12. Bragg X-ray Scattering from a Crystal

Introduction

You probably are familiar with the x-ray picture taken of your teeth in a dentist office or bone when having a chest x-ray.  X-rays are a form of electromagnetic radiation, like light but having a shorter wavelength of  about (10^(-8)) cm. The calcium material in bone absorbs x-rays and casts a shadow much like your body casts a shadow when you are in the sun.  The shape of the shadows in both cases can be explained by assuming x-rays (or light) has ray characteristics and it is natural to assume photons which are particles of x-rays or light travel along the rays. (You also were introduced to the particle aspect of light in the photoelectric experiment.)  Here you are introduced to the wave properties of x-rays.

Crystal Structure

Ordinary table salt is sodium chloride or NaCl and under a microscope the salt has a cubic shape.  This kind of solid is called crystalline since the atoms are in regular arrays.  You can "grow" larger (or smaller) crystals by dissolving salt in warm water and then wait for the water to cool to room temperature.  In this process, crystals of salt form on the bottom the container and these crystals all have the crystalline shape albeit the crystals are of different size.   This observation leads to the idea that salt crystals are made out of fundamental, small building blocks called a "unit cell" something like pictured below:

The distance d between atoms in the cubic arrangement of salt is

d = 2.82 ×(10^(-10)) meters

Determination of d:  Distance between Atoms

It might be helpful to understand how d can be determined from the density.  The  ρ=mass/volume for NaCl is

ρ = 2.26×(10^3) kg/(m^3)

and the molecular weight M of NaCl

M = 58.46×(10^(-3)) kg/mole.

Recall Avogadro's Number is

A = 
      6.02×(10^23) molecules/mole.

so the mass of NaCl is

m = (M/A)

The volume of a unit cell of salt is

V = (d^3)

so

ρ = (m/V) = ((M/A)/(2  (d^3)))

The 2 comes from the fact that NaCl has two particles per molecule.  Solving equation (7) for d yields

d = (((M/(2  ρ   A)))^(1/3))

and numerically

d = ((((58.46*(10^(-3)))/(2*2.26*(10^3)*6.02*(10^23))))^(1/3))

where this answer for d is in meters.

Bragg X-ray Scattering

When x-rays meet an individual atom of the salt crystal, the x-rays are scattered in all directions.  When you have a regular array of atoms then there is constructive and destructive from the waves scattered by different atoms and in effect the x-rays are "reflected" from the crystal.  Pictured below is a side view of a crystal with a beam of x-rays coming from above

The line AD (which represents points of common phase on the two incident rays) is perpendicular to AB because the incident waves along the two rays are in phase (by assumption).   Similarly, The line CD (which represents points of common phase on the two reflected rays) is perpendicular to BC because the "reflected" waves along the two rays are in phase to have constructive interference. This observation leads to the result that the angel of incidence θ   equals the angle of reflection θ' (with a little trigonometry).  Also the two angles θ indicated in the figure above are equal since when they are combined with the common angle α one gets right angles. As mentioned before, the two rays combine constructively in the reflected wave and they have a difference in path length of AB+BC=2×AB.  This difference in path length must be an integer n multiple of the wavelength λ of the x-rays in order to have constructive interference. Thus

2 × AB =  nλ

The definition of the sine function yields

sinθ = (AB/d)

Combining equations (9) and (10) yields

sinθ = (( nλ)/(2  d))

with n=1, 2, 3, ... and integer.  Equation (11)  is Bragg's relation for scattering of x-rays from a crystal.  You can use the above relationship together with the angle θ where the scattered x-rays are a maximum to determine the wavelength λ of the x-rays.  Incidentally equation (9) holds only for waves so the Bragg relationship (11)  is a result of a wave picture of x-rays.  "Reflection" of x-rays from a crystal is quite different from reflection of light from a mirror since in the later case θ can have any value.

Wavelength of the X-rays

The Tel-X-Ometer (described later) has an x-ray tube which produces x-rays of two different wavelengths which are close together.  These two wavelengths are called copper Kα and copper Kβ since they are associated with the K quantum number.  α and β are used to designate the two different wavelengths.  You will find one peak of x-rays at (Subscript[θ, α])=16°00' and another peak of x-rays at about (Subscript[θ, β])=14°20'  so they are fairly close together.  We can use this information together with the Bragg relation to calculate the corresponding wavelengths (Subscript[λ, α]) and (Subscript[λ, α]).  For example, the Bragg relation for the α peak is

\(\(\(\ \)\(sinθ\_α = \(\(\(\ \)\(λ\_α\)\)\/\(2  d\)\ \ \ \ with\ \ \ \ n = 1\)\)\)\)

(with n=1 for the first peak) and it follows that

(Subscript[λ, α]) = 
      2  dsin(Subscript[θ, α]).

Substitution for the angle  (Subscript[θ, α])=16°00' and d=2.82×(10^(-10)) yields

(Subscript[λ, α]) = 2*2.8*(10^(-10))*Sin[16.0*(π/180)]

or a wavelength (Subscript[λ, α])=1.54×(10^(-10)) m.  (Note we converted from angular measure in degrees to angular measure in radians in the argument of the Sin function.  Also, the Tel-X-Ometer measures angles in degrees and minutes and these should be converted to decimal degrees to use in the Bragg formula.)

