# Simple Metal and Binary Alloy Phases Based on the fcc Structure: Electronic Origin of Distortions, Superlattices and Vacancies

^{*}

## Abstract

**:**

## 1. Introduction

_{Ewald}. Another significant crystal energy term is the band structure energy, E

_{BS}, defined by ion–electron interactions for a given structure.

_{5}Zn

_{8}-cI52 and CuZn

_{3}-hP3 that are related to bcc through superlattices, distortions and vacancies. In our previous papers, we have considered metallic phases based on the bcc and hcp structures [4,5].

## 2. Theoretical Background and Method of Analysis

_{F}), values of reciprocal wave vectors of BZ planes (q

_{hkl}) and volumes of BZ and FS. The BZ planes are selected to match the condition q

_{hkl}≈ 2k

_{F}that have a significant structure factor. In this case, an energy gap is opened on the BZ plane leading to the lowering of the electron band energy. The ratio of ½q

_{hkl}to k

_{F}is usually less than 1 and equals ~0.95; it is called a “truncation” factor. In the FS–BZ presentations by the BRIZ program, the BZ planes cross the FS, whereas in the real system, the Fermi sphere is deformed and accommodated inside BZ. The “truncation” factor has a characteristic value of ~0.95 and corresponds to a decrease in the electron energy on the BZ plane.

_{at}). The valence electron concentration (z) is the average number of valence electrons per atom that gives the value of the Fermi sphere radius k

_{F}= (3π

^{2}z/V

_{at})

^{1/3}. Further structure characterization parameters are the number of BZ planes that are in contact with the FS, the degree of “truncation” factor and the value of BZ filling by electronic states, defined as a ratio of the volumes of FS and BZ.

_{1h}*cos30°, b = a

_{2h}* and c* = c

_{h}*. Structural data for binary phases considered in this paper have been found in the Pauling File [19] and in recent papers cited in the corresponding sections of this paper.

## 3. Results and Discussion

#### 3.1. Simple sp Elements with the fcc-Based Structures

#### 3.1.1. fcc in the Simple sp Elements

_{Ewald}and gaining of E

_{BS}. The common view of the Fermi surface for Cu, Ag and Au is a sphere with necks extending to the Brillouin planes of the (111) type.

#### 3.1.2. Distorted fcc Structures of the IIB Element Hg

**q**

_{003}=

**q**

_{012}resulting in c/a = 1.936 for the rhombohedral cell in the hexagonal setting. This value is much lower than the ideal c/a = 2.45 for fcc. Hg-hR1 is obtained from fcc through contraction along one of the [111] axes as shown on Figure 2b (right panel). Rhombohedral cell of the Hg-hR1 structure has α = 70.52° in comparison to 60° of the fcc. At low temperature and high pressure, the β-Hg phase is stable.

_{F}~ ½q

_{012}gives the value z ~ 2.17, which corresponds to ~8–9 at. % of a tetravalent metal and to ~17 at. % of a trivalent metal. The latter case is realized in the Hg–In system, where the α-Hg(In) solution has been experimentally observed to extend to ~19 at. % In. Stability of the α-Hg(Sn) phase was observed to 30 GPa.

^{3}, respectively).

#### 3.1.3. Rhombohedral Distortions of fcc in Group IA Element Li

#### 3.1.4. Distortion of fcc in Group IIIB Elements

#### 3.1.5. High Pressure fcc Phase in Polyvalent sp Elements

_{220}, which is usual for the Hume–Rothery phases.

#### 3.2. Binary Alloy Phases with the fcc Based Structures

#### 3.2.1. Long-Period Superlattice in the CuAu Alloy

_{hkl})/k

_{F}for ideal FS is usually ~0.95 as was considered by Sato and Toth [38] and is called a “truncation” factor. In our paper, we use the reciprocal value to characterize the FS–BZ interaction in all considered cases.

#### 3.2.2. Binary Alloy Au–Zn Phases Based on the fcc Structure

_{4}Zn-oP96 structure, the superlattice cell is related to the basic fcc cell (with a

_{f}parameter) with a model: a = 6a

_{f}, b = a

_{f}and c = 4a

_{f}[32]. Atomic positions are varied corresponding to the lattice modulation. In Figure 2b, the new BZ boundaries of the (115) and (710) types are shown crossing the FS, whereas for the basic BZ (shown right), there is no contact with FS. The phase stabilization is considered to be due to the formation of the new BZ planes in contact with the FS, which is reducing the crystal energy.

