Cartoons showing the use of the Ewald sphere in diffraction
Matlab codes
% Program to generate 2D schematic movie of rotation method
clear; close all;
vid = VideoWriter('MXexperimental2D.mp4','MPEG-4');
vid.Quality = 100;
vid.FrameRate = 60;
open(vid);
figure('units','pixels','position',[0 0 1920 1080],'ToolBar','none');
set(0,'defaultfigurecolor',[1 1 1]);
set(gca,'linewidth',7);
kV = 1/1; % Radius of Ewald sphere, here 1 AA^-1
ewaldCol = [1.0 0.83 0]; % Yellow colour of Ewald sphere
% Create a set of diffraction maxima with separations of 1/(2*nBP) AA^-1 out to
% 2 AA (0.5 AA^-1)
bpCol = [0.3 0.28 0.55]; % Color of Bragg peaks (if not on surface of Ewald sphere (ES))
rBP = 0.0053; % Radius of Bragg peak as plotted on ES
numBP = 7; % Number of Bragg peaks to edge of diffraction pattern at 0.5 AA^-1
% Vertices of drawn crystal
rhSize = 0.025;
v1 = [rhSize,0,0];
v2 = [0,rhSize,0];
v3 = [0,0,rhSize*1.25];
v4 = [-rhSize,0,0];
v5 = [0,-rhSize,0];
v6 = [0,0,-rhSize*1.25];
[x,y,z] = sphere; % Create 3D array to plot spheres
[a,b,c] = sphere; % Create 3D array to plot spheres
for phi = 0:0.125:179.875 % Rotate Bragg peaks around y-axis in steps of 0.25 degrees
hold off
% Plot semitransparent Ewald sphere at (-kV, 0, 0)
[x,y,z] = sphere(70); surf(kV*x-kV,kV*y,kV*z,'FaceAlpha',0.16,'FaceColor',...
ewaldCol,'LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
hold on
% Draw crystal
f1 = fill3(v1,v2,v3,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f2 = fill3(v4,v2,v3,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f3 = fill3(v1,v2,v6,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f4 = fill3(v4,v2,v6,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
rotate(f1,[0 1 0],phi,[0,0,0]);
rotate(f2,[0 1 0],phi,[0,0,0]);
rotate(f3,[0 1 0],phi,[0,0,0]);
rotate(f4,[0 1 0],phi,[0,0,0]);
% Loop through hkl space
for h = -0.5:0.5/numBP:0.5
for l = -0.5:0.5/numBP:0.5
hl = (h^2 + l^2)^0.5;
if (hl <= 0.5) % Only plot out Bragg peaks in reciprocal lattice
% within radius of 0.5 AA^-1.
hold on
LL = 1.01; % Defines size of plotted figure
xlim([-LL LL]);
ylim([-LL LL]);
zlim([-LL LL]);
% Rotate reciprocal lattice to angle phi
% and calculate new absolute (h k l) after rotation
hnew = h*cosd(phi) + l*sind(phi);
lnew = l*cosd(phi) - h*sind(phi);
% Calculate Delk, absolute distance from centre of Ewald sphere to hkl Bragg peak
Delk = ((kV + hnew)^2 + (lnew)^2)^0.5;
% Difference between centre of Bragg peak and Ewald sphere surface
Del = abs(kV - Delk);
if (Del > 2.0*rBP) % Undetected Bragg peaks if centre more than 2 x rBP from ES
H = surf(rBP*x+h,rBP*y,rBP*z+l,'FaceAlpha',(0.5 - hl)/0.5,...
'FaceColor',bpCol,'LineStyle','none');
rotate(H,[0 1 0],phi,[0,0,0]);
elseif (hl == 0) % (000) direct beam
apos = 0.78;
else % Detected Bragg peaks lying on Ewald sphere surface
H = surf(2*rBP*x+h,2*rBP*y,2*rBP*z+l,'FaceAlpha',...
((0.5 - (hl)^2)/0.5)*(1 - Del/(2*rBP))^0.5,...
