In order to understand the physics behind the surface properties and nanoscale phenomena, we are motivated first to investigate the inner bond strengths as well as the effect of number of neighboring atoms and their relative distance in addition to space positions (crystallography). Therefore, in order to study the effect of the nature of metallic bond on their physicochemical properties, we first tried to investigate and introduce a mathematical model for transforming the bulk molar cohesion energy into microscopic bond strengths between atoms. Then an algorithm for estimating the nature of bond type including the materials properties and lattice scale “cutoff” has been proposed. This leads to a new fundamental energy scale free from the crystallography and number of atoms. The results of our model in case of fundamental energy scale of metals not only perfectly describe the inter relation between binding and melting phenomena but also adequately reproduce the bond strength for different bond types with respect to other estimations reported in literatures. The generalized algorithm and calculation methodology introduced here by us are suggested to be used for developing energy scale of bulk crystal materials to explain or predict any particular materials properties related to bond strengths of metallic elements.
Binding strength inside the material which is mainly expressed by cohesion energy is classified as a bulk thermodynamics property. Such energy scale is believed to rule most of the physicochemical and mechanical properties of material [
In researches including bulk metal properties not only the macroscopic properties such as cohesion energy but also the number of neighboring atoms and crystallography of material along with the effective length of interaction energies between atoms play dominant roles in fundamental theories [
Moreover, in recent years bond energy calculation has been applied in estimation of surface properties through (atomic scale) broken bond model; thus the bond strength should be known between different neighboring atoms. In latest works of Fu et al. in [
Therefore, we are motivated to closely study the structural effect of different crystals on their cohesion energy and evaluate a structural free energy scale which defines the materials properties. In this report, we study the effect of crystallography and number of atomic neighbors on the cohesive energy in order to define a fundamental energy scale of material. This information not only enables us to evaluate the real interatomic bond strength inside the material but also can be used as a grand energy scale to investigate other fundamental properties of elements. This knowledge gives us a tool to evaluate the strength energy from a particular atom inside the material. Also we develop free software using which anyone can compute the bond strength scale at different distances as function of number of atoms and their relative distances for any crystal materials by knowing some simple input data without any adjusting parameters.
Most problems in solid state science require detailed study of energies along with the atomistic structural information inside the material. A detail study of crystallography effect and its related features on inner cohesive strength of pure metals needs a fundamental model which not only evolves macroscopic cohesion properties but also includes the number of atoms (those which are affected by cohesive force of an arbitrary inner atom) and their relative distances inside the materials. In order to fulfill such conditions we need to develop a model for distribution of cohesion energy between the (effected) atoms as a function of their number and relative distances from an arbitrary inner atom.
In most of theoretical investigation in materials science dealing with thermodynamics description of material, the dimension of cohesion energy is described as a molar quantity. Therefore, theoretically by applying that amount of energy to the material we can dissociate one mole of material into its atoms [
Regarding Figure
Schematic representation for (a) number of bonds related to number of atoms and (b) position of equal distance atomic groups and their distances from an arbitrary (inner) atoms in bcc lattice.
Therefore, during the transformation of molar cohesive energy of material into the atomic bond strength scale, we have to apply a 1/2 coefficient in order to avoid counting of multiple bonds between two particular atoms. Considering a simple case of Figure
However, in reality for an arbitrary inner atom we have more than one type of cohesive strength as Figure
We also suppose that the distribution of cohesive energy is homogenous between all atoms in each group ccn; therefore, regarding the coefficient of atomic bond scale in (
Simple schematic to illustrate how the number of atoms and their relative distances evolve in philosophy behind (
However, as we fundamentally subtract out the effect of number of neighbors by introducing the
Further on, by introducing (
In (
The
If we measure the distances of each atom from an arbitrary inner atom (equicentral spheres), the first coordination distance
As it can be seen from (
Therefore, if we would be able to construct the mathematical algorithms of all geometrical positions in fcc, bcc, and hcp crystallographic structures, then we would be able to extract out the
Equation (
Before any analytical investigation, we have to study the crystallography of different structures and evaluate the required mathematical series which represent the equal atomic distances
There were already some attempts in literature to evaluate the effect of number of neighboring atoms and in some cases the effect of their relative atomic distances mostly for estimation of surface properties of pure metals in fcc, bcc, and hcp structures in [
In their reports Zhang et al
Unfortunately neither the application of above literature resources nor their mathematical formulations can provide a generalized mathematical series for estimating the number of neighboring atoms and their relative distances. However, Sloane and Teo in [
However, not only our different viewpoint of such series but also our freesoftware program (based on information appearing in Appendix
Schematic view of first three atomic groups and their relative distances from an arbitrary inner atom in (a) fcc, (b) bcc, and (c) hcp structures.
Regarding (
First 15 numbers of neighboring atoms and their relative distances for each atomic group in fcc, bcc, and hcp crystallography while the relations


