How to calculate multiplicity

Symmetry

How to calculate multiplicity
=========

Yell can apply the Laue symmetry. This is a convenient feature since only the [independent cone](#Cone) of correlations have to be defined. 

Yell does not distinguish the interatomic pairs on special and average position, thus, the multiplicity of each interatomic (intermolecular) pair **must be provided manually**. If multiplicity is not provided, all the pairs except in the center of ∆PDF will get too little intensity:

![enter image description here][1]

If the multiplicity is provided, the Laue symmetry is applied properly:

![enter image description here][2]

Definition
---------

By *multiplicity* of an interatomic (intermolecular) pair we mean the number of interatomic (intermolecular) pairs which are symmetry related to the current pair. 

By symmetry we mean two distinct symmetries: the crystal space group symmetry and [combinatorial symmetry](#Combinatorial) which relates pair (A,B) to (B,A).

First way to determine multiplicity
-----------

Multiplicity can be determined by counting the symmetry equivalent pairs. 

### Example 
This example is performed in two dimensions, but results generalize trivially to three dimensions.

Assume a simple two-dimensional square crystal with a space group $p4mm$ and one atom in the unit cell:

![][3]

The zeroth neighbor connects an atom with itself. There is 1 such pair:

![][4]

The first neighbor with interatomic vector (1,0) has 4 symmetry equivalents:

![][5]

The neighbor (1,1) also has 4 symmetry equivalents:

![][6]

also the neighbor (2,0):

![enter image description here][7]

The neighbor (2,1) has 8 equivalents:

![enter image description here][8]

The neighbor (2,2) has again 4 equivalents:

![enter image description here][9]

The list of all pairs (without actual correlations) in Yell format will be the following:

    Correlations [
      [(0,0,0)
       Multiplicity 1
       ...]
       
       [(1,0,0)
       Multiplicity 4
       ...]
       
       [(1,1,0)
       Multiplicity 4
       ...]
       
       [(2,0,0)
       Multiplicity 4
       ...]
       
       [(2,1,0)
       Multiplicity 8
       ...]
       
       [(2,2,0)
       Multiplicity 4
       ...]
     ]
       
Second way to determine multiplicity {#FromSymmetry}
-----------

Sometimes it is complicated to count the number of symmetry related pairs. In such a case, the multiplicity can be calculated from the internal symmetry of a single pair.

Assume:

* $p$ is a pair
* $|G|$ is the number of symmetry elements in the crystal space group (space group order)
* $Int(p)$ is the number of symmetry elements in the internal symmetry of a pair (internal symmetry order)

Then, number of symmetry equivalent pairs is equal to:

$N_p = 2\frac{|G|}{|Int(p)|}$ if the pair connects different atoms( molecules) and
$N_p = \frac{|G|}{|Int(p)|}$ for zeroth neighbor.

The above mentioned formula is a consequence of the [orbit-stabilizer theorem](http://www.proofwiki.org/wiki/Orbit-Stabilizer_Theorem). The factor 2 appears in the formula due to [compinatorial symmetry](#Combinatorial). 

### Example

Again, assume a simple square crystal. The crystal has a plane group $p4mm$ (No. 11 in International Tables of Crystallography):

![enter image description here][10]

The plane group $p4mm$ has following symmetry operations:

$$
\begin{array}{llll}
\text{(1) $x,y$} & \text{(2) $\bar x,\bar y$} & \text{(3) $\bar y,x$} & \text{(4) $y,\bar x$} \\
\text{(5) $\bar x,y$} & \text{(6) $x,\bar y$} & \text{(7) $y,x$} & \text{(8) $\bar y,\bar x$}
\end{array}
$$

Totally there are 8 operations, thus $|G|=8$.

The zeroth neighbor has internal symmetry $4mm$:

![enter image description here][11]

The order of internal symmetry group $4mm$ is 8; this pair is a zeroth neighbor:

$$
N_{(0,0)} = \frac{|G|}{|Int(p_{(0,0)})|} = \frac{8}{8}=1
$$

Hence there is 1 such pair.

The neighbor (1,0) has internal symmetry $2mm$:

![enter image description here][12]

On the image the glide planes which are marked orange belong to the space group of the crystal, but not to the internal symmetry of the pair.

The order of $2mm$ is 4, thus the number of such pairs is:

$$
N_{(1,0)} = 2\frac{|G|}{|Int(p_{(1,0)})|} = 2\frac{8}{4}=4
$$

The neighbor (1,1) also has internal symmetry $2mm$

![enter image description here][13]

The average crystal has two additional planes and a four fold axis (marked orange) passing through coordinates (0.5,0.5), which do not belong to the internal symmetry of the pair. They are not counted.

