% find crytallographic direction that has known angle w.r.t. given
% direction. Given scattering angle (blocking dip position), scattering plane, and incident
% direction, find the unknown outgoing direction on the plane.

% S passivated Ge (111), Sep1108. The scattering plane is (1-10), detector 
% is at 90 degree position. The sample is R1 rotated 35 degrees.
% According to stereographic projection of fcc(111), now the incident
% direction is either [110] or [114]. The 2D spectrum shows two blocking
% dips at approximately 84 degrees and 96 degrees.

% If incident is [110], according to fcc(110) stereographic projection,
% there seems no major blocking at 84 degrees on (1-10) plane. Note that
% the pattern is symmetric about [110] on (1-10) plane, which indicates
% that if there is a blocking dip at theta (0<theta<90), there always another dip at
% 180 - theta.

% If incident is [114], since there is no stereographic projection for this
% direction, one needs to use fcc(110) stereographic projection for plane
% (1-10). The angle b/w [114] and [110] about 70 degrees, one needs to find a spot
% around 16 degrees one the other side. There IS a spot around 16 degrees,
% denoted by [11-x], where 0<x<1. Then the problem reduces to solve
% following equation: cos(theta) = a*b/(|a||b|), where a is given direction
% [114], b is the unknown [11-x], theta = 84 degrees.

clear all;

f = @(x)cos(84*pi/180)*sqrt(2+x^2)*sqrt(18)-(2-4*x);

x = fzero(f,1);

% result shows x = 0.3388. This means this direction is [11-1/3] or
% [33-1]. Actually the angle b/w [114] and [33-1] is 83.8 degrees.

% Therefore the incident is [114] and blocking is [33-1], giving scattering
% angle 96.2 degrees. 

% Another blocking dip is at scattering 84 degrees, corresponding angle w.r.t. [114]
% is 180-84=96 degrees. Let [11-y] be unknown direction:

g = @(y)cos(96*pi/180)*sqrt(2+y^2)*sqrt(18)-(2-4*y);

y = fzero(g,1);

% Result gives y = 0.6737, corresponding to [11-2/3] or [33-2] direction.
% This results in blocking dip at scattering angle of 84.2 degree.

% Note that these two blocking dips, at 84.2 degrees and 83.8 degrees, seem
% to indicate the pattern is symmetric about [114], therefore symmetric about
% the perpendicular direction to [114], i.e., 90 degrees scattering angle. 
% But this is not the case, just coincidance. Actually, scattering pattern
% does not show any symmetry about 90 degrees at all. 

% [11-1/2] or [22-1] should also appear as a blocking dip, at exactly 90
% degrees. 2D spectrum shows week blocking at 90 degrees.
% does not show any symmetry about 90 degree.