The Physics of the Hume-Rothery Electron Concentration Rule
Abstract
:1. Introduction
1.1. e/a versus Valence in the Hume-Rothery Electron Concentration Rule
1.2. Historical Survey on the e/a Issue for Transition Metals (TM) and Their Compounds
1.3. e/a-Dependent Hume-Rothery-Type Stabilization Mechanism
1.4. Stoichiometric Compounds versus Chemically Disordered Solid Solutions
2. The FLAPW-Fourier Theory and Its Application to Elements in the Periodic Table
2.1. WIEN2k-FLAPW Program Package
2.2. Representations of Quantities in Reciprocal Space
2.3. The FLAPW-Fourier Theory
2.4. FLAPW-Fourier Spectra
2.4.1. Na (cI2)
2.4.2. Al (cF4)
2.4.3. Si (cF8)
2.4.4. P (oC8)
2.4.5. Insulating S (mP28) versus High-Pressure Metallic S (hR1)
2.4.6. Insulating Solid Cl (oC8)
2.4.7. High-Pressure Metallic solid Br (oI2)
2.4.8. α-Mn (cI58)
2.5. Hume-Rothery Plot
2.6. Criterion to Judge Itinerancy of Electrons at the Fermi Level
2.7. Why Can the Hume-Rothery Plot Generate Dispersion Relations in the Extended Zone Scheme?
2.8. Determination of and e/a for Representative Elements in the Periodic Table
2.8.1. Na (cI2)
2.8.2. Al (cF4)
2.8.3. Si (cF8)
2.8.4. P (oC8)
2.8.5. Insulating Phase S (mP28) versus High-Pressure Metallic Phase S (hR1)
2.8.6. Insulating Solid Cl (oC8)
2.8.7. High-Pressure Metallic Br (oI2)
2.8.8. α-Mn (cI58)
2.9. e/a Determination for 54 Elements in the Periodic Table
3. Bond-Type Classification of Compounds on Van Arkel-Ketelaar Triangle Map
3.1. Scaling of Van Arkel-Ketelaar Triangle in Terms of Allen’s Electronegativity
3.2. The Physics behind the Allen Electronegativity
4. Hume-Rothery Electron Concentration Rule in Al-, Zn- and Cd-Based Alloy Systems
4.1. How to Explore Systems Obeying a New Hume-Rothery Electron Concentration Rule?
4.2. Gamma-Brasses (cI52 or cP52)
4.3. RT-Type 1/1-1/1-1/1 Approximants (cI160 or cI162)
4.4. RT-Type 2/1-2/1-2/1 Approximants (cP680)
4.5. MI-Type 1/1-1/1-1/1 Approximants (cP138, cP144)
4.6. Tsai-Type 1/1-1/1-1/1 Approximants (cP168, mC336)
4.7. Samson Compound Al3Mg2 (cF1178) and Al12Mg17 (cI58)
4.8. e/a Determination for TM-Al Binary Compounds
4.9. Interference Condition for Al-, Zn- and Cd-Compounds
5. Hume-Rothery Electron Concentration Rule in Zintl Compounds
5.1. I-III-Type Zintl Compounds (cF32)
5.2. I-II-Type Zintl Compounds (cF32)
5.3. Hume-Rothery Electron Concentration Rule
6. Hume-Rothery Electron Concentration Rule in Phosphorus Compounds
6.1. Equiatomic Phosphorus Compounds on the Van Arkel-Ketelaar Triangle Map
