Index

Image dictionary (imgCIF) version 1.5.4

Category 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.

 
         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

 
        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

 
        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

 
        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

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: