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## X-ray and neutron diffraction

Lesson for 16-19

Students can apply their understanding of diffraction to X-ray and neutron diffraction studies of the structure of matter.

This topic could extend a study of diffraction of waves, or be part of a study of material structures, or of atomic physics.

## Episode 529: Preparation for X-ray and neutron diffraction topic

Teaching Guidance for 16-19

- Level Advanced

This topic could extend a study of diffraction of waves, or be part of a study of material structures, or of atomic physics.

#### Main aims of this topic

Students will:

- relate diffraction effects to the spacing of gratings etc and to the wavelength of the radiation involved
- state some uses of X-ray and neutron diffraction
- apply Bragg’s law

#### Prior knowledge

Students should be familiar with diffraction due to a standard (transmission) grating
*d*sin( θ ) = n × *λ*
,
de Broglie’s formula
*λ* = *h**p*.
It will also help if they are aware that, at (absolute) temperature *T*, typical energy ~ *k**T*.

#### Where this leads

This topic could lead into a study of the structures and properties of solid materials.

### Up next

### X-ray diffraction

## Episode 530: X-ray diffraction

Lesson for 16-19

- Activity time 170 minutes
- Level Advanced

Most schools will not have an X-ray set, but there are plenty of effective analogue demonstrations using other types of waves (laser light, microwaves, and water waves).

Lesson Summary

- Discussion and student questions: Students’ knowledge of X-rays (30 minutes)
- Discussion: Diffraction and the limits of resolution (20 minutes)
- Demonstrations: Diffraction of laser light; crystal models (20 minutes)
- Discussion – optional: Deriving Bragg’s law (20 minutes)
- Demonstration: Crystal spacing by X-rays (30 minutes)
- Demonstrations: Various analogues of X-ray diffraction (30 minutes)
- Student activity: Chemical composition by X-ray analysis (20 minutes)

Note that students who are also studying chemistry may already have come across Bragg’s law.

#### Discussion and student questions: Students’ knowledge of X-rays

Rehearse students’ assumed knowledge of X-rays. If a typical wavelength is 1 nm , what is the frequency of such an X-ray?

(*f* = *c** λ *

*f* = 3 × 10^{8} m s^{-1}1 × 10^{-9} m

*f* = 3 × 10^{17} Hz)

Students can learn about how X-rays were discovered, and how they are used.

Episode 530-1: What are X-rays? (Word, 28 KB)

#### Discussion: Diffraction and the limits of resolution

The shortest wavelength of visible light ~ 450 nm (450 × 10^{-9} m) sets a limit for the smallest thing that can be seen

using visible light. This limit comes about because of diffraction effects, when the wavelength is comparable to physical dimensions. To investigate matter on a smaller scale requires that we look at it

using shorter wavelengths. X-ray wavelengths are < 1 nm.

The wavelengths of X-rays are comparable to the atomic spacing in solid matter. Hence X-rays will be diffracted by planes of atoms in crystalline solids.

Show some X-ray diffraction patterns. Emphasise the idea that, the narrower the spacing, the greater the diffraction. This means that diffraction patterns can be used to determine the arrangement of atoms within a solid, and their separations.

You could also point out that a single crystal gives a pattern of discrete dots; a polycrystalline material or powder gives rings (because all orientations are present), and an amorphous material gives blurred rings or dots.

#### Demonstration: Diffraction of laser light; crystal models

Shine a laser beam at normal incidence onto a grating and note the separation of the diffracted beams. Now rotate the grating about a vertical axis. Observe that the separations of the diffracted beams increase as the effective slit width decreases. (Alternatively, use gratings with different spacing.)

#### Safety

Provided the laser is class 2 (less than 1 mW of visible light), the warning ‘Do not stare down the beam' is sufficient. Avoid specular reflections.

If you have crystal models handy (ask your chemistry department), look through in different directions. Many planes

of atoms reveal themselves, each with its own separation *d*.

#### Discussion – optional: Deriving Bragg’s law

You may have to derive Bragg’s law. Beware of potential confusion: students will have met the formula for diffraction by a (transmission) grating. In diffraction from crystals the angle is defined differently, and the crystal is acting as a reflection grating. Furthermore, the theory uses reflection rather than diffraction!