'FaceColor','r','LineStyle','none','FaceLighting','Flat');
rotate(H,[0 1 0],phi,[0,0,0]);
% Plot Bragg peak rays if BP on or close to Ewald sphere
apos = 0.78;
bpos = 0;
cpos = lnew * (1.8/(1+hnew));
BPpos = cpos; % Radius of position of BP on detector
BPline = plot3([-1 apos],[0 0],[0 cpos],'color','red', 'LineWidth', 1.4);
BPline.Color(4) = ((0.5 - (hl)^2)/0.5)*(1 - Del/(2*rBP))^0.5;
DetSurfLine = plot3([apos+0.02 apos+0.02],[0 0],[-1 1],'color',[0,0,0], 'LineWidth', 14);
end % End of finding out if BP on ES
end % End of selecting only hl values within radius of 0.5 r.l.u.
end % End of l-loop
end % End of h-loop
% End of drawing diffraction maxima
% Cross at ES origin and dotted axis
line1 = plot3([-1 -1],[-0.05 0.05],[0 0],'color','red', 'LineWidth', 1.4);
line2 = plot3([-1 -1],[0 0],[-0.05 0.05],'color','red', 'LineWidth', 1.4);
line3 = plot3([-1.05 0.78],[0 0],[0 0],'color','red', 'LineWidth', 1.4, 'LineStyle','-.');
set(gca, 'Projection','perspective');
set(gca,'View',[0,0]);
lp = [-0.7 -0.5 0.5];
light('Position',lp,'Style','infinite');
axis equal
axis off
LL = 1.01;
xlim([-LL LL]);
ylim([-LL LL]);
zlim([-LL LL]);
% Store the frame
frame = getframe(gcf);
writeVideo(vid,frame);
end % End of phi-rotation loop
% Output the movie as an mpg file
close(vid);
% Program to generate 2D schematic movie of rotation method
clear; close all;
vid = VideoWriter('MXexperimental2D.mp4','MPEG-4');
vid.Quality = 100;
vid.FrameRate = 60;
open(vid);
figure('units','pixels','position',[0 0 1920 1080],'ToolBar','none');
set(0,'defaultfigurecolor',[1 1 1]);
set(gca,'linewidth',7);
kV = 1/1; % Radius of Ewald sphere, here 1 AA^-1
ewaldCol = [1.0 0.83 0]; % Yellow colour of Ewald sphere
% Create a set of diffraction maxima with separations of 1/(2*nBP) AA^-1 out to
% 2 AA (0.5 AA^-1)
bpCol = [0.3 0.28 0.55]; % Color of Bragg peaks (if not on surface of Ewald sphere (ES))
rBP = 0.0053; % Radius of Bragg peak as plotted on ES
numBP = 7; % Number of Bragg peaks to edge of diffraction pattern at 0.5 AA^-1
% Vertices of drawn crystal
rhSize = 0.025;
v1 = [rhSize,0,0];
v2 = [0,rhSize,0];
v3 = [0,0,rhSize*1.25];
v4 = [-rhSize,0,0];
v5 = [0,-rhSize,0];
v6 = [0,0,-rhSize*1.25];
[x,y,z] = sphere; % Create 3D array to plot spheres
[a,b,c] = sphere; % Create 3D array to plot spheres
for phi = 0:0.125:179.875 % Rotate Bragg peaks around y-axis in steps of 0.25 degrees
hold off
% Plot semitransparent Ewald sphere at (-kV, 0, 0)
[x,y,z] = sphere(70); surf(kV*x-kV,kV*y,kV*z,'FaceAlpha',0.16,'FaceColor',...
ewaldCol,'LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
hold on
% Draw crystal
f1 = fill3(v1,v2,v3,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f2 = fill3(v4,v2,v3,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f3 = fill3(v1,v2,v6,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f4 = fill3(v4,v2,v6,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
rotate(f1,[0 1 0],phi,[0,0,0]);
rotate(f2,[0 1 0],phi,[0,0,0]);
rotate(f3,[0 1 0],phi,[0,0,0]);
rotate(f4,[0 1 0],phi,[0,0,0]);
% Loop through hkl space
for h = -0.5:0.5/numBP:0.5
for l = -0.5:0.5/numBP:0.5
hl = (h^2 + l^2)^0.5;
if (hl <= 0.5) % Only plot out Bragg peaks in reciprocal lattice
% within radius of 0.5 AA^-1.
