X-Ray Diffraction - Final Version

7/31/2019 X-Ray Diffraction - Final Version

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X-Ray Diffraction

David SirajuddinNuclear Engineering & Radiological Sciences

February 8, 2007


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Sirajuddin, David Lab 2 - X-Ray Diffraction

Contents

1 Abstract 2

2 Introduction 2

3 Theory 33.1 X Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Braggs Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3 X-Ray Diffractometry (XRD) . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 Reciprocal Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.5.1 Face-Centered Cubic (FCC) . . . . . . . . . . . . . . . . . . . . . . . 93.5.2 Body-Centered Cubic (BCC) . . . . . . . . . . . . . . . . . . . . . . 103.5.3 Hexagonal Close-Packed (HCP) . . . . . . . . . . . . . . . . . . . . . 11

3.6 Diffraction Pattern Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.6.1 Identifying K and K Peaks . . . . . . . . . . . . . . . . . . . . . . 113.6.2 Change of Variables and Curve Fitting . . . . . . . . . . . . . . . . . 12

3.7 Depth of Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.8 Density Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.9 Concentration of Substitutional Copper Imputurities in the Bulk Material . . 153.10 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Procedure 16

5 Results and Analysis 175.1 Surface of Penny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.1.1 Diffraction Pattern Analysis . . . . . . . . . . . . . . . . . . . . . . . 175.1.2 Depth of Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2 Bulk of Penny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Error Propagation 24

7 Summary and Conclusions 25

8 References 26

9 Appendix 27

9.1 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.2 Data Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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1 Abstract

The crystal structure of a United States one cent coin was investigated by X-ray Diffrac-tion. Due to the nonuniformity of the composition, two data sets were taken. Initially, thesurface of the penny was analyzed for angular values of 2 between 12 and 15 degrees. The

penny was then sanded down so as to expose the core material, and values of 2 between10 and 60 degrees were examined. The resulting diffraction patterns were used to determinecharacteristic quantities of the lattice structure including the lattice type, lattice parameter,interplanar spacing, number density, and the mass density. Additionally, these quantitiesaided in labelling the crystallographic planes in regards to Miller indices on the diffractionpattern.

The data from the Copper surface indicated the material to Face-Centered Cubic (FCC).Accuracy in the analysis of the diffraction pattern was accomplished by a change of variablesfrom an angular dependence to a quantity s, defined to operate in reciprocal space. Eachpeak was fitted to a function using Kaleidagraph, and the centroid of each peak was found byway of interpolation. Interplanar spacings for each peak were calculated from these centroid

values and used in the assignment of crystallographic plane indexing. Identifying the materialto be FCC, and discerning the appropriate {h,k,l} values allowed for a calculation of theaverage lattice parameter. This value was found to be 3.622 A. Furthermore, a numberdensity of 8.419 1022 atoms/cm3, and a mass density of 8.884 g/cm3 was calculated. Thesevalues were within one percent of those from literature.

The bulk material of the penny was also analyzed, and known to be composed of byand large Zinc in a Hexagonal Close-Packed (HCP) structure. Crystallographic planes inthe diffraction pattern were assigned by matching corresponding interplanar spacings withgiven values of{h,k,l}. All peaks were identifiable except five. This discrepency was taken tobe due to the presence of susbtitutional Copper impurities. The number density of the bulk

material was found to be 1.9347 1022

atoms/cm3

, and the concentration of Copper impurityin the material was calculated to be 3.729 atom percent. For both data sets, error was takeninto account for the centroid s of each peak, and the interplanar spacing d. The standarddeviations for both quantities calculated were found to generally be on the order of 104 -103. This error is minute, and it can be said that the error in the experimental proceduredid not substantially perturb the results. In all, there was a close agreement among theory,literature, and experiment.

