||

http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list

下面是从其他网站转的，读过后，对wyckoff position 可以有大致的了解了。From examination of a space group in “The International Tables for Crystallography” Vol. A, you should be able to ascertain the following information:

· Herman-Mauguin (HM) Symbol (Long, Short)

· Point Group (HM, Schoenflies)

· Locate and identify symmetry elements

· Understand Wyckoff site multiplicity and symmetry

· Distinguish general and special positions

· Extinction conditions

· Identify possible subgroups and supergroups

Understanding the Herman-Mauguin Space Group Symbol

Space groups are typically identified by their short Herman-Mauguin symbol (i.e. Pnma, I4/mmm, etc.). The symmetry elements contained in the short symbol are the minimum number needed to generate all of the remaining symmetry elements. This symbolism is very efficient, condensed form of noting all of the symmetry present in a given space group. We won’t go into all of the details of the space group symbol, but I will expect you to be able to determine the Crystal system, Bravais Lattice and Point group from the short H-M symbol. You should also be able to determine the presence and orientation of certain symmetry elements from the short H-M symbol and vice versa.

The HM space group symbol can be derived from the symmetry elements present using the following logic.

The first letter identifies the centering of the lattice, I will hereafter refer to this as the lattice descriptor :

· P ® Primitive

· I ® Body centered

· F ® Face centered

· C ® C-centered

· B ® B-centered

· A ® A-centered

The next three symbols denote symmetry elements present in certain directions, those directions are as follows:

Crystal System | Symmetry Direction | ||

Primary | Secondary | Tertiary | |

Triclinic | None | ||

Monoclinic | [010] | ||

Orthorhombic | [100] | [010] | [001] |

Tetragonal | [001] | [100]/[010] | [110] |

Hexagonal/ Trigonal | [001] | [100]/[010] | [120]/[1`1 0] |

Cubic | [100]/[010]/ [001] | [111] | [110] |

[100] – Axis parallel or plane perpendicular to the x-axis.

[010] – Axis parallel or plane perpendicular to the y-axis.

[001] – Axis parallel or plane perpendicular to the z-axis.

[110] – Axis parallel or plane perpendicular to the line running at 45° to the x and y axes.

[1`1 0] – Axis parallel or plane perpendicular to the long face diagonal of the ab face of a hexagonal cell.

[111] – Axis parallel or plane perpendicular to the body diagonal.

For a better understanding see specific examples from class notes. However, with no knowledge of the symmetry diagram we can identify the crystal system from the space group symbol.

· Cubic – The secondary symmetry symbol will always be either 3 or –3 (i.e. Ia3, Pm3m, Fd3m)

· Tetragonal – The primary symmetry symbol will always be either 4, (-4), 4_{1}, 4_{2} or 4_{3} (i.e. P4_{1}2_{1}2, I4/m, P4/mcc)

· Hexagonal – The primary symmetry symbol will always be a 6, (-6), 6_{1}, 6_{2}, 6_{3}, 6_{4} or 6_{5} (i.e. P6mm, P6_{3}/mcm)

· Trigonal – The primary symmetry symbol will always be a 3, (-3) 3_{1} or 3_{2} (i.e P31m, R3, R3c, P312)

· Orthorhombic – All three symbols following the lattice descriptor will be either mirror planes, glide planes, 2-fold rotation or screw axes (i.e. Pnma, Cmc2_{1}, Pnc2)

· Monoclinic – The lattice descriptor will be followed by either a single mirror plane, glide plane, 2-fold rotation or screw axis or an axis/plane symbol (i.e. Cc, P2, P2_{1}/n)

· Triclinic – The lattice descriptor will be followed by either a 1 or a (-1).

