Workflows

Command line tools for data reduction using garnet reduction.

The main command for setting up and running reduction is reduce.sh. It is a tool that when given a reduction flag, parses through an input reduction plan YAML file and executes the reduction as a parallel job.

Running a reduction using reduce.sh.

/SNS/software/scd/reduce.sh -i[n] INPUT.yaml PROCESSES

• n \(=\mathrm{normalization}\)

• i \(=\mathrm{integration}\)

It is possible to generate a template input file for a given instrument using the command reduce.sh. The file must not exist previously.

Creating a template input file.

/SNS/software/scd/reduce.sh -t INPUT.yaml INSTRUMENT

Valid instrument input option names.

Instrument	Name	Beamline	Example configuration files
WAND²	wand2	hb2c	/HFIR/HB2C/shared/benchmark/*/wand2_reduction_plan.yaml
DEMAND	demand	hb3a	/HFIR/HB3A/shared/benchmark/*/demand_reduction_plan.yaml
SNAP	snap	bl3	/SNS/SNAP/shared/benchmark/*/snap_reduction_plan.yaml
CORELLI	corelli	bl9	/SNS/CORELLI/shared/benchmark/*/corelli_reduction_plan.yaml
MANDI	mandi	bl11b	/SNS/MANDI/shared/benchmark/*/mandi_reduction_plan.yaml
TOPAZ	topaz	bl12	/SNS/TOPAZ/shared/benchmark/*/topaz_reduction_plan.yaml

Parameters

To setup and run any reduction workflow, it is necessary to define some common parameters relevant to all. Bold indicates required parameter.

• Instrument name: string

• Proposal IPTS number: integer

• Run/scan number(s): string

• UB matrix file: string

Initial configuration: instrument, proposal number, run number, and UB matrix file.

Instrument: "INSTRUMENT" # "TOPAZ", "CORELLI", "MANDI", "SNAP", "DEMAND", "WAND²"
IPTS: 12345
Runs: "10000:10020,10025:10030"
UBFile: null # .mat file required

Run numbers are specified using a shorthand run number string.

Note

The shorthand notation involves : for range and , for separations in the sequence. For example, 123:130,132:135 is equivalent to a list 123,124,125,…,130,132,133,…,135.

For DEMAND, an experiment number should also be specified.

• Experiment: integer

DEMAND-only experiment number.

Experiment: 100

A vanadium file must be provided for all instruments. An optional background file may also be specified.

• Vanadium file: string

• Background file: string

For time-of-flight (TOF) data, the vanadium (also sometimes called the solid-angle) is the integrated detector counts from a vanadium measurement. The vanadium file is similar for monochromatic data. However, rather than being measured in events with counts vs detector ID, it is a detector image of counts and must be integrated over all gonionmeter angles. The background file is similar to the vanadium file and need not necessarily be normalized to counting statistics (e.g. proton charge or counting time).

Vanadium data.

VanadiumFile: /FACILITY/INSTRUMENT/shared/path/to/vanadium.nxs
BackgroundFile: /FACILITY/INSTRUMENT/shared/path/to/background.nxs

Additional calibration files only for TOF data can also be specified. For TOF data, the flux file must be in units of momentum.

• Flux file: string

• Masking file: string

• Detector calibration: string

Calibration data only for TOF instruments.

FluxFile: /FACILITY/INSTRUMENT/shared/path/to/flux.nxs
MaskFile: /FACILITY/INSTRUMENT/shared/path/to/mask.xml
DetectorCalibration: /FACILITY/INSTRUMENT/shared/path/to/calibration.xml # .xml or .DetCal

A CORELLI-only tube calibration is available that adjust the positions of tubes within a sixteenpack.

CORELLI-only tube calibration.

TubeCalibration: /SNS/CORELLI/shared/calibration/tube/calibration_corelli_20200109.nxs.h5

For CORELLI, it is also possible to load and/or generate the elastic-only data from the cross-correlated data. If the elastic option is provided, a time offset for the correlation chopper may also be provided to recalculate the elastic signal. Otherwise, it is obtained from the auto-reduced data.

• Elastic signal: boolean

• Time offset: integer

CORELLI-only elastic-only cross-correlation data.

Elastic: true # false
TimeOffset: None

Note

Elastic options may only be used for CORELLI data with the correlation chopper running.

Normalization

The normalization procedure (re)-constructs a reciprocal space map binned with appropriate normalization and in reciprocal lattice units. The flag is -n.

Input

It is necessary to provide the binning parameters that include;

• Projections: lists of integers

• Extents: lists of floats

• Bins: list of integers

Binning parameters.

Projections: [[1, 0, 0], [0, 1, 0], [0, 0, 1]] # columns must form a definite matrix
Extents: [[-16, 16], [-16, 16], [-16, 16]] # (min, max) pair of bin-centers for each dimension
Bins: [321, 321, 321] # number of bins

The projection vectors define the axes direction and must form a definite matrix from stacking the vectors by columns.

Common projections with type indicated by first two axes.

Type	Projections
\(hk0\)	[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
\(hhl\)	[[1, 1, 0], [0, 0, 1], [-1, 1, 0]]
\(h0l\)	[[1, 0, 0], [0, 0, 1], [0, 1, 0]]
\(0kl\)	[[0, 1, 0], [0, 0, 1], [1, 0, 0]]

The extents are defined as the minimum and maximum of the the bin-center values along that axis. The bin size is calculated as \((\mathrm{max}-\mathrm{min})/(\mathrm{bins}-1)\). An extra bin +1 is required to account for both edges. A bin size of 1 is also valid and the bin size is \((\mathrm{max}-\mathrm{min})\).

Symmetry operations from a space or point group corresponding to the underlying Laue group can be applied by providing a valid symbol. Options are listed below.

• Symmetry: string

Symmetry to apply.

 Symmetry: null # null, space/point group symbol

Symmetry is not required to be applied and can be specified accordingly with null.

Output

Given the configuration file as OUTPUT.yaml, a minimum of three files will be created. The files will be appended with "symm", the Laue class symbol.

Reciprocal space (re)-construction output files.

