- Email: [email protected]
Structural details of decagonal AlNiCo studied by molecular dynamics
]O[ll~blllL OF
ELSEVIER
Journal of Non-CrystallineSolids 223 (1998) 152-157
Structural details of decagonal AlNiCo studied by molecular dynamics J~Srg Stadler, Hans-Rainer Trebin * Institut fdr Theoretische und Angewandte Physik, Universitiit Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
Received 19 February 1997; revised 8 July 1997
Abstract The structure of the decagonal quasicrystalline phase of AlNiCo is studied by molecular (MD) dynamics simulations. An initial structure model, based on X-ray data obtained by Steurer et al., is equilibrated in MD runs with suitable two-body potentials. We investigate whether the decagonal layers are fiat and where the short A1-A1 distances found in the X-ray data originate. Some atoms move to positions in between the decagonal layers to form icosahedral clusters. The formation of split-positions is observed which explains the short A1-A1 separations observed in experiment. © 1998 Elsevier Science B.V.
1. Introduction Molecular dynamics simulations are a powerful tool for examining the structure and the properties of solids [1]. Here we employ the method to investigate the structure of the quasicrystalline decagonal phase of AlNiCo. This alloy has been studied intensively in the past due to the availability of high quality single crystal specimens [2]. It is possible to grow the phase from a liquid in millimeter-sized, thermodynamically stable single crystals [2]. High resolution X-ray studies were performed and several models were suggested for the structure [2,8,9]. From the data it has been derived that AlNiCo consists of two plane decagonal quasicrystalline layers, which are alternately stacked along a perpendicular tenfold screw (105 ) axis.
* Correspondingauthor. Tel.: + 49-711 685 5255; fax: + 49-711 685 5271; e-mail: [email protected].
However, questions concerning structural details remain unresolved. For example, in the evaluation of the X-ray data, flatness of the layers has been assumed, which requires verification. We approach the questions by computer simulations as follows. First, we create a structural model from Steuerer's electron density data. Second, we fit a simple pair potential to the model structure and determine stability in relaxation calculations. Last, we run molecular dynamics simulations at different temperatures and analyze the evolution of the structure.
2. Simulation method and structure model 2.1. M e t h o d
In our molecular dynamics calculations the quasicrystal is modeled as a collection of point masses that interact via spherically symmetric pair poten-
0022-3093/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0022-3093(97)00434-1
J. Stadler, H.-R. Trebin // Journal of Non-CrystaUine Solids 223 (1998) 152-157
tials. We solve the equations of motion for this system and monitor the motion of the atoms. The equations of motion are modified such that they simulate a system at a fixed temperature and with a fixed external pressure, i.e. they generate an isothermal and isobaric ensemble [3,4]. We perform runs at a constant temperature, in order to find information about details of the structure because initially the energetically unfavourable local environments are rearranged. 2.2. Model The basis of our work is structural data obtained by Steuerer and co-workers [2] in X-ray diffraction measurements. The results are presented in terms of positions and peak-intensities of the electronic density distribution, p, of one asymmetric unit. The data are two-dimensional and, to derive a three-dimensional structure, we have to perform two steps. First, from the magnitude of a maximum of p we decide what type of atom to assign to a given position. The decision is ambiguous in many cases. In the plot of the density maxima (Fig. 1) only two
153
peaks appear instead of the three expected for a ternary alloy. Moreover, there are several finite values of p between these peaks. Thus, we can only decide that the atoms at small density maxima should be A1 and those at larger density maxima are either Ni or Co. We denote the latter simply TM (for transition metal). The numerical value of the density separating the A1 from the TM atoms is chosen as 0.77 (see dotted line in Fig. 1) such that the chemical composition (A170Ni~sCo15) of the model is correct. Thus we arrive at a model with only two different types of atoms, namely A1 and TM. Second, to generate a three-dimensional quasicrystal we apply the (105) screw operation [2] to all points of the asymmetric unit. There are two shortcomings in the resulting structure: it contains sites with unrealistically short A1-A1 distances and the z-coordinates are generated artificially. Thus, the structure cannot be correct and we retrieve it with computer simulations. In particular we direct our attention to the questions: what is the origin of the short A1-A1 distances and are the decagonal layers fiat?
