1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
|
/*
* relrod.c
*
* Calculate reflection positions via line-sphere intersection test
*
* (c) 2007-2009 Thomas White <thomas.white@desy.de>
*
* template_index - Indexing diffraction patterns by template matching
*
*/
#include <stdlib.h>
#include <math.h>
#include <stdio.h>
#include "image.h"
#include "utils.h"
#include "cell.h"
static void mapping_rotate(double x, double y, double z,
double *ddx, double *ddy, double *ddz,
double omega, double tilt)
{
double nx, ny, nz;
double x_temp, y_temp, z_temp;
/* First: rotate image clockwise until tilt axis is aligned
* horizontally. */
nx = x*cos(omega) + y*sin(omega);
ny = -x*sin(omega) + y*cos(omega);
nz = z;
/* Now, tilt about the x-axis ANTICLOCKWISE around +x, i.e. the
* "wrong" way. This is because the crystal is rotated in the
* experiment, not the Ewald sphere. */
x_temp = nx; y_temp = ny; z_temp = nz;
nx = x_temp;
ny = cos(tilt)*y_temp + sin(tilt)*z_temp;
nz = -sin(tilt)*y_temp + cos(tilt)*z_temp;
/* Finally, reverse the omega rotation to restore the location of the
* image in 3D space */
x_temp = nx; y_temp = ny; z_temp = nz;
nx = x_temp*cos(-omega) + y_temp*sin(-omega);
ny = -x_temp*sin(-omega) + y_temp*cos(-omega);
nz = z_temp;
*ddx = nx;
*ddy = ny;
*ddz = nz;
}
void get_reflections(struct image *image, UnitCell *cell)
{
ImageFeatureList *flist;
double smax = 0.01e9;
double tilt, omega, wavenumber;
double nx, ny, nz; /* "normal" vector */
double kx, ky, kz; /* Electron wavevector ("normal" times 1/lambda) */
double ux, uy, uz; /* "up" vector */
double rx, ry, rz; /* "right" vector */
double asx, asy, asz; /* "a*" lattice parameter */
double bsx, bsy, bsz; /* "b*" lattice parameter */
double csx, csy, csz; /* "c*" lattice parameter */
signed int h, k, l;
double res_max;
/* Get the reciprocal unit cell */
cell_get_reciprocal(cell, &asx, &asy, &asz,
&bsx, &bsy, &bsz,
&csx, &csy, &csz);
/* Prepare list and some parameters */
flist = image_feature_list_new();
tilt = image->tilt;
omega = image->omega;
wavenumber = 1.0/image->lambda;
/* Calculate (roughly) the maximum resolution */
if ( image->fmode == FORMULATION_CLEN ) {
double w2, h2;
w2 = image->width/2; h2 = image->height/2;
w2 = pow(w2, 2.0); h2 = pow(h2, 2.0);
res_max = sqrt(w2 + h2) / image->resolution;
res_max *= (wavenumber / image->camera_len);
} else {
fprintf(stderr,
"Unrecognised formulation mode in get_reflections"
" (resolution cutoff calculation)\n");
return;
}
printf("Resolution cutoff is %5.2f nm^-1\n", res_max/1e9);
res_max = pow(res_max, 2.0);
/* Calculate the (normalised) incident electron wavevector */
mapping_rotate(0.0, 0.0, 1.0, &nx, &ny, &nz, omega, tilt);
kx = nx / image->lambda;
ky = ny / image->lambda;
kz = nz / image->lambda; /* This is the centre of the Ewald sphere */
/* Determine where "up" is */
mapping_rotate(0.0, 1.0, 0.0, &ux, &uy, &uz, omega, tilt);
/* Determine where "right" is */
mapping_rotate(1.0, 0.0, 0.0, &rx, &ry, &rz, omega, tilt);
for ( h=-50; h<50; h++ ) {
for ( k=-50; k<50; k++ ) {
for ( l=-50; l<50; l++ ) {
double xl, yl, zl;
double a, b, c;
double s1, s2, s, t;
double g_sq, gn;
/* Get the coordinates of the reciprocal lattice point */
xl = h*asx + k*bsx + l*csx;
yl = h*asy + k*bsy + l*csy;
zl = h*asz + k*bsz + l*csz;
g_sq = modulus_squared(xl, yl, zl);
gn = xl*nx + yl*ny + zl*nz;
/* Early bailout if resolution is clearly too high */
if ( g_sq > res_max ) continue;
/* Next, solve the relrod equation to calculate
* the excitation error */
a = 1.0;
b = 2.0*(wavenumber + gn);
c = -2.0*gn*wavenumber + g_sq;
t = -0.5*(b + sign(b)*sqrt(b*b - 4.0*a*c));
s1 = t/a;
s2 = c/t;
if ( fabs(s1) < fabs(s2) ) s = s1; else s = s2;
/* Skip this reflection if s is large */
if ( fabs(s) <= smax ) {
double xi, yi, zi;
double gx, gy, gz;
double theta;
double x, y;
double dx, dy, psi;
/* Determine the intersection point */
xi = xl + s*nx; yi = yl + s*ny; zi = zl + s*nz;
/* Calculate Bragg angle */
gx = xi - kx;
gy = yi - ky;
gz = zi - kz; /* This is the vector from the centre of
* the sphere to the intersection */
theta = angle_between(-kx, -ky, -kz, gx, gy, gz);
/* Calculate azimuth of point in image
* (anticlockwise from +x) */
dx = xi*rx + yi*ry + zi*rz;
dy = xi*ux + yi*uy + zi*uz;
psi = atan2(dy, dx);
/* Get image coordinates from polar
* representation */
if ( image->fmode == FORMULATION_CLEN ) {
x = image->camera_len*tan(theta)*cos(psi);
y = image->camera_len*tan(theta)*sin(psi);
x *= image->resolution;
y *= image->resolution;
} else if ( image->fmode==FORMULATION_PIXELSIZE ) {
x = tan(theta)*cos(psi) / image->lambda;
y = tan(theta)*sin(psi) / image->lambda;
x /= image->pixel_size;
y /= image->pixel_size;
} else {
fprintf(stderr,
"Unrecognised formulation mode "
"in get_reflections\n");
return;
}
x += image->x_centre;
y += image->y_centre;
/* Sanity check */
if ( (x>=0) && (x<image->width)
&& (y>=0) && (y<image->height) ) {
/* Record the reflection.
* Intensity should be multiplied by relrod
* spike function, except reprojected
* reflections aren't used quantitatively for
* anything. */
image_add_feature(flist, x, y, image, 1.0);
} /* else it's outside the picture somewhere */
} /* else reflection is not excited in this orientation */
}
}
}
image->rflist = flist;
}
|