_{3+}Zn-tI64 phase is related to fcc in the following way: $a=\sqrt{2}{a}_{f}$, $c=8{a}_{f}$. Atomic shifts produce extra diffraction reflections [42], resulting in the BZ boundaries enveloping the FS (Figure 2c).

_{3}Zu

_{+}-oC34 phase occurs through a phase transition from fcc in a similar way with b and c ~ $\sqrt{2}{a}_{f}$ and a = 4a

_{f}(standardized data are used). FS–BZ construction is given in Figure 2d showing relation of two Au

_{3}Zn phases with slight difference in composition.

_{5}Zn

_{3}-oI128 phase has also a relation to fcc and occurs at the alloy concentration boundary for the fcc stability. Cell parameters are produced from fcc cell in the following way: $a=\sqrt{2}{a}_{f}$, $b=2\sqrt{2}{a}_{f}$ and c = 8a

_{f}, resulting in 32 fcc cells with 128 atoms (Figure 2e). Thus, in the Au – Zn alloy system in the composition range up to ~40 at. % Zn, there are several complex phases that are superlattices based on fcc. Formation of these superstructures is related to the Hume–Rothery mechanism.

#### 3.2.3. Defect Structures Based on fcc

_{2}Sb-tP6 is related to the tetragonal distortion of the fcc with the double cell along the c-axis and with the absence of Cu atoms in the middle of c. The resulting number of atoms in the cell is N = 6 and the constructed FS–BZ configuration satisfies the Hume–Rothery mechanism (Figure 2f). Similar tI6 structure is found in Cu

_{2}As.

_{2}Sb-tP6 type structure is formed in the Cu–Te alloy at the composition of 41.7 at. % Te [19]. Because Te is an element from the next group (VIB) after the group of Sb (V B), there are more vacancies of Cu atoms for basic Cu positions 2c and 2a with occupation 0.7, which gives the resulting number of atoms N = 4.8 in the unit cell instead of N = 6 for Cu

_{2}Sb. This is the response of the crystal structure to FS–BZ interactions for the phase Cu

_{1.4}Te.

_{2}-tP3 with two bcc cells along the c-axis and with missing Cu atoms in the middle of c. This is a way to adjust structure to the number of valence electrons in the cell.

#### 3.3. Miscibility Gap of Two fcc Phases in the Al–Zn Alloys

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Calculated diffraction patterns for selected phases from Table 1 (left panel) and corresponding Brillouin–Jones (BZ) zones with the Fermi spheres (FS) (right panel). The position of 2k

_{F}and the hkl indices of the planes used for the BZ construction are indicated on the diffraction patterns. Constructions of BZ–FS are given in common projection (first column) and in special projection (second column) (see text).

**Figure 2.**Calculated diffraction patterns for selected phases from Table 2 (left panel) and corresponding Brillouin–Jones zones with the inscribed Fermi spheres (right panel). The position of 2k

_{F}and the hkl indices of the planes used for the BZ construction (first column) are indicated on the diffraction patterns. Constructions of BZ planes corresponding to the strong reflections (second column) show structural relationship to the basic cells (see text). For BZ of oC32 (

**d**, third column) the projection is given along

**b*** to show relation to oP128 (

**e**). (See text for description of (

**a**–

**f**) sub-figures).

**Figure 3.**The phase diagram Al–Zn [19]. The Al(Zn) solid solution region with the fcc structure contains a miscibility gap with two-phase region (colored). The boundary phases with 17 and 59 at. % Zn constructions of FS–BZ are given with a common view and a view along a* (see text).

**Table 1.**Structure parameters of several metal phases with the fcc and fcc-based structures. Fermi sphere radius k

_{F}, ratios of k

_{F}to distances of Brillouin zone planes ½q

_{hkl}and the filling degree of Brillouin zones (BZ) by electron states V

_{FS}/V

_{BZ}are calculated using the program BRIZ [18].