Distance in fcc as 
Distance in fcc as 

Distance in bcc as 
Distance in bcc as 

Distance in hcp as 
Distance in hcp as 

1  12  1  4/2  8  3/3  4/3  12  3/3  3/3 
2  6  2  4/4  6  3/4  4/4  6  3/6  3/6 
3  24  3  4/6  12  3/8  4/8  2  3/8  3/8 
4  12  4  4/8  24  3/11  4/11  18  3/9  3/9 
5  24  5  4/10  8  3/12  4/12  12  3/11  3/11 
6  8  6  4/12  6  3/16  4/16  6  3/12  3/12 
7  48  7  4/14  24  3/19  4/19  12  3/15  3/15 
8  6  8  4/16  24  3/20  4/20  12  3/17  3/17 
9  36  9  4/18  24  3/24  4/24  6  3/18  3/18 
10  24  10  4/20  32  3/27  4/27  6  3/19  3/19 
11  24  11  4/22  12  3/32  4/32  12  3/20  3/20 
12  24  12  4/24  48  3/35  4/35  24  3/21  3/21 
13  72  13  4/26  30  3/36  4/36  6  3/22  3/22 
14  48  15  4/30  24  3/40  4/40  12  3/25  3/25 
15  12  16  4/32  24  3/43  4/43  12  3/27  3/27 
To facilitate calculation of
Algorithm of calculating
After clicking the calculate button, three kinds of information are shown as output of the software. First, the value of
Regarding the aforementioned hypothesis in (
One of the advantages of modeling the nature of bond strength by introducing the dimensionless function
For sake of simplicity we will classify all metallic elements into three main lattice groups (fcc, bcc, and hcp) and try to evaluate the quantity of
Obviously, by increasing the value of
Calculated results of
As it can be seen in Figure
In classical literatures, value of
Figure
(a) Calculated results of
Proposed algorithm: regarding the behaviors of simple metals, we can overview the following simple three boundary conditions (two plus one) using which the acceptable values of
As
In addition to the above two conditions, we consider a limitation of
By applying the condition (a), while searching with condition (c) (
in fcc by
in bcc by
in hcp by
Therefore, considering the condition (b), using mathematical series in Appendix
By factorizing out the crystallographic aspect of total cohesive energy regarding the classical or corrected cohesion scales [
Obviously, regarding the temperature dependency of relation (
Calculated structural free cohesive energy scale
fcc*  at 


