10. Symmetry Operations

10.1. Symmetry Operators

Symmetry operators or “sym op”s define the transformation between sets of symmetry related sets. In Horace these are implemented as the Symop class, which is subclassed to represent the three basic forms of symmetry operations:

• SymopReflection A reflection across a plane

• SymopRotation A rotation about an axis

• SymopGeneral A general matrix transform which may be the product of a series of reflections and rotations

Note

Symmetry operators are, by definition, non-scaling transformations and as such must have a determinant of 1 (Rotation) or -1 (Reflection).

10.1.1. Reflections

Reflections are implemented as the SymopReflection class and are defined in Horace by three 3-vectors; two vectors which define the plane of reflection, and a vector which defines a point on the plane (the offset). The constructor for a SymopReflection is as follows:

SymopReflection(v1, v2, offset)
>> sym = SymopReflection([1 0 0], [0 1 0], [0 0 0]); % Reflection across the Z axis

Note

For any Symop constructor the offset can be omitted and it will default to [0 0 0].

10.1.2. Rotations

Rotations are implemented as the SymopRotation class and are defined by two 3-vectors and a scalar; these are the axis of rotation, the angle (in degrees) of rotation and a point on the axis (the offset). The constructor for a SymopRotation is as follows:

SymopRotation(axis, angle, offset)
>> sym = SymopRotation([0 0 1], 60, [0 0 0]); % Rotation of 60 degrees about th Z axis

SymopRotation also provides a convenience method for generating the appropriate set of symmetry operations for cutting/reducing an n-Fold rotationally symmetric dataset about an axis. This takes a scalar integer and two 3-vectors; the number of reductions (for an angle of 360/nFold each time) and the axis and offset of the rotation as above.

SymopRotation.fold(nFold, axis, offset)
>> sym = SymopRotation.fold(4, [0 0 1], [0 0 0])   % Ready to cut from a 4-fold rotationally symmetric dataset about Z

sym =

  4x1 cell array

    {1x1 SymopIdentity}
    {1x1 SymopRotation}
    {1x1 SymopRotation}
    {1x1 SymopRotation}

>> celldisp(sym)

sym{1} =

Identity operator (no symmetrisation)


sym{2} =

Rotation operator:
       axis (rlu): [0;0;1]
      angle (deg): 90
     offset (rlu): [0;0;0]


sym{3} =

Rotation operator:
       axis (rlu): [0;0;1]
      angle (deg): 180
     offset (rlu): [0;0;0]


sym{4} =

Rotation operator:
       axis (rlu): [0;0;1]
      angle (deg): 270
     offset (rlu): [0;0;0]

10.1.3. General Transformations

Generalised matrix transforms are implemented as the SymopGeneral class and are defined by a 3x3 matrix and a 3-vector. These are the transform itself and the offset. The constructor for a SymopGeneral is as follows:

SymopGeneral(matrix, offset)
>> sym = SymopGeneral([0 1 0
                       1 0 0
                       0 0 1], [0 0 0]); % Reflection across y=x

Warning

The matrix defining a SymopGeneral must have a determinant of 1 or -1 or else this will result in an error.

It should be noted that it is possible to get the general transformation from any of the other transformation types by applying the transform to the identity (for which R is a convenience property), though this does not consider offsets.

>> sym = SymopRotation([0 1 0], 90, [0 0 0]);
>> sym.R

 ans =

     0.0000         0    1.0000
          0    1.0000         0
    -1.0000         0    0.0000

>> sym.transform_vec(eye(3))

 ans =

     0.0000         0    1.0000
          0    1.0000         0
    -1.0000         0    0.0000

10.1.4. Groups of symmetry operators

For a more complex transformation involving a series of rotations and reflections it is possible to construct an array of transformations to be applied in sequence (as a series of pre-multiplications, i.e. applied in the reverse order of the list).