Symmetry:	null	"symm"
Normalized result:	OUTPUT.nxs	OUTPUT_symm.nxs
Raw counts:	OUTPUT_data.nxs	OUTPUT_symm_data.nxs
Inverse weights:	OUTPUT_norm.nxs	OUTPUT_symm_norm.nxs

Additional files will be created if an optional background file is provided.

Reciprocal space (re)-construction background files.

Symmetry:	null	"symm"
Subtracted result:	OUTPUT_sub_bkg.nxs	OUTPUT_symm_sub_bkg.nxs
Background result:	OUTPUT_bkg.nxs	OUTPUT_bkg_symm.nxs
Background counts:	OUTPUT_bkg_data.nxs	OUTPUT_symm_bkg_data.nxs
Background inverse weights:	OUTPUT_bkg_norm.nxs	OUTPUT_symm_bkg_norm.nxs

Note

The data output corresponds to the raw counts while the norm relates to the normalization factors (proton charge, counting time, and other corrections measured with vanadium). The result of dividing data with norm is the normalized data. The prior outputs are intermediate. Additional measurements can be added before dividing to obtain a new updated normalized data. The corresponding outputs are similar for background data. The subtracted result is the normalized background subtracted from the normalized data.

Examples

The following provides examples illustrating how the output file can be visualized. Several convenience functions are defined that makes allows for reading the file with h5py, calculating the appropriate skew transforms, and plotting with matplotlib.

Utilities for plotting slices from three-dimensional histograms of reciprocal space data.

 import h5py

 import matplotlib.pyplot as plt
 import matplotlib.transforms

 import numpy as np
 import scipy.linalg

 def load_histograms(filename):

     with h5py.File(filename, mode='r') as f:

         exp_info = f['MDHistoWorkspace/experiment0']
         UB = exp_info['sample/oriented_lattice/orientation_matrix'][()].reshape(3,3)
         W = exp_info['logs/W_MATRIX/value'][()].reshape(3,3)

         data = f['MDHistoWorkspace/data']
         signal = data['signal'][()].T

         axes, labels, = [], []

         for i, key in enumerate(data):
             if key.startswith('D') or key.startswith('Q'):
                 ax = data[key]
                 axes.append((ax[1:]+ax[:-1])/2)
                 name = ax.attrs['long_name'].decode('utf-8')
                 units = ax.attrs['units'].decode('utf-8')
                 labels.append('{} ({})'.format(name, units))

     return signal, axes, labels, UB, W

 def slice_transforms(signal, axes, labels, UB, W, normal=[0,0,1], value=0):

     G = np.dot(UB.T, UB)

     B = scipy.linalg.cholesky(G, lower=False)
     Bp = np.dot(B, W)

     Q, R = scipy.linalg.qr(Bp)

     ind = np.array(normal) != 1

     axes_ind = np.arange(3)[ind]
     slice_ind = np.arange(3)[~ind][0]

     coords = [axes[i] for i in axes_ind]
     titles = [labels[i] for i in axes_ind]

     z = axes[slice_ind]
     i = np.argmin(np.abs(z-value))

     titles.append('{} = {:.2f}'.format(labels[slice_ind], z[i]))

     if slice_ind == 0:
         data = signal[i,:,:].T
     elif slice_ind == 1:
         data = signal[:,i,:].T
     else:
         data = signal[:,:,i].T

     v = scipy.linalg.cholesky(np.dot(R.T, R)[ind][:,ind], lower=False)
     v /= v[0,0]

     T = np.eye(3)
     T[:2,:2] = v

     s = np.diag(T).copy()
     T[1,1] = 1
     T[0,2] = -T[0,1]*coords[1].min()

     return coords, data, titles, T, s[1]

 def plot_slice(coords, data, T=[[1,0,0],[0,1,0],[0,0,1]], aspect=1, titles=['','','']):

     x, y = coords

     fig, ax = plt.subplots(1, 1)

     transform = matplotlib.transforms.Affine2D(T)+ax.transData

     im = ax.pcolormesh(x,
                        y,
                        data,
                        norm='log',
                        shading='nearest',
                        transform=transform)

     ax.set_aspect(aspect)
     ax.set_xlabel(titles[0])
     ax.set_ylabel(titles[1])
     ax.set_title(titles[2])
     ax.minorticks_on()

     ax.xaxis.get_major_locator().set_params(integer=True)
     ax.yaxis.get_major_locator().set_params(integer=True)

     cb = fig.colorbar(im, ax=ax)
     cb.minorticks_on()

The convience functions can be reused as needed.

filename = '/HFIR/HB2C/shared/benchmark/norm/normalization/wand2_reduction_plan_m-3m.nxs'

signal, axes, labels, UB, W = load_histograms(filename)
coords, data, titles, T, aspect = slice_transforms(signal, axes, labels, UB, W, normal=[0,0,1], value=0)
plot_slice(coords, data, T, aspect, titles)

WAND² NaCl data symmetrized and plotted at \((hk0)\).

filename = '/HFIR/HB3A/shared/benchmark/norm/normalization/demand_reduction_plan_mmm.nxs'

signal, axes, labels, UB, W = load_histograms(filename)

coords, data, titles, T, aspect = slice_transforms(signal, axes, labels, UB, W, normal=[0,0,1], value=0)
plot_slice(coords, data, T, aspect, titles)

DEMAND LiFePO₄ data symmetrized and plotted at \((hk0)\).

filename = '/HFIR/HB3A/shared/benchmark/norm/normalization/demand_reduction_plan_mmm.nxs'

signal, axes, labels, UB, W = load_histograms(filename)

coords, data, titles, T, aspect = slice_transforms(signal, axes, labels, UB, W, normal=[0,1,0], value=0)
plot_slice(coords, data, T, aspect, titles)

DEMAND LiFePO₄ data symmetrized and plotted at \((h0l)\).

filename = '/HFIR/HB3A/shared/benchmark/norm/normalization/demand_reduction_plan_mmm.nxs'

signal, axes, labels, UB, W = load_histograms(filename)

coords, data, titles, T, aspect = slice_transforms(signal, axes, labels, UB, W, normal=[1,0,0], value=0)
plot_slice(coords, data, T, aspect, titles)

DEMAND LiFePO₄ data symmetrized and plotted at \((0kl)\).