Episode 530-2: Bragg reflection (Word, 27 KB)

#### Demonstration: Crystal spacing by X-rays

There are problems with using X-rays in school. However, some schools do have X-ray sets. A good alternative is to arrange a visit to a university department (Physics, Chemistry or Materials Science) to see one in use, and to learn about contemporary applications.

You could arrange to show a determination of the crystal plane spacing in alkali halides.

#### Demonstration: Various analogues of X-ray diffraction

Here are some further analogues, for you to choose from (perhaps dependent on the equipment available).

Using laser light and diffraction gratings: Two crossed

diffraction gratings represent the atomic planes, and give an array of diffracted spots. The fact that many solids are polycrystalline and are made up of many small crystallites orientated randomly can be simulated by slowly rotating the crossed gratings. The array of diffracted spots rotates too. What would the diffraction pattern look like if the crossed gratings were rotated quite quickly to simulate all the possible orientations? (A series of rings.) Rig up two crossed gratings with an electric motor to spin them round while diffracting the laser beam. This always makes a visual impact.

Episode 530-3: Simulating X-ray diffraction (Word, 27 KB)

Finding the structure of DNA is perhaps the best-known use of X-ray diffraction. The iconic X

shaped diffraction pattern from the helix can be simulated by diffracting a laser beam off a fine bolt thread. Using bolts of different pitches alters the angle of the X

.

(See School Science Review vol 85 (312) pp 18-19.)

Use a ripple tank

Episode 530-4: Diffraction with water waves (Word, 42 KB)

Try using microwaves and /or a ripple tank with an array of pins

Episode 530-5: Diffraction by crystals (Word, 28 KB)

#### Student activity: Chemical composition by X-ray analysis

An interesting use of the Bragg equation is to find the wavelength of X-rays emitted by a substance or object, so that information about its chemical composition can be found.

Episode 530-6: Where has it been? (Word, 44 KB)

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### Up next

### Neutron diffraction

## Episode 531: Neutron diffraction

Lesson for 16-19

- Activity time 30 minutes
- Level Advanced

This episode takes a very brief look at the topic of neutron diffraction.

Lesson Summary

- Discussion and worked example: de Broglie wavelength of a neutron (15 minutes)
- Discussion: Sources and uses of neutrons (15 minutes)

#### Discussion and worked example: de Broglie wavelength of a neutron

Neutrons are very penetrating, typically 10 cm in lead. Why might this be? (Neutrons have no electric charge, so they do not interact with the atomic electrons.) However, neutrons do have a magnetic moment that allows them to interact with magnetic nuclei.

Neutron diffraction was proposed in 1934, to exploit de Broglie’s hypothesis about the wave nature of matter. Use de Broglie’s equation * λ * =

*h*

*p*to calculate the momentum and energy of a neutron whose wavelength is comparable to atomic spacing,

say 1.8 × 10^{-10} m.

*p* = *h* *λ*

*p* = 6.6 × 10^{-34} J s1.8 × 10^{-10} m

*p* = 3.7 × 10^{-24} kg m s^{-1}

*E*_{K} = *p*^{ 2}2 *m*

*E*_{K} = 3.7 × 10^{-24} kg m s^{-1}2 × 1.67 × 10^{-27} kg

*E*_{K} = 4.0 × 10^{-21} J

If we equate this energy to *k**T*, we get:

*T* = 4.0 × 10^{-21} J1.38 × 10^{-23} J

*T* = 290 K

which is roughly ambient temperature

.

(It is useful to be able to recall that 140 eV is the typical energy of a particle at room temperature.)

The neutrons released in nuclear fission are fast

neutrons, i.e. much more energetic than this. Will their wavelengths be greater or less than atomic dimensions? (Less)

So we need to slow down the fast neutrons to thermalise

them. How is this achieved in a nuclear power reactor? (By including a moderator.)

#### Discussion: Sources and uses of neutrons

Neutrons – where do we get them from? (Nuclear reactors; spallation sources, which use accelerators to crash ions into heavy nuclei.)

#### What can they tell us?

For *elastic* interactions where the neutron energy is not changed, the Bragg equation allows the determination of the separation of atomic planes.

In an *inelastic* process, the neutron gives energy to, or receives energy from, the vibrating ions that make up the crystal lattice. This leads to information about the forces between atoms in the crystal lattice.

The interaction of magnetic moments gives information about magnetic properties of materials – e.g. the position of the domain boundaries, their size and orientation, the magnetic moments of the atoms etc.