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Date of experiment: 13/11/2012
Date of report: 14/12/2012

Investigation of Crystal Structures and X-Ray
Diffraction

Experiment conducted by Sultan LeMarc
Partner: Navjeevan Soor
Supervisor: Dr. W. Dickson
Abstract
This experiment investigated the characteristics of X-rays by measuring the count rate of Xrays reflected off alkali halide crystals at varying angles of incidence and using the principles
of Bragg’s law. The experiment probes into crystal structures using X-ray diffractometry and
deduces the lattice constants and ionic radii using the Miller index notation. The experiment
successfully computes the characteristic wavelengths of Copper and clearly demonstrates the
effect of filters on spectrum intensities. The interpretation of Miller indices and diffraction
patterns are effectively used to analyse crystalline structures and compare lattice
arrangements.
Introduction
X-rays were discovered in 1895 by German physicist Wilhelm Röntgen, who received
the first Nobel Prize in Physics in 1901 in recognition of his work on electromagnetic
radiation. As Röntgen refused to patent the discovery in order to allow further
development and practical applications, the discovery prompted further study of the
radiation. [1]
Nikola Tesla proceeded to devise experimental setups which produced high energy Xrays. Tesla also alerted the scientific community about the safe operation of X-ray
equipment and the biological hazards associated with exposure.
Thomas Edison led the advances in radiology and medical X-ray examinations by
developing a fluoroscope in 1896. Since the initial pioneering works of the late 19th
century, many important discoveries have been made using X-rays which are now
used in various applications. This includes the establishment of X-ray crystallography,
an experimental technique which allows the dimensional arrangement of atoms within
crystal structures to be determined by the interpretation of diffraction patterns. [2]
The field was developed by the pioneering research of physicist William Bragg in
1913 who won a joint Nobel Prize in Physics in 1915 with his father for their work on
crystallography. X-ray crystallography has since made significant contributions to
chemistry and material science. Today, X-rays are most notably used for medical
imaging, microscopy and astronomy.
Crystals were first scientifically investigated in the 17th century for their regularity and
symmetry, most notably by Danish geologist Nicolas Steno, who successfully
confirmed this symmetry by showing that the angles between the faces are the same
for a given type of crystal. This was followed by the introduction of a notation system
for planes in crystal lattices called the Miller indices by William Miller in 1839, which
is still used today to identify crystals. [3]
The aim of this experiment was to investigate the production and emission
characteristics of X-rays. This was achieved by measuring the count rate of X-rays
reflected off alkali halide crystal at varying angles of incidence in order to find the
characteristic peaks of a copper (Cu) target using the principles of Bragg’s law.
In addition, the experiment aimed to examine how the Bragg scattering of X-rays can
be used to determine the crystal structures and their lattice constants. There was also a
consideration of energy loss by observing the varying intensities of characteristic
wavelengths.

1
1. Theory
X-rays are a form of ionising electromagnetic radiation found in the short wavelength and high
energy end of the spectrum with wavelengths ranging from 0.01nm to 10nm. This range of
wavelengths of X-rays is of the order of distances between molecules and crystal lattices,
making them ideal for spectroscopic techniques for characterisation of the elemental
composition of materials. The corresponding energies of this range are 120 eV to 120 keV
respectively.
There are two commonly used methods to produce X-rays, by an X-ray tube which is a
vacuum tube that linearly accelerates charged particles, or by a synchrotron which is a cyclic
particle accelerator that applies electric and magnetic fields. Most natural sources of X-rays
are extra-terrestrial such as the Sun and black holes but they are also emitted by the decay of
unstable nuclei on Earth. [4]
An X-ray tube functions as a typical vacuum tube which uses the potential difference
between a cathode and an anode to accelerate charged particles. In an X-ray tube a metal
filament is the cathode and is heated by a low voltage current in order to emit electrons in a
process called thermionic emission. As a stream of electrons are released into the vacuum, a
large electric potential is applied between the cathode and the anode; a metal target. This
accelerates the electrons towards the anode due to electrostatic attraction.
There are two principle mechanisms by which X-rays are produced, by bremsstrahlung or Kshell emission. In the first, the emission of X-rays takes place when high-energy electrons (or
charged particles) are decelerated, in speed or direction, by bombarding targets. In accordance
with Maxwell’s equations, the electrons emit electromagnetic radiation upon deceleration, a
process called bremsstrahlung. [5]
In the second mechanism, X-rays are produced by the excitation
and ionisation of the atoms in the target. In this process
transitions of electrons between atomic orbit shells take place as
the bombardment of electrons can excite and eject inner
electrons from the target atoms, provided the incident electron
has sufficient energy.
This leaves a vacant space in the inner shell and is filled by an
electron at the higher level outer shell. In the course of this
transition, the higher level electron is losing kinetic energy as it
fills the vacant inner shell.

Figure 1: The emission
spectra of a heavy metal xray source. [6]

Under the principles of conservation of energy, the transition is
accompanied by the emission of an X-ray photon with unique
energy corresponding to the energy difference between the two
shells i.e. the energy lost by the shifting electron. These X-rays
are called characteristic X-rays with wavelengths that are
distinctive for each particular element and transition.

The innermost shell from which the incident electron has dislodged an atomic electron is
called the K shell. When the vacancy is filled by an electron from the next higher shell, the L
shell, the photon emitted has an energy corresponding to the Kα characteristic X-ray line on
2
the emission spectra. When the vacancy in the K shell is filled by an
electron from the M shell, the next higher shell after the L shell, the Kβ
characteristic X-ray line is produced on the emission spectra. [7]
Diffracted waves from different atoms within the lattice can interfere
with each other, leading to a modulated resultant intensity distribution.
Interference occurs when two waves meet and superimpose either
constructively or destructively.
Constructive interference takes place when two identical phases meet
and superimpose into a wave with combined amplitude. Destructive
interference takes place when two waves are completely out of phase
with each other, resulting to the cancelation of each other. As the atoms
are in a period arrangement, the diffracted waves interfere to produce
diffraction patterns that reflect the symmetry of the distribution of the
crystal lattice.