filename = '/SNS/CORELLI/shared/benchmark/norm/normalization/corelli_reduction_plan_m-3m.nxs'

signal, axes, labels, UB, W = load_histograms(filename)

coords, data, titles, T, aspect = slice_transforms(signal, axes, labels, UB, W, normal=[0,0,1], value=0)
plot_slice(coords, data, T, aspect, titles)

filename = '/SNS/TOPAZ/shared/benchmark/norm/normalization/topaz_reduction_plan_(h,k,0)_[-10,10]_[-10,10]_[-10,10]_201x201x201_2_m.nxs'

signal, axes, labels, UB, W = load_histograms(filename)

coords, data, titles, T, aspect = slice_transforms(signal, axes, labels, UB, W, normal=[0,0,1], value=0)
plot_slice(coords, data, T, aspect, titles)

filename = '/SNS/TOPAZ/shared/benchmark/norm/normalization/topaz_reduction_plan_(h,k,0)_[-10,10]_[-10,10]_[-10,10]_201x201x201_2_m.nxs'

signal, axes, labels, UB, W = load_histograms(filename)

coords, data, titles, T, aspect = slice_transforms(signal, axes, labels, UB, W, normal=[0,1,0], value=0)
plot_slice(coords, data, T, aspect, titles)

filename = '/SNS/TOPAZ/shared/benchmark/norm/normalization/topaz_reduction_plan_(h,k,0)_[-10,10]_[-10,10]_[-10,10]_201x201x201_2_m.nxs'

signal, axes, labels, UB, W = load_histograms(filename)

coords, data, titles, T, aspect = slice_transforms(signal, axes, labels, UB, W, normal=[1,0,0], value=0)
plot_slice(coords, data, T, aspect, titles)

Integration

Integrating the peaks from a single crystal diffraction experiment is invoked with the -i flag.

Input

The workflow uses the UB to predict peaks based on the reflection condition.

• Cell: string

• Centering string

• Intial radius: float

• Minimum d-spacing: float

Integration parameters.

Cell: "Triclinic" # "Monoclinic", "Orthorhombic", "Tetragonal", "Rhombohedral", "Hexagonal", "Cubic"
Centering: "P" # "I", "F", "R(obv)", "R(rev)", "A", "B", "C"
Radius: 0.25
MinD: 0.7

The cell corresponds to the lattice system and is used to constrain the refinement of the lattice parameters.

Parameter constraints for each lattice system.

Lattice system	Lengths	Angles	Point Group
Cubic	\(a=b=c\)	\(\alpha=\beta=\gamma=90^\circ\)	\(m\bar{3}m\)
Hexagonal	\(a=b\)	\(\alpha=\beta=90^\circ\), \(\gamma=120^\circ\)	\(6/mmm\)
Rhombohedral	\(a=b=c\)	\(\alpha=\beta=\gamma\)	\(\bar{3}m\)
Tetragonal	\(a=b\)	\(\alpha=\beta=\gamma=90^\circ\)	\(4/mmm\)
Orthorhombic	None	\(\alpha=\beta=\gamma=90^\circ\)	\(mmm\)
Monoclinic	None	\(\alpha=\gamma=90^\circ\)	\(2/m\)
Triclinic	None	None	\(\bar{1}\)

The centering symbol only predicts peaks that correspond to the reflection condition.

Integral reflection conditions for centered cells.

Primitive	Base			Body	Face	Rhombohedral
\(Pm\bar{3}m\)				\(Im\bar{3}m\)	\(Fm\bar{3}m\)
\(P6/mmm\)						\(R\bar{3}m\)
\(R\bar{3}m\)
\(P4/mmm\)				\(I4/mmm\)
\(Pmmm\)	\(Cmmm\)	\(Ammm\)	\(Bmmm\)	\(Immm\)	\(Fmmm\)
\(P2/m\)	\(C2/m\)	\(A2/m\)	\(B2/m\)
\(P\bar{1}\)
	\(h+k=2n\)	\(k+l=2n\)	\(l+h=2n\)	\(h+k+l=2n\)	\(h,k,l\) unmixed	\(-h+k+l=3n\) or \(h-k+l=3n\)

Note

For a Rhombohedral lattice, use R (equivalent to P) for the reflection condition. This is not to be confused with a Hexagonal cell with R(obverse) or R(reverse) centering.

Both the minimum \(d\)-spacing and initial peak radius are given in units of \(\mathrm{Å}\). It is also possible to include a minimum modulation \(d\)-spacing that is greater than or equal to the overall minimum \(d\)-spacing. If unset, it defaults to the minimum overall \(d\)-spacing. This parameter is desirable if the intensity of the satellite peaks quickly falls off in \(Q\) commonly observed for magnetic materials.

• Minimum modulation \(d\)-spacing: float

• Modulation vector 1/2/3: list

• Max order: integer

• Cross terms: boolean

Modulation parameters.

ModVec1: [0, 0, 0]
ModVec2: [0, 0, 0]
ModVec3: [0, 0, 0]
MaxOrder: 0
CrossTerms: false

Modulation vector fractional offsets.

Vector 1 \(\times(\pm m)\)	Vector 2 \(\times(\pm n)\)	Vector 3 \(\times(\pm p)\)
\(\Delta{h}_1\)	\(\Delta{h}_2\)	\(\Delta{h}_3\)
\(\Delta{k}_1\)	\(\Delta{k}_2\)	\(\Delta{k}_3\)
\(\Delta{l}_1\)	\(\Delta{l}_2\)	\(\Delta{l}_3\)

The modulation vectors are the offsets in reciprocal lattice units. They are predicted away from a parent nuclear Bragg peak so that it follows the provided centering reflection condition. The opposite of \((\Delta{h}_i,\Delta{k}_i,\Delta{l}_i)\) is always included in the search \((-\Delta{h}_i,-\Delta{k}_i,-\Delta{l}_i)\). Maximum order must be greater than zero to predict any satellite peak. This value is the maximum absolute integer of \(m\), \(n\), and \(p\). Cross terms (typically false) if true, include \((m,n,p)\) without constraint that two components are zero. Othewise, only one component is allowed to be nonzero at a time: \((m,0,0)\), \((0,n,0)\), and \((0,0,p)\).

Symbols

Some options of the YAML file rely on knowledge of certain crystallographic symbols. The tables below lists look up symbols for space groups. See Crystallographic Space Group Diagrams and Tables.