% Rotate 90 deg about X, Reflect across X, Rotate back 90 deg about X
>> big_sym = [SymopRotation([1 0 0], 90), SymopReflection([0 1 0], [0 0 1]), SymopRotation([1 0 0], -90)];

10.1.5. Irreducible region

Symop transformations on pixels take what we call the irreducible region into account when transforming. The irreducible region exists to ensure that symmetry reductions reduce the data, rather than mapping the data across the symmetry transformation.

Warning

This is currently only defined for SymopReflection and SymopRotation (which is why SymopGeneral is not currently permitted for symmetric reductions).

The irreducible region for SymopReflection is defined as the the positive half-volume with respect to the normal vector of the plane of reflection. Mathematically this is defined as:

\[\lbrace{}\vec{q} \in{} Q ~|~ \vec{q} \cdot{} (\vec{u}\times{}\vec{v}) > 0 \rbrace{}\]

where \(Q\) is the set of coordinates to be transformed and \(\vec{u}\) and \(\vec{v}\) are the vectors defining the plane of reflection.

The irreducible region for SymopRotation is defined as the wedge bounded in the upper-right (positive) quadrant in the q-coordinate space by the planes defined by the absolute (relative to the q-coordinates) x-axis and the axis of rotation; and the transformed x-axis and the axis of rotation.

Note

In the special case of rotation about the x-axis, the y-axis is used to define the wedge instead of the x-axis.

Mathematically, this is defined as:

\[\lbrace{}\vec{q} \in{} Q ~|~ \vec{q} \cdot{} (\vec{n} \times{} \vec{u}) > 0 \wedge{} \vec{q} \cdot{} (\vec{v} \times{} \vec{n}) > 0 \rbrace{}, \textrm{where}~~ \vec{u}, \vec{n} \textrm{ and } \vec{v} \textrm{ are not co-planar}\]

where \(Q\) is the set of coordinates to be transformed, \(\vec{n}\) is the axis of rotation, \(\vec{u}\) is the x- (or y-) axis (as above) and \(\vec{v}\) is the transformed \(\vec{u}\).

Note

For an angle > 90 degrees or folds < 4, this will cover the positive quadrant and some of a negative domain.

10.2. Commands for cuts and slices

In Horace it is possible to symmetrise by 3 methods:

• symmetrise whole S(Q, \(\omega{}\)) objects using symmetrise_sqw

• symmetrise and extract subsets of S(Q, \(\omega{}\)) objects using cut

• equivalently to cut with symmetry, it is possible to use symmetrise_sqw and then cut

Note

While symmetrise_sqw then cut is possible, it is not recommended unless the intermediate symmetrised S(Q, \(\omega{}\)) is required. This approach has the overhead of transforming all pixels in S(Q, \(\omega{}\)), while cut has optimisations to transform only those that might contribute to the result.

Warning

Symmetrisation maps the pixels outside the irreducible region into their respective symmetry related sites. This means that subsequent binning/cutting of the sqw object will see these pixels as being on the symmetry related site rather than their original location.

Symmetrising an S(Q, \(\omega{}\)) is an irreversible operation and overwriting saved .sqw files may lead to loss of information.

10.3. Symmetrising

10.3.1. symmetrise_sqw

It is possible to reduce an entire dataset at once by symmetry, transforming all pixels according to the symmetry operations and accumulating the transformed pixels into the bins appropriately. This is done through the symmetrise_sqw function, the signature for which is below:

Warning

Due to restrictions related to the irreducible region, symmetrise_sqw is only defined for SymopReflection and SymopRotation and NOT for SymopGeneral.