2 Introduction

The purpose of the experiment was to use X rays as a diagnostic tool to reveal keyparameters about the crystal structure. Motivated by the knowledge of the heterogeneityin composition of the penny, two sets of data were conducted and analyzed: the surface ofthe coin, and the core material (hereafter referred to as the bulk). In both data sets, thepenny was bombarded with X rays, and an analysis of the diffraction pattern allowed for thedetermination of various parameters relating to the specific crystal structure, such as thelattice parameter, lattice type, and number density. The surface of the penny was taken tobe made of pure Copper, and was known to have a cubic type lattice. The crystallographic

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planes were labelled according to the Miller indexing scheme, and the lattice parameter, andboth the mass and number densities were determined. The bulk material, presupposed tobe made of essentially Zinc and to have a hexagonal close-packed structure, was juxtaposedto literature data for pure Zinc to aid in the labelling of the crystallographic planes in theexperimental data. Inconsistencies between the literature values and experiment were used

to pinpoint defects in the material, which were taken to be only substitutional impurities inthe form of copper atoms. The number densities of pure Zn with that of the bulk materialin the experiment were compared in order to estimate the percentage of copper impurities.

3 Theory

3.1 X Rays

X Rays are high frequency photons (30-30,000 PHz) emitted via electron transitions inatoms. Due to their high frequency, and thusly high energy, X rays are highly penetrable

and are readily applicable to probing the microstructure of materials. Particularly of use,is their low wavelength (10-0.01 nm), which can be utilized in tandem with Braggs Law(Section 3.2) to construct diffraction patterns that reveal key parameters of the crystallinelattice structure of solids such as the structures lattice parameter, and its lattice type.

3.2 Braggs Law

Braggs law expresses the criterion needed for two beams to emerge after interactionwith constructive interference. The derivation is elementary, but it is instructive to present.Without loss of generality, the expression can be formulated in two dimensions for the simplecase of two beams of common angles of incidence.

Figure 3.1 - Two X-ray beams deflecting off of neighboring atomic planes. Beam travels a distance of

AB + BC farther than beam .

In order for constructive interference to occur, two waves must superimpose after theirdeflection to cause an increase in amplitude. This occurs only when two waves add thatare in phase with each other. It follows from Figure 3.1, that beam travels a distance ofAB + BC further than the beam . In order to stay in phase with each other, the distancetravelled by beam must be an integral multiple of the wavelength, such that:

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n = AB + CD (1)

where denotes the wavelength of the beams, and n is an element of the integers. An

analysis of a triangle formed between the two atomic planes allows for the distance AB +BCto be determined.

Figure 3.2 - A right triangle constructed involving the interplanar spacing d reveals the sidelength AB tobe equal to dsin. By inspection, AB = BC, yielding the total excess distance travelled by beam to be

simply 2AB = 2dsin.

Defining the distance between the two planes as the interplanar spacing d, the excessdistance travelled by beam can be formally recognized as simply:

AB + CD = 2dsin

Inputting this result into Eqn. (1) yields

n = 2dsin (2)

which is exactly that of Braggs Law. It is then apparent that the requisite for meetingBraggs Law is dependent on the interplanar spacing, or equivalently, the orientation of theplanes with respect to the incident beams. For all intended purposes, the value of n can betaken to be unity, so that Eqn. (2) takes the form

= 2dsin (3)

yielding the formulation of Braggs Law that is used throughout the analysis.

3.3 X-Ray Diffractometry (XRD)

X-Ray Diffractometry (XRD) involves the bombardment of a material with X rays ofa single wavelength. The wavelength is chosen so as to be on the order of the interatomicspacing of the atoms that make up the crystal lattice in order to yield data with tractableinformation. The apparatus operates on the principle of X-ray photon interactions with

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the crystal lattice of the material. X rays interact with the lattice atoms and the scatteredphotons are recorded by a counter situated at a certain angle with respect to the X-rayemitter. The data gathered is used to construct a representative history of the X ray-materialinteractions based upon the intensity of the counts called a diffraction pattern. The resolutionof the spectrum will be highest with the incident X rays being as monochromatic as possible.