Triclinic

Space group, point group, lattice system, and crystal systems.

Space Group Number	Space Group Symbol	Point Group	Lattice System	Crystal System
1	P1	1	Triclinic	Triclinic
2	P-1	-1	Triclinic	Triclinic

Monoclinic

Space group, point group, lattice system, and crystal systems.

Space Group Number	Space Group Symbol	Point Group	Lattice System	Crystal System
3	P112	112	Monoclinic	Monoclinic
3	P121	2	Monoclinic	Monoclinic
4	P1121	112	Monoclinic	Monoclinic
4	P1211	2	Monoclinic	Monoclinic
5	A112	112	Monoclinic	Monoclinic
5	A121	2	Monoclinic	Monoclinic
5	B112	112	Monoclinic	Monoclinic
5	C121	2	Monoclinic	Monoclinic
5	I112	112	Monoclinic	Monoclinic
5	I121	2	Monoclinic	Monoclinic
6	P11m	11m	Monoclinic	Monoclinic
6	P1m1	m	Monoclinic	Monoclinic
7	P11a	11m	Monoclinic	Monoclinic
7	P11b	11m	Monoclinic	Monoclinic
7	P11n	11m	Monoclinic	Monoclinic
7	P1a1	m	Monoclinic	Monoclinic
7	P1c1	m	Monoclinic	Monoclinic
7	P1n1	m	Monoclinic	Monoclinic
8	A11m	11m	Monoclinic	Monoclinic
8	A1m1	m	Monoclinic	Monoclinic
8	B11m	11m	Monoclinic	Monoclinic
8	C1m1	m	Monoclinic	Monoclinic
8	I11m	11m	Monoclinic	Monoclinic
8	I1m1	m	Monoclinic	Monoclinic
9	A11a	11m	Monoclinic	Monoclinic
9	A1n1	m	Monoclinic	Monoclinic
9	B11n	11m	Monoclinic	Monoclinic
9	C1c1	m	Monoclinic	Monoclinic
9	I11b	11m	Monoclinic	Monoclinic
9	I1a1	m	Monoclinic	Monoclinic
10	P112/m	112/m	Monoclinic	Monoclinic
10	P12/m1	2/m	Monoclinic	Monoclinic
11	P1121/m	112/m	Monoclinic	Monoclinic
11	P121/m1	2/m	Monoclinic	Monoclinic
12	A112/m	112/m	Monoclinic	Monoclinic
12	A12/m1	2/m	Monoclinic	Monoclinic
12	B112/m	112/m	Monoclinic	Monoclinic
12	C12/m1	2/m	Monoclinic	Monoclinic
12	I112/m	112/m	Monoclinic	Monoclinic
12	I12/m1	2/m	Monoclinic	Monoclinic
13	P112/a	112/m	Monoclinic	Monoclinic
13	P112/b	112/m	Monoclinic	Monoclinic
13	P112/n	112/m	Monoclinic	Monoclinic
13	P12/a1	2/m	Monoclinic	Monoclinic
13	P12/c1	2/m	Monoclinic	Monoclinic
13	P12/n1	2/m	Monoclinic	Monoclinic
14	P1121/a	112/m	Monoclinic	Monoclinic
14	P1121/b	112/m	Monoclinic	Monoclinic
14	P1121/n	112/m	Monoclinic	Monoclinic
14	P121/a1	2/m	Monoclinic	Monoclinic
14	P121/c1	2/m	Monoclinic	Monoclinic
14	P121/n1	2/m	Monoclinic	Monoclinic
15	A112/a	112/m	Monoclinic	Monoclinic
15	A12/n1	2/m	Monoclinic	Monoclinic
15	B112/n	112/m	Monoclinic	Monoclinic
15	C12/c1	2/m	Monoclinic	Monoclinic
15	I112/b	112/m	Monoclinic	Monoclinic
15	I12/a1	2/m	Monoclinic	Monoclinic

Orthorhombic

Space group, point group, lattice system, and crystal systems.

