# A Higher Dimensional Description of the Structure of β-Mn

^{1}

^{2}

^{*}

## Abstract

**:**

_{1}32. The pseudo 8-fold nature of the 4

_{1}axes makes it constitute an approximant to the octagonal quasicrystals. In this paper we analyze why a five-dimensional super space group containing mutually perpendicular 8-fold axes cannot generate P4

_{1}32 on projection to 3-d space and how this may instead be accomplished from a six-dimensional model. A procedure for generating the actual structure of β-Mn lifted to six-dimensional space is given.

## 1. Introduction

**Figure 1.**The structure of β-Mn represented as a packing of three mutually perpendicular sets of tetrahelices. Each tetrahelix has a symmetry close to 8

_{3}.

_{1}32). By lifting β-Mn to higher space, we may generate a structure where the three mutually perpendicular, pseudo 8-fold axes are truly 8-fold. An axial QC would be expected to be describable as a projection from a five-dimensional, hyperspace. As it turns out, however, a simple analysis shows that this cannot be done for β-Mn. As we attempt to show in this paper, this conundrum is resolved by recourse to a six-dimensional, tri-iso-octagonal orthogonal super space group that contains both the five dimensional super space group that projects onto the oQC and the three dimensional cubic space group P4

_{1}32 as sub periodic groups. The principle is simple: In six dimensional space, it is possible to generate three mutually perpendicular 8 fold axes, and on projection to a five dimensional subspace, one of these may be preserved, while a projection to three-dimensional space may preserve only 4 fold axis. The procedure we are using owns a lot to that used by Lee et al. in a paper dealing with the unexpected occurrence of mutually perpendicular 5 fold axis in large cubic intermetallic structures [7].

## 2. Results and Discussion

#### 2.1. Preliminary Considerations is Five Dimensional Space

^{5}× 5! = 3840. This means that all permutations of indices are realized and half of the permutations of sign (those with even parity):

_{1}32 from any five-dimensional super space group containing orthogonal 8-fold axes.

#### 2.2. Preliminary Considerations is Six Dimensional Space

^{6}× 6! possible symmetry operations (=46080), a rather unwieldy number. Nevertheless, a full analysis of the character table for this group has been published [8].

_{6}contains five distinct sets of 8 fold operations. Basically these are quite similar. The 8-fold operation in six dimensional space is a hyper rotation that permutes four indices out of six, and changes the sign of one of those to produce an 8 fold operation. Simultaneously, the remaining two indices may be involved in another operation, creating a double rotation. The different possibilities are classified in the paper by Deonarine and Birman [8] as shown in Table 1.

Class | 43 | 45 | 47 | 49 | 51 |
---|---|---|---|---|---|

Operation in12 plane | Invariant | Reflection in 1 or 2 | C_{2} rotation perp 12 | C_{2} rotation within 12 | C_{4} rotation within 12 |

