DRAFT DICTIONARY CBF/imgCIF Extensions Dictionary | Draft version 1.5 for comment

#

Index

##### Image dictionary (imgCIF) version 1.5.4

# Category AXIS

Name:
`'axis'`

Description:

        Data items in the AXIS category record the information required
         to describe the various goniometer, detector, source and other
         axes needed to specify a data collection or the axes defining the
         coordinate system of an image.

         The location of each axis is specified by two vectors: the axis
         itself, given by a  unit vector in the direction of the axis, and
         an offset to the base of the unit vector.

         The vectors defining an axis are referenced to an appropriate
         coordinate system.  The axis vector, itself, is a dimensionless
         unit vector.  Where meaningful, the offset vector is given in
         millimetres.  In coordinate systems not measured in metres,
         the offset is not specified and is taken as zero.

         The available coordinate systems are:

             The imgCIF standard laboratory coordinate system
             The direct lattice (fractional atomic coordinates)
             The orthogonal Cartesian coordinate system (real space)
             The reciprocal lattice
             An abstract orthogonal Cartesian coordinate frame

         For consistency in this discussion, we call the three coordinate
         system axes X, Y and Z.  This is appropriate for the imgCIF
         standard laboratory coordinate system, and last two Cartesian
         coordinate systems, but for the direct lattice, X corresponds
         to a, Y to b and Z to c, while for the reciprocal lattice,
         X corresponds to a*, Y to b* and Z to c*.

         For purposes of visualization, all the coordinate systems are
         taken as right-handed, i.e., using the convention that the extended
         thumb of a right hand could point along the first (X) axis, the
         straightened pointer finger could point along the second (Y) axis
         and the middle finger folded inward could point along the third (Z)
         axis.

         THE IMGCIF STANDARD LABORATORY COORDINATE SYSTEM

         The imgCIF standard laboratory coordinate system is a right-handed
         orthogonal coordinate similar to the MOSFLM coordinate system,
         but imgCIF puts Z along the X-ray beam, rather than putting X along the
         X-ray beam as in MOSFLM.

         The vectors for the imgCIF standard laboratory coordinate system
         form a right-handed Cartesian coordinate system with its origin
         in the sample or specimen.  The origin of the axis system should,
         if possible, be defined in terms of mechanically stable axes to be
         be both in the sample and in the beam.  If the sample goniometer or other
         sample positioner has two axes the intersection of which defines a
         unique point at which the sample should be mounted to be bathed
         by the beam, that will be the origin of the axis system.  If no such
         point is defined, then the midpoint of the line of intersection
         between the sample and the center of the beam will define the origin.
         For this definition the sample positioning system will be set at
         its initial reference position for the experiment.

                                 | Y (to complete right-handed system)
                                 |
                                 |
                                 |
                                 |
                                 |
                                 |________________X
                                /       principal goniometer axis
                               /
                              /
                             /
                            /
                           /Z (to source)

         Axis 1 (X): The X-axis is aligned to the mechanical axis pointing from
         the sample or specimen along the  principal axis of the goniometer or
         sample positioning system if the sample positioning system has an axis
         that intersects the origin and which form an angle of more than 22.5
         degrees with the beam axis.

         Axis 2 (Y): The Y-axis completes an orthogonal right-handed system
         defined by the X-axis and the Z-axis (see below).

         Axis 3 (Z): The Z-axis is derived from the source axis which goes from
         the sample to the source.  The Z-axis is the component of the source axis
         in the direction of the source orthogonal to the X-axis in the plane
         defined by the X-axis and the source axis.

         If the conditions for the X-axis can be met, the coordinate system
         will be based on the goniometer or other sample positioning system
         and the beam and not on the orientation of the detector, gravity etc.
         The vectors necessary to specify all other axes are given by sets of
         three components in the order (X, Y, Z).
         If the axis involved is a rotation axis, it is right-handed, i.e. as
         one views the object to be rotated from the origin (the tail) of the
         unit vector, the rotation is clockwise.  If a translation axis is
         specified, the direction of the unit vector specifies the sense of
         positive translation.

