The program describes the dawn of reciprocal space building and geometrical explanation of diffraction: how does different type of sample and detector moving scans reciprocal space  see figures
1 
3. Correct accessible region in reciprocal space for both Bragg and Laue geometries could be calculated for any crystal with any normal orientation and for any plane of incidence  see figures
4 
10. Formulas for calculation of this figures could be found in appendix
A. For visualization of accessible reciprocal nodes beampasses could be plotted fig.
11. Also the program shows basis of reciprocal space mapping for 1layer and 2layered systems. The program is very dynamic in 3 dimensions  user can change many parameters (what does every control do is described as Hint).
The idea of building such program arises at XTOP 2006 tutorial, where D.Lubbert tried to show such things, but there were only static pictures which were understandable only for people who already know all of it.
The program was shown with explanation at the conference in V.Novgorod (may 2007) and it had great success  most young scientists told that it was very interesting and understandable and even such well known people as Bushuev, Kutt, Punegov, Suvorov, Shulpina, Lomov etc. liked the program.
This program hopefully will help young scientists of the world to understand XRay diffraction better.
Final release of this program with some features improved and help file will be placed on this cite a little bit later. Maybe ready presentation will be made and placed here as well.
A Main formulas in Cartesian coordinates
For a given reciprocal point with vector
Q (
Q = Q < = 2 K) and given plane of incidence with its normal
P, incident beam
K_{0} could be found by solving equations:
 
((Q_{y} P_{z} P_{y} Q_{z})^{2}+(Q_{z} P_{x}  P_{z} Q_{x})^{2}+(Q_{y} P_{x}  P_{y} Q_{x})^{2}) K_{0x}^{2} 

 
 
 Q^{2} (Q_{z} P_{x} P_{z} + Q_{y} P_{x} P_{y}  Q_{x} (P_{y}^{2}+P_{z}^{2})) K_{0x} + 

 
 
Q^{4} (P_{y}^{2}+P_{z}^{2}) /4  K^{4} (Q_{y} P_{z}  P_{y} Q_{z})^{2} = 0 ; 

  (1) 

K_{0y} = 
 P_{z} Q^{2} /2 + (P_{x} Q_{z}  P_{z} H_{x}) K_{0x}
P_{z} H_{y}  P_{y} H_{z}

; 
 (2) 
K_{0z} = 
 P_{y} Q^{2} /2 + (P_{x} Q_{y}  P_{y} H_{x}) K_{0x}
P_{y} H_{z}  P_{z} H_{y}

. 
 (3) 
Diffracted beam
K_{h} is then:
K_{hx} = K_{0x} + Q_{x} ; K_{hy} = K_{0y} + Q_{y} ; K_{hz} = K_{0z} + Q_{z} . 
 (4) 
If considering simplified coordinate system, where vectors:
N = (0,0,N
_{z}),
P = (0,P
_{y},0),
K_{0} = (K
_{0x},0,K
_{0z}), equations (
1)(
3) take the form:
(Q_{z}^{2} + Q_{x}^{2}) K_{0x}^{2} + Q^{2} Q_{x}^{2} K_{0x} +Q^{4} /4 + K^{4} Q_{z}^{2} = 0 ; 
 (5) 
K_{0y} = 0 ; K_{0z} = ±(K^{2}  K_{0x}^{2})^{1/2} . 
 (6) 
In described coordinate system equation for fundamentally accessible reciprocal points (a half of torus for Q
_{z} > 0 on fig.
6, yellow surface on fig.
7 and green for Q
_{z} > = 0 on fig.
8) takes the form (here and below
 2 K < = Q
_{x} < = 2 K and
 K < = Q
_{y} < = K are set and Q
_{z} is calculated):
Q_{z}^{4} + 2 (Q_{x}^{2} + Q_{y}^{2}  2 K^{2}) Q_{z}^{2} + (Q_{x}^{2} +Q_{y}^{2})^{2}  4 K^{2} Q_{x}^{2} = 0 . 
 (7) 
Only positive Q
_{z} should be considered, because surface for Q
_{z} < 0 is determined by equation K
_{0z}=0  it is described below.
Equation for surface, where
K_{h} is parallel to the crystal surface (K
_{hz} = 0) is:
Q_{z}^{4} + 2 (Q_{x}^{2}  Q_{y}^{2}) Q_{z}^{2} + (Q_{x}^{2} +Q_{y}^{2})^{2}  4 K^{2} Q_{x}^{2} = 0 . 
 (8) 
One solution of this equation is plotted in yellow in fig.
8 and the other in purple in fig.
7. Two other roots are negative and nonaccessible due to the presence of the surface, described below.
Equation for surface, where
K_{0} is parallel to the crystal surface (K
_{0z} = 0) is:
Q_{z}^{2} =  Q_{x}^{2}  Q_{y}^{2} + 2 K Q_{x} . 
 (9) 
The roots are shown in green in fig.
7 (Q
_{z} > = 0) and in fig.
8 (Q
_{z} < = 0).
Some figures:
Omega/2Theta

Omega



Figure 1: Moving of beams for Omega/2Theta scan. Incident beam K_{0} is red and diffracted K_{h} is green.

Figure 2: Moving of beams for Omega scan. Incident beam K_{0} is red and diffracted K_{h} is green.

2Theta



Figure 3: Moving of beams for 2Theta scan. Incident beam K_{0} is red and diffracted K_{h} is green.

Picture

Description


Figure 4: Plotted in some books accessible region for noncoplanar Bragg case. (218kb)


Figure 5: Possible ends of vector K_{h} is a sphere. K_{h}  green, K_{0}  red. (373kb)


Figure 6: All fundamentally accessible for current l reciprocal points lie inside the torus. (445kb)


Figure 7: Right accessible reciprocal region for Bragg geometry in general case. (539kb)


Figure 8: Right accessible reciprocal region for Laue geometry in general case. (842kb)


Figure 9: Right region (one quarter) for Bragg geometry for Si [001] crystal using structure factor. (214kb)


Figure 10: Right region for Bragg geometry for InAs [001] crystal using structure factor. (415kb)


Figure 11: Two possible beampathes for reaching 333 reciprocal node (blue point) for Si [001] crystal and incident beam belongs to (010) plane. Incident beam  red, diffracted  green. (75kb)

Download:
XVis v.1.1 (08.08.2008)  The program with some files needed for its correct work. Bug reports and question address to developers (Email is in "About" dialog of the program).
XVis sources The sources for Borland TurboC and Borland Builder 6. Compilation instructions in readme.txt.
Article about the program O. Yefanov, V. Kladko, M. Slobodyan, Yu. Polischuk // Journal of Applied Crystallography, 2008. V.41. Part 3. P.647652. doi:10.1107/S0021889808008625
©
Yefanov O.M, scientific supervisor
Kladko V.P..
V. Lashkarev Institute of Semiconductor Physics, NAS of Ukraine