A polychromatic X-ray beam would yield too many peaks, hiding any information about thecrystal structure, and eliminate any usefulness of the experiment. These diffraction patternsreveal characteristic peaks of higher intensity at certain values of the angle of reflection 2,as per Figure 3.3. Physically, the angle 2 is the angle measured between the undiffractedpath of the X ray to that of the diffracted X ray being collected at receiving slit. It thenfollows that, according to the setup, this angle is the angular value of 2 as described in thediscussion Braggs Law in section 3.2.

The apparatus used in the laboratory was an X-Ray Diffractometer (Figure 3.3)

Figure 3.3 - Schematic of an X-ray diffractometer [Ref. 3]

A powder sample is affixed to the flat plate in the center of the diffractometer circle. Xrays are generated via an X-ray tube (Figure 3.4). A vacuum chamber containing a filamentis heated by way of passing a current through the filament which then thermionically emitselectrons. Electrons are accelerated in the vacuum tube, and are targeted at a Molybdenumanode. The accelerated electrons contact the anode, and excite or knock off electrons fromthe Molybdenum lattice sites. When these electrons decay, they emit characteristic X rays.Most apparent are the K and the K emissions.

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Figure 3.4 - Diagram of the X-ray tube used to generate X rays in a diffractometry experiment [Ref. 3]

These X-ray photons enter the divergent slit and are incident on the target center. Uponinteracting with the material, certain diffracted X rays are collected at the receiver slit,situated at an angle 2 as identified by the graduated scale along the circumference of the

circle, and finally counted. The process is customarily automated, such that the X rays areallowed to interact with the material at incremental angles of 2 for a fixed amount of time,both of which are specifiable by the operator. The flat plate is situated on a rotatable axis.Also, the specimen and receiver slits are mechanically coupled so that for a rotation of thecounter by some angle 2x, the powder sample will rotate by an angle x. This adjustmentaccounts for potential geometrical complications and any nonuniformities of interactions thatcould incur from the X rays and the material. In this setup, the coupled joints allow forthe angle of reflection to be equal to the angle of incidence, in accordance with Braggs Law(Section 3.2).

The diffraction pattern is constructed based upon the counts received and the respectiveangle according to the diffractometer. Coupling the specimen and the receiving slit provides

for the angle of incidence to be equal to the angle of reflection. Only peaks that constructivelyinterfere, or equivalently stated, satisfy Braggs law (Eqn. 3) are counted. This idea isillustrated in Figure 3.5.

Figure 3.5 - Two X-ray beams are depicted as interacting with the atomic planes of a crystal lattice. In(a) the beam satisfies Braggs Law allowing for the outgoing beams to constructively interfere and be

counted by the detector, (b) The beam does not satisfy Braggs Law, the outgoing beams emerge out ofphase yielding destructive interference and are unable to be counted despite having the proper orientation

[adapted from Ref. 6].

Only X rays that undergo constructive interference, and those outgoing rays that are

oriented in the direction of the detector will be counted, as in Figure (3.5a). The depictionon the right (Figure (3.5b) illustrates the idea of destructive interference not being countedby the detector. Patterns are fashioned from the data collected, and upon analysis canbe used to discern information about the microstructure of the material.

Owing to its high penetrability, an X ray incident on a material sees a crystal latticeconsisting of regularly spaced atoms. Specifically, the X ray sees planes of atoms in the mi-crostructure of the material. The X-ray photons primarily interact with the bound electronsof these atoms. Diffraction results upon X-ray interaction with an atomic plane in a manner

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that the incident X ray becomes modulated such that a redistribution of energy, and henceintensity, of the X-ray wave occurs in a distinct manner by consequence of the interferencebetween X rays. The specimen used is a powder sample, a fine substrate composed of crys-tal lattice structures oriented randomly. As the the apparatus is rotated so that the X raysinteract with different areas of the material, the photons necessarily interact with the atoms

of the lattice at different angles, and also interact with different atomic planes. By rotatingthe components of the apparatus in this fashion, Braggs law will be periodically satisfiedwhenever atomic planes are in the proper orientation. This variation creates more peaks sothat a diffraction pattern is developed. Recalling that atoms in a lattice have essentially uni-form spacing, the X-ray wave packets, or photons, counted by the detector are geometricallycharacteristic of the structure of the material as the angle is varied.