>> w1 = sqw(data);

>> sym = SymopReflection([0 0 1], [1 1 0]); % Reflect about X-axis
>> w2 = symmetrise_sqw(w1, sym);

We can also combine symmetry operations:

sym = SymopReflection([0 0 1], [1 1 0]);
sym2 = SymopReflection([0 0 1], [-1 1 0]);
sym_comb = [sym, sym2];
w2 = symmetrise_sqw(wa, sym_comb);

It is also possible to reduce data through a rotationally symmetric operation:

% Perform a 6-fold rotational reduction about Z
% The resulting wedge with be a 60 degree segment
>> w1 = sqw(data);

A SymopRotation maps pixels into the Irreducible region)

>> sym = SymopRotation([0 0 1], 60);
>> w2 = symmetrise_sqw(w1, sym);

Note

Equally we could have folded the data through:

>> sym = SymopRotation.fold(6, [0 0 1]); % Same as above
>> w3 = symmetrise_sqw(w1, sym);

And they would be equivalent

>> equal_to_tol(w2, w3);

ans =

   logical

   1

10.3.2. gen_sqw

If you need to symmetrise a large sqw object, it can also be done during sqw generation, i.e. during generation of the sqw file, rather than after the object has been created. The gen_sqw function has a special option transform_sqw which can be used with any method, transforming an S(Q, \(\omega{}\)) at generation time.

For example:

sym = SymopReflection(v1, v2, offset);
gen_sqw (spefile, par_file, sym_sqw_file, efix, emode, alatt, angdeg,...  u, v, psi, omega, dpsi, gl,
         gs, 'transform_sqw', @(x)(symmetrise_sqw(x,sym)))

or, more generally:

gen_sqw (spefile, par_file, sym_sqw_file, efix, emode, alatt, angdeg,...  u, v, psi, omega, dpsi, gl,
         gs, 'transform_sqw', @user_symmetrisation_routine)

where spefile, par_file, etc… are the options used during initial sqw file generation (see Generating SQW files). The first gen_sqw would build a .sqw file reflected as in the example for the reflection above. In the second, more general, case the user defined function (in a .m-file on the Matlab path) can define multiple symmetrisation operations that are applied sequentially to the entire data. An example is as follows, which folds a cubic system so that all eight of the symmetrically equivalent regions are folded onto each other:

function wout = user_symmetrisation_routine(win)

%fold about line (1,1,0) in HK plane
wout = symmetrise_sqw(win, SymopReflection([1,1,0], [0,0,1]));
%fold about line (-1,1,0) in HK plane
wout = symmetrise_sqw(wout,SymopReflection([-1,1,0],[0,0,1]));
%fold about line (1,0,1) in HL plane
wout = symmetrise_sqw(wout,SymopReflection([1,0,1], [0,1,0]));
%fold about line (1,0,-1) in HL plane
wout = symmetrise_sqw(wout,SymopReflection([1,0,-1],[0,1,0]));

end

Warning

When defining the function to apply the symmetrisation (as above) one can only use symmetry operations supported by symmetrise_sqw. Any other transformations may modify the data ranges in unexpected ways, making the resulting transformed sqw file into complete nonsense!

Note

Due to a quirk in MATLAB’s function loading, in order to work with parallel Horace (c.f. Running Horace in Parallel) it is necessary that the symmetrisation function is in the same folder as the generation script.

if isempty(which('my_additional_user_routine'))
    addpath('/home/myFedID/path_to_my_additional_user_routine');
end

Alternatively with an array of Symop objects this could be done in one step as:

sym = [SymopReflection([1,1,0], [0,0,1])
       SymopReflection([-1,1,0],[0,0,1])
       SymopReflection([1,0,1], [0,1,0])
       SymopReflection([1,0,-1],[0,1,0])];
gen_sqw (spefile, par_file, sym_sqw_file, efix, emode, alatt, angdeg,...  u, v, psi, omega, dpsi, gl,
         gs, 'transform_sqw', @(x)(symmetrise_sqw(x,sym)))

10.4. Cutting

In order to do a symmetrised cut, the ordinary cut function (see cut) is used with the appropriate symmetry operations additionally passed into the function as an argument after the bin axes specification (see example below). The cut operation will then use the symmetry operations to compute the transformations of the given projection, accumulate the symmetrically-related pixels into the primary binning axes (the cut region specified in the cut operation) and transform their pixel coordinates according to the symmetry operations as though the S(Q, \(\omega{}\)) had been symmetrised.