The number of (1,1) pairs is:

$$
N_{(1,1)} = 2\frac{|G|}{|Int(p_{(1,1)})|} = 2\frac{8}{4}=4
$$

Neighbor (2,0) again has the symmetry $2mm$

![enter image description here][14]

and $N_{(2,0)} = 2*8/4=4$

Neighbor (2,1) has the symmetry $2$:

![enter image description here][15]

Thus the number of such pairs is $N_{(2,0)} = 2*8/2=8$

Neighbor (2,2) has symmetry $2mm$:

![enter image description here][16]

And thus there are 4 such pairs.

From symmetry analysis it is clearly seen:

* The zeroth neighbor has symmetry $4mm$ and only 1 equivalent.
* Neighbors $(x,0)$ and $(x,x)$ have symmetry $2mm$ and thus 4 equivalents.
* All the other neighbors $(x,y)$ have symmetry $2$ and thus 8 equivalents.

Combinatorial symmetry {#Combinatorial}
-----------

By *combinatorial equivalent* of an (ordered) pair (A,B) we mean the pair (B,A). In PDF, combinatorial pairs are obtained by reverting interatomic vectors.

Assume we have a planar NaCl-type structure:

![enter image description here][17]

The pair Na-Cl with interatomic vector (0.5,0.5) has a combinatorially equivalent pair Cl-Na with vector (-0.5,-0.5). Thus the multiplicity of such pair is 8:

![enter image description here][18]



When the multiplicity is calculated using  [the second method](#FromSymmetry), combinatorially symmetric pairs are automatically counted. The pair Na-Cl (0.5,0.5) has the internal symmetry $m$:

![][19]

The multiplicity is equal to $N_{NaCl} = 2*8/2=8$

Independent cones for all Laue groups {#Cone}
-----------

The following table summarizes the independent parts of each Laue group:

$$
\begin{array}{ccc}
\text{Laue group} &\text{Group order} &\text{Independent cone conditions} \\
m\bar 3m& 48& x \geq y \geq z \geq 0 \\
m\bar 3 & 24 & x \geq z,\; y \geq z,\; z \geq 0 \\
6/mmm & 24 & x \geq 2y \geq 0,\; z\geq0 \\
6/m & 12 & x\geq y\geq 0,\; z \geq 0 \\
\bar 3m:H &  12  &  x \geq y \geq 0,\; z \geq 0 \\
\bar 3m:R &  12  &  z \geq y \geq x,\; x+y+z \geq 0 \\
\bar 3:H &  6  & x \geq 0,\; y \geq 0,\; z \geq 0 \\
\bar 3:R & 6  & x \geq y,\; x \geq z ,\; x+y+z \geq 0 \\
4/mmm  &  16  & z \geq 0,\; x \geq y \geq 0 \\
4/m  &  8  &  x \geq 0,\; y \geq 0,\; z \geq 0 \\
mmm   &  8  &  x \geq 0,\; y \geq 0,\; z \geq 0 \\
2/m  &  4  &  z \geq 0,\; y \geq 0 \\
2/m:b  &  4  & z \geq 0,\; y \geq 0 \\
\bar 1  &  2  &  z \geq 0
\end{array}
$$


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  [9]: https://lh3.googleusercontent.com/XZrzH28SR3XrQSl2nJlB0aoUkflqMM9bV1XfL93Dlo4=s300
  [10]: https://lh4.googleusercontent.com/_q2pPCcpWjywdCnL391OhM7GeeiT7X9nnLCRsGIYBhA=s600
  [11]: https://lh3.googleusercontent.com/QH2LQbeljajEbxPlJlc5lmaa7bssClaQ--4pO3gxuyA=s600
  [12]: https://lh6.googleusercontent.com/p2Y_0uI0QJ-E4sfOTb_oMpHPghPcogAU7H7DlMM3RiM=s600
  [13]: https://lh5.googleusercontent.com/XPNTxskn1FCCARzcPrYk6JZbs696oMoWr0VeiixeZ40=s600
  [14]: https://lh4.googleusercontent.com/Xx0l08giXHN-azKwZQlFE7joCcg0U0-AGHsKXehGCBE=s600
  [15]: https://lh5.googleusercontent.com/xYBJYzKWc1meOpN1KcmHq77KihMfuMGMuSdEufRNFFw=s600
  [16]: https://lh3.googleusercontent.com/wHBFjQQuDMyJhj0ZizRvT0fvtpndqoTIIQBdZyAnarc=s600
  [17]: https://lh5.googleusercontent.com/AIBvkOR4j_t9K1sr0HifYynBOGsd59xy29vopSox2h0=s300
  [18]: https://lh4.googleusercontent.com/YcF870Bckp7yCAvJPswlwTb1jstT6OkFaCwTOWl9fH4=s300
  [19]: https://lh4.googleusercontent.com/-GHrkdfqmj8jFQbvINVpzuTfppmu8Md6aR3uhoiRxvI=s450