Figure 2: Transitions
from higher to lower
energy levels. [8]

The diffraction patterns of X-rays show that at most of the incident angles the incident X-rays
are scattered coherently to interfere destructively as the combining waves are out of phase
and thus have no resultant energy. However at some key incident angles the X-rays are
scattered coherently to interfere constructively, resulting in well-defined X-ray beams leaving
the crystal in various directions. As a result, a diffracted beam can be considered as a beam
composed of a large number of scattered rays mutually reinforcing each other.
The condition to be satisfied for there to be constructive interference in X-ray diffraction is
mathematically defined by Bragg’s law, which relates the incident wavelength λ, incident
angle θ, and the spacing between the planes in the atomic lattice d:

[9]
Bragg’s law states that intense peaks of diffracted radiation (Bragg peaks) are produced by
constructive interference of scattered waves at specific wavelengths and incident angles. The
scattering angle 2θ is defined as the angle between the incident and scattered rays.
It is important to note that Bragg's Law applies to scattering centres consisting of any
periodic distribution of electron density. In this case, a crystal is used as the scattering centre
to separate the different wavelengths of incident X-rays and scattering them to specific
angles.
The literature value of the Kβ and Kα wavelengths for copper are λβ = 0.1392 nm and λα =
0.1542 nm respectively.[10]

3
The Miller index notation, denoted by (h,k,l), is a
convention used in crystallography in order to define the
orientation of a crystal plane with respect to a main
crystallographic axis. They are a set of numbers which
quantify the intercepts and can be used to uniquely
identify planes of lattice structures, as shown in Figure 3.
Miller indices are sufficient to specify both the
orientation and the spacing of a set of parallel planes.
For cubic crystals, the relationship between the lattice Figure 3: Planes with different Miller
indices in cubic crystals. [11]
constant, a, which refers to the constant difference between
unit cells in a lattice, the miller indices, and the interplanar spacing, d, is given by:
√
[12]
The literature values of the lattice constant and the radius of the anions of the alkali halide
crystals being investigated in this experiment are:
Crystal

Lattice Constant (nm)

Anion radius (nm)

Lithium Fluoride (LiF)
Sodium Chloride (NaCl)
Potassium Chloride (KCl)
Rubidium Chloride (RbCl)

0.403
0.565
0.629
0.670

0.102
0.076
0.138
0.152

Table 1: Accepted values for the lattice constants and anion radii of the alkali halide crystals. [13]

Equations (1) and (2) were combined to give:

This allows the miller indices to be deduced once the lattice constant is found using equation
(2) from the Bragg angles for (2,0,0) and (4,0,0) reflections.

4
2. Experimental Procedure
In order to observe the characteristics peaks of the copper target, readings for the count-rate
of X-rays over a range of angles 2θ were taken using a Tel-X-Ometer and a Geiger counter.
A Tel-X-Ometer is an X-ray diffraction system device which is
used to detect the absorption and reflection of X-rays i.e. a
spectrometer. The device employs the fundamental principle of an
X-ray tube to produce X-rays and accelerate them towards the
target, which is mounted as the anode. The Tel-X-Ometer used in
this investigation had two settings for the accelerating voltage,
also known as extra high voltage (EHT), at 20kV and 30kV. The
filament current could also be adjusted, and was not allowed to
exceed 80µA to prevent damage.
The device allows crystal to be mounted in the centre of the
device to reflect the X-rays for detection. It also allows numerous
collimator slides to be mounted at different numbered positions
on a carriage arm assembly.

Figure 3: Tel-X-Ometer.
[14]

A Geiger counter is a particle detector that measured ionising radiation. It is able to detect
radiation by the ionisation produced in a low-pressure inert gas in a Geiger-Muller tube that
contains electrodes which accelerates the electrons released by the ionisation. Each detected
particle produces a pulse of current, and an audible click, which is translated to give a reading
of the intensity as the number of pulses per second i.e. the count rate.
A Geiger counter was mounted at the end of the carriage arm which was then moved around
the device from a fixed pivot point at the centre, allowing it to be placed at various scattering
angles. In accordance with Bragg’s law, the mounted crystal in the centre rotates to an
incident angle θ when the arm rotates at a (scattering) angle 2θ. This 1:2 ratio of the angular
displacement is maintained by gears at the central pivot point.
The base of the device had the degree scale of angles, like a large protractor, covering the
entire 360o circular base. The carriage arm’s minimum setting was at 12o to a maximum of
124o at each side. Due to the limited sensitivity of this scale, with the smallest interval being
1o, the carriage arm also had a miniature scale at its tip that can be operated and finely
calibrated accordingly by a thumb-wheel. This had a higher sensitivity, with the smallest
interval being 0.2o.
Risk Assessment
The entire Tel-X-Ometer apparatus is enclosed by a transparent plastic scatter shield which is
fitted with an aluminium and lead back-stop directly aligned with the X-ray source. The
plastic contains a higher percentage of chlorine to absorb radiation. For health and safety
purposes, the shield must be locked and centralised into place in order to turn on the voltage
and produce X-rays. The device comes with a timer which can be adjusted to automatically
stop the emission of X-rays after a specific period of time.

5
Part 1: Measurements on Lithium Fluoride
A LiF crystal was mounted on a Tel-X-Ometer with
the roughened side towards the crystal post, in the
reflecting position. This side of the crystal can be
identified as having a flat matt appearance.
Precaution should be taken in avoiding contact with
this surface.
Working on the left-hand side of the beam, the
counter was slowly scanned through increasing
Figure 4: Schematic diagram of
scattering angles, with corresponding count rates
Tel-X-Ometer setup (made by Sultan LeMarc).
being recorded at each angle.
The recorded data was constantly evaluated to establish the two closely spaced peaks in the
intensity distribution for the Kα and Kβ peaks of the copper target. The recorded angles
represented 2θ on the scale of the Tel-X-Ometer. Given that Kα radiation has a wavelength of
λα = 0.1542nm for the copper target, the wavelength of the Kβ radiation was found using
Bragg’s law equation (1).
The first set of Kα and Kβ corresponded to peaks for (2,0,0) plane, the angles at which the
(4,0,0) peaks occur were pre-determined using equations (1) and (2) and then measured at
those positions.
Part 2: Effect of Nickel filter
A nickel (Ni) filter was placed on positioned slots fixed on the carriage arm in front of the
Geiger counter. The filter was placed into the path of the X-ray beam to observe the effect on
intensities of emission peaks through the measurement of count-rates at varying scattering
angles.
Part 3: Measurements of Sodium Chloride, Potassium Chloride and Rubidium Chloride
crystals
The procedure of part 1 was repeated to measure the (2,0,0) and (4,0,0) peaks for each of
these three alkali halide crystals. Only the data for the stronger Kα radiation was recorded.
Part 4: Measurements on periclase (Magnesium Oxide)
Note: Measurement of peaks for the magnesium oxide (MgO) single crystal sample could not
be made as the crystal was not provided as part of the apparatus.
A photograph of the diffraction pattern from the
powdered MgO was provided. As the distance from the
centre of hole B to the centre of hole C was 100 ± 0.1mm
and corresponded to a Bragg angle of 180o, the angle per
unit distance was calculated, 10 ± 0.01mm = 18 o. This
allowed the Bragg angles for all visible lines in the
pattern to be determined by measuring the distance of
each line relative to hole B and converting it to the angle.
6