>> w1 = sqw(data);

% Take 2D cut from w1
>> sym = SymopReflection([0 1 0], [0 0 1]);
>> w3 = cut(w1, ortho_proj([1 0 0], [0 1 0]), [0.2 0.1 0.8], [32 2 70], [-inf inf], [-inf inf], sym)

Representation of w3’s cut. The primary axes are within the rectangle specified by the two corners (0.2,32) and (0.8, 70). The reflection about the Y-axis captures the data in the region between (-0.2, 32) and (-0.8, 70) which are transformed by the symmetry operation into the primary axes and accumulated into the cut.

w = sqw(...)

wout = cut(w, ...);

% 2 cuts (identity always included), 2 quadrants
sym = {SymopReflection([1 1 0], [0 0 1])}
wout = cut(w, ...);

% 3 cuts, 3 quadrants
sym = {SymopReflection([1 1 0], [0 0 1]), ...
       SymopReflection([-1 1 0], [0 0 1])}
w_out = cut(w, ..., sym)

% Cut all 4 quadrants and combine
sym = {SymopReflection([1 1 0], [0 0 1]), ...
       SymopReflection([-1 1 0], [0 0 1]), ...
       [SymopReflection([1 1 0], [0 0 1]), ...
        SymopReflection([-1 1 0], [0 0 1])]}
w_out = cut(w, ..., sym)

10.5. Combining

wout=combine_sqw(win)

Combine two sqw objects (w1 and w2) of the same dimensionality into a single sqw object in order to improve statistics.

Note

The output object will have a combined value for the integration range e.g. combining two 2d slices taken at L=1 and L=2 will result in an output for which the stated value of L is L=1.5.

Note

Two objects which use different projection axes can be combined. The output object will have the projection axes of w1.

10.6. Rebinning

Resize the bin boundaries along one or more axes, and rebin the data accordingly. There are several possibilities for the input format:

wout = rebin_sqw(win,step1,step2,...)

Rebin the sqw object win with bins along the first axis that have width step1, bins along the second axis (if there is one) with width step2, and so on. The original limits of the axes will be retained. To leave an axis unaltered, the corresponding step argument can be set to 0.

wout = rebin_sqw(win,[lo1,step1,hi1],[lo2,step2,hi2],...)

As above, but specifying new upper and lower limits along each of the axes to be rebinned.

wout = rebin_sqw(win,w2)

Rebin the sqw object win with the boundaries (and projection axes) of the template object w2.

10.7. Symmetrise data, then unfold back to original range

Warning

For producing plots only, any analysis on these results will be invalid due to multiple counting of data.

Below we show a script that uses the symmetrise_sqw and combine_sqw commands to reduce a dataset and then unfold it. In the example we have a constant energy slice in the (h,k)-plane which we reduce twice to obtain the positive quadrant. We then reflect the result in the opposite direction and combine with the positive quadrant, then reflect this and combine. This produces an image which covers all four quadrants of the original with the reduced dataset (thereby increasing the counts four-fold).

%The original data
proj2 = ortho_proj([1,0,0], [0,1,0]);
hkplane = cut_sqw(sqw_file,proj2,[-2,0.05,2],[-2,0.05,2],[-0.05,0.05],[13,16]);
plot(smooth(d2d(hkplane)));

%Fold twice to get into a quadrant. Note order of vectors
sym = [SymopReflection([0,0,1],[0,1,0])
       SymopReflection([1,0,0],[0,0,1])];
fold2 = symmetrise_sqw(hkplane,sym);

%Check the result
plot(smooth(d2d(fold2)));

%Fold this back again (reverse order of vectors in first fold)
sym = SymopReflection([0,1,0],[0,0,1]);
fold2a = symmetrise_sqw(fold2,sym);
plot(smooth(d2d(fold2a)))

%Combine with what you started with
combi1 = combine_sqw(fold2,fold2a);
plot(smooth(d2d(combi1)));

%Fold back again (reverse order of vectors in second fold)
sym = SymopReflection([0,0,1],[1,0,0]);
fold3a = symmetrise_sqw(combi1, sym);
plot(fold3a)

%Combine and plot
combi2 = combine_sqw(combi1,fold3a);
plot(smooth(d2d(combi2)));

10.8. Limitations

• At present symmetrise_sqw, combine_sqw, and rebin_sqw work ONLY for sqw objects, since they require access to individual detector pixel information. The functions will work for any dimensionality of object, however.