hold on
LL = 1.01; % Defines size of plotted figure
xlim([-LL LL]);
ylim([-LL LL]);
zlim([-LL LL]);
% Rotate reciprocal lattice to angle phi
% and calculate new absolute (h k l) after rotation
hnew = h*cosd(phi) + l*sind(phi);
lnew = l*cosd(phi) - h*sind(phi);
% Calculate Delk, absolute distance from centre of Ewald sphere to hkl Bragg peak
Delk = ((kV + hnew)^2 + (lnew)^2)^0.5;
% Difference between centre of Bragg peak and Ewald sphere surface
Del = abs(kV - Delk);
if (Del > 2.0*rBP) % Undetected Bragg peaks if centre more than 2 x rBP from ES
H = surf(rBP*x+h,rBP*y,rBP*z+l,'FaceAlpha',(0.5 - hl)/0.5,...
'FaceColor',bpCol,'LineStyle','none');
rotate(H,[0 1 0],phi,[0,0,0]);
elseif (hl == 0) % (000) direct beam
apos = 0.78;
else % Detected Bragg peaks lying on Ewald sphere surface
H = surf(2*rBP*x+h,2*rBP*y,2*rBP*z+l,'FaceAlpha',...
((0.5 - (hl)^2)/0.5)*(1 - Del/(2*rBP))^0.5,...
'FaceColor','r','LineStyle','none','FaceLighting','Flat');
rotate(H,[0 1 0],phi,[0,0,0]);
% Plot Bragg peak rays if BP on or close to Ewald sphere
apos = 0.78;
bpos = 0;
cpos = lnew * (1.8/(1+hnew));
BPpos = cpos; % Radius of position of BP on detector
BPline = plot3([-1 apos],[0 0],[0 cpos],'color','red', 'LineWidth', 1.4);
BPline.Color(4) = ((0.5 - (hl)^2)/0.5)*(1 - Del/(2*rBP))^0.5;
DetSurfLine = plot3([apos+0.02 apos+0.02],[0 0],[-1 1],'color',[0,0,0], 'LineWidth', 14);
end % End of finding out if BP on ES
end % End of selecting only hl values within radius of 0.5 r.l.u.
end % End of l-loop
end % End of h-loop
% End of drawing diffraction maxima
% Cross at ES origin and dotted axis
line1 = plot3([-1 -1],[-0.05 0.05],[0 0],'color','red', 'LineWidth', 1.4);
line2 = plot3([-1 -1],[0 0],[-0.05 0.05],'color','red', 'LineWidth', 1.4);
line3 = plot3([-1.05 0.78],[0 0],[0 0],'color','red', 'LineWidth', 1.4, 'LineStyle','-.');
set(gca, 'Projection','perspective');
set(gca,'View',[0,0]);
lp = [-0.7 -0.5 0.5];
light('Position',lp,'Style','infinite');
axis equal
axis off
LL = 1.01;
xlim([-LL LL]);
ylim([-LL LL]);
zlim([-LL LL]);
% Store the frame
frame = getframe(gcf);
writeVideo(vid,frame);
end % End of phi-rotation loop
% Output the movie as an mpg file
close(vid);
% Program to generate 3D schematic movie of rotation method used in MX
clear; close all;
figure('units','pixels','position',[0 0 1920 1080],'ToolBar','none');
vid = VideoWriter('MXexperimental2.mp4','MPEG-4');
vid.Quality = 100;
vid.FrameRate = 60;
open(vid);
set(0,'defaultfigurecolor',[1 1 1]);
set(gca,'linewidth',7);
% Create semitransparent Ewald sphere centered around (-kV,0,0) with 1
% reciprocal Angstrom radius
kV = 1/1; % Radius of Ewald sphere, here 1 AA^-1
ewaldCol = [1.0 0.83 0]; % Yellow colour of Ewald sphere
% End of drawing Ewald sphere
% Create a set of diffraction maxima with separations of 1/40 AA^-1 out to
% 2 A (0.5 A^-1)
bpCol = [0.3 0.28 0.55]; % Color of Bragg peaks (if not on surface of Ewald sphere (ES))
lyel = [1 1 0.64];
rBP = 0.0053; % Radius of Bragg peak
numBP = 12;
ewaldRad = 0.5;
stepSize = ewaldRad/numBP;
% Vertices of drawn crystal
rhSize = 0.05;
X1 = [0 0 rhSize]; % x-coordinates of vertices of first face of octahedron
Y1 = [0,-rhSize,0]; % y-coordinates of vertices of first face of octahedron
Z1 = [rhSize,0 0]; % z-coordinates of vertices of first face of octahedron
X2 = [0 0 rhSize]; % etc, etc...