         Note:  This choice of coordinate system is similar to but significantly
         different from the choice in MOSFLM (Leslie & Powell, 2004).  In MOSFLM,
         X is along the X-ray beam (the CBF/imgCIF Z axis) and Z is along the
         rotation axis.

         In some experimental techniques, there is no goniometer or the principal
         axis of the goniometer is at a small acute angle with respect to
         the source axis.  In such cases, other reference axes are needed
         to define a useful coordinate system.  The order of priority in
         defining directions in such cases is to use the detector, then
         gravity, then north.

         If the X-axis cannot be defined as above, then the
         direction (not the origin) of the X-axis should be parallel to the axis
         of the primary detector element corresponding to the most rapidly
         varying dimension of that detector element's data array, with its
         positive sense corresponding to increasing values of the index for
         that dimension.  If the detector is such that such a direction cannot
         be defined (as with a point detector) or that direction forms an
         angle of less than 22.5 degrees with respect to the source axis, then
         the X-axis should be chosen so that if the Y-axis is chosen
         in the direction of gravity, and the Z-axis is chosen to be along
         the source axis, a right-handed orthogonal coordinate system is chosen.
         In the case of a vertical source axis, as a last resort, the
         X-axis should be chosen to point North.

         All rotations are given in degrees and all translations are given in mm.

         Axes may be dependent on one another.  The X-axis is the only goniometer
         axis the direction of which is strictly connected to the hardware.  All
         other axes are specified by the positions they would assume when the
         axes upon which they depend are at their zero points.

         When specifying detector axes, the axis is given to the beam centre.
         The location of the beam centre on the detector should be given in the
         DIFFRN_DETECTOR category in distortion-corrected millimetres from
         the (0,0) corner of the detector.

         It should be noted that many different origins arise in the definition
         of an experiment.  In particular, as noted above, it is necessary to
         specify the location of the beam centre on the detector in terms
         of the origin of the detector, which is, of course, not coincident
         with the centre of the sample.

         The unit cell, reciprocal cell and crystallographic orthogonal
         Cartesian coordinate system are defined by the CELL and the matrices
         in the ATOM_SITES category.

         THE DIRECT LATTICE (FRACTIONAL COORDINATES)

         The direct lattice coordinate system is a system of fractional
         coordinates aligned to the crystal, rather than to the laboratory.
         This is a natural coordinate system for maps and atomic coordinates.
         It is the simplest coordinate system in which to apply symmetry.
         The axes are determined by the cell edges, and are not necessarily
         othogonal.  This coordinate system is not uniquely defined and
         depends on the cell parameters in the CELL category and the
         settings chosen to index the crystal.

         Molecules in a crystal studied by X-ray diffracraction are organized
         into a repeating regular array of unit cells.  Each unit cell is defined
         by three vectors, a, b and c.  To quote from Drenth,

         "The choice of the unit cell is not unique and therefore, guidelines
         have been established for selecting the standard basis vectors and
         the origin.  They are based on symmetry and metric considerations:

          "(1)  The axial system should be right handed.
           (2)  The basis vectors should coincide as much as possible with
           directions of highest symmetry."
           (3)  The cell taken should be the smallest one that satisfies
           condition (2)
           (4)  Of all the lattice vectors, none is shorter than a.
           (5)  Of those not directed along a, none is shorter than b.
           (6)  Of those not lying in the ab plane, none is shorter than c.
           (7)  The three angles between the basis vectors a, b and c are
           either all acute (<90\%) or all obtuse (>=90\%)."

         These rules do not produce a unique result that is stable under
         the assumption of experimental errors, and the the resulting cell
         may not be primitive.

         In this coordinate system, the vector (.5, .5, .5) is in the middle
         of the given unit cell.

