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 X-

rays 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.

1

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 X-

rays. 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.

2

Figure 1: The emission

spectra of a heavy metal x-

ray source. [6]

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 K-

shell 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.

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

3

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.

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]

Figure 2: Transitions

from higher to lower

energy levels. [8]

4

Figure 3: Planes with different Miller

indices in cubic crystals. [11]

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

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) 0.403 0.102

Sodium Chloride (NaCl) 0.565 0.076

Potassium Chloride (KCl) 0.629 0.138

Rubidium Chloride (RbCl) 0.670 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.

5

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.

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 12

o 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.

Figure 3: Tel-X-Ometer.

[14]

6

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

scattering angles, with corresponding count rates

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.

Figure 4: Schematic diagram of

Tel-X-Ometer setup (made by Sultan LeMarc).

Figure 5: Photograph of powdered MgO diffraction pattern (by Navjeevan Soor).

7

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]

8

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)

9

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

Literature

value

(nm)

(2,0,0) lattice

constant (nm)

% difference

from value

(4,0,0) lattice

constant (nm)

% difference

from value

NaCl 0.564 0.54 ± 0.18 3.6% 0.56 ± 0.09 0.7%

KCl 0.629 0.56 ± 0.20 13.2% 0.64 ± 0.12 1.6%

RbCl 0.670 0.64 ± 0.26 4.5% 0.68 ± 0.12 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-X-

Ometer 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

10

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 Lattice constant

(nm)

Cation radius

(nm)

Anion radius

(nm)

Literature

value (nm)

% difference

from value

LiF 0.403 0.123 ± 0.04 0.079 ± 0.04 0.076 3.8%

NaCl 0.564 0.166 ± 0.08 0.116 ± 0.08 0.102 12.0%

KCl 0.629 0.166 ± 0.08 0.149 ± 0.08 0.138 7.4%

RbCl 0.670 0.166 ± 0.08 0.169 ± 0.08 0.152 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 + k

2+ l

2

Possible permutation

(h,k,l)

1 18.5 3 (1,1,1)

2 21.0 4 (2,0,0)

3 31.0 8 (2,2,0)

4 37.0 11 (3,1,1)

5 39.0 12 (2,2,2)

6 46.5 14 (3,2,1)

7 50.0 18 (3,3,0)

8 54.5 20 (4,2,0)

9 61.0 24 (4,2,2)

10 71.5 27 (5,1,1) Table 4: Part 4 results - deducing one possible permutation of Miller indices for lines of MgO powder

pattern.

The h2 + k

2+ l

2 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.

11

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 X-

rays 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.

12

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 X-

rays 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.

13

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