What’s a Crystal?
It’s a form of the matter in which the atoms
are arranged in a periodic form.
What’s the unit cell?
It is the basic unit that by simple translations
can generate the complete crystal.
a3
a2
a1
Any physical property of the crystal is
invariant under a translation t(u,v,w)
defined as
t(u,v,w) = u a1 + v a2 + w a3
u, v, w

Z
BRAGG’S LAW
q
q
An X-ray will be diffracted by a set
of crystal planes with a spacing d if
2d sin q = nl
 12.39852
l (A) 
E (keV )
What the hell does h k l mean?
h k l represents a family of parallel planes
with one of them cutting the unit cell vectors
a1, a2, a3 at a1/h, a2/k, a3/l
(equivalent to give the canonical equation of
the plane in direct lattice coordinates)
And reciprocal space?
Instead of using the reference system provided
by the unit cell, one can define other one with
the vectors b1, b2, b3 defined in such a way that
h b1 + k b3 + l b3 give a vector normal to the
family of h k l planes
The reciprocal cell, with no physical existence,
is the one formed by the vectors b1, b2, b3
What is expected from us?
To provide a program that takes into
account all the geometrical calculations
to make their life as easy as just to type
“br 4 4 0” in order to orient crystal and
detectors to detect the 4 4 0 reflection
Why there are so many SPEC geometries?
Let’s try to answer ...
Geometry Handling
A vector v can be written in terms of the
reciprocal lattice vectors as
3
v   vi bi
i 1
However, it is more convenient to work
with a set of cartesian axes attached to
the crystal choosing
x axis parallel to b1
y axis in the plane of b1 and b2
z axis perpendicular to b1 and b2
b3
b2
z
y
x
b1
The B matrix
b3
b2
1
z
y
3
x
2
b1
vc = B v
 b b cos 

b
cos

 1 2


3
2


B   0 b2 sin   b3 sin 2 cos  1 
 0

0
1 / a3


sin  
cos a3  b1 b2
Cell Volume
cos 1 
cos  2 cos    cos  1
sin  2 sin  
The U Matrix
Now, we associate a set of cartesian axes
to each of the circles of the diffractometer
in such a way that are all coincident with
the laboratory system when all the angles
are zero
The matrix U is the one transforming the
coordinates from the crystal cartesian axes
to the system mounted on the first circle
(called phi in common geometries)
That matrix is obtained experimentally from
the positions of two reflections
vf = U vc = UB v
Rotations
Now that the vector is expressed in terms
of the systems fixed to one circle, all what
we need to pass to the laboratory system
is to know the positions of the motors and
the rotation axis of each circle in the lab
system
Assuming angles are positive counter clockwise
Rotation around X axis
1
0

 0 cos 

 0 sin

0 

 sin 

cos  
Rotation around Y axis
 cos  0  sin 


 0
1
0 


 sin 0

cos



Rotation around Z axis
 cos 

 sin

 0

 sin
cos 
0
0

0

1 
The SPEC FOURC Geometry
z
X-ray beam
x
y
F 1st circle, rotates around z axis
c 2nd circle, CLOCKWISE around y axis
Q 3rd circle, rotates around z axis
vlab = Q C F UB v
v = (UB)-1 F-1 c-1 Q-1 vlab
The ID20 Patch
The System
fourc geometry with an additional circle mounted
on the phi circle. Its rotation axis is on the x axis
in the lab system when all angles are set to zero.
The rotation is positive when moving clockwise.
The Problem
When that circle, called rho, is moved, SPEC had
no means of finding the reflections
The Pain
A diffraction peak obtained at rho=0 is lost when
moving the motor and has to be found back by
hand
The (simple) solution
The actual transformation is
vlab = Q C F r UB v
rho corresponds to a clockwise rotation
around an x axis
1

0

0

0
cos r
 sinr


sinr 


cos r 
0
The solution consists on replacing the
matrix UB calculated by SPEC by the
matrix (r UB) at each movement of rho
This is done by entering the values in the
associative array UB and calling userUB
Conclusion
Only two circles are needed to orient a vector
in space. Other circles serve to put constraints
Each instrument is in a certain sense optimized
for a set of experiments. Matrix product is not
commutative, therefore, the slightest change
implies a new set of equations
There is no unified criterion about the reference
systems nor about the positive sense of the
rotations. To set a new instrument can be a real
nightmare unless a detailed description of the
code is available