40
35
30
25 E 20 E 15
10
i
0.5
1.0 electron density at maxima [relative units]
1.5
Fig. 1. Number of peaks in the two-dimensional electron-density maps of Steurer [2] versus the peak height. Peaks left of the broken line are associated to A1 atoms, these on the right to TM ( = Ni, Co) atoms.
154
J. Stadler, H.-R. Trebin/ Journal of Non-Crystalline Solids 223 (1998) 152-157 maxima of the respective partial radial distribution function. To investigate which features of the potential are responsible for the stability of the decagonal phase, we truncated the AI-TM and T M - T M potentials after the first minimum. In spite of this simplification the decagonal phase remains stable. Therefore, the detailed form of the AI-A1 interaction seems to be most important for stability. The resulting effective pair potentials are displayed in Fig. 2. Their single most important feature are the two nearby minima in the A1-A1 interaction that allow for A1-AI bonds of 0.31 and 0.26 nm. These are essential for the stability of the structure. To further assess the potential, we performed relaxation calculations on the Burkov-Zeger model of d-A1CuCo [8,9] and the approximate phase of A14Fe13 [10]. We obtained stable structures in both cases. Atoms can be trapped in shallow minima of the potential energy during relaxation calculations, which simulate the crystal at a very low temperature. We use molecular dynamics simulations at fixed temperature to obtain structural data, since there energetically unfavourable atom configurations are more likely to rearrange.
2.3. Pair potentials
We performed relaxation calculations with the method of steepest descent to find a pair potential stabilizing the structure [5]. With a Lennard-Jones type interaction, the decagonal phase relaxes to an amorphous state. In general, effective pair potentials in metallic alloys depend on the details of the electronic structure [6]. According to Harrison [7] however, all these interactions have common features which are a repulsive core described by a Yukawa potential a Vrep(r) = --exp( - br) (1) r
and a long range term of Friedel oscillations that can be modelled by V,r(r)=c
cos( er + f ) (er+f) 3
(2)
For a binary model, there is a potential Vq(r) for each of the possible pairs of atoms AI-A1, T M - T M and A1-TM. All are assumed to have the above form. The parameters are chosen in such a way that the minima of each potential coincides with the 2
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radius / [0.1 rim] Fig. 2. Effectiveinteractionpotential.The AI-AI interactionis givenby the full, TM-TM by the dashed and AI-TM by the dottedline.
J. Stadler, H.-R. Trebin / Journal of Non-Crystalline Solids 223 (1998) 152-157 60
3. Calculations As initial configuration for the simulation we take a section of our quasicrystal of about 6 X 6 X 1.22 nm size (see Fig. 3). This corresponds to roughly 20 atoms per edge of the quasicrystalline plane and to 6 layers stacked along the z-direction. In simulations using the isothermal-isobaric ensemble, the volume of the simulation cell is changed in order to keep the pressure constant [4]. One can either use moving walls or periodic boundary conditions. We chose the latter. Since the sample is periodic in the z-direction, but quasiperiodic in the x- and y-directions, the boundary conditions introduce a stacking fault. Hence, we can only collect structural information from the central, undisturbed part of the simulation cell. In a constant temperature simulation with this sample we obtain the following results.
50
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10
0 0
In the molecular-dynamics simulation the equations of motion for the system are integrated for 200 000 timesteps. The temperature is fixed at 0.25. The melting temperature of the system, which was
50
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Fig. 4. Final configuration a~er 2 0 0 0 0 M D s t e p s . Projection and scale as Fig. 3.