Figure 5: Photograph of powdered MgO
diffraction pattern (by Navjeevan Soor).
3. Results & Discussion

Figure 6: Graph of results for parts 1 and 2 of the experimental procedure providing a direct comparison of
the LiF crystal results with and without the Ni filter.
(Note: Error bars were too small to be represented on the scale of the graph, ∆θ = 0.1, ∆counts/sec = 0.5)

The results in Figure 6 showed that Ni filters monochromatise the radiation from the Cu
target by only allowing the characteristic Kα radiation to pass while absorbing the Kβ
radiation. This is because the maximum wavelength that Ni absorbs is between the two peaks.
The filter also removes much of the high energy background radiation.
Figure 7 shows the X-ray spectrum for a Cu target with the absorption curve for Ni. When
Ni is used as a filter, the resultant intensity of the transmitted Cu radiation will be the Cu
spectrum minus the Ni absorption curve.

Figure 7: Attenuation of the Cu Kβ radiation from absorption by Ni filter. [15]

7
The wavelength of the Kβ radiation was calculated using equation (1) to be λβ = 0.1402 ± 0.08
nm from Figure 6 as the Kβ peak appears at θβ = 20 ± 0.1o. The interplanar spacing of the
crystal was calculated using equation (1) to be d = 0.2 ± 0.05 nm given that the literature
value for Kα wavelength of Cu was λα = 0.1542 nm and that it appears in Figure 6 at θα = 22 ±
0.1o.
The errors on these values were found using the following propagation equations:

Using the measured Bragg angle for the (2,0,0) peaks, θβ = 20 ± 0.1o and θα = 22 ± 0.1o, the
angles at which the (4,0,0) peaks should theoretically appear were predicted by using
equations (1) and (2).
For (4,0,0), the interplanar spacing was calculated using equation (2) and (5) to be d = 0.1 nm
(errors could not be propagated as calculation used literature values with no associated
errors). Using the values of the experimental value of λβ and the accepted value of λα with this
value of d, the corresponding angle was θα2 = 48.8. This suggested that the peaks for (4,0,0)
plane occurred at angles that were approximately twice those at which the (2,0,0) peaks
occurred.
The result in Figure 6 confirms these predictions as the second pair of the peaks did indeed
occur at around twice the angles as the first pair.

Figure 8: Graph of results for part 3 of the experimental procedure providing a direct comparison of the
NaCl, KCl and RbCl crystals.
(Note: Error bars were too small to be represented on the scale of the graph, ∆θ = 0.1, ∆counts/sec = 0.5)

8
The results show that each crystal diffracts the X-ray beam differently, depending on its
structure and orientation. As the crystal structures contain unique planes of atoms, each plane
will reflect incident X-rays differently.
The results reflect the relative sizes of the lattice constants (and hence the interplanar
spacing) of the crystals through the positions of the peaks of each curve. The crystal with the
smallest lattice constant, NaCl, has peaks at higher angles than the others. As the lattice
constant increases, the positions of the peaks shift to the left to lower angles. Thus, RbCl,
with the lowest lattice constant of the three has peaks at the lowest angles, whilst KCl, with
the intermediate lattice constant has peaks at the intermediate angles.
The experimental values of the lattice constants for the alkali halide crystals were calculated
using the angles of the peaks in Figure 8 and applying equations (1) and (2) with the accepted
value of λα = 0.1542 nm.

Crystal
NaCl
KCl
RbCl

Literature
value
(nm)
0.564
0.629
0.670

(2,0,0) lattice
constant (nm)

% difference
from value

(4,0,0) lattice
constant (nm)

% difference
from value

0.54 ± 0.18
0.56 ± 0.20
0.64 ± 0.26

3.6%
13.2%
4.5%

0.56 ± 0.09
0.64 ± 0.12
0.68 ± 0.12

0.7%
1.6%
1.5%

Table 2: Comparison of experimental values of lattice constants for alkali halide crystals with accepted
values. [13]

The comparative analysis in Table 2 showed that all values agree with the literature values as
they were within the error range. The errors of calculated lattice constants were found using:
|√

|

The lattice constant values for the NaCl crystal had the lowest relative errors and were also
the most accurate of all relative to the respective literature values. This was because the peaks
for NaCl occurred at greater angles than KCl and RbCl due to the smaller interplanar spacing
of the crystal. The greater angles meant that the relative instrumental error of the Tel-XOmeter is smaller, thus allowing for more accurate experimental values to be found.
This is also why the measurements made with the reflections from the (4,0,0) planes were
more accurate than those made using (2,0,0) planes. The interplanar spacing is smaller for the
(4,0,0) planes resulting to peaks at higher angles with smaller relative instrumental errors. In
addition, the radiation is reflected more times in a given distance when scattering through
planes with the smaller interplanar spacing i.e. (4,0,0).
As the lattice constant represents the constant distance between unit cells in a crystal lattice,
X-ray diffraction was used to determine the characteristic radii by measuring the ionic radii
of the LiF, NaCl, KCl and RbCl crystals.
Although neither atoms nor ions have sharp boundaries, they can be treated as spheres with
radii such that the sum of ionic radii of the cation and anion gives the distance between the
molecules in a crystal lattice i.e. lattice constant. Therefore the ionic radii were found with
9
the assumption that the crystalline composition was such that the ions can be considered to be
in contact with each other.
For example, as the accepted value for the lattice constant (length of each edge of a unit cell)
for NaCl is a = 0.564nm, each edge may be considered to have atoms arranged as Na+…Cl..Na+. Thus, the lattice constant is twice the Na-Cl separation. The radii of the Li+, Na+, K+
and Rb+ anions were derived by using the accepted values of the lattice constant and cation
radii.
Crystal
LiF
NaCl
KCl
RbCl