Phase | Cu(Zn) | α-Hg | β-Hg | Li | In | Te |
---|---|---|---|---|---|---|

Pearson symbol | cF4 | hR1 | tI2 | hR3 | tI2 | cF4 |

Structural data ^{a} | ||||||

Space group | Fm$\overline{3}$m | R$\overline{3}$m (h) | I4/mmm | R$\overline{3}$m | I4/mmm | Fm$\overline{3}$m |

T, P conditions | 227K | 77K | 4.2K | Ambient conditions | 255GPa | |

lattice parameters (Å) | a = 3.698 | a = 3.470 | a = 3.995 | a = 3.111 | a = 3.248 | a = 3.757 |

c = 6.719 | c = 2.825 | c = 22.86 | c = 4.946 | |||

c/a = 1.936 | c/a = 0.707 | c/a = 1.523 | ||||

FS–BZ data from the BRIZ program | ||||||

Z (number of valence electrons per atom) | 1.364 | 2 | 2 | 1 | 3 | 6 |

k_{F} (Å^{−1}) | 1.473 | 1.364 | 1.380 | 1.116 | 1.504 | 2.375 |

Total number | 14 | 14 | 12 | 26 | 14 | 12 |

BZ planes | ||||||

k_{F}/(½q_{hkl}) | ||||||

max | 1.001 | 1.191 | 1.241 | 0.951 | 1.300 | 1.049 |

min | 0.867 | 0.972 | 1.013 | 0.825 | 1.099 | |

V_{FS}/V_{BZ} | 0.682 | 1.0 | 1.0 | 0.590 | 1.5 | 0.750 |

**Table 2.**Structure parameters of several binary fcc-based phases from the literature data. Fermi sphere radius k

_{F}, ratios of k

_{F}to distances of Brillouin zone planes ½q

_{hkl}and the filling degree of Brillouin zones by electron states V

_{FS}/V

_{BZ}are calculated using the program BRIZ [18].

Phase | CuAu | Au_{4}Zn | Au_{3}Zn (Au+) | Au_{3}Zn (Zn+) | Au_{5}Zn_{3} | Cu_{2}Sb |
---|---|---|---|---|---|---|

Pearson symbol | oI40 | oP96 | tI64 | oC32 | oI128 | tP6 |

Structural data ^{a} | ||||||

Space group | Imma | Pnmn | I4_{1}/acd | Cmca | Ibam | P4/nmm |

T, P conditions | Ambient conditions | |||||

lattice parameters (Å) | a = 3.676 | a = 24.216 | a = 5.586 | a = 16.603 | a = 5.510 | a = 4.001 |

b= 3.956 | b= 4.025 | c = 33.40 | b = 5.581 | b = 11.020 | c = 6.104 | |

c = 39.72 | c = 16.244 | c = 5.581 | c = 33.620 | |||

FS–BZ data from the BRIZ program | ||||||

Z (number of valence electrons per atom) | 1 | 1.19 | 1.235 | 1.26 | 1.375 | 2.33 |

k_{F} (Å^{–1}) | 1.27 | 1.288 | 1.310 | 1.322 | 1.367 | 1.618 |

Total number | 14 | 28 | 50 | 34 | 34 | 22 |

BZ planes | ||||||

k_{F}/(½q_{hkl}) | ||||||

max | 1.0078 | 1.0756 | 1.0635 | 1.069 | 1.046 | 1.0686 |

min | 0.7996 | 0.825 | 0.8705 | 0.873 | 0.914 | 0.979 |

V_{FS}/V_{BZ} | 0.600 | 0.875 | 0.972 | 0.968 | 0.921 | 0.916 |

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**MDPI and ACS Style**

Degtyareva, V.F.; Afonikova, N.S.
Simple Metal and Binary Alloy Phases Based on the *fcc* Structure: Electronic Origin of Distortions, Superlattices and Vacancies. *Crystals* **2017**, *7*, 34.
https://doi.org/10.3390/cryst7020034

**AMA Style**

Degtyareva VF, Afonikova NS.
Simple Metal and Binary Alloy Phases Based on the *fcc* Structure: Electronic Origin of Distortions, Superlattices and Vacancies. *Crystals*. 2017; 7(2):34.
https://doi.org/10.3390/cryst7020034

**Chicago/Turabian Style**

Degtyareva, Valentina F., and Nataliya S. Afonikova.
2017. "Simple Metal and Binary Alloy Phases Based on the *fcc* Structure: Electronic Origin of Distortions, Superlattices and Vacancies" *Crystals* 7, no. 2: 34.
https://doi.org/10.3390/cryst7020034