Optimization Methodology of Empirical Electronic Theory by Employing Statistical Model

Donghu Yang, Qunbo Fan, Fuchi Wang, Lu Wang and Tiejian Su

(School of Material Science and Engineering, Beijing Institute of Technology, Beijing 100081, China)

Abstract: An optimization methodology of empirical electronic theory (EET) for solids and molecules has been developed by employing a statistical model in this study. The current paper calculates the hybridization states of different atoms in some crystal structures and succeeds in predicting valence states of atoms. The prediction of γ-Fe hybridization states based on statistics is found to be in reasonable agreement with early measurements. Through calculating Cr alloy austenite and Cr alloy martensite, the statistical results proved stable for each atom, and it is found that the valence electron structure of an atom depends on its element type and location in the crystal cell; finally, wear resistant steel with 1 wt% C is designed by using a statistical model which is consistent with traditional empirical design.

Key words: empirical electronic theory (EET); statistical model; valence electron structure (VES); material design

Empirical electron theory (EET) for solids and molecules[1-5]was initially put forward by Mr. Yu Ruihuang (1906-1997), a famous physicist and academician working at Chinese Academy of Sciences, through years of experiments and in the way of induction approach. This theory starts from two fundamental atom-states, describes the hybridization sites of different atoms in a crystal, and reveals the covalence electron distribution, thus developing a method to predict the valence electron structure and macrophysical or mechanical properties of crystals and molecules. Since its establishment, numerous academic institutes and universities paid great attention to it and managed to solve a lot of problems that were extremely difficult if solved in other ways.

Based on EET, Wang Huanrong et al.[6]calculated the valence electron structure (VES) of TiC. The calculated results show that with the increase of temperature, the number of common electrons of TiC increases, which indicates that TiC has a good thermal stability. They also found that there was a close relationship between hardness and brittleness of TiC.

Zheng Yong and co-workers[7]calculated the valence electron structure of the rim phase in Ti(C,N)-based cermets by using EET, and the relationship between the VES and plasticity was determined. The results indicate that the plasticity of the rim phase in a Ti(C,N)-based cermet can be defined using the sum of thenA values for the covalent bonds, and that chromium dissolution in the rim phase improves the plasticity of the rim phase. Based on the results, they developed a new cermet which has more than twice the transverse rupture strength of a typical cermet.

Derived from the Ab initio calculations by using variational formulations, Cheng Kaijia[8-9]brought forward another completely new theory called improved Thomas-Fermi-Dirac (TFD) and compared it with covalence electrons, lattice electrons, as well as equivalent electrons in EET. The results were found to be surprisingly consistent, thus verifying the fact that EET’s criterion was just the foregone conclusion of the Ab initio calculations.

1 Basic Ideas and Deficiencies of EET

According to EET, each atom in crystals is hybridized from its two basic states, head (h) state and tail (t) state, and thus called binary hybridization. Here, “hybridization” means a mixture of different atom states, instead of different atomic orbits which are popular terms in Energy Band Theory. So, the actual hybridization state of a certain atom can be determined if only the initial h state and t state are given. In this way, it is not necessary to solve the complex secular equation any longer. Calculation results of EET mathematical formulations show that there are totally 18 different hybridization states at the most for a certain atom, including the h state and the t state. For instance, atom C has totally 6 hybridization states, and atom Fe has totally 18 hybridization states. With different h states and t states, the corresponding 18 hybridization states will also change. For example, Atom Fe has different binary hybridizations, like A-type, B-type, C-type, etc.

(1)

where, the superscriptsuandvare the two bonding atoms, respectively;αrefers to the bond sequence, which is usually sequenced according to the bond length, like A, B, C, …, and A is the shortest and usually also the strongest one;nαdenotes the number of the covalent electron pairs in bondα.

However, the precision of the minimum value of |ΔDnA| is not satisfying, and |ΔDnA| is intrinsically a value with poor precisions since the value itself approaches zero. Usually, it leads to large errors. According to EET, however, this minimum value is a key criterion for further calculating the hybridization stateσandnA, the number of covalent electron pairs of the strongest bond A. Tab.1 lists the calculated results of Fe-C-Cr martensite structure of 30Cr steel by Liu Zhilin[10]. For comparison, Tab.1 also lists the calculated results of the same structure by the author. It can be seen from the comparison that the numerical methods and initial data are completely the same, and the corresponding |ΔDnA| values are also approximately equal as one can expect, but the calculation process itself produces different errors, thus inevitably leading to different calculated hybridization states, and differentnAvalues for the same atoms. It might be noted that all the data in this paper are processed in double precision during calculation, which has been far beyond the precision employed by Liu’s.