Space Group = I`4c2 ® Point Group =`4m2

Space Group = P4_{2}/n ® Point Group = 4/m

Wyckoff Sites

Multiplicity | Wyckoff Letter | Site Symmetry | Coordinates |

2 | c | 1 | (1) x,y,z (2) x,-y,z |

1 | b | m | x,½,z |

1 | a | m | x,0,z |

The multiplicity tells us how many atoms are generated by symmetry if we place a single atom at that position. In this case for every atom we insert at an arbitrary position (x,y,z) in the unit cell a second atom will be generated by the mirror plane at x,-y,z. This corresponds to the uppermost Wyckoff position 2c. The letter is simply a label and has no physical meaning. They are assigned alphabetically from the bottom up. The symmetry tells us what symmetry elements the atom resides upon. The uppermost Wyckoff position, corresponding to an atom at an arbitrary position never resides upon any symmetry elements. This Wyckoff position is called the general position. The coordinates column tells us the coordinates of all of the symmetry related atoms (two in this case).

All of the remaining Wyckoff positions are called special positions. They correspond to atoms which lie upon one of more symmetry elements, because of this they always have a smaller multiplicity than the general position. Furthermore, one or more of their fractional coordinates must be fixed. In this case the y value must be either 0 or ½ or the atom would no longer lie on the mirror plane.

From the space group tables we see that the atoms are located on the following Wyckoff sites

Sr ® 8c

Al ® 4a

Ta ® 4b

O ® 24e

The number associated with the Wyckoff sites tells us how many atoms of that type there are in the unit cell. In this

So there are 40 atoms in the unit cell, with stoichiometry Sr_{8}Al_{4}Ta_{4}O_{24} which reduces to the empirical formula Sr_{2}AlTaO_{6}. Since the number of atoms in the unit cell is four times the number of atoms in the formula unit, we say that Z = 4.

Using the face centering generators (0,0,0), (½,½,0), (½,0,½), (0,½,½) together with the coordinates of each Wyckoff site we can generate the fractional coordinates of all atoms in the unit cell:

1:(0.25,0.25,0.25), 2:(0.75,0.75,0.25), 3:(0.75,0.25,0.75), 4:(0.25,0.75,0.75)

5:(0.25,0.25,0.75), 6:(0.75,0.75,0.75), 7:(0.75,0.25,0.25), 8:(0.25,0.75,0.25)

Al

1:(0.0,0.0,0.0), 2:(0.5,0.5,0.0), 3:(0.5,0.0,0.5), 4:(0.0,0.5,0.5)

Ta

1:(0.5,0.5,0.5), 2:(0.0,0.0,0.5), 3:(0.0,0.5,0.0), 4:(0.5,0.0,0.0)

O

1:(0.24,0.0,0.0), 2:(0.74,0.5,0.0), 3:(0.74,0.0,0.5), 4:(0.24,0.5,0.5)

5:(0.76,0.0,0.0), 6:(0.26,0.5,0.0), 7:(0.26,0.0,0.5), 8:(0.76,0.5,0.5)

9:(0.0,0.24,0.0), 10:(0.5,0.74,0.0), 11:(0.5,0.24,0.5), 12:(0.0,0.74,0.5)

13:(0.0,0.76,0.0), 14:(0.5,0.26,0.0), 15:(0.5,0.76,0.5), 16:(0.0,0.26,0.5)

17:(0.0,0.0,0.24), 18:(0.5,0.5,0.24), 19:(0.5,0.0,0.74), 20:(0.0,0.5,0.74)

21:(0.0,0.0,0.76), 22:(0.5,0.5,0.76), 23:(0.5,0.0,0.26), 24:(0.0,0.5,0.26)

We can also work out bond distances from this information. The first Al ion is octahedrally coordinated by six oxygens (1,5,9,13,17,21) and the Al-O distance is :

d = 7.80′[(0.24-0.0)^{2} + (0.0-0.0)^{2} + (0.0-0.0)^{2}]^{1/2} = 1.87Å

while the first Ta ion is also surrounded by 6 oxygens (4,8,11,15,18,22) at a distance of

d = 7.80′[(0.24-0.5)^{2} + (0.5-0.5)^{2} + (0.5-0.5)^{2}]^{1/2} = 2.03Å

and Sr is surrounded by 12 oxygens (1,4,6,7,9,11,14,16,17,18,23,24) at a distance of

d = 7.80′[(0.24-0.25)^{2} + (0.0-0.25)^{2} + (0.0-0.25)^{2}]^{1/2} = 2.76Å

**Determining a Crystal Structure from Symmetry & Composition**

Another use is that given the stoichiometry, space group and unit cell size (which can typically be determined from diffraction techniques) and the density of a compound we can often deduce the crystal structure of relatively simple compounds.