bcc*  at 


hcp*  at 



Ca  bcc  49.610  55.800  Li  bcc  25.087  22.700  Be  bcc  69.309  78.070 
Sr  bcc  51.712  52.540  Na  bcc  20.821  18.560  Mg  hcp  41.618  36.590 
Rh  fcc  100.593  86.400  K  bcc  18.977  16.830  Sc  bcc  83.824  90.680 
Ir  fcc  122.614  105.09  Rb  bcc  17.721  15.640  Y  bcc  82.411  90.030 
Ni  fcc  77.813  66.86  Cs  bcc  17.112  15.090  Ti  bcc  88.974  97.030 
Pd  fcc  81.294  70.610  Ba  bcc  58.575  50.040  Zr  bcc  97.193  106.340 
Pt  fcc  91.404  79.130  V  bcc  126.739  109.600  Hf  bcc  116.398  125.110 
Cu  fcc  59.976  52.540  Nb  bcc  159.683  137.620  Re  hcp  159.975  136.880 
Ag  fcc  54.518  47.740  Ta  bcc  188.851  163.050  Ru  hcp  116.029  100.010 
Au  fcc  59.255  51.770  Cr  bcc  123.929  106.590  Os  hcp  150.348  130.820 
Al  fcc  41.231  36.110  Mo  bcc  169.613  144.930  Co  fcc  81.932  68.410 
Pb  fcc  26.422  23.230  W  bcc  216.034  184.160  Tl  bcc  25.929  28.870 
In  fcc  18.741  16.610  Fe  bcc  106.917  90.530  Zn  hcp  31.225  27.460 
Cd  hcp  26.757  23.540 
fcc
The bulk melting phenomenon in crystalline materials takes place when the bindings of crystallographic cells lose its structure; thus the best verification properties for cohesive energy scale are believed to be the melting point [
However, regarding the logic mentioned in (
Verification of structural free cohesive scale with their melting points for fcc, bcc, and hcp crystals.
As our calculation results show, the correlation in three different crystallographic forms is perfectly represented by
Applying the aforementioned computational algorithm in (
The latest reported about the applications of bond strengths were for estimating the surface properties of pure metals in works of Fu et al. in [
Comparison between our
fcc  ^{ 1}b. t. 

[ 
^{
2}Equations ( 

fcc 
^{ 1}b. t. 

[ 
^{
2}Equations ( 


Au  A or 1  12  31.34491  29.627500  31.51  Pt  1  12  40.31798  45.702000  48.61 
B or 2  6  0.22635  1.851719  0.1230  2  6  0.34635  2.856375  0.1898  
C or 3  24  0.00548  0.365772  0.0048  3  24  0.00959  0.564222  0.0074  
4  12  0.115732  4  12  0.178523  


Ag  1  12  25.88074  27.259000  28.99  Pd  1  12  49.24018  40.647000  43.23 
2  6  0.18539  1.703688  0.1132  2  6  0.43890  2.540438  0.1688  
3  24  0.00446  0.336531  0.0044  3  24  0.01250  0.501815  0.0065  
4  12  0.106480  4  12  0.158777  


Al  1  12  25.50269  20.615500  21.92  Ni  1  12  48.16400  38.906500  41.38 
2  6  0.38510  1.288469  0.0856  2  6  0.64762  2.431656  0.1616  
3  24  0.01643  0.254512  0.0033  3  24  0.02529  0.480327  0.0063  
4  12  0.080529  4  12  0.151979  


Cu  1  12  37.39334  29.988000  31.89  
2  6  0.45492  1.874250  0.1245  
3  24  0.01644  0.370222  0.00486  
4  12  0.117141 
Comparison between our
bcc  ^{ 1}b. t. 

[ 
^{
2}Equations ( 

bcc  ^{ 1}b. t. 

[ 
^{
2}Equations ( 


Ba  A or 1  8  26.94813  29.287500  34.27  Cr  1  8  76.53049  61.964500  72.51 
B or 2  6  2.63391  8.025239  2.573  2  6  15.03984  16.979255  5.444  
C or 3  12  0.354669  0.0050  3  12  0.10841  0.750384  0.0106  
4  24  0.084618  4  24  0.179028  


V  1  8  53.13139  63.369500  74.16  Mo  1  8  91.04682  84.806500  99.25 
2  6  9.70330  17.364247  5.568  2  6  15.64323  23.238325  7.452  
3  12  0.05577  0.767399  0.0010  3  12  0.07440  1.026999  0.0145  
4  24  0.183088  4  24  0.245023  


Nb  1  8  95.48674  79.841500  93.44  W  1  8  107.09495  108.017000  126.41 
2  6  15.15142  21.877837  7.015  2  6  18.22915  29.598370  9.491  
3  12  0.966873  0.0137  3  12  0.08422  1.308076  0.0185  
4  24  0.230679  4  24  0.312083  


Ta  1  8  109.20519  94.425500  110.5  Fe  1  8  59.67579  53.458500  62.56 
2  6  17.31214  25.874084  8.296  2  6  11.84176  14.648476  4.697  
3  12  1.143484  0.0162  3  12  0.08796  0.647377  0.0091  
4  24  0.272815  4  24  0.154453 
Fu et al. in [
Comparison between our
hcp  ^{ 1}b. t. 

