Experiment 2Bench 2MZU-PE / XRD-2Shutter closed

X-ray Bragg Diffraction

θ–2θ Powder Diffractometer

Run a θ–2θ scan with a Cu Kα tube and locate the Bragg reflections on the chart recorder.

Duration: 3 hours
Kittel: Chapter 2
Topics: Bragg's lawX-ray diffractionPowder method

Goniometer — plan view

Copper Kα, λ = 1.5406 Å · specimen: Sodium chloride powder

HVBeam

Standby

Chart recorder

θ–2θ scan · 10° to 140° · 0.25° step

ONOFF

hkl

Standby

No trace — press START SCAN

19 allowed reflections

Detector electronics

Scintillation counter + ratemeter

Detector angle 2θ

deg

Specimen angle θ

deg

Count rate

c/sPK

d from Bragg

Wavelength λ

sin θ

Rate

Indexing

Nearest reflection

(111)

h²+k²+l²

d = a/√(h²+k²+l²)

3.2564Å

a at 293 K

5.6402Å

Rock salt. The all-odd reflections are weak because the Na and Cl scattering factors subtract.

Diffractometer control

MZU-PE / XRD-2

Standby

Ionising radiation. The shutter will not open until the tube high voltage is on.

ONOFF

Tube HV

OPENSHUT

Shutter

Tube high voltage is off

X-ray tube

Anode

Voltage40 kV

Current30 mA

Specimen stage

Sample rack

Specimen temp293 K

Count time1.00 s

Goniometer drive

Detector angle 2θ28.00 °

Key Equations

2d_{hkl}\sin\theta = n\lambda

Constructive interference occurs only when the path difference between rays from adjacent planes is a whole number of wavelengths. Diffraction is therefore impossible unless $\lambda \le 2d$ .

d_{hkl}=\frac{a}{\sqrt{h^2+k^2+l^2}}

Systematic absences come from the structure factor: BCC keeps only $h+k+l$ even, FCC only $h,k,l$ all even or all odd, and diamond additionally kills all-even reflections unless $h+k+l = 4n$ .

Procedure

0/4

ABring the beam up and take a scanFit the Cu tube, load the NaCl powder, set 40 kV / 30 mA. Switch the tube HV on, open the shutter, then press START SCAN. Wait for the pen to reach 140°.
BIndex the reflectionsDrive the goniometer to each peak with the 2θ knob and press PEAK SEARCH to sit exactly on it. Record 2θ, θ, sin θ and the d-spacing the instrument reports. Then check d against a/√(h²+k²+l²).
CChange the wavelengthFit the Mo tube (short λ) and then the Cr tube (long λ) and rescan the same specimen. Find the reflection that disappears with Cr and explain it using λ ≤ 2d.
DAbsences, glass and temperatureScan Fe, Si and fused silica in turn and compare which peaks are present. Then take NaCl from 100 K to 900 K and watch the peaks shift and shrink.

Table 2 — Bragg reflections

0 entries

Sit on a peak with PEAK SEARCH and let the ratemeter settle before you record.

#	Specimen	Tube	λ/ Å	hkl	2θ/ °	θ/ °	sin θ	d (Bragg)/ Å	Rate/ c/s	T/ K
No readings yet — set the controls, then press “Record reading”.

Analysis & Reflection

Analysis questions

Plot $\sin\theta$ against $1/d$ for the reflections you indexed in Task B. What is the slope, and what physical quantity does it represent?
Visible light has $\lambda \approx 5000\ \text{Å}$ and a typical plane spacing is $d \approx 3\ \text{Å}$ . Use the condition you found in Task C to show why visible light cannot diffract from a crystal.
Compare the Fe and Al scans. Fe is missing every reflection with h+k+l odd and Al is missing every mixed-parity reflection. Explain both patterns of absences in terms of a plane of atoms halfway between the ones you are indexing.
Silicon is missing (200) and (222) even though both satisfy the FCC rule. What does that tell you about the second atom in the diamond basis?
When you heated the NaCl specimen, the peaks moved to lower 2θ and the high-angle ones lost height. Account for both effects.

Physics problems

Aluminium is FCC with $a = 4.05\ \text{Å}$ . Calculate $d_{111}$ and the Bragg angle for the (111) reflection with Cu Kα, then check both against the instrument.
Neutrons of wavelength 1.80 Å can be used instead of X-rays. What extra information do neutrons give that X-rays cannot? Think about magnetic as well as nuclear scattering.
A crystal expands by 0.5% on heating. Use $2d\sin\theta = \lambda$ to estimate the shift, in degrees, of a peak that sat at 2θ = 45°.
Describe in plain language what the Ewald sphere represents, and state the condition under which a reciprocal lattice point lies on it.