4. Results
60
155
10
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50
60
Fig. 3. Initial configuration. Six layers are projected along the decagonal axis. No distinction is made between the different types of atoms. Distances are given in 0.1 rim.
determined by heating simulations, is approximately at 0.35 (in Lennard-Jones units). When comparing initial and final configurations (Figs. 3 and 4) we recognize that the structural details of the sample (most notably the layers and the ten-rings) are conserved. The inspection also shows that some atoms have left the layers and some of the ten-rings are deformed. Additional disorder in the final configuration originates from the artificial boundary conditions. The most prominent motifs of the decagonal phase are the ten-rings. These consist of pentagons that are rotated by 36 ° in alternating layers and form decaprismatic, columnar clusters (see Fig. 5). The center of some of the pentagons is occupied by an atom. While the pentagons consist of either A1 or TM atoms, the central atoms are always A1. At most every second pentagon in a column can be filled. Occupation of all of them would yield an unrealistic A1-A1 distance. The occurrence of the filled pentagons explains why there are two distances required in the A1-A1 potential. The longer distance is associated with the bonds along an edge of such a pentagon, the shorter one connects the central A1 with the peripheral atoms.
J. Stadler, H.-R. Trebin / Journal of Non-Crystalline Solids 223 (1998) 152-157
156
O
Fig. 5. Initial columnar cluster, filled an non-filled pentagons alternate in different layers. A1 and TM atoms form the pentagons while the central atoms are AI.
Some of the central atoms leave the decagonal layers during the simulation and move to a position between the layers. This motion changes short A1-AI bonds to long bonds. Together with the pentagon atoms these interlayer atoms form one half of an
•
•
•
•
•
•
•
O0
Fig. 6, Columnar cluster after 20000 MD steps. Central A1 atoms have left decagonal layers and form an icosahedral cluster together with the pentagons in the decagonal layers.
Fig. 7. Newly built split atom viewed from the top. The columnar cluster is projected along the decagonal axis. One A1 atom has snapped inward. In the projection the two marked atoms are too close.
icosahedron. It occurs at some places in the final configuration that all the central atoms on a vertical line leave their initial position and move in between the layers. In this case a column of stacked, distorted icosahedra is formed (see Fig. 6). Not only can short bonds change into long ones, but the opposite case can occur in empty pentagons and transforms a regular pentagon into a distorted one. This particular atomic motion represents flips predicted in a recent study of phason kinetics [9]. When considering the projection of the configuration into the decagonal plane (see Fig. 7) we observe two occupied nearby positions. The configuration explains the very short distance found earlier in the experimental data. There are two possible places for A1 atoms on the edges of a pentagon that can alternatively be occupied in different layers. The X-ray diffraction experiments average over several layers and both positions are observed [2].
5. Conclusion
We have shown that decagonal quasicrystals can be stabilized by isotropic pair potentials. Relaxation calculations require that the A1-A1 potential must
J. Stadler, H.-R. Trebin / Journal of Non-Crystalline Solids 223 (1998) 152-157
allow for two A1-A1 nearest neighbour distances. Due to this property, A1 atoms in the center of the columnar clusters can leave the decagonal layers. When two atoms do so in a correlated way, icosahedra are formed. The edges of pentagons can move inward, when an initially long A1-A1 bond changes into a short one and lead to the so called split-atoms, which are observed in the experiment.
Acknowledgements The authors wish to thank Professor W. Steurer and Dr T. Haibach for their X-ray data files and inspiring discussions.
157
References [1] M. Allen, D. Tildesley, Computer Simulation of Liquids, Clarendon, 1992. [2] W. Steurer, T. Haibach, B. Zhang, S. Kek, R. LiJck, Acta Crystallogr. B 46 (1993) 661. [3] S. Toxvaerd, Phys. Rev. E 47 (1) (1993) 343. [4] S. Nose, Molecular Dynamics at Constant Temperature and Pressure, Kluwer Academic, 1991, pp. 21-41. [5] J. Roth, J. Stadler, R. Schilling, H.-R. Trebin, J. Non-Cryst. Solids 153&154 (1993) 536. [6] J. Hafner, From Hamiltonians to Phase Diagrams, Springer, 1987. [7] W.A. Harrison, Electronic Structure and the Properties of Solids, Freeman, 1980. [8] S.E. Burkov, Phys. Rev. Lett. 67 (1991) 614. [9] G. Zeger, H.-R. Trebin, Phys. Rev. B 54 (1996) R720. [10] P.J. Black, Acta. Crystallogr. 8 (1955) 1954.