Exercise #1

You should look for this peak of x-rays on the Tel-X-Ometer at about (Subscript[θ, α])=32°00' .  IMPORTANT NOTE:  As will be discussed later, the angle shown on the Tel-X-Ometer is actually  (Subscript[θ, α])=2×θ  so you must take the measured angle and divide by two to get the Bragg angle θ.    As said before, there is another peak of x-rays at about (Subscript[θ, β])=28°30' and you should look for this peak of x-rays on the Tel-X-Ometer.    Calculate the wavelength of x-rays associated with this second peak.  You will NOT get the same angles as given above due to experimental error.  Write your results in the table below:

({{X - 
            Ray, Theory Angle θ, Experimental Angle θ}, {α, 16.0  °, \[Placeholder]}, {β, 14.3  °, \[Placeholder]}})

Your results should be almost repeatable. Repeat measurement (at the same angle) of the counts/10 sec will NOT be the same. This is because n=counts/10 sec are distributed according to a Poisson distribution and the uncertainty is (Sqrt[n]).  Once you locate a peak roughly you should make a fine (say every 10 minutes of angle) sweep about 1° on either side of the peak.  Record your results in your lab notebook (to help you find the peak more accurately) but for the report just give the result for the angle of the peak.  What is the experimental error in your wavelength measurement (Theoretical Angle minus Experimental Angle)? This error is largely the result of misalignment of the angular measuring device.

Determination of n=2, n=3 peaks

Now that we know the two wavelengths of the x-rays produced by our tube we can use the Bragg relationship to determined the angular location of higher order peaks.   For example, there should be an n=2 peak associated with (Subscript[λ, α]) and you can determine its location (subject to instrument error) as follows.  First

sin(Subscript[θ, α])=(( 2  (Subscript[λ, α]))/(2  d)) =(( (Subscript[λ, α]))/d) =(( 1.54×(10^(-10)) m)/(2.82×(10^(-10)) m))  with    n=2

thus computationally

θ = 
      ArcSin[(( 1.54*(10^(-10)) )/(2.82*(10^(-10))))]

which is the angle in radians.  Converting to degrees

θ = 0.577*(180/π)

0.0597*60

so there should be an n=2 peak θ=33° 3.5'  or at about (Subscript[θ, α]) = 66  ° 7'.  Find this peak on the Tel-X-Ometer.

Exercises #2

Follow the procedure above and calculate all the n=2 and n=3 x-ray peaks.  Fill in the blanks in the table below by calculating the theoretical value from the Bragg formula equation (11).  Also included your observed or experimental values of the radiation peaks from the Tel-X-Ometer:

({{\[Placeholder], n = 1, n = 1, n = 2, n = 2, n = 
            3, n = 3}, {x - 
            ray λ, Theo .   θ, Exp .  θ, Theo .   θ, Exp .  θ, Theo .   θ, Exp .  θ}, {α, 16  °00', \[Placeholder], 33  ° 7', \[Placeholder], \[Placeholder], \[Placeholder]}, {β, 14  °10', \[Placeholder], \[Placeholder], \[Placeholder], \[Placeholder], \[Placeholder]}})

Were you able to find all the peaks?  Is the instrument error about the same for all of the above?

The Tel-X-Ometer

The Tel-X-Ometer x-ray unit is pictured below:

The copper x-ray tube is the circular structure near the top and midway across.  Free electrons are created from a filament in the x-ray tube much like the filament in an ordinary light bulb.  The electrons get so hot they boil off the surface of the filament.  The copper plater of the x-ray tube is negative (about 30 kilovolts) relative the filament so the electrons are accelerated toward the plate.  When the electons strike the copper plate they cause some of the electrons of the copper atoms to rise to a higher energy.  The electrons of the copper atoms quickly return to the ground state and when they do they give off x-rays of a two characteristic wavelengths: (Subscript[λ, α])=1.54×(10^(-10)) m  and  (Subscript[λ, β])=1.38×(10^(-10)) m.   The crystal is mounted at the center of the Tel-x-Ometer as indicated below:

IMPORTANT:  When tightening the clamping screw make sure you do it NOT too tightly or the pressure will break the crystal.  Also do NOT leave the crystal in the mount overnight since Sodium Chloride absorbs water vapor especially on damp days and the moisture corrodes the metal.

Geometry of Bragg Scattering

Below is a schematic drawing of the x-ray tube, crystal, geiger tube and path the x-rays take in Bragg scattering:

Also indicated is the path of the x-rays if the crystal is removed.  The gearing in the crystal holder and geiger tube mount is designed to that when the crystal rotates through an angle,  the geiger tube moves through and angle 2θ.

Exercise #3

The x-ray spectra of the NaCl crystal is pictured below.  Peak 2 and the envelop is due to Bremsstrahlung or "Braking Radiation" and is not of interest to us.  The features 3 -8 are of interest as they are the Bragg peaks.  Collect data from the Tel-X-ometer and graph your results.  Compare agreement or disagreement with the graph below.

HIINT:  Take counts/10 sec every degree except near a peak.  Near a peak, take data every 10 minutes of arc since the counts rate varies rapidly near a peak.