M^{1} | 1 2 4 5 6-3 | 1-2 4 5 6-3 | -1-2 4 5 6-3 | 2 1 4 5 6-3 | 2-1 4 5 6-3 |

M^{2} | 1 2 5 6-3-4 | 1 2 5 6-3-4 | 1 2 5 6-3-4 | 1 2 5 6-3-4 | -1-2 5 6-3-4 |

M^{3} | 1 2 6-3-4-5 | 1-2 6-3-4-5 | -1-2 6-3-4-5 | 2 1 6-3-4-5 | -2 1 6-3-4-5 |

M^{4} | 1 2-3-4-5-6 | 1 2-3-4-5-6 | 1 2-3-4-5-6 | 1 2-3-4-5-6 | 1 2-3-4-5-6 |

M^{5} | 1 2-4-5-6 3 | 1-2-4-5-6 3 | -1-2-4-5-6 3 | 2 1-4-5-6 3 | 2-1-4-5-6 3 |

M^{6} | 1 2-5-6 3 4 | 1 2-5-6 3 4 | 1 2-5-6 3 4 | 1 2-5-6 3 4 | -1-2-5-6 3 4 |

M^{7} | 1 2-6 3 4 5 | 1-2-6 3 4 5 | -1-2-6 3 4 5 | 2 1-6 3 4 5 | -2 1-6 3 4 5 |

M^{8} | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 |

_{6->5}should result in an axial 5 dimensional group with 8-fold rotational symmetry. For class 49 this requires that A + B = B + A, which is trivially true. For class 51, that requirement is instead A + B = B − A = − A − B = − B + A which is equally trivially incompatible with any other solution than A = B = 0, rendering the class 51 option unhelpful.

_{3}operations as reported for the oQC, we simply add a translational part to the point group operations so that the three generators become

_{x}), double rotation (C

_{x}, C

_{y}), etc., the number of such operations in the group, the typical form from which the invariants are easily found, the translational part of the operations given as multiples of eighths along the six axes in super space, and finally a generation of the operation from the basic operations M

_{1}, M

_{2}, M

_{3 }are all given in Table 2 below. The generating form is a sequence of basis operations needed to generate a particular operation. Example: Since the 3-fold double rotation that permutes the even indices and the odd indices simultaneously, (class 39) is generated by the sequence of operations M

_{2}(M

_{1}(M

_{1}(M

_{3}))), the generating form of that operation is given as 2113.

Deonarine-Birman class | Type | Number | Typical form | Translational part (* 1/8) | Generating form
M_{1}, M_{2}, M_{3} |
---|---|---|---|---|---|

50 | C_{4}, C_{4} | 144 | 2-1 4 5 6 3 | 3 7 3 3 3 7 | 111111212 |

49 | C_{8}, C_{2} | 144 | 2 1 4 5 6-3 | 3 3 3 7 3 7 | 1 |

39 | C_{3}, C_{3} | 64 | 3 4 5 6 1 2 | 0 0 0 0 0 0 | 2113 |

35 | C_{3}, C_{2} | 48 | 1 4-3 6-5 2 | 4 0 4 0 0 0 | 2111 |

31 | C_{3} | 16 | 1 4 3 6 5 2 | 0 0 0 0 0 0 | 112112112111 |

24 | C_{2}, C_{2 },C_{2} | 36 | -1-2 5 6 3 4 | 2 4 6 6 2 2 | 222113 |

22 | C_{4}, C_{2}, | 72 | 1-2-5 6 3 4 | 2 4 2 6 6 2 | 212111 |

20 | C_{4}, C_{4} | 36 | 1 2 5 6-3-4 | 6 6 2 2 2 4 | 11 |

5 | C_{2},C_{2} | 9 | 1 2-3-4-5-6 | 4 4 4 6 0 2 | 1111 |

3 | C_{2} | 6 | 1 2 3-4 5-6 | 0 4 0 6 0 2 | 21111121 |

1 | C_{1} | 1 | 1 2 3 4 5 6 | 0 0 0 0 0 0 | 11111111 |

#### 2.3. Projection to 3D

Classes | Typical form | Translational part eights | 3D symmetry |
---|---|---|---|

1,3,31 | 1 3 5 | 0 0 0 | x y z |

5,35 | 1-3-5 | 4 4 0 | ½ + x ½-y-z |

20 | 1 5-3 | 6 2 2 | ¾ + x ¼ + z ¼-y |

22 | 1-5 3 | 2 2 6 | ¼ + x ¼-z ¾ + y |

24 | -1 5 3 | 2 6 2 | ¼- x ¾ + z ¼ + y |

39 | 3 5 1 | 0 0 0 | y z x |

49,50 | 2 4 6 | 3 3 3 | Creation of a second independent position |

_{1}32. Each position in 6D space will generate two separate orbits depending on the values of the odd and even coordinates respectively.