The diffraction pattern is formed as a result of X-ray photons reaching the detector. TheX rays can only reach the detector if the outgoing waves are directed towards the counter, andalso only if the wave has not been significantly destructively interfered with. Characteristicis the interactions between the respective outgoing waves. This interference can either reduceor enhance the intensity peak recorded by way of destructive and constructive interferencerespectively. Since the X rays primarily interact with the electrons, a higher probability ofinteraction incurs in regions of higher electron densities. These regions of higher density occurin the vicinity of the atoms in the lattice. Thus, if a material with regularly spaced atomsin a crystal lattice is bombarded with X rays, the detector will record a characteristic X-raydiffraction pattern with sharp peaks of constructive interference that carries informationwith the same symmetry as the distribution of atoms in the crystal lattice of the material.From the diffraction pattern, given that the wavelength of the emitted K X ray is known,a number of interplanar spacings can be calculated for identified values of 2 correspondingto the observed K peaks via Braggs law. This interplanar spacing d is fixed, and can thenbe used to calculate the angles of where the respective K peaks should lie. This interplanar

spacing is presumed to be constant and a corresponding angle 2 can be found using the valueof d calculated from the preliminary analysis done using peaks granted the wavelength ofthe K emissions is used in Eqn. (3). In practice, the pattern will appear with significantbackground radiation superimposed upon the shark peaks. This background consists ofa Bremsstrahlung emission continuum resulting from the X-ray photons decelerating uponinteraction with the material. Different materials are composed of different phases and atomsgiving rise to a different diffraction pattern due to their different atomic spacings, electrondensities, and lattice structures. Extracting information from the diffraction patterns aboutthe symmetry of the distribution of atoms, identifying the lattice type, and discerning keyparameters about the lattice structure is the basis for X-ray Diffractometry.

3.4 Reciprocal Space

The X rays interact with electrons in the crystal lattice structure. Thus, the diffractionpattern yielded is a reflection of the symmetry of the electron density in the structure of thematerial. Given the symmetry of the lattice, it follows that the electron density is periodicalin 3-dimensional space. This density depends on the electrons per atom, the binding type,and also the lattice structure. The periodicity of the electron density function n(r) suggeststhat it can be expanded in the form of a Fourier series, such that

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n(r) =G

nG ei(Gr) (4)

whereG is a Reciprocal Lattice Vector (RLV), r is a lattice point position vector , andnG are the coefficients in the Fourier expansion.

The lattice is then defined in terms of reciprocal space. The electron density function ofEqn. (4) allows for an extractable definition of a reciprocal lattice. The reciprocal lattice can

be thought of as the set of all vectors G such that eiGr = 1. Alternatively, this corresponding

reciprocal space lattice can be envisaged as the set of imaginary points constructed so thatthe direction of an arbitrary vector coincides with the direction of a normal to the real spaceplanes and the magnitude of that vector is equal to the reciprocal of the real interplanarspacing d.

The vector G is formulated as a set of volume normalized primitive vectors of real space,such that G can be expressed in the general form of

G = h A + k B + l C, (5)

where A, B, and C are the previously described primitive vectors, now operating inreciprocal lattice. The magnitude of the reciprocal lattice vector |G| is customarily definedas the product of the reciprocal of the interplanar spacing d and 2, i.e. |G| = 2/d. Thisfactor of 2 changes the units of G from inverse length, to radians per unit length. Thisadjustment helps to make a direct comparison among G and the wave vector k = 2/ ofeither the incident, or outgoing X ray.

