Lecture 2

1. Look at homework problem2. Look at coordinate systems—Chapter 2 in

Massa

Optical Goniometer

Polarizing Microscope

An Introduction to Crystal Coordiantes

I have been frequently asked in the past how to work with crystallographic coordinates.

If the results of a crystal structure are to be used for further studies than it is important to be able to do this

This is also important in deriving bond distances and angles from the atomic positions.

Vectors

Vectors have length and direction We will use bold v to represent a vector. |v| is the magnitude (length) of a vector. The dot product of two vectors is as follows

v1 · v

2 = |v

1| x |v

2| x cos θ where theta is the

angle between the vectors. The dot product is a scalar.

Right Handed Coordinates

Since space has no preferred arrangementthere are two coordinate systems possible.Always use a right handed system!

Cartesian Coordinates

|a|, |b|, and |c| all equal 1 α, β, and γ all 90º A reminder cos(0)=1 cos (90)=0 sin(0)=0

sin(90)=1 The symbols x, y, and z represent positions

along a, b, and c A general vector is given by

v=xa + yb + zc Since the basis vectors magnitude is 1 they are

ignored!

Some Useful Facts

v1 = (x

1,y

1,z

1) [shorthand for x

1a+y

1b+z

1c]

v1 · v

2 = |v

1| x |v

2| x cos(θ)=x

1x

2 +y

1y

2+z

1z

2

v · v = |v| x |v| x cos(0)=x2 +y2+z2

|v| = (x2 +y2+z2)1/2

Non-Cartesian Coordinates

INSIDE LENGTH 39'5”

INSIDE WIDTH 7'8''

INSIDE HEIGHT 7'10”

Let's make the basis vectors the length, width,

and height of the container. Therefore |a|=39'5” |b|=7'8” and |c|=7'10” The basis vectors are orthogonal Place the origin at the door bottom left Now the coordinates inside the box are

fractional. v=xa + yb + zc

Non-orthogonal Coordinates

There is no rule requiring the basis vectors make right angles! It is just very convenient if they do

If the basis vectors have a magnitude of 1 then a·a = b·b = c·c = 1

a·b = cos(γ) b·c = cos(α) a·c = cos(β) If the basis vectors are orthogonal than the dot

products are all zero.

Crystallographic Coordinates

These provide the worst of all possible worlds.They frequently are non-orthogonalThe do not have unit vectors as the basis vectors.The coordinates system is defined by the edges of the unit cell.

Dot Product in random coordinates

Assume |a|, |b|, and |c| not equal to one.

Assume the vectors are not orthogonal

a·b=x1x

2|a|2+y

1x

2|a||b|cosγ+z

1x

2|a||c|cosβ+

x1y

2|a||b|cosγ+y

1y

2|b|2+z

1y

2|b||c|cosα+

x1z

2|a||c|cosβ+y

1z

2|b||c|cosα+z

1z

2|c|2

Magnitude of a Vector

This can be derived from the dot product formula by making x1 and x2 into x, etc.

|v|2=x2|a|2+y2|b|2+z2|c|2+2xy|a||b|cosγ+

2xz|a||c|cosβ+2yz|b||c|cosα

Angles

• The dot product contains the angle between the vectors V·W=|V||W|cos(Θ) then

• Θ=cos-1(V·W/|V||W|)• Note to do this we need to take three dot

products: V·V W·W and V·W • This is a good deal of arithmetic for non-

orthogonal coordinates and much easier for Cartesian coordinates.

Homework Problem #2

Using the cargo containerfor reference coordinates(l=39'5”,w=7'8”, h=7'10”)calculate the distancebetween 2 packageslocated at (0.33,0.16,0.45)and (0.52,0.43,0.23).

Is it valid for a coordinate tobe greater or less thanone?

What does it mean whenthe z (height) coordinate is