi own very small

g parameters, which can be explained from the little differences in the metallic radii of their componential elements: Fe 1.26 Å, Co 1.25 Å, Cr 1.28 Å, Ni 1.24 Å, and Mn 1.27 Å. The metallic radius can be served as an effective criterion for the composition design of MPE alloys.

**Figure 5.**
Relaxed atomic structure models of four typical MPE alloys. The Al, Fe, Co, Cr, Ni, Mn, and Cu atoms are in red, magenta, green, blue, cyan, yellow, and grey colors respectively.

**Figure 6.**
Illustration of the local lattice constant calculation for a deformed cubic cell. The local lattice constant a_{l} near the yellow atom is calculated by the averaged length of its three neighboring edges a_{l}=(a1+a2+a3)/3 along the three forward axis-directions.