For a cubic structure, the interplanar spacing can be explicitly represented, and an ex-pression for |G| can be arrived at in the following fashion:

|G| = 2d

|G| = 2a

h2 + k2 + l2 (6)

where h, k, and l denote the integral multiples of unit vectors defining the lattice. An-other convenient quantity used to describe in analog with the reciprocal lattice vector G isthe structure factor SG, which describes the manner in which radiation scatters off of the

structure.

SG =j

fjei(krj) (7)

where j represents the jth atom, fj is each atoms form factor (a property of the atom type,and independent of its position in the lattice), rj is the position of the j

th atom in the lattice,

and k is the difference between the wave vectors before and after interaction. That is to say,8


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k = k k, such that k and k are the wave vectors of the incident and outgoing radiationrespectively. It can be shown that the condition for constructive interference demands thatthis vector |k| = |G| = 4 sin, which is identical to the quantity s, used in the change ofvariables of Section 3.6.2.

The above expression can be expressed more explicitly if a general reciprocal lattice vector

(Eqn. (5)) is considered, along with a general position vector rj. Under this representation,the dot product k rj in the exponential of Eqn. (7) can be expressed as

k rj = 2(hxj + kyj + lzj) (8)where a factor of 2 has been factored out of the notation, and the constants have been

consolidated.Using this explicit representation, the structure factor of Eqn. (7) can be revaluated as

SG = i fje2i(hxj+kyj+lzj) (9)

Different lattice types will hold different structure factors. It is then immediately recog-nizable that the structure factor can be used to determine the lattice type.

3.5 Crystal Structure

The atoms in material solids are arranged in a unique geometry according to its crystalstructure. A crystal structure is comprised of three dimensionally tesselated unit cells, orlattice structures. The unit cells are comprised of one or more atoms arranged in space,such that translational operators applied to the basis can accomplish the mathematicaltesselation. A unit cell holds a characteristic length known as the lattice parameter a. These

measurements differ depending on the type of lattice, and the lattices constituent atoms.In this experiment, three types of lattices were of relevance: Face-Centered Cubic, Body-Centered Cubic, and Hexagonal-Close Packed.

3.5.1 Face-Centered Cubic (FCC)

The Face-Centered Cubic lattice structure contains 4 atoms in one unit cell (Figure 3.6).An atom is located at each of the vertices of the cube, and one is placed in the center ofeach of the faces. These atoms are situated in the unit cell such that only 1/8 of an atom isactually contained within the unit cell at each of the vertices, and only 1/2 of the atom iscontained on each of the faces.

Figure 3.6 - A diagram of the face-centered cubic unit cell. [Ref. 6]9


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Mathematically, an origin can be placed at the center of the cube, r0 = 0, with atomsplaced at r1 = (a/2)(x + y), r2 = (a/2)(y + z), r3 = (a/2)(x + z), where a is the latticeparameter of the cell. Noting that the form factor fj is only dependent on the type of atom,it follows that for a monatomic lattice, all of the form factors fj are identical (fj = f forall j). Inputting this into the structure factor formulation of Eqn. (9), yields the following

result:

SG = fei

G0 + eiG(a/2)(x+y) + ei

G(a/2)(y+z) + eiG(a/2)(x+z)

= f

1 + (1)h+k + (1)k+l + (1)h+l

it follows from elementary number theory that any two even or odd integers will yield an

even sum, and any mixed parity will yield an odd number, giving way to the general result:

SG =

4f if h, k, l are all even or odd0 if h, k, l are of mixed parity

Thus, if the values of a set of indices h, k, l are known, whether or not these correspond toFCC can be ascertained from a simple crosscheck between the indices and the above result.