Space Group Number	Space Group Symbol	Point Group	Lattice System	Crystal System
16	P222	222	Orthorhombic	Orthorhombic
17	P2221	222	Orthorhombic	Orthorhombic
17	P2212	222	Orthorhombic	Orthorhombic
17	P2122	222	Orthorhombic	Orthorhombic
18	P22121	222	Orthorhombic	Orthorhombic
18	P21221	222	Orthorhombic	Orthorhombic
18	P21212	222	Orthorhombic	Orthorhombic
19	P212121	222	Orthorhombic	Orthorhombic
20	A2122	222	Orthorhombic	Orthorhombic
20	B2212	222	Orthorhombic	Orthorhombic
20	C2221	222	Orthorhombic	Orthorhombic
21	A222	222	Orthorhombic	Orthorhombic
21	B222	222	Orthorhombic	Orthorhombic
21	C222	222	Orthorhombic	Orthorhombic
22	F222	222	Orthorhombic	Orthorhombic
23	I222	222	Orthorhombic	Orthorhombic
24	I212121	222	Orthorhombic	Orthorhombic
25	P2mm	2mm	Orthorhombic	Orthorhombic
25	Pm2m	m2m	Orthorhombic	Orthorhombic
25	Pmm2	mm2	Orthorhombic	Orthorhombic
26	P21am	2mm	Orthorhombic	Orthorhombic
26	P21ma	2mm	Orthorhombic	Orthorhombic
26	Pb21m	m2m	Orthorhombic	Orthorhombic
26	Pcm21	mm2	Orthorhombic	Orthorhombic
26	Pm21b	m2m	Orthorhombic	Orthorhombic
26	Pmc21	mm2	Orthorhombic	Orthorhombic
27	P2aa	2mm	Orthorhombic	Orthorhombic
27	Pb2b	m2m	Orthorhombic	Orthorhombic
27	Pcc2	mm2	Orthorhombic	Orthorhombic
28	P2cm	2mm	Orthorhombic	Orthorhombic
28	P2mb	2mm	Orthorhombic	Orthorhombic
28	Pbm2	mm2	Orthorhombic	Orthorhombic
28	Pc2m	m2m	Orthorhombic	Orthorhombic
28	Pm2a	m2m	Orthorhombic	Orthorhombic
28	Pma2	mm2	Orthorhombic	Orthorhombic
29	P21ab	2mm	Orthorhombic	Orthorhombic
29	P21ca	2mm	Orthorhombic	Orthorhombic
29	Pb21a	m2m	Orthorhombic	Orthorhombic
29	Pbc21	mm2	Orthorhombic	Orthorhombic
29	Pc21b	m2m	Orthorhombic	Orthorhombic
29	Pca21	mm2	Orthorhombic	Orthorhombic
30	P2an	2mm	Orthorhombic	Orthorhombic
30	P2na	2mm	Orthorhombic	Orthorhombic
30	Pb2n	m2m	Orthorhombic	Orthorhombic
30	Pcn2	mm2	Orthorhombic	Orthorhombic
30	Pn2b	m2m	Orthorhombic	Orthorhombic
30	Pnc2	mm2	Orthorhombic	Orthorhombic
31	P21mn	2mm	Orthorhombic	Orthorhombic
31	P21nm	2mm	Orthorhombic	Orthorhombic
31	Pm21n	m2m	Orthorhombic	Orthorhombic
31	Pmn21	mm2	Orthorhombic	Orthorhombic
31	Pn21m	m2m	Orthorhombic	Orthorhombic
31	Pnm21	mm2	Orthorhombic	Orthorhombic
32	P2cb	2mm	Orthorhombic	Orthorhombic
32	Pba2	mm2	Orthorhombic	Orthorhombic
32	Pc2a	m2m	Orthorhombic	Orthorhombic
33	P21cn	2mm	Orthorhombic	Orthorhombic
33	P21nb	2mm	Orthorhombic	Orthorhombic
33	Pbn21	mm2	Orthorhombic	Orthorhombic
33	Pc21n	m2m	Orthorhombic	Orthorhombic
33	Pn21a	m2m	Orthorhombic	Orthorhombic
33	Pna21	mm2	Orthorhombic	Orthorhombic
34	P2nn	2mm	Orthorhombic	Orthorhombic
34	Pn2n	m2m	Orthorhombic	Orthorhombic
34	Pnn2	mm2	Orthorhombic	Orthorhombic
35	A2mm	2mm	Orthorhombic	Orthorhombic
35	Bm2m	m2m	Orthorhombic	Orthorhombic
35	Cmm2	mm2	Orthorhombic	Orthorhombic
36	A21am	2mm	Orthorhombic	Orthorhombic
36	A21ma	2mm	Orthorhombic	Orthorhombic
36	Bb21m	m2m	Orthorhombic	Orthorhombic
36	Bm21b	m2m	Orthorhombic	Orthorhombic
36	Ccm21	mm2	Orthorhombic	Orthorhombic
36	Cmc21	mm2	Orthorhombic	Orthorhombic
37	A2aa	2mm	Orthorhombic	Orthorhombic
37	Bb2b	m2m	Orthorhombic	Orthorhombic
37	Ccc2	mm2	Orthorhombic	Orthorhombic
38	Am2m	m2m	Orthorhombic	Orthorhombic
38	Amm2	mm2	Orthorhombic	Orthorhombic
38	B2mm	2mm	Orthorhombic	Orthorhombic
38	Bmm2	mm2	Orthorhombic	Orthorhombic
38	C2mm	2mm	Orthorhombic	Orthorhombic
38	Cm2m	m2m	Orthorhombic	Orthorhombic
39	Ae2m	m2m	Orthorhombic	Orthorhombic
39	Aem2	mm2	Orthorhombic	Orthorhombic
39	B2em	2mm	Orthorhombic	Orthorhombic
39	Bme2	mm2	Orthorhombic	Orthorhombic
39	C2me	2mm	Orthorhombic	Orthorhombic
39	Cm2e	m2m	Orthorhombic	Orthorhombic
40	Am2a	m2m	Orthorhombic	Orthorhombic
40	Ama2	mm2	Orthorhombic	Orthorhombic
40	B2mb	2mm	Orthorhombic	Orthorhombic
40	Bbm2	mm2	Orthorhombic	Orthorhombic
40	C2cm	2mm	Orthorhombic	Orthorhombic
40	Cc2m	m2m	Orthorhombic	Orthorhombic
41	Ae2a	m2m	Orthorhombic	Orthorhombic
41	Aea2	mm2	Orthorhombic	Orthorhombic
41	B2eb	2mm	Orthorhombic	Orthorhombic
41	Bbe2	mm2	Orthorhombic	Orthorhombic
41	C2ce	2mm	Orthorhombic	Orthorhombic
41	Cc2e	m2m	Orthorhombic	Orthorhombic
42	F2mm	2mm	Orthorhombic	Orthorhombic
42	Fm2m	m2m	Orthorhombic	Orthorhombic
42	Fmm2	mm2	Orthorhombic	Orthorhombic
43	F2dd	2mm	Orthorhombic	Orthorhombic
43	Fd2d	m2m	Orthorhombic	Orthorhombic
43	Fdd2	mm2	Orthorhombic	Orthorhombic
44	I2mm	2mm	Orthorhombic	Orthorhombic
44	Im2m	m2m	Orthorhombic	Orthorhombic
44	Imm2	mm2	Orthorhombic	Orthorhombic
45	I2cb	2mm	Orthorhombic	Orthorhombic
45	Iba2	mm2	Orthorhombic	Orthorhombic
45	Ic2a	m2m	Orthorhombic	Orthorhombic