Lattice constant
(nm)

Cation radius
(nm)

Anion radius
(nm)

Literature
value (nm)

% difference
from value

0.403
0.564
0.629
0.670

0.123 ± 0.04
0.166 ± 0.08
0.166 ± 0.08
0.166 ± 0.08

0.079 ± 0.04
0.116 ± 0.08
0.149 ± 0.08
0.169 ± 0.08

0.076
0.102
0.138
0.152

3.8%
12.0%
7.4%
10.0%

Table 3: Determining ionic radii from X-ray diffraction. [16]

The results of Table 3 correctly showed that the radii of the anions are smaller than those of
the cations. In terms of electronic configuration, this is because the cations lose an electron
therefore having one less shell reduces the overall radii. In contrast, anions gain one electron
in the molecular bond, shielding the outer electrons from the nuclear attraction and thus
increasing the overall radii.
Line

Angle (degrees)

h2 + k2+ l2

1
2
3
4
5
6
7
8
9
10

18.5
21.0
31.0
37.0
39.0
46.5
50.0
54.5
61.0
71.5

3
4
8
11
12
14
18
20
24
27

Possible permutation
(h,k,l)
(1,1,1)
(2,0,0)
(2,2,0)
(3,1,1)
(2,2,2)
(3,2,1)
(3,3,0)
(4,2,0)
(4,2,2)
(5,1,1)

Table 4: Part 4 results - deducing one possible permutation of Miller indices for lines of MgO powder
pattern.

The h2 + k2+ l2 values were calculated using equation (3), with λα = 0.1542 nm and the
literature value for the lattice constant of MgO as a = 0.421nm [17]. The lattice constant of
MgO could not be determined experimentally due to reasons discussed in experimental
procedure. The pattern in the numerical values for the indices is characteristic of the cubic
structures possessed by the alkali halide crystals examined in part 3 of this experiment.

10
4. Conclusion
Overall the investigation successfully supported the theoretical principles of X-ray diffraction
and their application to examine crystalline structures. It met its aims to examine the
diffraction of X-rays as well as their unique emission characteristics.
The experimental data supported the validity of the methods used in the experiment,
particularly the use of crystal structures to scatter the X-rays by separating the different
wavelengths at varying incident angles. The plots of the emission spectra reproduced the
typical features of an X-ray spectrum, as shown in Figure 1, such as the bremsstrahlung
continuum and the characteristics peaks of Kα and Kβ.
The application of Bragg’s law in order to deduce the characteristics of lattice structures was
reliably effective as the experimental values for lattice constants closely matched the
literature values for the respective quantities. The closest experimental value obtained was
that of the NaCl crystal, which had a lattice constant that was 0.7% different from the
accepted value which also lied in its error range. It was also established that the reflections
from (4,0,0) planes gave more accurate calculations than those from (2,0,0) planes.
Finally, the investigation successfully allowed the ionic radii to be deduced from the data as
all experimental values were credible approximations. The characteristics of cubic structures
were effectively probed further in order to deduce the Miller indices corresponding to lines of
a powdered MgO crystal diffraction pattern. This allowed the similarities between cubic
structures to be realised.
The main source of error in this investigation came from the measurement of the angle which
was possibly due to specimen displacement, instrument misalignment and error in the zero 2θ
position. An additional source of error was the Geiger counter due to its limited sensitivity.
The measurement of the count-rate lacked the precision because of the counter’s high
instrumental error. The scale of the counter had a sensitivity that was insufficient to detect Xrays accurately as it was restricted by large intervals on the scale, making it difficult to prove
into the ranges where there were fine and sharp changes, compromising the detection of
peaks.
The Geiger counter also had a short time-constant which meant that readings were highly
fluctuant over a wide range of readings, making it increasingly difficult to obtain readings.
Moreover, the reading was taken on an analogue scale with a pointer which introduced
random parallax error.
An improvement of this particular error can be made by using a high sensitivity electronic
counter. The modern counters with a long and adjustable time-constant, allowing readings to
be taken without as many fluctuations whilst also removing parallax error.
Repeat measurements for the count-rate and angles in parts 1, 2 and 3 would increase the
accuracy of the results as it would allow an average to be taken. A major way to improve the
experiment by using a modern Tel-X-Ometer with a fitted Tel-X-Driver which allows the
unit to be controlled with a computer. It would measure and record the data digitally resulting
to a better resolution in the measurement of the angles.
11
Direct extensions of the tasks in this investigation include the analysis of different metal
targets to provide comparisons of characteristic spectra in order to deduce their atomic
properties through the interpretation of transition energies. Furthermore, the scattering of Xrays could be considered in terms of their polarization and representing them as vector waves
in Fourier transform in order to relate the amplitude with the measured intensities.
In addition, the experimentation of the wavelength’s inverse relationship with the atomic
number of the target could be investigated through the principles of Mosley’s law. Part 2 of
the experiment could be expanded through the consideration of the filter thickness and using
the mass-absorption law to determine the optimum thickness for attenuation of desired
wavelengths.
In conclusion, the experiment was considered a success as a whole as it had achieved the
primary objective to probe into the fundamental principles of X-rays through the use of
classically verified experimentation methods, producing data that agreed with the
hypothesised theory.

12
References
[1] http://www.nobelprize.org/educational/physics/x-rays/
[2] http://www.britannica.com/EBchecked/topic/179233/Thomas-Alva-Edison
[3] http://www.britannica.com/EBchecked/topic/565304/Stenos-law
[4] http://www.britannica.com/EBchecked/topic/650351/X-ray
[5] http://www.britannica.com/EBchecked/topic/78784/bremsstrahlung
[6] R. Serway. Physics for Scientists and Engineers, p. 1277, 2010.
[7] R. Serway. Physics for Scientists and Engineers, p. 1278, 2010.
[8] R. Serway. Physics for Scientists and Engineers, p. 1278, 2010.
[9] H. Young. University Physics, p. 1206, 2012.
[10] University of Florida. X-Ray Diffraction and Absorption, XDA 1, Jan 2012.
[11] http://upload.wikimedia.org/wikipedia/commons/9/98/Miller_Indices_Cubes2.svg
[12] King’s College London. Second Year Physics Laboratory Manual, 111-112, Oct 2012
[13] http://www.almazoptics.com/
[14] http://www.telatomic.com/x-ray/
[15] R. Jenkins and J. L. de Vries. An Introduction to Powder Diffractometry, p. 14, 1977
[16] http://www.chemguide.co.uk/atoms/properties/atradius.html
[17] http://www.2spi.com/catalog/submat/magnesium-oxide.shtml