3.5.2 Body-Centered Cubic (BCC)

The body centered cubic has 2 atoms in the unit cell, one located at the unit cells center,and an atom is located at each of the vertices. The entirety of the atom is located at thecenter, and 1/8 of the atoms on the vertices are contained within the cell giving way to atotal of 2 atoms per unit cell in a BCC structure.

Figure 3.7 - A diagram of the body-centered cubic unit cell [Ref. 6]

Mathematically, if the origin is located at the center, r0 = 0, then this describes thelocation of the center atom, and r1 = (a/2)(x + y + z), where a is the lattice parameter of

the unit cell. A similar treatment, as per the FCC analysis, of the structure factor yieldsthe result:

SG =

2f if h + k + l is even0 if h + k + l is odd

The differences between the conditions for the integral multiples h, k, and l in regards tothe structure factor SG in both cubic structures allow for lattice structure to be characterizedbased upon whichever set of indices is consistent. That is the say, both cubic lattices have a

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constant lattice parameter a, which is related to the interplanar spacing d and the indices h,k, and l via Eqn. (9). Deducing the structure of the material by analysis of the diffractionpattern can be accomplished by selecting the appropriate values of h, k, ad l that allow forthe lattice parameter to remain constant. Upon finding the proper set of values of h, k, l,the structure can be discerned from a juxtaposition of the indices with each lattices criteria

for the structure factor.

3.5.3 Hexagonal Close-Packed (HCP)

The Hexagonal Close-Packed (HCP) unit cell contains 6 atoms situated as shown inFigure 3.8.

Figure 3.8 - A diagram of the hexagonal closely packed unit cell [Ref. 6]

The structure consists of six triangular prisms conjoined so as to form a hexagonal prism.The characteristic lengths associated with the HCP structure include its height c, and theside length of the basal triangles is the its lattice parameter a. The triangles that composethe bases of the hexagonal prism are equilateral of length a. Thus, all that need be known to

calculate the volume of the cell is the height c, and the side length a. Deriving an expressionfor the volume VHCP of the HCP unit cell is elementary, and thusly not shown in this report;however, the equation that results is simply:

VHCP = 6

3

4a2c (10)

3.6 Diffraction Pattern Analysis

The general methods described above are now explicitly represented.

3.6.1 Identifying K and K Peaks

It is apparent that whether the analysis of the pattern is done with exclusively the K orthe K peaks is completely arbitrary (as both will yield the same information of the crystalstructure). However, an observation of the diffraction pattern reveals that the K peaksare much more visible due to their higher amplitudes. Thus, it is more reliable to analyzethe pattern based upon these peaks rather than the lower amplitude K peaks (which canoften be hidden amidst the background noise of the spectrum). It is important to analyze

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either only the , or only the peaks. Examining both would be redundant, and analyzinga mix of the two would give untrue information about the symmetry of the lattice. It isthen necessary to distinguish these K peaks from the K peaks on the spectrum from oneanother for a consistent analysis of only the K peaks. This is done in a two step process.

As a first step, the K peaks can be identified by inspection of the diffraction pattern.

The angle 2 is denoted, and the interplanar spacing d is calculated via Braggs Law (Eqn.(4)):

d =K

2sin(11)

Where K is the characteristic wavelength the K The interplanar spacing is a constant,and should remain the same whether the or peaks are used. Thus, after arriving at avalue for d, the angle 2K can be calculated for the emission, ifK is used in Eqn. (4):

2K = 2sin1

K2d

(12)

The second step involves a cross-checking of peaks. Since the identification of the Kpeaks was done by inspection, it is necessary to see if any of the K peaks previouslyidentified are in fact K peaks. Should this be the case for any peak, it should beremoved from further analysis to ensure consistent calculations throughout. Once the Kpeaks have been assigned, the K peaks can then be discarded as the remainder of theanalysis will be conducted on the K peaks.