         Grid coordinates are an important variation on fractional coordinates
         used when working with maps.  In imgCIF, the conversion from
         fractional to grid coordinates is implicit in the array indexing
         specified by _array_structure_list.dimension.  Note that this
         implicit grid-coordinate scheme is 1-based, not zero-based, i.e.
         the origin of the cell for axes along the cell edges with no
         specified _array_structure_list_axis.displacement will have
         grid coordinates of (1,1,1), i.e. array indices of (1,1,1).

         THE ORTHOGONAL CARTESIAN COORDINATE SYSTEM (REAL SPACE)

         The orthogonal Cartesian coordinate system is a transformation of
         the direct lattice to the actual physical coordinates of atoms in
         space.  It is similar to the laboratory coordinate system, but
         is anchored to and moves with the crystal, rather than being
         schored to the laboratory.  The transformation from fractional
         to orthogonal cartesian coordinates is given by the
                  _atom_sites.Cartn_transf_matrix[i][j]  and
                  _atom_sites.Cartn_transf_vector[i]
         tags.  A common choice for the matrix of the transformation is
         given in the 1992 PDB format document

                  | a      b cos(\g)   c cos(\b)                            |
                  | 0      b sin(\g)   c (cos(\a) - cos(\b)cos(\g))/sin(\g) |
                  | 0      0           V/(a b sin(\g))                      |

         This is a convenient coordinate system in which to do fitting
         of models to maps and in which to understand the chemistry of
         a molecule.

         THE RECIPROCAL LATTICE

         The reciprocal lattice coordinate system is used for diffraction
         intensitities.  It is based on the reciprocal cell, the dual of the cell,
         in which reciprocal cell edges are derived from direct cell faces:

            a* = bc sin(\a)/V  b* = ac sin(\b)/V  c* = ab sin(\g)/V
            cos(\a*) = (cos(\b) cos(\g) - cos(\a))/(sin(\b)  sin(\g))
            cos(\b*) = (cos(\a) cos(\g) - cos(\b))/(sin(\a) sin(\g))
            cos(\g*) = (cos(\a) cos(\b) - cos(\g))/(sin(\a) sin(\b))
            V = abc  SQRT(1 - cos(\a)^2^
                            - cos(\b)^2^
                            - cos(\g)^2^
                            + 2 cos(\a) cos(\b) cos(\g) )

         In this form the dimensions of the reciprocal lattice are in reciprocal
         \%Angstroms (\%A^-1^).  A dimensionless form can be obtained by
         multiplying by the wavelength.  Reflections are commonly indexed against
         this coordinate system as (h, k, l) triples.

         References:

         Drenth, J., "Introduction to basic crystallography." chapter
         2.1 in Rossmann, M. G. and Arnold, E. "Crystallography of
         biological macromolecules", Volume F of the IUCr's "International
         tables for crystallography", Kluwer, Dordrecht 2001, pp 44 -- 63

         Leslie, A. G. W. and Powell, H. (2004). MOSFLM v6.11.
         MRC Laboratory of Molecular Biology, Hills Road, Cambridge, England.
         http://www.CCP4.ac.uk/dist/X-windows/Mosflm/.

         Stout, G. H. and Jensen, L. H., "X-ray structure determination",
         2nd ed., Wiley, New York, 1989, 453 pp.

         __, "PROTEIN DATA BANK ATOMIC COORDINATE AND BIBLIOGRAPHIC ENTRY
         FORMAT DESCRIPTION," Brookhaven National Laboratory, February 1992.

Examples:

Example 1 - This example shows the axis specification of the axes of a kappa- geometry goniometer [see Stout, G. H. & Jensen, L. H. (1989). X-ray structure determination. A practical guide, 2nd ed. p. 134. New York: Wiley Interscience]. There are three axes specified, and no offsets. The outermost axis, omega, is pointed along the X axis. The next innermost axis, kappa, is at a 50 degree angle to the X axis, pointed away from the source. The innermost axis, phi, aligns with the X axis when omega and phi are at their zero points. If T-omega, T-kappa and T-phi are the transformation matrices derived from the axis settings, the complete transformation would be: X' = (T-omega) (T-kappa) (T-phi) X

             loop_
            _axis.id
            _axis.type
            _axis.equipment
            _axis.depends_on
            _axis.vector[1] _axis.vector[2] _axis.vector[3]
            omega rotation goniometer     .    1        0        0
            kappa rotation goniometer omega    -.64279  0       -.76604
            phi   rotation goniometer kappa    1        0        0

