EXPERIMENT 2
Bragg Diffraction and Measurements of Crystal Unit Cells
NOTICE: The X-ray apparatus must first be explained and
powered by the demonstrator before you use it.
Introduction
The study of X-rays and their interactions has played a very significant role in the development of
atomic physics over the last 100 years. Nowadays, X-rays have many routine practical applications
in medicine and industry. Because they can penetrate several centimeters of solid matter, they can
be used to visualize the interiors of materials that are opaque to ordinary light, such as broken
bones or defects in structural steel. The object to be visualized is placed between an X-ray source
and a large sheet of photographic film; the darkening of the film is proportional to the radiation
exposure. A crack or air bubble allows greater transmission and shows as a dark area. Bones appear
lighter than the surrounding soft tissue because they contain greater proportions of elements with
higher atomic number (and hence greater absorption); in the soft tissue the light elements carbon,
hydrogen, and oxygen predominate.
Recently, several vastly improved X-ray techniques have been developed. One widely used
system is computerized axial tomography; the corresponding instrument is called a CAT scanner.
The X-ray source produces a thin, fan-shaped beam that is detected on the opposite side of the
subject by an array of several hundred detectors in a line. Each detector measure absorption along
a thin line through the subject. The entire apparatus is rotated around the subject in the plane of
the beam during a few seconds. The changing photon-counting rates of the detectors are recorded
digitally; a computer processes this information and reconstructs a picture of absorption over an
entire section of the subject. Differences as small as 1% can be detected with CAT scans, and
tumours and other anomalies that are much too small to be seen with older X-ray techniques can
be detected.
Naturally, to correctly interpret the results obtained from such measurements, a complete understanding of the behaviour of X-rays is required. In this and a subsequent practical you will carry
out a range of experiments aimed at helping you understand:
• The production and properties of X-rays
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Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
Figure 2.1: The distribution by wavelength of the X-rays produced when 35-keV electrons strike a
molybdenum target. Note the sharp peaks standing out above a continuous background.
• The factors governing the appearance of X-ray spectra
• X-ray (Bragg) diffraction and crystal structures
• The processes of X-ray scattering and absorption
These experiments are highly relevant to course material covered in EP2.3 and EP2.4.
Continuous and Characteristic X-ray Spectra
X-rays are produced when energetic electrons strike a solid target and are brought to rest in it.
Fig. 2.1 shows the wavelength spectrum of the X-rays that are produced when a beam of 35 keV
electrons strikes a molybdenum anode. It consists of a broad spectrum of radiation (continuous
spectrum) on which are superimposed peaks of sharply defined wavelengths (characteristic X-ray
spectrum).
The continuous spectrum arises from the deceleration of the electrons in the target. An electron
may well lose an amount of energy in its encounter with the target nucleus which will appear as
the energy of an X-ray that is radiated away. This process is called bremsstrahlung (or ‘braking
radiation’) and has been discussed in first year lectures.
A prominent feature of the continuous spectrum of Fig. 2.1 is the sharply defined cutoff wavelength λmin , below which the continuous spectrum does not exist. This minimum wavelength
corresponds to an encounter in which one of the incident electrons, still with its initial kinetic energy eV, loses all this energy in a single encounter, radiating it away as a single X-ray photon. The
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Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
wavelength associated with this photon, the minimum possible X-ray wavelength, is found from
eV = hν =
hc
,
λmin
(2.1)
which yields
hc
(cutoff wavelength)
(2.2)
eV
The cutoff wavelength is totally independent of the target material. If you were to switch from
a molybdenum to a copper target, for example, all features of the X-ray spectrum of Fig. 2.1
would change except the cutoff wavelength. It follows from Eq. 2.2 that Planck’s constant can be
determined if the cutoff wavelength is known. This provides a very accurate method of measuring
this fundamental physical constant and details of the measurements required are given below.
The peaks labeled Kα and Kβ in Fig. 2.1 and any other peaks that may occur at longer wavelengths are characteristic of the target material and form what we call the characteristic X-ray
spectrum. The X-ray photons that produce these peaks are created as follows:
λmin =
1. An energetic incoming electron strikes an atom in the target and knocks out one of the deep
lying electrons. If the electron is in the shell with n = 1 (called, for historical reasons, the K
shell), there remains a vacancy or a hole in this shell.
2. One of the outer electrons moves in to fill this hole and in the process emits an X-ray photon
of characteristic wavelength.