13

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Crystal Structures and X-Ray Diffraction - Sultan LeMarc

  • 1. Date of experiment: 13/11/2012 Date of report: 14/12/2012 Investigation of Crystal Structures and X-Ray Diffraction Experiment conducted by Sultan LeMarc Partner: Navjeevan Soor Supervisor: Dr. W. Dickson
  • 2. Abstract This experiment investigated the characteristics of X-rays by measuring the count rate of Xrays reflected off alkali halide crystals at varying angles of incidence and using the principles of Bragg’s law. The experiment probes into crystal structures using X-ray diffractometry and deduces the lattice constants and ionic radii using the Miller index notation. The experiment successfully computes the characteristic wavelengths of Copper and clearly demonstrates the effect of filters on spectrum intensities. The interpretation of Miller indices and diffraction patterns are effectively used to analyse crystalline structures and compare lattice arrangements.
  • 3. Introduction X-rays were discovered in 1895 by German physicist Wilhelm Röntgen, who received the first Nobel Prize in Physics in 1901 in recognition of his work on electromagnetic radiation. As Röntgen refused to patent the discovery in order to allow further development and practical applications, the discovery prompted further study of the radiation. [1] Nikola Tesla proceeded to devise experimental setups which produced high energy Xrays. Tesla also alerted the scientific community about the safe operation of X-ray equipment and the biological hazards associated with exposure. Thomas Edison led the advances in radiology and medical X-ray examinations by developing a fluoroscope in 1896. Since the initial pioneering works of the late 19th century, many important discoveries have been made using X-rays which are now used in various applications. This includes the establishment of X-ray crystallography, an experimental technique which allows the dimensional arrangement of atoms within crystal structures to be determined by the interpretation of diffraction patterns. [2] The field was developed by the pioneering research of physicist William Bragg in 1913 who won a joint Nobel Prize in Physics in 1915 with his father for their work on crystallography. X-ray crystallography has since made significant contributions to chemistry and material science. Today, X-rays are most notably used for medical imaging, microscopy and astronomy. Crystals were first scientifically investigated in the 17th century for their regularity and symmetry, most notably by Danish geologist Nicolas Steno, who successfully confirmed this symmetry by showing that the angles between the faces are the same for a given type of crystal. This was followed by the introduction of a notation system for planes in crystal lattices called the Miller indices by William Miller in 1839, which is still used today to identify crystals. [3] The aim of this experiment was to investigate the production and emission characteristics of X-rays. This was achieved by measuring the count rate of X-rays reflected off alkali halide crystal at varying angles of incidence in order to find the characteristic peaks of a copper (Cu) target using the principles of Bragg’s law. In addition, the experiment aimed to examine how the Bragg scattering of X-rays can be used to determine the crystal structures and their lattice constants. There was also a consideration of energy loss by observing the varying intensities of characteristic wavelengths. 1
  • 4. 1. Theory X-rays are a form of ionising electromagnetic radiation found in the short wavelength and high energy end of the spectrum with wavelengths ranging from 0.01nm to 10nm. This range of wavelengths of X-rays is of the order of distances between molecules and crystal lattices, making them ideal for spectroscopic techniques for characterisation of the elemental composition of materials. The corresponding energies of this range are 120 eV to 120 keV respectively. There are two commonly used methods to produce X-rays, by an X-ray tube which is a vacuum tube that linearly accelerates charged particles, or by a synchrotron which is a cyclic particle accelerator that applies electric and magnetic fields. Most natural sources of X-rays are extra-terrestrial such as the Sun and black holes but they are also emitted by the decay of unstable nuclei on Earth. [4] An X-ray tube functions as a typical vacuum tube which uses the potential difference between a cathode and an anode to accelerate charged particles. In an X-ray tube a metal filament is the cathode and is heated by a low voltage current in order to emit electrons in a process called thermionic emission. As a stream of electrons are released into the vacuum, a large electric potential is applied between the cathode and the anode; a metal target. This accelerates the electrons towards the anode due to electrostatic attraction. There are two principle mechanisms by which X-rays are produced, by bremsstrahlung or Kshell emission. In the first, the emission of X-rays takes place when high-energy electrons (or charged particles) are decelerated, in speed or direction, by bombarding targets. In accordance with Maxwell’s equations, the electrons emit electromagnetic radiation upon deceleration, a process called bremsstrahlung. [5] In the second mechanism, X-rays are produced by the excitation and ionisation of the atoms in the target. In this process transitions of electrons between atomic orbit shells take place as the bombardment of electrons can excite and eject inner electrons from the target atoms, provided the incident electron has sufficient energy. This leaves a vacant space in the inner shell and is filled by an electron at the higher level outer shell. In the course of this transition, the higher level electron is losing kinetic energy as it fills the vacant inner shell. Figure 1: The emission spectra of a heavy metal xray source. [6] Under the principles of conservation of energy, the transition is accompanied by the emission of an X-ray photon with unique energy corresponding to the energy difference between the two shells i.e. the energy lost by the shifting electron. These X-rays are called characteristic X-rays with wavelengths that are distinctive for each particular element and transition. The innermost shell from which the incident electron has dislodged an atomic electron is called the K shell. When the vacancy is filled by an electron from the next higher shell, the L shell, the photon emitted has an energy corresponding to the Kα characteristic X-ray line on 2