Y2 = [0,rhSize,0];
Z2 = [rhSize,0 0];
X3 = [0 0 -rhSize];
Y3 = [0,-rhSize,0];
Z3 = [rhSize,0 0];
X4 = [0 0 -rhSize];
Y4 = [0,rhSize,0];
Z4 = [rhSize,0 0];
X5 = [0 0 rhSize];
Y5 = [0,-rhSize,0];
Z5 = [-rhSize,0 0];
X6 = [0 0 rhSize];
Y6 = [0,rhSize,0];
Z6 = [-rhSize,0 0];
X7 = [0 0 -rhSize];
Y7 = [0,-rhSize,0];
Z7 = [-rhSize,0 0];
X8 = [0 0 -rhSize];
Y8 = [0,rhSize,0];
Z8 = [-rhSize,0 0];
[x,y,z] = sphere;
[a,b,c] = sphere;
[Z,Y,X] = cylinder(1,180);
detector = zeros(3,101);
th = 0:pi/50:2*pi;
detector(1,:) = zeros(1,101) + 0.8;
detector(2,:) = 1*cos(th);
detector(3,:) = 1*sin(th);
for phi = 0:0.25:89.75 % Rotate Bragg peaks around y-axis
%newplot;
phi
hold off
[x,y,z] = sphere(70); surf(kV*x-kV,kV*y,kV*z,'FaceAlpha',0.16,'FaceColor',...
ewaldCol,'LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
hold on
% Draw crystal
f1 = fill3(X1,Y1,Z1,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f2 = fill3(X2,Y2,Z2,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f3 = fill3(X3,Y3,Z3,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f4 = fill3(X4,Y4,Z4,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f5 = fill3(X5,Y5,Z5,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f6 = fill3(X6,Y6,Z6,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f7 = fill3(X7,Y7,Z7,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
f8 = fill3(X8,Y8,Z8,'green','LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
rotate(f1,[0 1 0],phi,[0,0,0]);
rotate(f2,[0 1 0],phi,[0,0,0]);
rotate(f3,[0 1 0],phi,[0,0,0]);
rotate(f4,[0 1 0],phi,[0,0,0]);
rotate(f5,[0 1 0],phi,[0,0,0]);
rotate(f6,[0 1 0],phi,[0,0,0]);
rotate(f7,[0 1 0],phi,[0,0,0]);
rotate(f8,[0 1 0],phi,[0,0,0]);
for h = -0.5:0.5/numBP:0.5
for k = -0.5:0.5/numBP:0.5
for l = -0.5:0.5/numBP:0.5
hkl = (h^2 + k^2 + l^2)^0.5;
if (hkl <= 0.5) % Plot out Bragg peaks within radius of 0.5 r.l.u.
hold on
LL = 1.05;
xlim([-LL LL]);
ylim([-LL LL]);
zlim([-LL LL]);
% Calculate new absolute h k l after rotation
hnew = h*cosd(phi) + l*sind(phi);
knew = k;
lnew = l*cosd(phi) - h*sind(phi);
% Calculate Delk, absolute distance from centre of Ewald sphere to hkl Bragg peak
Delk = ((kV + hnew)^2 + (knew)^2 + (lnew)^2)^0.5;
Del = abs(kV - Delk); % Difference between centre of Bragg peak and Ewald sphere surface
if (Del > 2.0*rBP) % Undetected Bragg peaks
H = surf(rBP*x+h,rBP*y+k,rBP*z+l,'FaceAlpha',(0.5 - hkl)/0.5,...