Example 2 - This example shows the axis specification of the axes of a detector, source and gravity. The order has been changed as a reminder that the ordering of presentation of tokens is not significant. The centre of rotation of the detector has been taken to be 68 millimetres in the direction away from the source.

            loop_
            _axis.id
            _axis.type
            _axis.equipment
            _axis.depends_on
            _axis.vector[1] _axis.vector[2] _axis.vector[3]
            _axis.offset[1] _axis.offset[2] _axis.offset[3]
            source       .        source     .       0     0     1   . . .
            gravity      .        gravity    .       0    -1     0   . . .
            tranz     translation detector rotz      0     0     1   0 0 -68
            twotheta  rotation    detector   .       1     0     0   . . .
            roty      rotation    detector twotheta  0     1     0   0 0 -68
            rotz      rotation    detector roty      0     0     1   0 0 -68

Example 3 - This example show the axis specification of the axes for a map, using fractional coordinates. Each cell edge has been divided into a grid of 50 divisions in the ARRAY_STRUCTURE_LIST_AXIS category. The map is using only the first octant of the grid in the ARRAY_STRUCTURE_LIST category. The fastest changing axis is the gris along A, then along B, and the slowest is along C. The map sampling is being done in the middle of each grid division

            loop_
            _axis.id
            _axis.system
            _axis.vector[1] _axis.vector[2] _axis.vector[3]
            CELL_A_AXIS    fractional       1 0 0
            CELL_B_AXIS    fractional       0 1 0
            CELL_C_AXIS    fractional       0 0 1

            loop_
            _array_structure_list.array_id
            _array_structure_list.index
            _array_structure_list.dimension
            _array_structure_list.precedence
            _array_structure_list.direction
            _array_structure_list.axis_id
            MAP 1 25 1 increasing CELL_A_AXIS
            MAP 1 25 2 increasing CELL_B_AXIS
            MAP 1 25 3 increasing CELL_C_AXIS

            loop_
            _array_structure_list_axis.axis_id
            _array_structure_list_axis.fract_displacement
            _array_structure_list_axis.fract_displacement_increment
            CELL_A_AXIS 0.01 0.02
            CELL_B_AXIS 0.01 0.02
            CELL_C_AXIS 0.01 0.02

Example 4 - This example show the axis specification of the axes for a map, this time as orthogonal \%Angstroms, using the same coordinate system as for the atomic coordinates. The map is sampling every 1.5 \%Angstroms (1.5e-7 millimeters) in a map segment 37.5 \%Angstroms on a side.

            loop_
            _axis.id
            _axis.system
            _axis.vector[1] _axis.vector[2] _axis.vector[3]
            X    orthogonal       1 0 0
            Y    orthogonal       0 1 0
            Z    orthogonal       0 0 1

                    loop_
            _array_structure_list.array_id
            _array_structure_list.index
            _array_structure_list.dimension
            _array_structure_list.precedence
            _array_structure_list.direction
            _array_structure_list.axis_id
            MAP 1 25 1 increasing X
            MAP 2 25 2 increasing Y
            MAP 3 25 3 increasing Z

            loop_
            _array_structure_list_axis.axis_id
            _array_structure_list_axis.displacement
            _array_structure_list_axis.displacement_increment
            X 7.5e-8 1.5e-7
            Y 7.5e-8 1.5e-7
            Z 7.5e-8 1.5e-7

Category groups:
`inclusive_group`
`axis_group`
`diffrn_group`
Category keys:
`_axis.id`
`_axis.equipment`

Mandatory category: no

#### HTML version of draft dictionary created by modified version of makedicthtml by B. McMahon from modified version of imgCIF dcitionary 1.3.2 which is subject to the following copyright:

Copyright © 2005 International Union of Crystallography