6.2. e/a Determination and Interference Condition for the Host Element P in Group 15
6.3. e/a Determination and Interference Condition of M-P Binary Compounds
6.3.1. SiP and GeP Compounds
6.3.2. TM-P (TM = Sc to Ni) Compounds
6.3.3. Mg-P, Ca-P, Zn-P and Cu-P Compounds
6.3.4. A-P (A = Li, Na, K, Rb and Cs) Compounds
6.4. Hume-Rothery Electron Concentration Rule in Phosphorus-Based Compounds
6.4.1. TMP3 (TM=Co, Ni, Rh and Ir), TMAs3 (TM=Co, Rh and Ir) and TMSb3 (TM=Co, Rh and Ir) Skutterudite Compounds (cI32)
6.4.2. (TM)P (TM = Cr, Mn, Fe and Co) Compounds (oP8)
6.4.3. TM3P (TM = Cr, Mn, Fe and Ni) Compounds (tI32)
6.5. Interference Phenomenon for Phosphorus Compounds
6.6. Summary
7. Hume-Rothery Electron Concentration Rule in Inter-Transition Metal Compounds
7.1. e/a Determination for Group 2 Elements Ca, Sr and Ba
7.2. e/a Determination for Group 3 Elements Sc, Y and La
7.3. Alloying Environment Effects and the Linear Interpolation Rule
7.3.1. Ca-, Sr-, Sc- and Y-Compounds with Simple Elements
7.3.2. Ba- and La-Compounds with Simple Elements
7.3.3. Ca-, Sr- and Ba-Compounds with TM Elements
7.3.4. Sc-, Y- and La-Compounds with TM Elements
7.3.5. Inter-TM (Two TMs Selected from Group 4 to 10) Compounds
7.4. Hume-Rothery Electron Concentration Rule in Laves Compounds (cF24)
7.5. Stabilization Mechanism of Laves Compounds
8. The FLAPW-Fourier Theory to Develop New Functional Materials
8.1. III-V Compounds
8.2. Development of New Electronic Functional Materials with e/a = 4
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A
References
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Subgroup | System | N | (2kF)2 | e/a | e/uc | |G|2c | |
---|---|---|---|---|---|---|---|
× (2π/a)2 | × (2π/a)2 | ||||||
gamma-brasses | 1 | Cu5Zn8 (cI52) | 52 | 18.5 ± 0.1 | 1.60 ± 0.02 | 83 | 18 |
Ag5Zn8 (cI52) | 52 | 18.5 ± 0.1 | |||||
Al4Cu9 (cP52) | 52 | 18.5 ± 0.1 | |||||
In4Ag9 (cP52) | 52 | 18.5 ± 0.1 | |||||
2 | Al8V5 (cI52) | 52 | 22.7 ± 1.1 | 2.1 ± 0.1 | 109 | 22 | |
Al8Cr5 (cI52) | 52 | 22.1 ± 1.1 | |||||
3 | Ag5Li8 (cI52) | 52 | 13.51 ± 0.27 | 1.00 ± 0.02 | 52 | 14 |
Subgroup | Approximants | Space Group | N | |G|2c × (2π/a)2 | a Å | 2d Å | (2kF)2 × (2π/a)2 | e/a | e/uc | |
---|---|---|---|---|---|---|---|---|---|---|
1/1 | 1 | Al3Mg4Zn3 | Im-3 | 160 | 50 | 14.355 | 4.06 | 49.8 | 2.30 | 368 |
Al9Mg8Ag3 | Im-3 | 160 | 50 | 14.4799 | 4.09 | 49.4 | 2.27 | 363 | ||
2 | Al21Li13Cu6 | Im-3 | 160 | 46 | 13.89 | 4.09 | 47.1 | 2.10 | 336 | |