^{
2,3}Reference [ 
^{
4}Equations ( 

hcp  ^{ 1}b. t. 

^{
2,3}Reference [ 
^{
4}Equations ( 


Be  A, B, 1  12  26.63648  34.654500  36.02  Hf  1  12  63.18858  58.199000  60.49 
C or 2  6  0.81081  1.531527  0.0703  2  6  0.2562  2.572057  0.1181  
D or 3  2  0.2014  0.419663  0.0052  3  2  0.02446  0.704784  0.0088  
E, F, 4  18  0.06083  0.247010  0.0018  4  18  0.00410  0.414830  0.0030  
G or 5  12  0.01951  0.100124  0.0003  5  12  0.00059  0.168149  0.0005  
6  6  0.067685  6  6  0.113670  


Mg  1  12  10.26661  20.809000  21.63  Re  1  12  77.23997  79.987500  83.14 
2  6  0.0961  0.919637  0.0422  2  6  0.66184  3.534981  0.1623  
3  2  0.0092  0.251995  0.0031  3  2  0.06507  0.968641  0.0121  
4  18  0.00285  0.148322  0.0010  4  18  0.01845  0.570133  0.0042  
5  12  0.00041  0.060121  0.0001  5  12  0.00272  0.231100  0.0006  
6  6  0.040643  6  6  0.156226  


Sc  1  12  30.80765  41.912000  43.56  Co  1  12  61.57229  40.966000  42.58 
2  6  0.24091  1.852266  0.0850  2  6  0.79776  1.810459  0.0831  
3  2  0.02786  0.507550  0.0063  3  2  0.09046  0.496094  0.0062  
4  18  0.00634  0.298740  0.0022  4  18  0.03042  0.291997  0.0021  
5  12  0.00107  0.121092  0.0003  5  12  0.00504  0.118359  0.0003  
6  6  0.081859  6  6  0.080012  


Y  1  12  33.92099  41.205500  42.83  Zn  1  12  13.52421  15.612500  16.22 
2  6  0.1627  1.821043  0.0836  2  6  0.06636  0.689982  0.0316  
3  2  0.01822  0.498995  0.0062  3  2  0.00438  0.189066  0.0023  
4  18  0.00298  0.293704  0.0021  4  18  0.00241  0.111282  0.0008  
5  12  0.00049  0.119051  0.0003  5  12  0.00008  0.045108  0.0001  
6  6  0.080479  6  6  0.030493  


Ti  1  12  41.89133  44.487000  46.24  Cd  1  12  15.34774  13.378500  13.9 
2  6  0.50957  1.966066  0.0903  2  6  0.03772  0.591252  0.0271  
3  2  0.07548  0.538733  0.0067  3  2  0.00206  0.162012  0.0020  
4  18  0.01884  0.317094  0.0023  4  18  0.00097  0.095359  0.0007  
5  12  0.00393  0.128532  0.0003  5  12  0.00002  0.038653  0.0001  
6  6  0.086889  6  6  0.026130  


Zr  1  12  44.28581  48.596500  50.51  
2  6  0.37696  2.147682  0.0986  
3  2  0.04472  0.588499  0.0074  
4  18  0.01056  0.346385  0.0025  
5  12  0.00183  0.140405  0.0004  
6  6  0.094915 
Value of
Metal 