10.9. Symop Methods - Advanced

Symop objects have methods to transform a variety of objects which may be related by symmetry. These are:

• transform_vec

• transform_pix

• transform_proj

Which transform numeric vectors, PixelDataBase objects and aProjection objects respectively.

10.9.1. transform_vec

transform_vec takes a 3xN list of 3-vectors to transform. This method can be applied directly from a single Symop or from an array (but not cell array) of Symop objects (see: Groups of symmetry operators).

>> sym = SymopReflection([1 0 0], [0 1 0])

sym =

Reflection operator:
 In-plane u (rlu): [1;0;0]
 In-plane v (rlu): [0;1;0]
     offset (rlu): [0;0;0]

>> sym.transform_vec([3; 6; 1])

ans =

     3
     6
    -1

>> big_sym = [SymopRotation([1 0 0], 90), SymopReflection([0 1 0], [0 0 1]), SymopRotation([1 0 0], -90)];
                         %v1|v2|v3|v4|v5
>> big_sym.transform_vec([1, 3, 5, 1, 3
                          2, 2, 4, 6, 1
                          6, 3, 1, 3, 6])

ans =
    %v1'| v2'|  v3'|  v4'|  v5'
    -1    -3    -5    -1    -3
     2     2     4     6     1
     6     3     1     3     6

10.9.2. transform_pix

transform_pix takes a PixelDataBase derived object and transforms the contained pixel q-coordinates according to the symmetry operations and returns a new object with the transformed pixels.

Note

transform_pix takes the Symop object’s Irreducible region into account and does not transform the pixels which are considered to be within the irreducible region.

>> sym = SymopReflection([1 0 0], [0 1 0]);
% 5 pixels in memory
>> pix = PixelDataMemory(rand(9, 5));
% Put pixels outside of "positive quadrant"
>> pix.q_coordinates(:, [1 2]) = -pix.q_coordinates(:, [1 2]);
>> pix_new = sym.transform_pix(pix);
>> pix.data

ans =

   -0.4898   -0.1190    0.6991    0.8143    0.8308 % q_x
   -0.4456   -0.4984    0.8909    0.2435    0.5853 % q_y
   -0.6463   -0.9597    0.9593    0.9293    0.5497 % q_z
    0.7094    0.3404    0.5472    0.3500    0.9172 % dE
    0.7547    0.5853    0.1386    0.1966    0.2858 % run_idx
    0.2760    0.2238    0.1493    0.2511    0.7572 % detector_idx
    0.6797    0.7513    0.2575    0.6160    0.7537 % energy_idx
    0.6551    0.2551    0.8407    0.4733    0.3804 % signal
    0.1626    0.5060    0.2543    0.3517    0.5678 % variance

>> pix_new.data

ans =

   -0.4898   -0.1190    0.6991    0.8143    0.8308 % q_x
   -0.4456   -0.4984    0.8909    0.2435    0.5853 % q_y
    0.6463    0.9597    0.9593    0.9293    0.5497 % q_z
    0.7094    0.3404    0.5472    0.3500    0.9172 % dE
    0.7547    0.5853    0.1386    0.1966    0.2858 % run_idx
    0.2760    0.2238    0.1493    0.2511    0.7572 % detector_idx
    0.6797    0.7513    0.2575    0.6160    0.7537 % energy_idx
    0.6551    0.2551    0.8407    0.4733    0.3804 % signal
    0.1626    0.5060    0.2543    0.3517    0.5678 % variance

10.9.3. transform_proj

transform_proj is used to transform subclasses of the aProjection type. It is an internal function which creates a new projection with the symmetries applied and is not normally needed by users, but is recorded here for completeness.