Ga21Li13Cu6 | Im-3 | 160 | 46 | 13.568 | 4.00 | 47.6 | 2.15 | 344 | ||
3 | Na26Au24Ga30 | Im-3 | 160 | 42 | 14.512 | 4.48 | 41.8 | 1.77 | 283 | |
Na26Au37Ge18 | Im-3 | 162 | 42 | 14.581 | 4.50 | 41.4 | 1.72 | 278 | ||
Na26Au37Sn18 | Im-3 | 162 | 42 | 15.009 | 4.63 | 41.1 | 1.70 | 275 | ||
Na26Cd40Pb6 | Im-3 | 160 | 42 | 15.992 | 4.94 | 42.3 | 1.80 | 288 | ||
2/1 | 1 | Al13Mg27Zn45 | Pa-3 | 680 | 126 | 23.0349 | 4.10 | 124.0 | 2.13 | 1448 |
2 | Na27Au27Ga31 | Pa-3 | 680 | 110 | 23.446 | 4.47 | 109.2 | 1.76 | 1197 |
Subgroup | Approximants | Space Group | N | |G|2c × (2π/a)2 | a Å | 2d Å | (2kF)2 × (2π/a)2 | e/a | e/uc | |
---|---|---|---|---|---|---|---|---|---|---|
1/1 | 1 | Al114Mn24 | Pm-3 | 138 | 50 | 12.68 | 3.59 | 48.9 | 2.59 | 357 |
Al114Re24 | Pm-3 | 138 | 50 | 12.86 | 3.64 | 48.6 | 2.57 | 355 | ||
Al102Re24Si12 | Pm-3 | 138 | 50 | 12.8603 | 3.64 | 51.0 | 2.76 | 381 | ||
Al108Cu6Fe24Si6 | Pm-3 | 144 | 50 | 12.48 | 3.53 | 50.0 | 2.57 | 370 | ||
Al108Cu6Ru24Si6 | Pm-3 | 144 | 50 | 12.6832 | 3.59 | 51.7 | 2.70 | 389 |
Subgroup | Approximants | Space Group | N | |G|2c × (2π/a)2 | a Å | 2d Å | (2kF)2 × (2π/a)2 | e/a | e/uc | |
---|---|---|---|---|---|---|---|---|---|---|
1/1 | 1 | Zn6Sc | C2/c | 336 | 80 | a = 19.47 b = 13.79 c = 19.55 β = 89.931o | 79.0 ± 0.2 | 2.18 | 732 | |
Cd6Ca | Pm-3 | 168 | 46 | 15.702 | 4.63 | 47.1 ± 1.0 | 2.01 | 338 | ||
Cd6Yb | Im-3 | 168 | 46 | 15.638 | 4.61 | 47.5 ± 1.0 | 2.04 | 343 |
Type | Compounds | Lattice Constant Å | N | |G|2c × (2π/a)2 | (2kF)2 × (2π/a)2 | e/a | e/uc |
---|---|---|---|---|---|---|---|
I-III cF32 Fd-3m | LiAl | 6.3757 | 32 | 16 | 16.29 ± 0.20 | 2.09 ± 0.06 | 67 |
LiGa | 6.150 | 32 | 16 | 15.97 ± 0.20 | |||
LiIn | 6.7920 | 32 | 16 | 15.75 ± 0.20 | |||
NaIn | 7.332 | 32 | 16 | 15.99 ± 0.20 | |||
NaTl | 7.473 | 32 | 16 | 16.12 ± 0.20 | |||
I-II cF32 Fd-3m | LiZn | 6.209 | 32 | 12 | 13.00 ± 0.20 | 1.53 ± 0.02 | 49 |
LiCd | 6.702 | 32 | 12 | 13.03 ± 0.20 |
System | Space Group | Pearson Symbol | |G|2c × (2π/a)2 | (2kF)2 × (2π/a)2 | e/a | e/uc |
---|---|---|---|---|---|---|
P | Cmca | oC8 | 10.56 | 11.30 ± 0.33 | 4.97 | 40 |
As | R-3m | hR2 | 4.74 | 4.45 ± 0.22 | 4.92 | 10 |
Sb | R-3m | hR2 | 4.54 | 4.49 ± 0.22 | 4.99 | 10 |
Bi | R-3m | hR2 | 4.53 | 4.46 ± 0.22 | 4.94 | 10 |
Family | Subgroup | System | N | |G|2c × [2π/(Ωuc)1/3]2 | (2kF)2 × (2π/a)2 | e/a | e/uc |
---|---|---|---|---|---|---|---|
skutterudites (TM)M3 (TM = Co, Rh, Ir, Ni) (M = P, As, Sb) | 1 | CoAs3 (cI32) | 32 | 26 | 25.2 ± 0.2 | 4.34 ± 0.20 | 139 |