ELSEVIER
Journal of Non-CrystallineSolids 223 (1998) 152-157
Structural details of decagonal AlNiCo studied by molecular dynamics J~Srg Stadler, Hans-Rainer Trebin * Institut fdr Theoretische und Angewandte Physik, Universitiit Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
Received 19 February 1997; revised 8 July 1997
Abstract The structure of the decagonal quasicrystalline phase of AlNiCo is studied by molecular (MD) dynamics simulations. An initial structure model, based on X-ray data obtained by Steurer et al., is equilibrated in MD runs with suitable two-body potentials. We investigate whether the decagonal layers are fiat and where the short A1-A1 distances found in the X-ray data originate. Some atoms move to positions in between the decagonal layers to form icosahedral clusters. The formation of split-positions is observed which explains the short A1-A1 separations observed in experiment. © 1998 Elsevier Science B.V.
1. Introduction Molecular dynamics simulations are a powerful tool for examining the structure and the properties of solids [1]. Here we employ the method to investigate the structure of the quasicrystalline decagonal phase of AlNiCo. This alloy has been studied intensively in the past due to the availability of high quality single crystal specimens [2]. It is possible to grow the phase from a liquid in millimeter-sized, thermodynamically stable single crystals [2]. High resolution X-ray studies were performed and several models were suggested for the structure [2,8,9]. From the data it has been derived that AlNiCo consists of two plane decagonal quasicrystalline layers, which are alternately stacked along a perpendicular tenfold screw (105 ) axis.
* Correspondingauthor. Tel.: + 49-711 685 5255; fax: + 49-711 685 5271; e-mail: [email protected].
However, questions concerning structural details remain unresolved. For example, in the evaluation of the X-ray data, flatness of the layers has been assumed, which requires verification. We approach the questions by computer simulations as follows. First, we create a structural model from Steuerer's electron density data. Second, we fit a simple pair potential to the model structure and determine stability in relaxation calculations. Last, we run molecular dynamics simulations at different temperatures and analyze the evolution of the structure.
2. Simulation method and structure model 2.1. M e t h o d
In our molecular dynamics calculations the quasicrystal is modeled as a collection of point masses that interact via spherically symmetric pair poten-
0022-3093/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0022-3093(97)00434-1
J. Stadler, H.-R. Trebin // Journal of Non-CrystaUine Solids 223 (1998) 152-157
tials. We solve the equations of motion for this system and monitor the motion of the atoms. The equations of motion are modified such that they simulate a system at a fixed temperature and with a fixed external pressure, i.e. they generate an isothermal and isobaric ensemble [3,4]. We perform runs at a constant temperature, in order to find information about details of the structure because initially the energetically unfavourable local environments are rearranged. 2.2. Model The basis of our work is structural data obtained by Steuerer and co-workers [2] in X-ray diffraction measurements. The results are presented in terms of positions and peak-intensities of the electronic density distribution, p, of one asymmetric unit. The data are two-dimensional and, to derive a three-dimensional structure, we have to perform two steps. First, from the magnitude of a maximum of p we decide what type of atom to assign to a given position. The decision is ambiguous in many cases. In the plot of the density maxima (Fig. 1) only two
153
peaks appear instead of the three expected for a ternary alloy. Moreover, there are several finite values of p between these peaks. Thus, we can only decide that the atoms at small density maxima should be A1 and those at larger density maxima are either Ni or Co. We denote the latter simply TM (for transition metal). The numerical value of the density separating the A1 from the TM atoms is chosen as 0.77 (see dotted line in Fig. 1) such that the chemical composition (A170Ni~sCo15) of the model is correct. Thus we arrive at a model with only two different types of atoms, namely A1 and TM. Second, to generate a three-dimensional quasicrystal we apply the (105) screw operation [2] to all points of the asymmetric unit. There are two shortcomings in the resulting structure: it contains sites with unrealistically short A1-A1 distances and the z-coordinates are generated artificially. Thus, the structure cannot be correct and we retrieve it with computer simulations. In particular we direct our attention to the questions: what is the origin of the short A1-A1 distances and are the decagonal layers fiat?
40
35
30
25 E 20 E 15
10
i
0.5
1.0 electron density at maxima [relative units]
1.5
Fig. 1. Number of peaks in the two-dimensional electron-density maps of Steurer [2] versus the peak height. Peaks left of the broken line are associated to A1 atoms, these on the right to TM ( = Ni, Co) atoms.
154
J. Stadler, H.-R. Trebin/ Journal of Non-Crystalline Solids 223 (1998) 152-157 maxima of the respective partial radial distribution function. To investigate which features of the potential are responsible for the stability of the decagonal phase, we truncated the AI-TM and T M - T M potentials after the first minimum. In spite of this simplification the decagonal phase remains stable. Therefore, the detailed form of the AI-A1 interaction seems to be most important for stability. The resulting effective pair potentials are displayed in Fig. 2. Their single most important feature are the two nearby minima in the A1-A1 interaction that allow for A1-AI bonds of 0.31 and 0.26 nm. These are essential for the stability of the structure. To further assess the potential, we performed relaxation calculations on the Burkov-Zeger model of d-A1CuCo [8,9] and the approximate phase of A14Fe13 [10]. We obtained stable structures in both cases. Atoms can be trapped in shallow minima of the potential energy during relaxation calculations, which simulate the crystal at a very low temperature. We use molecular dynamics simulations at fixed temperature to obtain structural data, since there energetically unfavourable atom configurations are more likely to rearrange.
2.3. Pair potentials
We performed relaxation calculations with the method of steepest descent to find a pair potential stabilizing the structure [5]. With a Lennard-Jones type interaction, the decagonal phase relaxes to an amorphous state. In general, effective pair potentials in metallic alloys depend on the details of the electronic structure [6]. According to Harrison [7] however, all these interactions have common features which are a repulsive core described by a Yukawa potential a Vrep(r) = --exp( - br) (1) r
and a long range term of Friedel oscillations that can be modelled by V,r(r)=c
cos( er + f ) (er+f) 3
(2)
For a binary model, there is a potential Vq(r) for each of the possible pairs of atoms AI-A1, T M - T M and A1-TM. All are assumed to have the above form. The parameters are chosen in such a way that the minima of each potential coincides with the 2
1.5
'~'
I
................ :~ :; ;I .
...............
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
.
.
.
.
.
.
.
.
.
.
.
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i 3
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radius / [0.1 rim] Fig. 2. Effectiveinteractionpotential.The AI-AI interactionis givenby the full, TM-TM by the dashed and AI-TM by the dottedline.
J. Stadler, H.-R. Trebin / Journal of Non-Crystalline Solids 223 (1998) 152-157 60
3. Calculations As initial configuration for the simulation we take a section of our quasicrystal of about 6 X 6 X 1.22 nm size (see Fig. 3). This corresponds to roughly 20 atoms per edge of the quasicrystalline plane and to 6 layers stacked along the z-direction. In simulations using the isothermal-isobaric ensemble, the volume of the simulation cell is changed in order to keep the pressure constant [4]. One can either use moving walls or periodic boundary conditions. We chose the latter. Since the sample is periodic in the z-direction, but quasiperiodic in the x- and y-directions, the boundary conditions introduce a stacking fault. Hence, we can only collect structural information from the central, undisturbed part of the simulation cell. In a constant temperature simulation with this sample we obtain the following results.
50
40
30
20
10
0 0
In the molecular-dynamics simulation the equations of motion for the system are integrated for 200 000 timesteps. The temperature is fixed at 0.25. The melting temperature of the system, which was
50
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Fig. 4. Final configuration a~er 2 0 0 0 0 M D s t e p s . Projection and scale as Fig. 3.