Tab.1 Comparison of calculation results reported by Liu et al.[10] and the authors of this paper

Fig.1 and Fig.2 show the calculation comparisons of hybridization states andnAvalues of 0.3wt% C martensite with different alloys, respectively. The letter A of the expression MA in Fig.1 represents element M of A type while the subscript A ofnAin Fig.2 means the strongest bond A in a crystal cell. From the two comparisons, it can be seen that some final results, such as hybridization states ornAvalues of MnA, NiA by different people are not identical or even approximate at all due to random errors during the process of calculation.

Fig.1 Comparison of different calculations for hybridization state in 0.3wt% martensite

Fig.2 Comparison of different calculations for nA in 0.3wt% martensite

Results shown in Fig.1 and Fig.2 indicate that during the process of calculation, errors are easily generated and transferred to thenAvalues finally. In a lot of efforts of material design, however,nA, the number of covalent electron pairs of the strongest bond A, is always used as an important parameter to evaluate the effects of added elements on some important macro properties, such as mechanical strength, hardness, wear resistance, or other macro properties. Hence, errors of |ΔDnA| will unavoidably mislead the final material design. The errors are just the deficiencies of EET.

2 Calculation by Employing Statistical Model

2.1 Basic assumption

2.2 Statistical model

Fig.3 Diagrammatic sketch of the statistical model

After calculation of A1 is completed, it is assumed that hybridization states of atom A1, A3, A4,… are already given, and hybridization state of atom A2 is calculated again. In the same way, the frequency of hybridization states of each atom can be determined.

3 Results and Discussion

3.1 Calculation results of γ-Fe crystal cell and comparison with experiments

By measuring the magnetic moment, Yu[1]found that the Fe atom in a γ-Fe crystal cell will be in the hybridization state of 11. To verify the accuracy of the statistical model, the current paper calculates the valence electron structure using the statistical model mentioned in section 2.2. Fig.4 represents a γ-Fe crystal cell with a reference atom O, which forms bonds in the cell together with atom A, atom B and atom C. The frequency numbers with respect to 18 hybridization states for atom A, atom B, atom C, and atom O are illustrated in Fig.5, indicating that the three face centered Fe atoms (A, B and C) appear similar regularities, gathering in the range of 10-18, while atom O shows a gradually rising tendency, that is higher hybridization states means higher frequencies. It can be seen from Fig.5 that in a crystal cell, even if the same kind of atom shows different statistical distribution due to different positions. Fig.5 also indicates that the arithmetic-averaged hybridization state of atom A, atom B, atom C and atom O are 10.8, 10.09, 10.66, and 11.82, respectively. Thus, the total averaged hybridization state of the four atoms is 11, which is in good agreement with Yu’s experimental results mentioned previously.

Fig.4 Crystal cell structure of γ-Fe

Fig.5 Frequency numbers with respect to hybridization states for four Fe atoms in a γ-Fe crystal cell

3.2 Stability of the statistical model

The structure of Fe-C-Cr austenite crystal cell and the structure of Fe-C-Cr martensite crystal cell are calculated in order to investigate the stability of the statistical model, so as to determine whether a pre-restricted value of a certain atom’s hybridization states would influence the hybridization states of other atoms.

Fig.6 shows the structure of the Fe-C-Cr austenite crystal cell on the basis of so-called “averaged crystal cell” model[10]. Fef and Fec in Fig.6 represent atom Fe at the face-centered position and the one at the cornered position, respectively. The positions of atom C and atom Cr are also shown in Fig.6. It can be seen from this figure that atom C is located in the body-centered position, while the other two atoms of Cr are located in the upper and bottom face-centered positions.

Fig.6 Structure of Fe-C-Cr austenite crystal cell

According to the “averaged crystal cell” model, because of the introduction of atom Cr, the original lattice parameter of the crystal cell will change, and there exists some relationship between the lattice parameter and the weight percentage of C. Therefore, when calculating hybridization states by employing statistical model, the content of carbon shall also be taken into account.

Fig.7 and Fig.8 indicate the frequency numbers of hybridization states in the 0.4wt% Fe-C-Cr austenite crystal structure by using a statistical model. In Fig.7, the hybridization state of atom C is not restricted and each state, from 1 to 6, corresponds to a frequency number. From Fig.7, it appears that the alloy atom Cr has a similar frequency in each hybridization state, and Fef shows approximately the same tendency as Fec, mainly distributing in the state of 13, 14, 15, 16, 17 and 18. The statistical model results show that the averaged hybridization states of Fef, Fec, Cr, and C are 15.96, 16.31, 7.07, and 3.71, respectively. In Fig.8, the hybridization state of atom C is restricted to 6, thus decreasing the total reasonable hybridization states obviously. As is shown in Fig.8, the statistical distributions of the atoms are similar to those in Fig.7 except for atom C. The averaged hybridization states of Fef, Fec, and Cr are 15.8, 15.95, and 8.06, respectively.