46	I2cm	2mm	Orthorhombic	Orthorhombic
46	I2mb	2mm	Orthorhombic	Orthorhombic
46	Ibm2	mm2	Orthorhombic	Orthorhombic
46	Ic2m	m2m	Orthorhombic	Orthorhombic
46	Im2a	m2m	Orthorhombic	Orthorhombic
46	Ima2	mm2	Orthorhombic	Orthorhombic
47	Pmmm	mmm	Orthorhombic	Orthorhombic
48	Pnnn	mmm	Orthorhombic	Orthorhombic
48	Pnnn:2	mmm	Orthorhombic	Orthorhombic
49	Pbmb	mmm	Orthorhombic	Orthorhombic
49	Pccm	mmm	Orthorhombic	Orthorhombic
49	Pmaa	mmm	Orthorhombic	Orthorhombic
50	Pban	mmm	Orthorhombic	Orthorhombic
50	Pban:2	mmm	Orthorhombic	Orthorhombic
50	Pcna	mmm	Orthorhombic	Orthorhombic
50	Pcna:2	mmm	Orthorhombic	Orthorhombic
50	Pncb	mmm	Orthorhombic	Orthorhombic
50	Pncb:2	mmm	Orthorhombic	Orthorhombic
51	Pbmm	mmm	Orthorhombic	Orthorhombic
51	Pcmm	mmm	Orthorhombic	Orthorhombic
51	Pmam	mmm	Orthorhombic	Orthorhombic
51	Pmcm	mmm	Orthorhombic	Orthorhombic
51	Pmma	mmm	Orthorhombic	Orthorhombic
51	Pmmb	mmm	Orthorhombic	Orthorhombic
52	Pbnn	mmm	Orthorhombic	Orthorhombic
52	Pcnn	mmm	Orthorhombic	Orthorhombic
52	Pnan	mmm	Orthorhombic	Orthorhombic
52	Pncn	mmm	Orthorhombic	Orthorhombic
52	Pnna	mmm	Orthorhombic	Orthorhombic
52	Pnnb	mmm	Orthorhombic	Orthorhombic
53	Pbmn	mmm	Orthorhombic	Orthorhombic
53	Pcnm	mmm	Orthorhombic	Orthorhombic
53	Pman	mmm	Orthorhombic	Orthorhombic
53	Pmna	mmm	Orthorhombic	Orthorhombic
53	Pncm	mmm	Orthorhombic	Orthorhombic
53	Pnmb	mmm	Orthorhombic	Orthorhombic
54	Pbaa	mmm	Orthorhombic	Orthorhombic
54	Pbab	mmm	Orthorhombic	Orthorhombic
54	Pbcb	mmm	Orthorhombic	Orthorhombic
54	Pcaa	mmm	Orthorhombic	Orthorhombic
54	Pcca	mmm	Orthorhombic	Orthorhombic
54	Pccb	mmm	Orthorhombic	Orthorhombic
55	Pbam	mmm	Orthorhombic	Orthorhombic
55	Pcma	mmm	Orthorhombic	Orthorhombic
55	Pmcb	mmm	Orthorhombic	Orthorhombic
56	Pbnb	mmm	Orthorhombic	Orthorhombic
56	Pccn	mmm	Orthorhombic	Orthorhombic
56	Pnaa	mmm	Orthorhombic	Orthorhombic
57	Pbcm	mmm	Orthorhombic	Orthorhombic
57	Pbma	mmm	Orthorhombic	Orthorhombic
57	Pcam	mmm	Orthorhombic	Orthorhombic
57	Pcmb	mmm	Orthorhombic	Orthorhombic
57	Pmab	mmm	Orthorhombic	Orthorhombic
57	Pmca	mmm	Orthorhombic	Orthorhombic
58	Pmnn	mmm	Orthorhombic	Orthorhombic
58	Pnmn	mmm	Orthorhombic	Orthorhombic
58	Pnnm	mmm	Orthorhombic	Orthorhombic
59	Pmmn	mmm	Orthorhombic	Orthorhombic
59	Pmmn:2	mmm	Orthorhombic	Orthorhombic
59	Pmnm	mmm	Orthorhombic	Orthorhombic
59	Pmnm:2	mmm	Orthorhombic	Orthorhombic
59	Pnmm	mmm	Orthorhombic	Orthorhombic
59	Pnmm:2	mmm	Orthorhombic	Orthorhombic
60	Pbcn	mmm	Orthorhombic	Orthorhombic
60	Pbna	mmm	Orthorhombic	Orthorhombic
60	Pcan	mmm	Orthorhombic	Orthorhombic
60	Pcnb	mmm	Orthorhombic	Orthorhombic
60	Pnab	mmm	Orthorhombic	Orthorhombic
60	Pnca	mmm	Orthorhombic	Orthorhombic
61	Pbca	mmm	Orthorhombic	Orthorhombic
61	Pcab	mmm	Orthorhombic	Orthorhombic
62	Pbnm	mmm	Orthorhombic	Orthorhombic
62	Pcmn	mmm	Orthorhombic	Orthorhombic
62	Pmcn	mmm	Orthorhombic	Orthorhombic
62	Pmnb	mmm	Orthorhombic	Orthorhombic
62	Pnam	mmm	Orthorhombic	Orthorhombic
62	Pnma	mmm	Orthorhombic	Orthorhombic
63	Amam	mmm	Orthorhombic	Orthorhombic
63	Amma	mmm	Orthorhombic	Orthorhombic
63	Bbmm	mmm	Orthorhombic	Orthorhombic
63	Bmmb	mmm	Orthorhombic	Orthorhombic
63	Ccmm	mmm	Orthorhombic	Orthorhombic
63	Cmcm	mmm	Orthorhombic	Orthorhombic
64	Aeam	mmm	Orthorhombic	Orthorhombic
64	Aema	mmm	Orthorhombic	Orthorhombic
64	Bbem	mmm	Orthorhombic	Orthorhombic
64	Bmeb	mmm	Orthorhombic	Orthorhombic
64	Ccme	mmm	Orthorhombic	Orthorhombic
64	Cmce	mmm	Orthorhombic	Orthorhombic
65	Ammm	mmm	Orthorhombic	Orthorhombic
65	Bmmm	mmm	Orthorhombic	Orthorhombic
65	Cmmm	mmm	Orthorhombic	Orthorhombic
66	Amaa	mmm	Orthorhombic	Orthorhombic
66	Bbmb	mmm	Orthorhombic	Orthorhombic
66	Cccm	mmm	Orthorhombic	Orthorhombic
67	Aemm	mmm	Orthorhombic	Orthorhombic
67	Bmem	mmm	Orthorhombic	Orthorhombic
67	Cmme	mmm	Orthorhombic	Orthorhombic
68	Aeaa	mmm	Orthorhombic	Orthorhombic
68	Bbeb	mmm	Orthorhombic	Orthorhombic
68	Ccce	mmm	Orthorhombic	Orthorhombic
69	Fmmm	mmm	Orthorhombic	Orthorhombic
70	Fddd	mmm	Orthorhombic	Orthorhombic
70	Fddd:2	mmm	Orthorhombic	Orthorhombic
71	Immm	mmm	Orthorhombic	Orthorhombic
72	Ibam	mmm	Orthorhombic	Orthorhombic
72	Icma	mmm	Orthorhombic	Orthorhombic
72	Imcb	mmm	Orthorhombic	Orthorhombic
73	Ibca	mmm	Orthorhombic	Orthorhombic
73	Icab	mmm	Orthorhombic	Orthorhombic
74	Ibmm	mmm	Orthorhombic	Orthorhombic
74	Icmm	mmm	Orthorhombic	Orthorhombic
74	Imam	mmm	Orthorhombic	Orthorhombic
74	Imcm	mmm	Orthorhombic	Orthorhombic
74	Imma	mmm	Orthorhombic	Orthorhombic
74	Immb	mmm	Orthorhombic	Orthorhombic