Metal 


metal 



Ca  275.99  317.35  Li  112.30  124.08  Be  386.14  432.76 
Sr  259.90  330.80  Na  91.83  102.98  Mg  228.47  259.86 
Rh  552.72  643.48  K  83.25  93.86  Sc  448.51  523.39 
Ir  672.28  784.35  Rb  77.39  87.65  Y  445.30  514.57 
Ni  427.72  497.76  Cs  74.64  84.64  Ti  479.95  555.55 
Pd  451.73  520.03  Ba  247.52  289.71  Zr  525.99  606.87 
Pt  506.19  584.70  V  542.08  626.85  Hf  618.81  726.78 
Cu  336.14  383.66  Nb  680.69  789.79  Re  854.70  998.87 
Ag  305.45  348.75  Ta  806.44  934.05  Ru  624.50  724.48 
Au  331.19  379.05  Cr  527.23  612.95  Os  816.83  938.76 
Al  231.05  263.75  Mo  716.83  838.90  Co  437.62  511.58 
Pb  148.66  169.02  W  910.89  1068.5  Tl  142.82  161.90 
In  106.26  119.89  Fe  447.77  528.81  Zn  171.46  194.97 
Cd  147.03  167.07 
Values of inner cohesion energy of other elements can be collected from [
In case of hcp metallic crystals we also evaluated a set of calculation with
A detailed investigation on the crystallography of fcc, bcc, and hcp lattices has been performed. This application of mathematical theta series gives us the information about the number of atoms and their relative equal distances from a central inner atom for up to 50th layers. Therefore, as long as the cutoff length (longest atomic distances from which an atom could sense the attraction of other one) is known, the maximum number of neighboring atoms and their distances from the central atom can be calculated.
Using these data, we introduced a model for transformations of bulk molar cohesion energy into inter atomic bond strengths inside the metallic elements. Obviously, in addition to number of neighboring atoms and their distances, the nature of bond type of particular material (as power in an inverse power potential) in given crystal structure should be known.
Therefore, an algorithm for estimating the value of neighboring atoms and distances from the central atom as function of crystal type and type of bond (nature of bond strength) has been proposed and for the sake of simplicity three collective results for all fcc, bcc, and hcp crystals have been evaluated and reported here.
The effect of number of neighboring atoms and their distance along with the physics of bond type inside the metallic elements has been investigated. This study leads to defining an energy scale free from crystallographic information. The aforementioned energy scale presented here is suggested to be used for scaling any fundamental properties of metallic elements which are interrelated to their inner cohesive feature.
We developed a freesoftware (IBSEVer1) which enables us to evaluate the number of neighboring atoms and their relative distance from a central atom in fcc, bcc, or hcp up to 50th layers. The freesoftware is able to calculate the structural free energy scale and interatomic bond strengths at different neighboring atomic distances depending on the total number of atoms, cutoff length, and the nature of bond type. The results of our calculation in case of structural free energy scale for the first time perfectly reproduced the expected classical hypothesis of linear dependency of bond energy and melting points. Also the results of bond strengths for different type of metallic elements adequately produced the tendencies reported in literature.
Obviously using the enclosed freesoftware more reliable values of structural free energy scale can be evaluated separately for each metallic element via considering better set of cutoff length and power of potential energy related to nature of bond types in particular crystal structure.
In next paper the mathematical series and other freesoftware for estimating the surface bond strength and related surface properties of all metallic elements at different crystal structures and crystal planes will be submitted.
Assuming the origin of coordinates at the center of an arbitrary atom in a close pack structure, theta series are able to calculate number of neighbors and their distance from this central atom. These series are used for 2D (circular) or 3D (spherical) close pack structures. Let
(a) 2D square lattice. (b) 2D hexagonal lattice.
Theta series for a close packing structure (
This equation is a power series in the variable
For example in 2D square lattice shown in Figure
This power series predicts that there are 4, 4, 4, 8, 4,… neighbors at distance of
Similarly, for hexagonal lattice shown in Figure
In the same way, all 2D and 3D close pack structures can be expressed using the following three special Jacobi theta series:
The Jacobi theta series satisfy large number of identities using which they can be simplified.
For some clusters the following series is also helpful:
For fcc structure, theta series are given below which can predict up to 10 numbers of neighbors and their distance from central atom:
Similarly for hcp
Sloane and Teo in [
For more details see Table
The authors declare that there is no conflict of interests regarding the publication of this paper.