If the electron falls from the shell with n = 2 (called the L−shell) into the K-shell we have the Kα
line of Fig. 2.1; if it falls from the shell with n = 3 (called the M −shell) we have the Kβ line; and
so on. Of course, such a transition will leave a hole in either the L or the M shell, but this will
be filled in by an electron from still farther out in the atom, causing the emission of yet another
characteristic X-ray photon.
Fig. 2.2 shows an X-ray energy level diagram for molybdenum, the element to which Fig. 2.1
refers. The base line (E = 0) represents, the neutral atom in its ground state. The level marked
K (at E = -20 keV) represents the energy of the molybdenum atom with a hole in its K-shell.
Similarly, the level marked L (at E = - 2.7 keV) represents the energy of the atom with a hole in its
L-shell, and so on. The transitions marked Kα and Kβ in Fig. 2.2 show the origins of the two sharp
X-ray lines in Fig. 2.1. The Kα line, for example, originates when an electron from the L-shell of
molybdenum fills a hole in the K-shell.
X-Ray Diffraction by Crystals
In 1912 Max von Laue (1879-1960) suggested an experiment that verified the wave nature of
X-rays. Von Laue pointed out that if X-rays have wavelengths λ that are about the same as the
spacing d between planes of atoms in crystals, then X-ray waves impinging on crystals would
exhibit interference effects. Recall that a transmission grating, because it consists of a regular
array of slits, causes light waves to exhibit strong constructive interference at a few particular
angles and almost complete cancellation at all other angles. To observe these interference effects,
the slit spacing must be almost as small as the wavelength. Similarly, a crystalline solid consists
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Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
Figure 2.2: An atomic energy level diagram for molybdenum, showing the transitions that give rise
to the characteristic X-rays of that element. (All levels except the K-level consist of a number of
closely lying components, not shown here.)
of a regular array of atoms. When a beam of X-rays impinges on a crystal, strong constructive
interference effects can be observed readily if the wavelength λ is somewhat smaller than the
interplanar spacing d in the crystal.
Fig. 2.3 shows a two-dimensional representation of a three-dimensional crystal; the rows of dots
portray planes of atoms. X-rays of a single wavelength are in phase before being scattered from the
atoms in plane A and the atoms in plane B . For constructive interference of the X-rays scattered
from each plane of atoms, the angle of incidence turns out to be equal to the angle of reflection.
To reach the detector, the waves scattered from the atoms in plane B travel a greater distance than
those scattered from the atoms in plane A by the amount 2(d sin θ). If the angle θn is given by the
relation
2d sin θn = nλ (n = 1, 2, . . .)
then the waves scattered from the atoms in plane A will arrive at the detector in phase with the
waves scattered from the atoms in plane B. Thus the waves will constructively interfere and produce an interference maximum. Similarly, constructive interference will occur for waves scattered
from the atoms of each of the many planes that are parallel to planes A and B. This relation was
first developed by W.L. Bragg (1890-1971) and is called Bragg0 s law. You will use this technique
with crystals of NaCl and LiF to record the continuous and characteristic spectrum emitted from
the copper target in the X-ray tube.
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Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
Incident Waves
Reflected Waves
m
m
Plane A
d
d Sin
d Sin
Plane B
Figure 2.3: To reach the detector, X-ray waves reflected from plane B must travel a distance
2(d sin θ) farther than those reflected from plane A. The waves constructively interfere at the detector when 2d sin θ = nλ.
The X-ray Apparatus
The X-ray apparatus is shown in Fig. 2.4. The high voltage can be set to 30,000 volts or 20,000
volts and the X-rays are produced when the accelerated electrons bombard a copper target, knocking out inner shell (K) electrons. L & M shell electrons drop into the K-shell vacancies producing
x-rays at characteristic energies : Kα and Kβ radiation. The X-ray emission from the tube is collimated at the lead glass dome to be a circular beam of 5mm diameter (Fig. 2.4), this primary beam
diverges from the Basic Port to a useful beam diameter at the Crystal Port of 15mm diameter, at
Experimental station (ES) 30mm diameter. A Geiger Muller tube, operated at a plateau voltage of
425 volts is used to detect the X-rays. An interlock safety system is used to ensure that the X-ray
tube cannot be powered unless the scatter shield is in place. An additional glass cover is used to
ensure that the residual X-ray level from the apparatus is below the background radiation level in
the laboratory.
Experimental Procedures
X-ray Diffraction : Wavelength Measurement with the Bragg Method
Sir Lawrence Bragg presumed that the atoms of a crystal such as Sodium Chloride were arranged
in a cubic and regular three-dimensional pattern.