  • 5. the emission spectra. When the vacancy in the K shell is filled by an electron from the M shell, the next higher shell after the L shell, the Kβ characteristic X-ray line is produced on the emission spectra. [7] Diffracted waves from different atoms within the lattice can interfere with each other, leading to a modulated resultant intensity distribution. Interference occurs when two waves meet and superimpose either constructively or destructively. Constructive interference takes place when two identical phases meet and superimpose into a wave with combined amplitude. Destructive interference takes place when two waves are completely out of phase with each other, resulting to the cancelation of each other. As the atoms are in a period arrangement, the diffracted waves interfere to produce diffraction patterns that reflect the symmetry of the distribution of the crystal lattice. Figure 2: Transitions from higher to lower energy levels. [8] The diffraction patterns of X-rays show that at most of the incident angles the incident X-rays are scattered coherently to interfere destructively as the combining waves are out of phase and thus have no resultant energy. However at some key incident angles the X-rays are scattered coherently to interfere constructively, resulting in well-defined X-ray beams leaving the crystal in various directions. As a result, a diffracted beam can be considered as a beam composed of a large number of scattered rays mutually reinforcing each other. The condition to be satisfied for there to be constructive interference in X-ray diffraction is mathematically defined by Bragg’s law, which relates the incident wavelength λ, incident angle θ, and the spacing between the planes in the atomic lattice d: [9] Bragg’s law states that intense peaks of diffracted radiation (Bragg peaks) are produced by constructive interference of scattered waves at specific wavelengths and incident angles. The scattering angle 2θ is defined as the angle between the incident and scattered rays. It is important to note that Bragg's Law applies to scattering centres consisting of any periodic distribution of electron density. In this case, a crystal is used as the scattering centre to separate the different wavelengths of incident X-rays and scattering them to specific angles. The literature value of the Kβ and Kα wavelengths for copper are λβ = 0.1392 nm and λα = 0.1542 nm respectively.[10] 3
  • 6. The Miller index notation, denoted by (h,k,l), is a convention used in crystallography in order to define the orientation of a crystal plane with respect to a main crystallographic axis. They are a set of numbers which quantify the intercepts and can be used to uniquely identify planes of lattice structures, as shown in Figure 3. Miller indices are sufficient to specify both the orientation and the spacing of a set of parallel planes. For cubic crystals, the relationship between the lattice Figure 3: Planes with different Miller indices in cubic crystals. [11] constant, a, which refers to the constant difference between unit cells in a lattice, the miller indices, and the interplanar spacing, d, is given by: √ [12] The literature values of the lattice constant and the radius of the anions of the alkali halide crystals being investigated in this experiment are: Crystal Lattice Constant (nm) Anion radius (nm) Lithium Fluoride (LiF) Sodium Chloride (NaCl) Potassium Chloride (KCl) Rubidium Chloride (RbCl) 0.403 0.565 0.629 0.670 0.102 0.076 0.138 0.152 Table 1: Accepted values for the lattice constants and anion radii of the alkali halide crystals. [13] Equations (1) and (2) were combined to give: This allows the miller indices to be deduced once the lattice constant is found using equation (2) from the Bragg angles for (2,0,0) and (4,0,0) reflections. 4
  • 7. 2. Experimental Procedure In order to observe the characteristics peaks of the copper target, readings for the count-rate of X-rays over a range of angles 2θ were taken using a Tel-X-Ometer and a Geiger counter. A Tel-X-Ometer is an X-ray diffraction system device which is used to detect the absorption and reflection of X-rays i.e. a spectrometer. The device employs the fundamental principle of an X-ray tube to produce X-rays and accelerate them towards the target, which is mounted as the anode. The Tel-X-Ometer used in this investigation had two settings for the accelerating voltage, also known as extra high voltage (EHT), at 20kV and 30kV. The filament current could also be adjusted, and was not allowed to exceed 80µA to prevent damage. The device allows crystal to be mounted in the centre of the device to reflect the X-rays for detection. It also allows numerous collimator slides to be mounted at different numbered positions on a carriage arm assembly. Figure 3: Tel-X-Ometer. [14] A Geiger counter is a particle detector that measured ionising radiation. It is able to detect radiation by the ionisation produced in a low-pressure inert gas in a Geiger-Muller tube that contains electrodes which accelerates the electrons released by the ionisation. Each detected particle produces a pulse of current, and an audible click, which is translated to give a reading of the intensity as the number of pulses per second i.e. the count rate. A Geiger counter was mounted at the end of the carriage arm which was then moved around the device from a fixed pivot point at the centre, allowing it to be placed at various scattering angles. In accordance with Bragg’s law, the mounted crystal in the centre rotates to an incident angle θ when the arm rotates at a (scattering) angle 2θ. This 1:2 ratio of the angular displacement is maintained by gears at the central pivot point. The base of the device had the degree scale of angles, like a large protractor, covering the entire 360o circular base. The carriage arm’s minimum setting was at 12o to a maximum of 124o at each side. Due to the limited sensitivity of this scale, with the smallest interval being 1o, the carriage arm also had a miniature scale at its tip that can be operated and finely calibrated accordingly by a thumb-wheel. This had a higher sensitivity, with the smallest interval being 0.2o. Risk Assessment The entire Tel-X-Ometer apparatus is enclosed by a transparent plastic scatter shield which is fitted with an aluminium and lead back-stop directly aligned with the X-ray source. The plastic contains a higher percentage of chlorine to absorb radiation. For health and safety purposes, the shield must be locked and centralised into place in order to turn on the voltage and produce X-rays. The device comes with a timer which can be adjusted to automatically stop the emission of X-rays after a specific period of time. 5