As an example consider the following information:

Stoichiometry = SrTiO_{3}

Space Group = Pm3m

a = 3.90 Å

Density = 5.1 g/cm^{3}

To derive the crystal structure from this information the first step is to calculate the number of formula units per unit cell :

Formula Weight SrTiO_{3} = 87.62 + 47.87 + 3′(16.00) = 183.49 g/mol

Unit Cell Volume = (3.90′10^{-8} cm)^{3} = 5.93′10^{-23} cm^{3}

(5.1 g/cm^{3})′(5.93′10^{-23} cm^{3})′(mol/183.49 g)′(6.022′10^{23}/mol) = 0.99

Thus there is one formula unit per unit cell (Z=1), and the number of atoms per unit cell is : 1 Sr, 1 Ti and 3 O.

Next we compare the number of atoms in the unit cell with the multiplicities of the Wyckoff sites.

· From the multiplicities of the special positions in space group Pm3m we see that Sr must occupy either the 1a or 1b positions (otherwise there would be more than one Sr in the unit cell)

· By the same reasoning Ti must also reside in either the 1a or 1b position, and, since there are no free positional parameters (x,y or z) in either 1a or 1b, the two ions cannot occupy the same site.

· To maintain 3 oxygen ions in the unit cell it must reside at either site 3c or 3d.

If we arbitrarily put Ti at the origin (1a), then by default Sr must go to 1b. To evaluate the prospects of putting O at either 3c or 3d we calculate the Ti-O bond distances:

D (O @ 3c) = 3.90′[(0-0)^{2} + (0-0.5)^{2} + (0-0.5)^{2}]^{1/2} = 2.76Å

D (O @ 3d) = 3.90′[(0-0.5)^{2} + (0-0)^{2} + (0-0)^{2}]^{1/2} = 1.95Å

Of these two the latter (3d) is obviously more appropriate for a Ti-O bond (consult tables of ionic radii to convince yourself of this statement).

Thus we obtain the structure of SrTiO_{3} to be

具体可以参考陈小明先生编写的《单晶结构解析－原理和实践》一书(www.crystalstar.org/soft 可下载pdf版本)。

Wyckoff位置（Wyckoff Positions）用来表示晶胞中等价原子的对称性的，包括多重度(Multiplicity)，字母(Wyckoff Letter), 位置的对称性(Site Symmetry)以及该位置的分数坐标。如所说的"1a 2b 3c 16f..."实际上就是Wyckoff字母,一般它是由a开始往下写：b,c,d,...一直到最大的字母。最大的一个字母所代表的那个位置上的多重度最大,因为这些位置上如果有原子，那么就会由对称操作转出最多数目的其他等价原子。这个最大的一个字母所代表的那个位置代表的是一般性的位置，故也称为"一般性位置"（General position）；其他多重度小的位置用较小序号的字母表示，这些位置都是特殊的几个位置(往往其分数坐标的一个或多个分量是特定值，这往往是对称元素的位置确定的)，故也称"特殊位置"(Special Position)。

http://www.chemistry.ohio-state.edu/~woodward/ch754/sym_itc.htm ;

对于某个具体的空间群（Space Group，S.G.），都对应有一套Wyckoff Positions。但是对于属于该空间群的某个特定的晶体，并不是可能具有所有可能的Wyckoff Positions的（往往只具有某一些）。如果知道了某个晶体的空间群和相应的Wyckoff Positions，我们就可以利用各个Wyckoff Positions上的Coordinates关系从非对称原子（其总数一般用Z表示）导出所有的等价原子，于是，整个晶胞中的所有原子我们就可以推导出来了。

http://www.chemistry.ohio-state.edu/~woodward/ch754/sym_itc.htm

http://blog.sciencenet.cn/blog-296555-935077.html

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