3.6.2 Change of Variables and Curve Fitting

In order to ease the analysis, a change of variables is introduced. The quantity s is usedinstead of the angular dependence, where

s = |k| = (4/)sin (13)

and has units of inverse length. The quantity s is defined to operate in reciprocal space.As noted in Section 3.4, the condition for constructive interference yields an equivalencebetween a corresponding s value to the reciprocal lattice vector G. Thus, given an angularvalue 2 that corresponds to the centroid of a K1 peak on the diffraction pattern, Eqn. (11)

can be rewritten in terms of the quantity s, such that

d =2

s(14)

In order to locate this centroid, a function I(s) is fitted to each peak. This function isplotted atop of the data points, and a centroid value can be interpolated from the function.The function I(s) used for the fitting is of the form:

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I(s) = As + B + c(2fs1,s(s) + fs2,s(s)) (15)

where

fsi,s(s) = exp

1

2

s si

s

2

and A, B, c, si, s are constants. The former part of Eqn. (5) is a linear approximationof the background noise, c is a constant that modulates the height Gaussian peak, s isthe Full Width at Half Maximum (FWHM) of the peak, si is the moment center of the ithpeak, where i can either be K1 or K2. The factor of two is introduced as a multiplicativefactor appended to the Gaussian approximation of the peak corresponding to the K1 X-rayinteraction due to its tendency to be approximately twice the amplitude of the K2 peak.Another reason for the change of variables is that the quantities s1 and s2 have a fixed

relationship, and an expression can therefore be formulated to constrain the fit even further.By invoking Braggs law and recognizing that the interplanar spacing d is the same for agiven pair of K1 and K2 peaks, relation between the two s values can be arrived at (Ref.2):

s2 = s1(K1/K2 ) (16)

The above relation reduces the number of fitting parameters in the function I(s) of Eqn.(15).

3.7 Depth of MaterialIn order to find the estimated depth of the material being probed, it is necessary to

discern how penetrating the X ray is with respect to the interacting medium. The surface ofthe penny is considered for this calculation, and is presumed to be made of pure Cu-29. TheX rays are taken to be targeted at the material as a collimated beam with a finite diameter.A one-dimensional treatment of the scenario is both a good approximation, and does notcause a loss of generality in the result. The intensity I of the beam through the materialcan be modeled as a differential equation to find an expression for the mass attenuationcoefficient . Specifically,

dIdx

= I

where x is a spatial coordinate along the path of the X-ray beam. The negative signfollows from the derivative on the LHS of the equation dI

dx< 0. The equation is routinely

solved to yield:

I(x) = I0ex (17)

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where I0 is the intensity before interaction with the material. Thus, the attenuationwith increasing distance x along the path of the X-ray beam is explicitly represented inEqn. (17). The above equation provides the mass attenuation coefficient , which providesall the information needed to estimate the depth of the material. Since the distribution isexponential, the mean of the distribution is simply the reciprocal of the mass attenuation

coefficient, which is physically recognized as the mean free path of an X ray in the material.Mass attenuation coefficients are tabulated in literature with respect to incident photon

energy, and the medium of interaction. The energy of the incident X-ray photon can befound by the usual idea of quantized energies:

E = h

E =hc

(18)

where and are the linear frequency and wavelength of the photon respectively. The

incident energy can thusly be extracted from this formulation, and the mass attenuationcoefficient can thereafter be found from literature.After acquiring the mass attenuation coefficient, there is still a matter to decipher how

the incident depth of the material varies as a function of the incident angle. Figure 3.9provides an illustration of the process.