  • 8. Part 1: Measurements on Lithium Fluoride A LiF crystal was mounted on a Tel-X-Ometer with the roughened side towards the crystal post, in the reflecting position. This side of the crystal can be identified as having a flat matt appearance. Precaution should be taken in avoiding contact with this surface. Working on the left-hand side of the beam, the counter was slowly scanned through increasing Figure 4: Schematic diagram of scattering angles, with corresponding count rates Tel-X-Ometer setup (made by Sultan LeMarc). being recorded at each angle. The recorded data was constantly evaluated to establish the two closely spaced peaks in the intensity distribution for the Kα and Kβ peaks of the copper target. The recorded angles represented 2θ on the scale of the Tel-X-Ometer. Given that Kα radiation has a wavelength of λα = 0.1542nm for the copper target, the wavelength of the Kβ radiation was found using Bragg’s law equation (1). The first set of Kα and Kβ corresponded to peaks for (2,0,0) plane, the angles at which the (4,0,0) peaks occur were pre-determined using equations (1) and (2) and then measured at those positions. Part 2: Effect of Nickel filter A nickel (Ni) filter was placed on positioned slots fixed on the carriage arm in front of the Geiger counter. The filter was placed into the path of the X-ray beam to observe the effect on intensities of emission peaks through the measurement of count-rates at varying scattering angles. Part 3: Measurements of Sodium Chloride, Potassium Chloride and Rubidium Chloride crystals The procedure of part 1 was repeated to measure the (2,0,0) and (4,0,0) peaks for each of these three alkali halide crystals. Only the data for the stronger Kα radiation was recorded. Part 4: Measurements on periclase (Magnesium Oxide) Note: Measurement of peaks for the magnesium oxide (MgO) single crystal sample could not be made as the crystal was not provided as part of the apparatus. A photograph of the diffraction pattern from the powdered MgO was provided. As the distance from the centre of hole B to the centre of hole C was 100 ± 0.1mm and corresponded to a Bragg angle of 180o, the angle per unit distance was calculated, 10 ± 0.01mm = 18 o. This allowed the Bragg angles for all visible lines in the pattern to be determined by measuring the distance of each line relative to hole B and converting it to the angle. 6 Figure 5: Photograph of powdered MgO diffraction pattern (by Navjeevan Soor).
  • 9. 3. Results & Discussion Figure 6: Graph of results for parts 1 and 2 of the experimental procedure providing a direct comparison of the LiF crystal results with and without the Ni filter. (Note: Error bars were too small to be represented on the scale of the graph, ∆θ = 0.1, ∆counts/sec = 0.5) The results in Figure 6 showed that Ni filters monochromatise the radiation from the Cu target by only allowing the characteristic Kα radiation to pass while absorbing the Kβ radiation. This is because the maximum wavelength that Ni absorbs is between the two peaks. The filter also removes much of the high energy background radiation. Figure 7 shows the X-ray spectrum for a Cu target with the absorption curve for Ni. When Ni is used as a filter, the resultant intensity of the transmitted Cu radiation will be the Cu spectrum minus the Ni absorption curve. Figure 7: Attenuation of the Cu Kβ radiation from absorption by Ni filter. [15] 7
  • 10. The wavelength of the Kβ radiation was calculated using equation (1) to be λβ = 0.1402 ± 0.08 nm from Figure 6 as the Kβ peak appears at θβ = 20 ± 0.1o. The interplanar spacing of the crystal was calculated using equation (1) to be d = 0.2 ± 0.05 nm given that the literature value for Kα wavelength of Cu was λα = 0.1542 nm and that it appears in Figure 6 at θα = 22 ± 0.1o. The errors on these values were found using the following propagation equations: Using the measured Bragg angle for the (2,0,0) peaks, θβ = 20 ± 0.1o and θα = 22 ± 0.1o, the angles at which the (4,0,0) peaks should theoretically appear were predicted by using equations (1) and (2). For (4,0,0), the interplanar spacing was calculated using equation (2) and (5) to be d = 0.1 nm (errors could not be propagated as calculation used literature values with no associated errors). Using the values of the experimental value of λβ and the accepted value of λα with this value of d, the corresponding angle was θα2 = 48.8. This suggested that the peaks for (4,0,0) plane occurred at angles that were approximately twice those at which the (2,0,0) peaks occurred. The result in Figure 6 confirms these predictions as the second pair of the peaks did indeed occur at around twice the angles as the first pair. Figure 8: Graph of results for part 3 of the experimental procedure providing a direct comparison of the NaCl, KCl and RbCl crystals. (Note: Error bars were too small to be represented on the scale of the graph, ∆θ = 0.1, ∆counts/sec = 0.5) 8
  • 11. The results show that each crystal diffracts the X-ray beam differently, depending on its structure and orientation. As the crystal structures contain unique planes of atoms, each plane will reflect incident X-rays differently. The results reflect the relative sizes of the lattice constants (and hence the interplanar spacing) of the crystals through the positions of the peaks of each curve. The crystal with the smallest lattice constant, NaCl, has peaks at higher angles than the others. As the lattice constant increases, the positions of the peaks shift to the left to lower angles. Thus, RbCl, with the lowest lattice constant of the three has peaks at the lowest angles, whilst KCl, with the intermediate lattice constant has peaks at the intermediate angles. The experimental values of the lattice constants for the alkali halide crystals were calculated using the angles of the peaks in Figure 8 and applying equations (1) and (2) with the accepted value of λα = 0.1542 nm. Crystal NaCl KCl RbCl Literature value (nm) 0.564 0.629 0.670 (2,0,0) lattice constant (nm) % difference from value (4,0,0) lattice constant (nm) % difference from value 0.54 ± 0.18 0.56 ± 0.20 0.64 ± 0.26 3.6% 13.2% 4.5% 0.56 ± 0.09 0.64 ± 0.12 0.68 ± 0.12 0.7% 1.6% 1.5% Table 2: Comparison of experimental values of lattice constants for alkali halide crystals with accepted values. [13] The comparative analysis in Table 2 showed that all values agree with the literature values as they were within the error range. The errors of calculated lattice constants were found using: |√ | The lattice constant values for the NaCl crystal had the lowest relative errors and were also the most accurate of all relative to the respective literature values. This was because the peaks for NaCl occurred at greater angles than KCl and RbCl due to the smaller interplanar spacing of the crystal. The greater angles meant that the relative instrumental error of the Tel-XOmeter is smaller, thus allowing for more accurate experimental values to be found. This is also why the measurements made with the reflections from the (4,0,0) planes were more accurate than those made using (2,0,0) planes. The interplanar spacing is smaller for the (4,0,0) planes resulting to peaks at higher angles with smaller relative instrumental errors. In addition, the radiation is reflected more times in a given distance when scattering through planes with the smaller interplanar spacing i.e. (4,0,0). As the lattice constant represents the constant distance between unit cells in a crystal lattice, X-ray diffraction was used to determine the characteristic radii by measuring the ionic radii of the LiF, NaCl, KCl and RbCl crystals. Although neither atoms nor ions have sharp boundaries, they can be treated as spheres with radii such that the sum of ionic radii of the cation and anion gives the distance between the molecules in a crystal lattice i.e. lattice constant. Therefore the ionic radii were found with 9