The mass of a molecule of NaCl is M/N Kg, where M is the molecular weight (58.46 × 10−3 kg
per mole) and N is Avogadro’s number (6.02 × 1023 molecules per mole).
The number of molecules per unit volume is ρ/ M
molecules per cubic metre, where ρ is the
N
density (2.16 × 103 kg m−3 ).
Since NaCl is diatomic the number of atoms per unit volume is 2ρN/M atoms per cubic metre.
The distance therefore between adjacent atoms, d in the lattice is derived from the equation
d3 =
1
2ρN/M
or
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d =3
q
m/2ρN
Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
Basic Port
Slave Plate
Crystal Post
Knurled clutch plate
Geiger Tube
ES spring
Clip
Carriage Arm
Manual Control
Thumb Wheel
Cubic Crystal
Chamfered Post
Clamping post
Clamping jaw
Figure 2.4: X-Ray Apparatus
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Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
Figure 2.5: Count rate as a function of angle (2θ) in a NaCl crystal for accelerating voltages of
20kV and 30kV
Work out ‘d’ for NaCl from this information.
* NB. Please ensure all items are returned to the box after use.
Set-up :
1. Mount the NaCl crystal, TEL 582.004, in the crystal post, ensuring that the major face having
“flat matt” appearance is in the reflecting position.
2. Locate Primary Beam Collimator 582.001 in the Basic Port with the 1mm slot vertical.
3. Mount Slide Collimator (3mm) 562.016 at Experimental Station(ES) 13 and Collimator
(1mm) 562.015 at ES 18.
4. Zero-set and lock the Slave Plate and the Carriage Arm cursor as precisely as possible.
5. Sight through the collimating slits and observe that the primary beam direction lies in the
surface of the crystal.
6. Mount the G.M tube and it’s holder at ES 26.
7. Select the 30keV and set the tube current at 50 µA. Monitor this current carefully throughout
the experiment. It should not vary by more than 10%. Any anomalies in the tube current
behaviour should be reported immediately. Using a Ratemeter track the Carriage Arm round
from its minimum setting (about 11◦ , 2θ) to maximum setting (about 124◦ , 2θ).
Plot on graph paper the count rate at 1◦ (2θ) intervals, using an integration time of 10-30s to
minimise statistical errors.
Where the count rate appears to peak, plot intervals of only 10’arc using the thumb-wheel
at each peak, measure and record the maximum count rate and the angle 2θ as precisely as
possible.
8. Select 20kV and repeat step 7 above.
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Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
Table 2.1: Results obtained for NaCl crystal
Feature
3
4
5
6
7
8
2θ
θ
sin θ
2d
nλ
n
Observe that the continuous spectra of “white” radiation exhibit peak intensities (Fig. 2.5.5
feature 2) and intercepts on the 2θ axis (feature 1) which vary with the voltage setting of the
X-ray tube.
The six peaks, features 3 to 8, superimposed on the continuous spectrum do not vary in angle
2θ with voltage setting, but only in amplitude.
9. Tabulate the results from the six superimposed peaks of the graph and calculate λ and n as
shown in Table 2.1.
Observe that the sharp peaks 3 and 4 are a pair of “emission lines” which re-appear in second
and third orders of diffraction.
The more energetic radiation, termed Kβ , is successively less intense than the longer wavelength, Kα , line.
10. Repeat the procedure with the LiF crystal in place of the NaCl.
Measurement of Atomic Sizes
1. Set up for Bragg reflection as in parts 1 - 6 of the previous section.
2. Minimise the CuKβ radiation by inserting the Ni filter TEL 564.004 at ES 17.
3. Increase the tube current to 70 µA. Monitor the current carefully throughout the experiment.
It should not exceed 80 µA at any time.
4. Using a Ratemeter, search and record the 2θ angle for the first diffraction peak for CuKα
radiation, λ = 0.154nm.
Tabulate and calculate d, assuming n = 1 in the Bragg equation nλ = 2d sin θ (see Table 2.2).
5. Repeat the experiment using the KCl crystal (Green).
6. Repeat the experiment using the RbCl crystal (Red).
Observe that the crystal spacing, d, increases with the Atomic Number of the Alkaline metals;
Na in the third period has eleven electrons at K, L and M levels; K in the fourth period has nineteen
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Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
Table 2.2: Tabulation of interatomic spacings for different crystals
Crystal
NaCl (yellow)
KCl (green)
RbCl (red)
2θ
θ
sin θ
λ/2
0.077
0.077
0.077
d nm
electrons but in the K, L, M and N states, whereas Rb has thirty seven electrons in the five energy
states, K, L, M, N and O. Chlorine is common to the three compounds and thus the evidence
suggests that the size of individual atoms increases with atomic number.