'FaceColor',bpCol,'LineStyle','none');
rotate(H,[0 1 0],phi,[0,0,0]);
elseif (hkl == 0) % (000) direct beam
H = surf(2*rBP*x+h,3.5*rBP*y+k,3.5*rBP*z+l,'FaceAlpha',...
(0.5 - hkl)/0.5,...
'FaceColor','r','LineStyle','none','FaceLighting','Flat');
[a,b,c] = sphere;
apos = 0.78;
BP = surf(0.02*rBP*a+apos,4*rBP*b,4*rBP*c,'FaceAlpha',1,...
'FaceColor',lyel,'LineStyle','none','FaceLighting','Flat');
else % Detected Bragg peaks lying on Ewald sphere surface
H = surf(2*rBP*x+h,2*rBP*y+k,2*rBP*z+l,'FaceAlpha',...
(0.5 - (hkl)^2)/0.5*(1 - Del/(2*rBP))^0.5,...
'FaceColor','r','LineStyle','none','FaceLighting','Flat');
rotate(H,[0 1 0],phi,[0,0,0]);
% Plot Bragg peaks detected on detector
apos = 0.78;
bpos = knew * (1.8/(1+hnew));
cpos = lnew * (1.8/(1+hnew));
BPpos = (bpos^2 + cpos^2)^0.5; % Radius of position of BP on detector
if (BPpos < 1.0)
BP = surf(0.02*rBP*a+apos,1.25*rBP*b+bpos,1.25*rBP*c+cpos,...
'FaceAlpha',(0.5 - (hkl)^2)/0.5*(1 - Del/(2*rBP))^0.5,...
'FaceColor',lyel,'LineStyle','none');
end % End of finding out if BP within detector
end % End of finding out if BP on ES
end % End of selecting only hkl values within radius of 0.5 r.l.u.
end % End of l-loop
end % End of k-loop
end % End of h-loop
% End of drawing diffraction maxima
% Side wall of detector
cyl = surf(0.8+X/5,Y*1.05,Z*1.05,'FaceAlpha',1,'FaceColor',[0.1 0.1 0.1],...
'LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
% Detector face
fill3(detector(1,:),1.05*detector(2,:),1.05*detector(3,:),[0 0 0]);
fill3(detector(1,:)-0.005,detector(2,:),detector(3,:),[0.2 0.1 0.1]);
% Side wall of beamstop
cyl = surf(0.7+X/12.5,0.03*Y,0.03*Z,'FaceAlpha',1,'FaceColor',[0.2 0.2 0.2],...
'LineStyle','none','FaceLighting','gouraud','DiffuseStrength',1);
% Beamstop face
fill3(detector(1,:)-0.1,0.03*detector(2,:),0.03*detector(3,:),[0.4 0.4 0.4],...
'LineStyle','none');
% Cross at ES origin and dotted axis
line1 = plot3([-1 -1],[-0.05 0.05],[0 0],'color','red', 'LineWidth', 1.4);
line2 = plot3([-1 -1],[0 0],[-0.05 0.05],'color','red', 'LineWidth', 1.4);
line3 = plot3([-1.05 0.7],[0 0],[0 0],'color','red', 'LineWidth', 1.4, 'LineStyle','-.');
set(gca, 'Projection','perspective');
lp = [-0.7 -0.5 0.5];
light('Position',lp,'Style','infinite');
if (phi == -0.5) % Set negative to skip the initial pan loop,
%if (phi == 0) % Set to 0 to include inital pan loop
for ang = 0:0.25:20
set(gca,'View',[-2*ang,ang]);
axis equal
axis off
LL = 1.05;
xlim([-LL LL]);
ylim([-LL LL]);
zlim([-LL LL]);
% Store the frame
frame = getframe(gcf);
writeVideo(vid,frame);
end
else % Rest of animation
set(gca,'View',[-40,20]);
axis equal
axis off
LL = 1.05;
xlim([-LL LL]);
ylim([-LL LL]);
zlim([-LL LL]);
% Store the frame
frame = getframe(gcf);
writeVideo(vid,frame);
end
end % End of phi-rotation loop
% Output the movie as an mpg file
close(vid);