Tetragonal

Space group, point group, lattice system, and crystal systems.

Space Group Number	Space Group Symbol	Point Group	Lattice System	Crystal System
75	P4	4	Tetragonal	Tetragonal
76	P41	4	Tetragonal	Tetragonal
77	P42	4	Tetragonal	Tetragonal
78	P43	4	Tetragonal	Tetragonal
79	I4	4	Tetragonal	Tetragonal
80	I41	4	Tetragonal	Tetragonal
81	P-4	-4	Tetragonal	Tetragonal
82	I-4	-4	Tetragonal	Tetragonal
83	P4/m	4/m	Tetragonal	Tetragonal
84	P42/m	4/m	Tetragonal	Tetragonal
85	P4/n	4/m	Tetragonal	Tetragonal
85	P4/n:2	4/m	Tetragonal	Tetragonal
86	P42/n	4/m	Tetragonal	Tetragonal
86	P42/n:2	4/m	Tetragonal	Tetragonal
87	I4/m	4/m	Tetragonal	Tetragonal
88	I41/a	4/m	Tetragonal	Tetragonal
88	I41/a:2	4/m	Tetragonal	Tetragonal
89	P422	422	Tetragonal	Tetragonal
90	P4212	422	Tetragonal	Tetragonal
91	P4122	422	Tetragonal	Tetragonal
92	P41212	422	Tetragonal	Tetragonal
93	P4222	422	Tetragonal	Tetragonal
94	P42212	422	Tetragonal	Tetragonal
95	P4322	422	Tetragonal	Tetragonal
96	P43212	422	Tetragonal	Tetragonal
97	I422	422	Tetragonal	Tetragonal
98	I4122	422	Tetragonal	Tetragonal
99	P4mm	4mm	Tetragonal	Tetragonal
100	P4bm	4mm	Tetragonal	Tetragonal
101	P42cm	4mm	Tetragonal	Tetragonal
102	P42nm	4mm	Tetragonal	Tetragonal
103	P4cc	4mm	Tetragonal	Tetragonal
104	P4nc	4mm	Tetragonal	Tetragonal
105	P42mc	4mm	Tetragonal	Tetragonal
106	P42bc	4mm	Tetragonal	Tetragonal
107	I4mm	4mm	Tetragonal	Tetragonal
108	I4cm	4mm	Tetragonal	Tetragonal
109	I41md	4mm	Tetragonal	Tetragonal
110	I41cd	4mm	Tetragonal	Tetragonal
111	P-42m	-42m	Tetragonal	Tetragonal
112	P-42c	-42m	Tetragonal	Tetragonal
113	P-421m	-42m	Tetragonal	Tetragonal
114	P-421c	-42m	Tetragonal	Tetragonal
115	P-4m2	-4m2	Tetragonal	Tetragonal
116	P-4c2	-4m2	Tetragonal	Tetragonal
117	P-4b2	-4m2	Tetragonal	Tetragonal
118	P-4n2	-4m2	Tetragonal	Tetragonal
119	I-4m2	-4m2	Tetragonal	Tetragonal
120	I-4c2	-4m2	Tetragonal	Tetragonal
121	I-42m	-42m	Tetragonal	Tetragonal
122	I-42d	-42m	Tetragonal	Tetragonal
123	P4/mmm	4/mmm	Tetragonal	Tetragonal
124	P4/mcc	4/mmm	Tetragonal	Tetragonal
125	P4/nbm	4/mmm	Tetragonal	Tetragonal
125	P4/nbm:2	4/mmm	Tetragonal	Tetragonal
126	P4/nnc	4/mmm	Tetragonal	Tetragonal
126	P4/nnc:2	4/mmm	Tetragonal	Tetragonal
127	P4/mbm	4/mmm	Tetragonal	Tetragonal
128	P4/mnc	4/mmm	Tetragonal	Tetragonal
129	P4/nmm	4/mmm	Tetragonal	Tetragonal
129	P4/nmm:2	4/mmm	Tetragonal	Tetragonal
130	P4/ncc	4/mmm	Tetragonal	Tetragonal
130	P4/ncc:2	4/mmm	Tetragonal	Tetragonal
131	P42/mmc	4/mmm	Tetragonal	Tetragonal
132	P42/mcm	4/mmm	Tetragonal	Tetragonal
133	P42/nbc	4/mmm	Tetragonal	Tetragonal
133	P42/nbc:2	4/mmm	Tetragonal	Tetragonal
134	P42/nnm	4/mmm	Tetragonal	Tetragonal
134	P42/nnm:2	4/mmm	Tetragonal	Tetragonal
135	P42/mbc	4/mmm	Tetragonal	Tetragonal
136	P42/mnm	4/mmm	Tetragonal	Tetragonal
137	P42/nmc	4/mmm	Tetragonal	Tetragonal
137	P42/nmc:2	4/mmm	Tetragonal	Tetragonal
138	P42/ncm	4/mmm	Tetragonal	Tetragonal
138	P42/ncm:2	4/mmm	Tetragonal	Tetragonal
139	I4/mmm	4/mmm	Tetragonal	Tetragonal
140	I4/mcm	4/mmm	Tetragonal	Tetragonal
141	I41/amd	4/mmm	Tetragonal	Tetragonal
141	I41/amd:2	4/mmm	Tetragonal	Tetragonal
142	I41/acd	4/mmm	Tetragonal	Tetragonal
142	I41/acd:2	4/mmm	Tetragonal	Tetragonal

Trigonal

Space group, point group, lattice system, and crystal systems.