Crystallography is not a ‘direct’ science; crystal structures are postulated after interpretation of
all the evidence available including the morphological, the optical and the chemical properties of
the specimen; the alkaline halides are both morphologically and chemically similar and the student
could expect that these salts will exhibit similar crystal structures. The size of atoms is of great
importance in metallurgy when studying crystal cohesion and in the development of alloys having
specific mechanical properties.
Unit Cell Calculations
Table 2.2 was drawn up from angular measurements of the first Bragg peaks and defines the largest
spacing within the three crystals which give rise to constructive interference.
Clearly, sets of planes can be chosen which have greater spacing, but if there are no Bragg
reflections, the interference must be destructive and not constructive.
A plan view of a model of NaCl exhibits a pattern of repetitive symmetry (Fig. 2.6) and one of
the first objectives in crystallography is to establish the linear repetition frequency of a symmetrical
pattern in each of three dimensions.
In the a-direction illustrated, the symmetrical structure bordered by layers a1 and a3 is repeated
by a3 and a5 and a7 etc.; an identical geometric form can be repeated by choosing a2 /a4 , a4 /a6 etc.
Accepting that NaCl has a cubic structure, a symmetrical pattern should be evident in both the
b- and c- axes and hence a “building block” can be defined as indicated by the bold line, the nature
of the atoms at the vertices being consistently either Cl or Na.
Earlier the distance between adjacent unlike atoms, Na and Cl, d was calculated to be 0.282 nm;
the distance between two like atoms, Cl and Cl is twice the spacing and hence the length of the
Table 2.3: Tabulation of Unit Cell Calculations.
Crystal
NaCl
KCl
RbCl
Unit Cell Side ‘a’ nm
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Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
side of the “building block” is 0.564 nm.
This repetitive geometric figure, when established in 3-dimensions, is defined as the UNIT
CELL. Where a choice exists with geometric forms of identical area then the form most clearly
approaching rectangular is usually adopted.
From Fig. 2.6 the length of the edge of the Unit Cell of NaCl in the arbitrary a- direction is 2d.
Similarly ‘a’ can be established for the Unit Cells of KCl and RbCl (Table 2.3).
Reflections are not observed for spacings equivalent to the side of the Unit Cell by virtue of
destructive interference.
When waves from layer a3 are one wavelength behind those from layer a1 , then waves from layer
a2 must be half a wavelength lagging on those from layer a1 and exactly opposite in phase; since
the layers throughout the crystal contain the same combination of atoms of Na and Cl, the electron
densities are the same and so the diffractive power of the layers is identical; the waves from layer
a2 therefore exactly cancel out those from a1 .
&O
1D
&O
D D D D D D D
Incident
X-Ray
Beam
Figure 2.6: A plan view of a model of NaCl
Determination of Minimum Wavelength and hence Planck’s Constant
In striking the Copper anode the majority of electrons experience nothing spectacular; they undergo sequential glancing collisions with particles of matter, lose their energy a little at a time and
merely increase the average kinetic energy of the particles in the target; the target gets hot. The
minority of electrons will undergo a variety of glancing collisions of varying severity; the electrons are decelerated imparting some of their energy to the target particle and some in the form of
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Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
electromagnetic radiation equivalent in energy to the energy loss experienced at each collision.
Since these collisions usually occur at a slight depth within the target the longer, less energetic,
wavelengths are absorbed within the target material.
This “bremsstrahlung” or “braking radiation” is thus a continuous spread of wavelengths, the
minimum wavelength (or maximum energy) being determined by the accelerating voltage of the
tube.
µ ¶
1
λmin = f
or V λmin = k
V
where V is the X-ray tube voltage selected.
1. Counts should be recorded over at least 10 second durations, the longer the counting period
the greater the accuracy of the results.
2. Mount the auxiliary slide carriage in horizontal mode using the 1mm slot Primary Beam
Collimator (vertical) as follows:
The hole in the end face of the Auxiliary Carriage is placed over the basic port in the glass
dome and then held in that position by one or other of the Primary collimators. In this mode
the axis of the centre of each experimental slide is horizontal and is transcribed by the X-ray
beam. Note that the carriage arm is now restricted to a maximum 2θ angle of 100◦ .
