What’s a Crystal?

It’s a form of the matter in which the atomsare arranged in a periodic form.

It is the basic unit that by simple translationscan generate the complete crystal.

What’s the unit cell?

1a

2a3a

Any physical property of the crystal isinvariant under a translation t(u,v,w)defined ast(u,v,w) = u a1 + v a2 + w a3

u, v, w Z

BRAGG’S LAW

An X-ray will be diffracted by a setof crystal planes with a spacing d if

2d sin = n

)(39852.12)A(keVE

What the hell does h k l mean?

h k l represents a family of parallel planeswith one of them cutting the unit cell vectorsa1, 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 providedby the unit cell, one can define other one withthe vectors b1, b2, b3 defined in such a way thath b1 + k b3 + l b3 give a vector normal to thefamily 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 intoaccount all the geometrical calculationsto make their life as easy as just to type “br 4 4 0” in order to orient crystal anddetectors 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 thereciprocal lattice vectors as

3

1iiibv v

However, it is more convenient to workwith a set of cartesian axes attached tothe crystal choosing x axis parallel to b1

y axis in the plane of b1 and b2

z axis perpendicular to b1 and b2

b1

b2

b3

z y

x

b1

b2

b3

z y

x

The B matrix

vc = B v

3

1232

2321

/100

cos0

coscos

a

sinbsinb

bbb

B

2

1

3

sinsin

coscoscoscos

2

121VolumeCell

bba sincos 21

3

The U Matrix

v = U vc = UB v

Now, we associate a set of cartesian axesto each of the circles of the diffractometerin such a way that are all coincident withthe laboratory system when all the angles are zero

The matrix U is the one transforming thecoordinates from the crystal cartesian axesto the system mounted on the first circle(called phi in common geometries)

That matrix is obtained experimentally fromthe positions of two reflections

Rotations

Now that the vector is expressed in terms of the systems fixed to one circle, all whatwe need to pass to the laboratory systemis to know the positions of the motors andthe rotation axis of each circle in the lab system

Assuming angles are positive counter clock-wise

Rotation around X axis

cos0

cos0

001

sin

sin

Rotation around Y axis

cos0

010

0cos

sin

sin

Rotation around Z axis

100

0cos

0cos

sin

sin

The SPEC FOURC Geometry

y

x

z

X-ray beam

1st circle, rotates around z axis 2nd circle, CLOCKWISE around y axis 3rd circle, rotates around z axis

vlab = UB v

v = (UB)-1 -1 -1 -1 vlab

The ID20 Patch

The System

fourc geometry with an additional circle mountedon the phi circle. Its rotation axis is on the x axisin 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 hadno means of finding the reflections

The Pain

A diffraction peak obtained at rho=0 is lost whenmoving the motor and has to be found back by hand

The (simple) solution

The actual transformation is

vlab = UB v

rho corresponds to a clockwise rotationaround an x axis

cos0

cos0

001

sin

sin

The solution consists on replacing thematrix UB calculated by SPEC by thematrix ( UB) at each movement of rho

This is done by entering the values in theassociative array UB and calling userUB

Conclusion

Only two circles are needed to orient a vectorin space. Other circles serve to put constraints

Each instrument is in a certain sense optimizedfor a set of experiments. Matrix product is notcommutative, therefore, the slightest changeimplies a new set of equations

There is no unified criterion about the referencesystems nor about the positive sense of therotations. To set a new instrument can be a realnightmare unless a detailed description of thecode is available