Dear Gert,

the answer to your question seems to be trivial: Referring to your example the planes x=1 or x=0.5 or x=0.25 correspond to Laue zones, the coordinates of atoms (1.0,y,z) or (0.5,y,z) or (0.25,y,z) correspond to reflections.

But these analogies may be extended: The more Laue zones are used in Fourier synthesis the "sharper" (more accurat) are the positions of atoms in the direction of the zone axis. Correspondingly: in the diffraction experiment the extension of reflexions becomes sharper in the direction of the beam (which is parallel to the zone axis) the more layers of the crystal contribute to the scattering phenomenon, i.e. the thicker the crystal is.

A more sophisticated link between Laue zones and direct space is the fact that the distance between subsequent Laue zones which may be considered as a "d-value" in reciprocal space and could be named 'd*' is 1/c, where c is the direct cell parameter for the setting of the cell as determined by the applied zone, i.e. by the choice of a* and b*; or: c = 1/d*. Since in TEM-work d* is directly accessible from any SAED-pattern which displays a FOLZ-ring and since V*=|a* x b*|.d* it is very easy to determine the volume V=1/V* of the reduced cell online.

With best wiches, yours

Gerhard Miehe

Dear Gerhard, this is mainly clear. My question was related to the fact that using HOLZ, FOLZ, SOLZ etc you have an tool to describe parallel and in reciprocal space equidistant planes which are all described by [uvw], the normal to these planes. 

Is there any term which enables the description of a specific parallel plane in crystal lattice? (hkl) defines all planes perpendicular to the reciprocal lattice direction (hkl), but one is crossing the origin, the next is defined by 1/h, 1/k, 1/l (Weißsche indizes), the next by 2/h, 2/k, 2/l etc.... I havn't heard about such term but this does not mean that it has been introduced somewhere.

Thanks,

Gert

Dear Gert,

now I believe I understood the problem.

Perhaps the following considerations are a little bit helpful:

While layers of ZOLZ- and HOLZ-reflections are metrically equivalent but not with respect to structure factors (reciprocal space has a point symmetry (point group), no translational symmetry), the layers in direct space with axis intercepts 1/h, 1/k, 1/l and 2/h,2/k,2/l etc. have the distance d(hkl) and are identical (translational symmetry, space group)). In reciprocal space the origin is 000, there is no choice. To describe the crystal structure in direct space you may choose the origin wherever you want, even 0.05nm distant from a center of symmetry. Therefore I believe there is no need to introduce a tool for describing a formation of parallel layers within a structure.

Greetings!

Gerhard

Good point. I simply forgot the translation symmetry. Geometrically as grid existant but not symmetry equivalent. Sorry that I cannot give you Two or more votes :-).

I have to think. You couldn't answer my question but perhaps it was a finally useless one. Thanks!

Dear Gert,

since your question concerning Laue zones appeares still in the

Q&A-list I suppose that it is still under discussion.

Pardon me if it is well known stuff what I am to reflect now:

Perhaps it is helpful to realize that for any zone [uvw] a setting

of the cell can be chosen with the property: All reflections

belonging to a Laue zone have one congruent index, say l (L, not

'one'), which is the order of the Laue zone of zone [uvw].

Therefore your problem may be reformulated:

"What is the real space analogon for a layer of reflections with l=const.?"

The above statement is trivial for zone [001]. It is not trivial

for a general zone. Let me show the procedure for the arbitrarily

chosen zone [1 2 4]:

(For centred cells a primitive setting is presumed.)

A primitive mesh by which zone [1 2 4] is spanned is e.g.

hkl1 : h1 k1 l1 = 2 -1 0 (new a*)

hkl2 : h2 k2 l2 = 0 -2 1 (new b*)

A 3rd vector hkl3 completes a 3x3-transformation-matrix M.

Condition for hkl3 is that the determinant(M) = 1,

i.e. that hkl3 can be used as a new c*, e.g.

hkl3 : h3 k3 l3 = 1 -1 0 (new c*)

Now transpose matrix M (M -> MT)

and invert matrix MT (MT -> MTI):

Matrix M      MT: M transposed     MTI: inverse of MT

 2 -1 0                   2 0 1                         1 1 2

 0 -2 1                 -1 -2 -1                       0 0 1

1 -1 0                   0  1 0                      -1 -2 -4

The cell we are looking for is: (a,b,c)new = (MTI)*(a,b,c)old

Examples for [1 2 4]:

                 a             b               c            al          be       ga        V

old:        6.0          7.0            8.0         65°        75°     85°     294

new: 22.1965 8.0000  41.81757  160.7° 172.8° 22.5°    294

or

old:      10.0      10.0        10.0          90°        90°      90°     1000

new: 24.495 10.0000 45.826    150.8°  168.5°   35.3°    1000

Indices hkl transform as cell parameters, therefore:

                  old cell -> new cell:

ZOLZ :a*     2 -1 0 -> 1 0 0,

            b*     0 -2 1 -> 0 1 0,

      a*+b*     2 -3 1 -> 1 1 0

3a*+2b*      6 -7 2 -> 3 2 0

HOLZ:  ZOLZ+H*(h3 k3 l3), e.g. FOLZ (H=1) and SOLZ (H=2)

FOLZ:   3 -2 0 -> 1 0 1      SOLZ:  4 -3 0 -> 1 0 2

(l=1)      1 -3 1 -> 0 1 1     (l=2)      2 -4 1 -> 0 1 2

              3 -4 1 -> 1 1 1                   4 -5 1 -> 1 1 2

              7 -8 2 -> 3 2 1                   8 -9 2 -> 3 2 2

Actually this somewhat longish remark doesn't answer your question,

but perhaps it offers a new aspect.

With best wishes,

Gerhard