Fig.7 Frequency numbers with respect to different hybridization states for atoms in 0.4wt% Fe-C-Cr austenite crystal structure (hybridization state of atom C is not restricted)

Fig.8 Frequency numbers with respect to different hybridization states for atoms in 0.4wt% Fe-C-Cr austenite crystal structure (hybridization state of atom C is restricted to 6)

Fig.9 shows the structure of Fe-C-Cr martensite crystal cell on the basis of the “averaged cell” model. Fe1 and Fe2 represent Fe atoms at different positions. The positions of atom C and atom Cr are also shown in Fig.9. In the crystal structure, the atom C is located in the octahedral interstice of the original martensite sublattice, and two Cr atoms fill the bottom and top positions of the octahedron. The original lattice parameter of the crystal cell will change due to the introduction of atom Cr like in the austenite crystal cell, and there also exists some relationship between the lattice parameter and the weight percentage of C.

Fig.9 Structure of Fe-C-Cr martensite crystal cell

Fig.10 Frequency numbers with respect to different hybridization states for atoms in 0.4wt% Fe-C-Cr martensite crystal structure (hybridization state of atom C is not restricted)

Fig.10 and Fig.11 indicate the frequency numbers of hybridization states in the 0.4wt% Fe-C-Cr martensite crystal structure using statistical model. In Fig.10, the hybridization state of atom C is not restricted and each state, from 1 to 6, corresponds to a frequency number. It can be seen from Fig.10 that, the frequency numbers for alloy element Cr are approximately equal, while Fe1 and Fe2 show different distribution tendencies. Frequencies for Fe2 seem to be similar for each hybridization state, while Fe1 are mainly gathered within the range of 6-18. The averaged hybridization states of Fe1, Fe2, Cr and C are 12.33, 10.20, 8.98 and 3.62, respectively. The different distribution for Fe1 and Fe2 indicate that atoms of the same element type probably have different valence electron numbers due to their different positions. This result can also be seen in γ-Fe. In Fig.11, the hybridization state of atom C is restricted to 6, thus the total reasonable hybridization states are decreased obviously. As is shown in Fig.11, the statistical distributions of the atoms are similar to those in Fig.10 except for atom C. The averaged hybridization states of Fe1, Fe2, and Cr are 11.45, 10.34, and 8.80, respectively.

Fig.11 Frequency numbers with respect to different hybridization states for atoms in 0.4wt% Fe-C-Cr martensite crystal structure (hybridization state of atom C is restricted to 6)

Comparing Fig.7 and Fig.8,and Fig.10 and Fig.11, it can be seen that when calculating hybridization states of atoms by employing the statistical model, the hybridization state of a certain atom would not be affected greatly due to the pre-restricted states of other atoms, revealing the excellent stability or robustness of the statistical model.

3.3 Design of wear resistant steel by employing statistical model

When designing a kind of new wear resistant steel, it shall be satisfied that the substrate has both high hardness and excellent toughness with the addition of primary alloy elements. In this current paper, the design approach of a statistical model based on EET for developing a new wear resistant steel with 1wt% C is shown. Fig.12 shows the number of covalent electron pairs of the strongest bond A,nA, in martensite and austenite, andnAis an arithmetic mean of values that are calculated by using statistical model. It can be seen from Fig.12 that, elements V, Nb, Ti, Si, Ni, Cr in martensite with 1.0wt% C have relatively largernAvalues, thus helping to raise the substrate’s hardness and wear resistance; while in austenite with 1.0wt%, V, Nb, Ti, Si, Ni also have relatively larger value, which would increase the residual quantity of austenite, and further decrease the hardness of martensite. Therefore, element Cr is selected to be the best element to strengthen the substrate. As a matter of fact that, in practical industrial applications, 10wt%-30wt% Cr is often added into high-carbon steel or cast iron to form various wear resistant materials.

Fig.12 Averaged nA values for different alloy elements in martensite and austenite

4 Conclusions

In this study, a statistical model as a new approach is developed to improve the traditional empirical electronic theory. The principal findings are summarized as follows.

① A newly statistical model is developed for the first time to improve EET, that is, within the restriction of a certain bond difference, the frequency numbers of hybridization states, their distribution and corresponding averaged values are used as the new material design criterion.

② Calculation results reveal the excellent stability or robustness of the statistical model, and the hybridization state of an atom would not be affected greatly by pre-restricted states of other atoms. Calculation results of γ-Fe are in good agreement with Yu’s early experiments.

③ Wear-resistant steel is designed by employing the statistical model. It is found that Cr shall be the primary added element. Calculated results are consistent with practical industrial applications.