What are high symmetry points in a crystal and how the values of values of various high symmetry points like Gamma, K, X, L, M etc. have been calculated for a particular crystal structure?
The term "high symmetry" refers to the fact that at such a local point you have more symmetry elements that copy this point onto itself. You need to distinguish direct and reciprocal space though. In your question you mix these up.
Take a simple othorhombic crystal with symmetry mm2. In this crystal you have a mirror plane normal to the a-axis and points with coordinates 0yz all are on this mirror plane. Similarly the second m refers to the mirror plane normal to the b axis through points with coordinates x0z. Finally you have the 2-fold axis parallel to the c-axis and all points with coordinates 00z are on this axis. (There are actually a few more symmetry elements present, but these are all caused by these three original symmetry operation )
A point with coordinates xyz will be copied by the symmetry operations onto the complete set of: x y z; -x y z; x -y z; -x -y z. No symmetry element copies the point x y z onto itself. The local symmetry of this point is 1, it sees a different environment in any direction.
The point 0 0 z will be copied onto itself by all both mirror planes and the two-fold axis. To this point the neighborhood along the positive a-axis and the negative a-axis will be the same , likewise along the b-axis. Along the c axis there is still a difference between the positive c axis and the negative c-axis. The local symmetry group is mm2.
A point with coordinates x0z will be copied by the symmetry operations onto the complete set of x 0 z; -x 0 z The mirror plane normal to the b-axis will copy x0z onto itself. Thus the local symmetry is .m.
In the space group mm2 there are two more mirror planes and three more two fold axes, all parallel to the one i mentioned before. Make a drawing of the unit cell, apply the symmetry operations mentioned and you should be able to work out the location of these secondary symmetry operations.
Similar rules can be derived for all the different Wyckoff positions in a given space group. So it is not just the "nice" coordinates but which symmetry operations of the space group copy an atom onto itself that make a site a low or high symmetry site. Within a space group there are usually several different Wyckoff positions that all have different local symmetries and as shown in the example above will have different symmetry, some with few symmetry operations that copy the atom onto itself, some with more symmetry operations that copy the atom onto itself. Thus there is not just a "high" symmetry but different degrees of "high".
Now in reciprocal space in the Brillouin zone the symmetry is that of the corresponding point group, not that of the space group. Still, there are points hkl in reciprocal space that are not copied onto themselves by any point group symmetry operation, while other point are copied onto themselves. In my example, a point hkl with arbitrary hkl (non-integer!) would not be copied onto itself by any symmetry operation. A point with indices h00 would be copied onto itself by the mirror plane normal to the b-axis, a point with indices 0k0 would be copied onto itself by the mirror plane normal to the a-axis and finally a point with indices 00l would be copied onto itself be either mirror plane and the two fold axis. In the Brilloiun zone the point Gamma refers to the point 0,0,0 in the Brillouin zone, other greek letters refer to special directions like along the a*, b* or c* directions, while capital latin letters refer to special points on the Brilloiun zone boundary.
A complete listing o the special "high" symmetry points in direct space can be found in the International tables for crystallography, on line see the Bilbao server at http://www.cryst.ehu.es/
As a summary you need to distinguish high symmetry == Wyckoff positions in direct space and special symmetry directions/ points in a Brilloiun zone ii.e. in reciprocal space. The Wyckoff positions are a property of the space group, not the crystal structure with that space group! The special direction in reciprocal space are a property of the corresponding point group.
I believe you are talking about crystallographic special points, like Wyckoff positions, and Brillouin zone points.
These are just coordinates in the unit cell (or reciprocal lattice) that have certain "nice" coordinates. Like (0, 0, 0) for the origin and Gamma. X would be (1, 0, 0) in the reciprocal lattice, if I'm not mistaken. The other points are similarly just points with simple integer or rational coordinates. Coordinates like (1/2, 1/2, 1/2) or (1/3, 1/3, 0), it's just different positions in the unit cell (or reciprocal lattice).
The specific coordinates that may be of interest will depend on the space group, setting and context...
Thank you. But i want to know that what is the significance of these points. Why these are called high symmetry points? On what basis these designations gamma, M, K etc. is given to some specific coordinates?
The term "high symmetry" refers to the fact that at such a local point you have more symmetry elements that copy this point onto itself. You need to distinguish direct and reciprocal space though. In your question you mix these up.
Take a simple othorhombic crystal with symmetry mm2. In this crystal you have a mirror plane normal to the a-axis and points with coordinates 0yz all are on this mirror plane. Similarly the second m refers to the mirror plane normal to the b axis through points with coordinates x0z. Finally you have the 2-fold axis parallel to the c-axis and all points with coordinates 00z are on this axis. (There are actually a few more symmetry elements present, but these are all caused by these three original symmetry operation )
A point with coordinates xyz will be copied by the symmetry operations onto the complete set of: x y z; -x y z; x -y z; -x -y z. No symmetry element copies the point x y z onto itself. The local symmetry of this point is 1, it sees a different environment in any direction.
The point 0 0 z will be copied onto itself by all both mirror planes and the two-fold axis. To this point the neighborhood along the positive a-axis and the negative a-axis will be the same , likewise along the b-axis. Along the c axis there is still a difference between the positive c axis and the negative c-axis. The local symmetry group is mm2.
A point with coordinates x0z will be copied by the symmetry operations onto the complete set of x 0 z; -x 0 z The mirror plane normal to the b-axis will copy x0z onto itself. Thus the local symmetry is .m.
In the space group mm2 there are two more mirror planes and three more two fold axes, all parallel to the one i mentioned before. Make a drawing of the unit cell, apply the symmetry operations mentioned and you should be able to work out the location of these secondary symmetry operations.
Similar rules can be derived for all the different Wyckoff positions in a given space group. So it is not just the "nice" coordinates but which symmetry operations of the space group copy an atom onto itself that make a site a low or high symmetry site. Within a space group there are usually several different Wyckoff positions that all have different local symmetries and as shown in the example above will have different symmetry, some with few symmetry operations that copy the atom onto itself, some with more symmetry operations that copy the atom onto itself. Thus there is not just a "high" symmetry but different degrees of "high".
Now in reciprocal space in the Brillouin zone the symmetry is that of the corresponding point group, not that of the space group. Still, there are points hkl in reciprocal space that are not copied onto themselves by any point group symmetry operation, while other point are copied onto themselves. In my example, a point hkl with arbitrary hkl (non-integer!) would not be copied onto itself by any symmetry operation. A point with indices h00 would be copied onto itself by the mirror plane normal to the b-axis, a point with indices 0k0 would be copied onto itself by the mirror plane normal to the a-axis and finally a point with indices 00l would be copied onto itself be either mirror plane and the two fold axis. In the Brilloiun zone the point Gamma refers to the point 0,0,0 in the Brillouin zone, other greek letters refer to special directions like along the a*, b* or c* directions, while capital latin letters refer to special points on the Brilloiun zone boundary.
A complete listing o the special "high" symmetry points in direct space can be found in the International tables for crystallography, on line see the Bilbao server at http://www.cryst.ehu.es/
As a summary you need to distinguish high symmetry == Wyckoff positions in direct space and special symmetry directions/ points in a Brilloiun zone ii.e. in reciprocal space. The Wyckoff positions are a property of the space group, not the crystal structure with that space group! The special direction in reciprocal space are a property of the corresponding point group.
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