#### 2.4. Projection to 5D

Operation | Rotational part | Translational part | Projection rotation part | Projection transational part |
---|---|---|---|---|

M_{1}^{1} | 2 1 4 5 6-3 | 3 3 3 7 3 7 | 1' 4 5 6-3 | 3 3 7 3 7 |

M_{1}^{2} | 1 2 5 6-3-4 | 6 6 2 2 2 4 | 1' 5 6-3-4 | 6 2 2 2 4 |

M_{1}^{3} | 2 1 6-3-4-5 | 1 1 5 1 7 5 | 1' 6-3-4-5 | 1 5 1 7 5 |

M_{1}^{4} | 1 2-3-4-5-6 | 4 4 4 6 0 2 | 1' -3-4-5-6 | 4 4 6 0 2 |

M_{1}^{5} | 2 1-4-5-6 3 | 7 7 1 7 5 3 | 1' -4-5-6 3 | 7 1 7 5 3 |

M_{1}^{6} | 1 2-5-6 3 4 | 2 2 2 4 6 6 | 1' 2-5-6 3 4 | 2 2 4 6 6 |

M_{1}^{7} | 2 1-6 3 4 5 | 5 5 7 5 1 5 | 1'-6 3 4 5 | 5 7 5 1 5 |

M_{1}^{8} | 1 2 3 4 5 6 | 0 0 0 0 0 0 | 1' 3 4 5 6 | 0 0 0 0 0 |

_{3}axis is preserved under the projection. Examining another 8-fold axis of rotation is enlightening. The orbit splits into 4 pairs of operations that leave the plane spanned by the basis vectors 3 and 4 invariant according to Table 5. It is notable that the translational part of the operations within that invariant plane is always ½ ½. Apart from this translational part, the operation contains an inversion of the other three indices. This means that the projection generates binary operations perpendicular to the unique axis in five-dimensional space.

## 3. Conclusions

Operation | Rotational part | Translational part | Projection rotation part | Projection translatinal part | New orbit |
---|---|---|---|---|---|

M_{2}^{1} | 2-5 4 3 6 1 | 3 7 3 3 3 7 | 1' 4 3 6 1 | 5 3 3 3 7 | I |

M_{2}^{2} | -5-6 3 4 1 2 | 2 4 6 6 2 2 | 2' 3 4 1 2 | 3 6 6 2 2 | II |

M_{2}^{3} | -6-1 4 3 2-5 | 7 5 1 1 5 1 | 3' 4 3 2-5 | 6 1 1 5 1 | II |

M_{2}^{4} | -1-2 3 4-5-6 | 0 2 4 4 4 6 | 4' 3 4-5-6 | 1 4 4 4 6 | IV |

M_{2}^{5} | -2 5 4 3-6-1 | 5 3 7 7 1 7 | -1' 4 3-6-1 | 4 7 7 1 7 | I |

M_{2}^{6} | 5 6 3 4-1-2 | 6 6 2 2 2 4 | -2' 3 4-1-2 | 6 2 2 2 4 | II |

M_{2}^{7} | 6 1 4 3-2 5 | 1 5 5 5 7 5 | -3' 4 3-2 5 | 3 5 5 7 5 | III |

M_{2}^{8} | 1 2 3 4 5 6 | 0 0 0 0 0 0 | -4' 3 4 5 6 | 0 0 0 0 0 | IV |

## Acknowledgments

## References and Notes

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Lidin, S.; Fredrickson, D.
A Higher Dimensional Description of the Structure of β-Mn. *Symmetry* **2012**, *4*, 537-544.
https://doi.org/10.3390/sym4030537

**AMA Style**

Lidin S, Fredrickson D.
A Higher Dimensional Description of the Structure of β-Mn. *Symmetry*. 2012; 4(3):537-544.
https://doi.org/10.3390/sym4030537

**Chicago/Turabian Style**

Lidin, Sven, and Daniel Fredrickson.
2012. "A Higher Dimensional Description of the Structure of β-Mn" *Symmetry* 4, no. 3: 537-544.
https://doi.org/10.3390/sym4030537