RhAs3 (cI32) | 32 | 26 | 26.3 ± 0.2 | ||||
IrAs3 (cI32) | 32 | 26 | 26.7 ± 0.2 | ||||
CoSb3 (cI32) | 32 | 26 | 26.5 ± 0.2 | ||||
RhSb3 (cI32) | 32 | 26 | 27.3 ± 0.2 | ||||
IrSb3 (cI32) | 32 | 26 | 27.8 ± 0.2 | ||||
NiP3 (cI32) | 32 | 26 | 25.1 ± 0.2 | ||||
CoP3 (cI32) | 32 | 26 | 25.0 ± 0.2 | ||||
RhP3 (cI32) | 32 | 26 | 25.9 ± 0.2 | ||||
IrP3 (cI32) | 32 | 26 | 26.7 ± 0.2 | ||||
(TM)P (TM = Cr, Mn, Fe, Co) | 1 | CrP (oP8) | 8 | 8.38 | 8.53 ± 0.43 | 3.25 ± 0.05 | 26 |
MnP (oP8) | 8 | 8.49 | 8.47 ± 0.42 | ||||
FeP (oP8) | 8 | 8.56 | 8.41 ± 0.42 | ||||
CoP (oP8) | 8 | 8.29 | 8.60 ± 0.83 | ||||
(TM)3P (TM = Cr, Mn, Fe, Ni) | 1 | Cr3P (tI32) | 32 | 16.3 | 16.67 ± 0.83 | 2.20 ± 0.05 | 70 |
Mn3P (tI32) | 32 | 16.4 | 16.67 ± 0.83 | ||||
Fe3P (tI32) | 32 | 16.2 | 16.47 ± 0.83 | ||||
Ni3P (tI32) | 32 | 16.2 | 16.56 ± 0.33 |
Compounds | Pearson Symbol | Energy Gap (eV) | (2kF)2 × (2π/a)2 | e/a | (e/a)av |
---|---|---|---|---|---|
SiP | oC48 | 1.2 | 35.2 | 4.55 | 4.56 |
GeP | tI4 | pseudogap | 6.66 | 4.50 | 4.56 |
ScP | cF8 | pseudogap | 10.02 | 4.15 | 4.03 |
TiP2 | oP12 | pseudogap | 13.19 | 4.18 | 3.72 |
VP2 | mC12 | pseudogap | 13.16 | 4.17 | 3.75 |
CrP2 | mC12 | pseudogap | 12.55 | 3.88 | 3.72 |
MnP4 | aP10 | 0.4 | 12.25 | 4.49 | 4.31 |
CoP3 | cI32 | pseudogap | 26.24 | 4.2 | 4.12 |
NiP3 | cI32 | pseudogap | 26.17 | 4.2 | 4.13 |
CuP2 | mP12 | 0.84 | 12.59 | 3.9 | 3.75 |
Cu2P7 | mC72 | 0.61 | 46.15 | 4.56 | 4.20 |
Compounds | Pearson Symbol | Energy Gap (eV) | (2kF)2 × (2π/a)2 | e/a | (e/a)av |
---|---|---|---|---|---|
ZnP2 | mP24 | pseudogap | 21.47 | 4.34 | 4.0 |
ZnP2 | tP24 | 1.63 | 19.28 | 3.69 | 4.0 |
ZnP4 | tP20 | 0.63 | 18.7 | 4.24 | 4.4 |
Mg3P2 | cI80 | 1.6 | 32.68 | 2.45 | 3.2 |
MgP4 | mP10 | 0.50 | 11.65 | 4.16 | 4.4 |
Ca5P8 | mC26 | 1.13 | 20.79 | 3.82 | 3.85 |
CaP3 | aP8 | pseudogap | 10.7 | 4.58 | 4.25 |
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Mizutani, U.; Sato, H. The Physics of the Hume-Rothery Electron Concentration Rule. Crystals 2017, 7, 9. https://doi.org/10.3390/cryst7010009
Mizutani U, Sato H. The Physics of the Hume-Rothery Electron Concentration Rule. Crystals. 2017; 7(1):9. https://doi.org/10.3390/cryst7010009
Chicago/Turabian StyleMizutani, Uichiro, and Hirokazu Sato. 2017. "The Physics of the Hume-Rothery Electron Concentration Rule" Crystals 7, no. 1: 9. https://doi.org/10.3390/cryst7010009
APA StyleMizutani, U., & Sato, H. (2017). The Physics of the Hume-Rothery Electron Concentration Rule. Crystals, 7(1), 9. https://doi.org/10.3390/cryst7010009