  • 12. the assumption that the crystalline composition was such that the ions can be considered to be in contact with each other. For example, as the accepted value for the lattice constant (length of each edge of a unit cell) for NaCl is a = 0.564nm, each edge may be considered to have atoms arranged as Na+…Cl..Na+. Thus, the lattice constant is twice the Na-Cl separation. The radii of the Li+, Na+, K+ and Rb+ anions were derived by using the accepted values of the lattice constant and cation radii. Crystal LiF NaCl KCl RbCl Lattice constant (nm) Cation radius (nm) Anion radius (nm) Literature value (nm) % difference from value 0.403 0.564 0.629 0.670 0.123 ± 0.04 0.166 ± 0.08 0.166 ± 0.08 0.166 ± 0.08 0.079 ± 0.04 0.116 ± 0.08 0.149 ± 0.08 0.169 ± 0.08 0.076 0.102 0.138 0.152 3.8% 12.0% 7.4% 10.0% Table 3: Determining ionic radii from X-ray diffraction. [16] The results of Table 3 correctly showed that the radii of the anions are smaller than those of the cations. In terms of electronic configuration, this is because the cations lose an electron therefore having one less shell reduces the overall radii. In contrast, anions gain one electron in the molecular bond, shielding the outer electrons from the nuclear attraction and thus increasing the overall radii. Line Angle (degrees) h2 + k2+ l2 1 2 3 4 5 6 7 8 9 10 18.5 21.0 31.0 37.0 39.0 46.5 50.0 54.5 61.0 71.5 3 4 8 11 12 14 18 20 24 27 Possible permutation (h,k,l) (1,1,1) (2,0,0) (2,2,0) (3,1,1) (2,2,2) (3,2,1) (3,3,0) (4,2,0) (4,2,2) (5,1,1) Table 4: Part 4 results - deducing one possible permutation of Miller indices for lines of MgO powder pattern. The h2 + k2+ l2 values were calculated using equation (3), with λα = 0.1542 nm and the literature value for the lattice constant of MgO as a = 0.421nm [17]. The lattice constant of MgO could not be determined experimentally due to reasons discussed in experimental procedure. The pattern in the numerical values for the indices is characteristic of the cubic structures possessed by the alkali halide crystals examined in part 3 of this experiment. 10
  • 13. 4. Conclusion Overall the investigation successfully supported the theoretical principles of X-ray diffraction and their application to examine crystalline structures. It met its aims to examine the diffraction of X-rays as well as their unique emission characteristics. The experimental data supported the validity of the methods used in the experiment, particularly the use of crystal structures to scatter the X-rays by separating the different wavelengths at varying incident angles. The plots of the emission spectra reproduced the typical features of an X-ray spectrum, as shown in Figure 1, such as the bremsstrahlung continuum and the characteristics peaks of Kα and Kβ. The application of Bragg’s law in order to deduce the characteristics of lattice structures was reliably effective as the experimental values for lattice constants closely matched the literature values for the respective quantities. The closest experimental value obtained was that of the NaCl crystal, which had a lattice constant that was 0.7% different from the accepted value which also lied in its error range. It was also established that the reflections from (4,0,0) planes gave more accurate calculations than those from (2,0,0) planes. Finally, the investigation successfully allowed the ionic radii to be deduced from the data as all experimental values were credible approximations. The characteristics of cubic structures were effectively probed further in order to deduce the Miller indices corresponding to lines of a powdered MgO crystal diffraction pattern. This allowed the similarities between cubic structures to be realised. The main source of error in this investigation came from the measurement of the angle which was possibly due to specimen displacement, instrument misalignment and error in the zero 2θ position. An additional source of error was the Geiger counter due to its limited sensitivity. The measurement of the count-rate lacked the precision because of the counter’s high instrumental error. The scale of the counter had a sensitivity that was insufficient to detect Xrays accurately as it was restricted by large intervals on the scale, making it difficult to prove into the ranges where there were fine and sharp changes, compromising the detection of peaks. The Geiger counter also had a short time-constant which meant that readings were highly fluctuant over a wide range of readings, making it increasingly difficult to obtain readings. Moreover, the reading was taken on an analogue scale with a pointer which introduced random parallax error. An improvement of this particular error can be made by using a high sensitivity electronic counter. The modern counters with a long and adjustable time-constant, allowing readings to be taken without as many fluctuations whilst also removing parallax error. Repeat measurements for the count-rate and angles in parts 1, 2 and 3 would increase the accuracy of the results as it would allow an average to be taken. A major way to improve the experiment by using a modern Tel-X-Ometer with a fitted Tel-X-Driver which allows the unit to be controlled with a computer. It would measure and record the data digitally resulting to a better resolution in the measurement of the angles. 11
  • 14. Direct extensions of the tasks in this investigation include the analysis of different metal targets to provide comparisons of characteristic spectra in order to deduce their atomic properties through the interpretation of transition energies. Furthermore, the scattering of Xrays could be considered in terms of their polarization and representing them as vector waves in Fourier transform in order to relate the amplitude with the measured intensities. In addition, the experimentation of the wavelength’s inverse relationship with the atomic number of the target could be investigated through the principles of Mosley’s law. Part 2 of the experiment could be expanded through the consideration of the filter thickness and using the mass-absorption law to determine the optimum thickness for attenuation of desired wavelengths. In conclusion, the experiment was considered a success as a whole as it had achieved the primary objective to probe into the fundamental principles of X-rays through the use of classically verified experimentation methods, producing data that agreed with the hypothesised theory. 12
  • 15. References [1] http://www.nobelprize.org/educational/physics/x-rays/ [2] http://www.britannica.com/EBchecked/topic/179233/Thomas-Alva-Edison [3] http://www.britannica.com/EBchecked/topic/565304/Stenos-law [4] http://www.britannica.com/EBchecked/topic/650351/X-ray [5] http://www.britannica.com/EBchecked/topic/78784/bremsstrahlung [6] R. Serway. Physics for Scientists and Engineers, p. 1277, 2010. [7] R. Serway. Physics for Scientists and Engineers, p. 1278, 2010. [8] R. Serway. Physics for Scientists and Engineers, p. 1278, 2010. [9] H. Young. University Physics, p. 1206, 2012. [10] University of Florida. X-Ray Diffraction and Absorption, XDA 1, Jan 2012. [11] http://upload.wikimedia.org/wikipedia/commons/9/98/Miller_Indices_Cubes2.svg [12] King’s College London. Second Year Physics Laboratory Manual, 111-112, Oct 2012 [13] http://www.almazoptics.com/ [14] http://www.telatomic.com/x-ray/ [15] R. Jenkins and J. L. de Vries. An Introduction to Powder Diffractometry, p. 14, 1977 [16] http://www.chemguide.co.uk/atoms/properties/atradius.html [17] http://www.2spi.com/catalog/submat/magnesium-oxide.shtml 13