4. Results
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Fig. 3. Initial configuration. Six layers are projected along the decagonal axis. No distinction is made between the different types of atoms. Distances are given in 0.1 rim.
determined by heating simulations, is approximately at 0.35 (in Lennard-Jones units). When comparing initial and final configurations (Figs. 3 and 4) we recognize that the structural details of the sample (most notably the layers and the ten-rings) are conserved. The inspection also shows that some atoms have left the layers and some of the ten-rings are deformed. Additional disorder in the final configuration originates from the artificial boundary conditions. The most prominent motifs of the decagonal phase are the ten-rings. These consist of pentagons that are rotated by 36 ° in alternating layers and form decaprismatic, columnar clusters (see Fig. 5). The center of some of the pentagons is occupied by an atom. While the pentagons consist of either A1 or TM atoms, the central atoms are always A1. At most every second pentagon in a column can be filled. Occupation of all of them would yield an unrealistic A1-A1 distance. The occurrence of the filled pentagons explains why there are two distances required in the A1-A1 potential. The longer distance is associated with the bonds along an edge of such a pentagon, the shorter one connects the central A1 with the peripheral atoms.
J. Stadler, H.-R. Trebin / Journal of Non-Crystalline Solids 223 (1998) 152-157
156
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Fig. 5. Initial columnar cluster, filled an non-filled pentagons alternate in different layers. A1 and TM atoms form the pentagons while the central atoms are AI.
Some of the central atoms leave the decagonal layers during the simulation and move to a position between the layers. This motion changes short A1-AI bonds to long bonds. Together with the pentagon atoms these interlayer atoms form one half of an
•
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Fig. 6, Columnar cluster after 20000 MD steps. Central A1 atoms have left decagonal layers and form an icosahedral cluster together with the pentagons in the decagonal layers.
Fig. 7. Newly built split atom viewed from the top. The columnar cluster is projected along the decagonal axis. One A1 atom has snapped inward. In the projection the two marked atoms are too close.
icosahedron. It occurs at some places in the final configuration that all the central atoms on a vertical line leave their initial position and move in between the layers. In this case a column of stacked, distorted icosahedra is formed (see Fig. 6). Not only can short bonds change into long ones, but the opposite case can occur in empty pentagons and transforms a regular pentagon into a distorted one. This particular atomic motion represents flips predicted in a recent study of phason kinetics [9]. When considering the projection of the configuration into the decagonal plane (see Fig. 7) we observe two occupied nearby positions. The configuration explains the very short distance found earlier in the experimental data. There are two possible places for A1 atoms on the edges of a pentagon that can alternatively be occupied in different layers. The X-ray diffraction experiments average over several layers and both positions are observed [2].
5. Conclusion
We have shown that decagonal quasicrystals can be stabilized by isotropic pair potentials. Relaxation calculations require that the A1-A1 potential must
J. Stadler, H.-R. Trebin / Journal of Non-Crystalline Solids 223 (1998) 152-157
allow for two A1-A1 nearest neighbour distances. Due to this property, A1 atoms in the center of the columnar clusters can leave the decagonal layers. When two atoms do so in a correlated way, icosahedra are formed. The edges of pentagons can move inward, when an initially long A1-A1 bond changes into a short one and lead to the so called split-atoms, which are observed in the experiment.
Acknowledgements The authors wish to thank Professor W. Steurer and Dr T. Haibach for their X-ray data files and inspiring discussions.
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References [1] M. Allen, D. Tildesley, Computer Simulation of Liquids, Clarendon, 1992. [2] W. Steurer, T. Haibach, B. Zhang, S. Kek, R. LiJck, Acta Crystallogr. B 46 (1993) 661. [3] S. Toxvaerd, Phys. Rev. E 47 (1) (1993) 343. [4] S. Nose, Molecular Dynamics at Constant Temperature and Pressure, Kluwer Academic, 1991, pp. 21-41. [5] J. Roth, J. Stadler, R. Schilling, H.-R. Trebin, J. Non-Cryst. Solids 153&154 (1993) 536. [6] J. Hafner, From Hamiltonians to Phase Diagrams, Springer, 1987. [7] W.A. Harrison, Electronic Structure and the Properties of Solids, Freeman, 1980. [8] S.E. Burkov, Phys. Rev. Lett. 67 (1991) 614. [9] G. Zeger, H.-R. Trebin, Phys. Rev. B 54 (1996) R720. [10] P.J. Black, Acta. Crystallogr. 8 (1955) 1954.