Space Group Number	Space Group Symbol	Point Group	Lattice System	Crystal System
143	P3	3	Hexagonal	Trigonal
144	P31	3	Hexagonal	Trigonal
145	P32	3	Hexagonal	Trigonal
146	R3	3	Hexagonal	Trigonal
146	R3:r	3r	Rhombohedral	Trigonal
147	P-3	-3	Hexagonal	Trigonal
148	R-3	-3	Hexagonal	Trigonal
148	R-3:r	-3r	Rhombohedral	Trigonal
149	P312	312	Hexagonal	Trigonal
150	P321	321	Hexagonal	Trigonal
151	P3112	312	Hexagonal	Trigonal
152	P3121	321	Hexagonal	Trigonal
153	P3212	312	Hexagonal	Trigonal
154	P3221	321	Hexagonal	Trigonal
155	R32	3	Hexagonal	Trigonal
155	R32:r	3r	Rhombohedral	Trigonal
156	P3m1	3m1	Hexagonal	Trigonal
157	P31m	31m	Hexagonal	Trigonal
158	P3c1	3m1	Hexagonal	Trigonal
159	P31c	31m	Hexagonal	Trigonal
160	R3m	3m	Hexagonal	Trigonal
160	R3m:r	3mr	Rhombohedral	Trigonal
161	R3c	3m	Hexagonal	Trigonal
161	R3c:r	3mr	Rhombohedral	Trigonal
162	P-31m	-31m	Hexagonal	Trigonal
163	P-31c	-31m	Hexagonal	Trigonal
164	P-3m1	-3m1	Hexagonal	Trigonal
165	P-3c1	-3m1	Hexagonal	Trigonal
166	R-3m	-3m	Hexagonal	Trigonal
166	R-3m:r	-3mr	Rhombohedral	Trigonal
167	R-3c	-3m	Hexagonal	Trigonal
167	R-3c:r	-3mr	Rhombohedral	Trigonal

Hexagonal

Space group, point group, lattice system, and crystal systems.

Space Group Number	Space Group Symbol	Point Group	Lattice System	Crystal System
168	P6	6	Hexagonal	Hexagonal
169	P61	6	Hexagonal	Hexagonal
170	P65	6	Hexagonal	Hexagonal
171	P62	6	Hexagonal	Hexagonal
172	P64	6	Hexagonal	Hexagonal
173	P63	6	Hexagonal	Hexagonal
174	P-6	-6	Hexagonal	Hexagonal
175	P6/m	6/m	Hexagonal	Hexagonal
176	P63/m	6/m	Hexagonal	Hexagonal
177	P622	622	Hexagonal	Hexagonal
178	P6122	622	Hexagonal	Hexagonal
179	P6522	622	Hexagonal	Hexagonal
180	P6222	622	Hexagonal	Hexagonal
181	P6422	622	Hexagonal	Hexagonal
182	P6322	622	Hexagonal	Hexagonal
183	P6mm	6mm	Hexagonal	Hexagonal
184	P6cc	6mm	Hexagonal	Hexagonal
185	P63cm	6mm	Hexagonal	Hexagonal
186	P63mc	6mm	Hexagonal	Hexagonal
187	P-6m2	-6m2	Hexagonal	Hexagonal
188	P-6c2	-6m2	Hexagonal	Hexagonal
189	P-62m	-62m	Hexagonal	Hexagonal
190	P-62c	-62m	Hexagonal	Hexagonal
191	P6/mmm	6/mmm	Hexagonal	Hexagonal
192	P6/mcc	6/mmm	Hexagonal	Hexagonal
193	P63/mcm	6/mmm	Hexagonal	Hexagonal
194	P63/mmc	6/mmm	Hexagonal	Hexagonal

Cubic

Space group, point group, lattice system, and crystal systems.

Space Group Number	Space Group Symbol	Point Group	Lattice System	Crystal System
195	P23	23	Cubic	Cubic
196	F23	23	Cubic	Cubic
197	I23	23	Cubic	Cubic
198	P213	23	Cubic	Cubic
199	I213	23	Cubic	Cubic
200	Pm-3	m-3	Cubic	Cubic
201	Pn-3	m-3	Cubic	Cubic
201	Pn-3:2	m-3	Cubic	Cubic
202	Fm-3	m-3	Cubic	Cubic
203	Fd-3	m-3	Cubic	Cubic
203	Fd-3:2	m-3	Cubic	Cubic
204	Im-3	m-3	Cubic	Cubic
205	Pa-3	m-3	Cubic	Cubic
206	Ia-3	m-3	Cubic	Cubic
207	P432	432	Cubic	Cubic
208	P4232	432	Cubic	Cubic
209	F432	432	Cubic	Cubic
210	F4132	432	Cubic	Cubic
211	I432	432	Cubic	Cubic
212	P4332	432	Cubic	Cubic
213	P4132	432	Cubic	Cubic
214	I4132	432	Cubic	Cubic
215	P-43m	-43m	Cubic	Cubic
216	F-43m	-43m	Cubic	Cubic
217	I-43m	-43m	Cubic	Cubic
218	P-43n	-43m	Cubic	Cubic
219	F-43c	-43m	Cubic	Cubic
220	I-43d	-43m	Cubic	Cubic
221	Pm-3m	m-3m	Cubic	Cubic
222	Pn-3n	m-3m	Cubic	Cubic
222	Pn-3n:2	m-3m	Cubic	Cubic
223	Pm-3n	m-3m	Cubic	Cubic
224	Pn-3m	m-3m	Cubic	Cubic
224	Pn-3m:2	m-3m	Cubic	Cubic
225	Fm-3m	m-3m	Cubic	Cubic
226	Fm-3c	m-3m	Cubic	Cubic
227	Fd-3m	m-3m	Cubic	Cubic
227	Fd-3m:2	m-3m	Cubic	Cubic
228	Fd-3c	m-3m	Cubic	Cubic
228	Fd-3c:2	m-3m	Cubic	Cubic
229	Im-3m	m-3m	Cubic	Cubic
230	Ia-3d	m-3m	Cubic	Cubic