3. Position the Slide Collimator, (1mm slot) 562.015 at ES 4 and Slide Collimator (3mm)
562.016 at ES 13.
4. With the LiF crystal mounted as before, set up as for parts 4-6 in Section X-ray Diffraction;
select 30kV.
5. Measure, tabulate and plot the count rate at every 300 arc, commencing at 110 300 , until the
“whale back” appears to fall off.
6. Repeat for 20 keV. Observe that the minimum setting of the Carriage Arm requires an extended extrapolation of the 30 keV curve to obtain and intercept on the x-axis. Observe that
the curves flatten out before intercepting the axis (Fig. 2.7), due to the contribution of the
general background radiation.
7. Extrapolate the theoretical intercepts and tabulate the results as in Table 2.4.
If the theory of the “inverse photoelectric effect” is valid then Einstein’s assumption of 1905,
that both emission and absorption are “quantised”, must be tested in relation to Planck’s
formula for photo-electron emission
E = hν joules
where E is the energy associated with each quanta, ν is the frequency of radiation and h is
Planck’s constant.
Since ν = cλ for electromagnetic radiation where c is the velocity of light, and E = eV is the
maximum energy that can be acquired by any electron within the X-ray tube system, then :
µ ¶
eV = hc/λ or h = V λ
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e
c
Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
Figure 2.7: Count rate as a function of angle (2θ) for NaCl and LiF showing the flattening of the
curves before they intercept the axis, due to the contribution from background
Table 2.4: Tabulate values to determine an average value for V x λ
Crystal
LiF
LiF
V
30 kV
20 kV
2θ
θ
sin θ
2d (nm)
0.403
0.403
λ (nm)
Vλ
Calculate the mean value of V λ from Table 2.4 and evaluate h. Compare this with the
internationally accepted value for h.
The difference between the accepted standard value and the evaluated result for h of about
5% is well within experimental limits and illustrates why the ‘inverse photoelectric effect’
is considered to be a very accurate method of determining the fundamental constant in the
Quantum Theory.
Summary
It is assumed that previous studies of optical spectra have established that “characteristic lines” in
the visible region of the electromagnetic spectrum are emitted from atomic energy-levels of high
principal quantum number, the N, O, P and Q levels; the relatively much shorter wavelengths of the
characteristic Kβ and Kα lines indicate that these shorter emissions are due to electron transitions
at energy levels of low principal quantum number. Any electron from the X-ray tube filament
having sufficient energy to eject a K electron in a collision process will ionise the Copper atom;
the ionised atom will revert to it’s stable state through electron transitions, each transition being
accompanied by the emission of a photon of equivalent energy.
By definition, the Kβ emission results from the N and M levels to the K level and Kα from
transitions from the L to the K level; the N and M levels have a greater energy difference with
respect to the K level than does the L level and hence the wavelength of the Kβ photon is shorter
and more energetic than that of Kα . But the closer proximity of the L and K levels results in more
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Experiment 2. Bragg Diffraction and Measurements of Crystal Unit Cells
frequent transitions than for the N or M levels and hence there is a greater “population” of Kα
exhibited by the relative intensities of the peaks 3 and 4 of Fig. 2.5.
The Bragg measurements established that a crystal can be used to demonstrate the co-operative
interference of X-rays; the wavelength limit of the continuous “white” spectrum is dependent
uniquely on the energy imposed upon the electrons by the potential difference between the electron emitting filament and the anode, regardless of its material; the “characteristic” line spectrum,
superimposed upon the white spectrum is due to the elemental composition of the anode and the
energy-levels associated with its individual electron system. The lines are unique to emission from
a Copper target and are thus termed CuKβ and CuKα emission lines.
Spectral analysis by the Bragg technique can accurately evaluate a) an unidentified voltage, using both a known crystal and anode material, b) an unknown crystal structure using an identified
voltage and anode material and c) the chemical composition of a material serving as an anode to
emit characteristic radiation, using an established crystal and an accurately defined voltage.
Questions
1. How does the spectrum of NaCl change with accelerating voltage? Why?
2. Is the LiF spectrum different to the NaCl spectrum for the same accelerating voltage? Why?
References
1. ‘Modern Physics’, Richtmyer, Kennard & Cooper
2. ‘X-rays in Atomic and Nuclear Physics’, N.A. Dyson
WWW :
• Crystal Structures - http://www.ill.fr/dif/3d-crystals
• Snow Crystals - http://www.snowcrystals.com
• Synchrotrons-Modern x-ray sources - http://www.esrf.fr
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