Figure 3.9 - An X ray is incident on a copper medium at an arbitrary angle . The average distance theX ray is expected to travel is equal to the mean free path = 1/. The depth D is a function of angle with a

magnitude ofD = (1/)sin()

If an X-ray photon of a given energy is incident on a material at an arbitrary angle, it

can be expected to travel approximately its mean free path (i.e. 1/). Thus, the relationshipbetween the depth D and the angle is quickly deduced as simply:

d() = (1/)sin()

As per the laboratory instructions, in order to plot the depth D as a function of 2, allthat need be done is to multiply the argument by a form of one (i.e. 2/2):

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Nb = (1 )NZn + NCu = Nb NZn

NCu

NZn

(22)

where Nb, NZn , and NCu are the number densities of the bulk material, Zinc, and Copperrespectively.

3.10 Error Analysis

The error attributed to the experiment came only from the value of the centroid s.This error, s was indicated among the fitting parameters outputted by Kaleidagraph wheneach peak was analyzed. Subsequent error was propagated in only the calculation of theinterplanar spacing d, where

d =2

s

A simple quadrature can be performed to account for the error in d, d, such that

dd

2=

22

2+

ss

2(23)

Noting that the variance of a constant is zero, the factor of 2 can be removed from Eqn.(24) to yield the following:

dd

2=

ss

2

dd

=

ss

d = ds

s (24)

where s is supplied by Kaleidagraph. These errors are listed among the fitting parame-ters in the Appendix. The ratio d/s is approximately equal to unity in the majority of thetabulated values, thus, this factor does not change the error significantly. The errors of thequantities s and d are both small.

4 Procedure

Material properties of a penny were investigated by performing XRD analysis on thepennys surface and bulk material. A powder specimen was placed on the fixed plate inthe X-ray diffractometer as per Figure 3.3, and an automated diffraction experiment wasconducted.

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The penny surface examined angular values of 2[12, 55] degrees in increments of 0.05degrees. Diffraction was analyzed at each angle for 10 seconds, yielding a total experimenttime of 8600 s, or approximately 2.38 hours.

The penny was then sanded down to the core in order to probe the bulk material, andplaced in the apparatus. Values of 2[10, 60] degrees were sampled in 0.05 degrees increments

at a sampling time of 30 seconds per increment for a total duration of experiment at 30000 s,or approximately 8.33 hours. The above listed parameters for the experiment are summarizedin Table 4.1, and the detection circuitry involved is included in the generic block diagram ofFigure 4.1.

Specimen Initial Angle Final Angle Step Size Time/Sample Total DurationCenter 10 60 0.05 30 2.38Surface 12 55 0.05 30 8.33

Table 4.1 - Experimental parameters for the XRD of the surface and center of the penny are summarized.

Figure 4.1 - Block diagram of signal chain for pulse counting of X rays.

Subsequent analysis of the diffraction patterns was conducted, in accordance with theTheory section.

5 Results and Analysis

5.1 Surface of Penny

5.1.1 Diffraction Pattern Analysis

The diffraction experiment produced a data set that recorded the intensity at respectivevalues of the angle 2. The raw data was plotted as intensity versus the angle 2 (attached

in the Appendix), and also with respect to the quantity s. The K peaks were isolated fromthe K peaks for later analysis by the procedure prescribed in Section 3.6.1. Kaleidagraphwas then used to fit the K peaks to function I(s) as per Eqn. (15) in Section 3.6.2. Tworepresentative plots along with the fitted function is shown in Figure 5.1.

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Figure 5.1 - Representative plots with corresponding fits to the first two peaks in the X-ray diffractionpattern of the surface of the penny. (a) corresponds to a centroid value of s = 2.993 A1, and (b)

corresponds to s = 3.4593 A1

The fitting parameters for all peaks fitted in Kaleidagraph can be found in the Appendix.The centroid values were found from the fitted function of each of the peaks, and a corre-sponding interplanar spacing d was calculated using Eqn. (14). These parameters are given

in Table 5.1. From these interplanar spacings, the lattice type was deduced from two setsof given candidate values of the indices h,k,l corresponding to either FCC or BCC lattices.Recognizing that the reciprocal lattice vector |G| s = (4/)sin for the case of construc-tive interference, the use of the indices {h,k,l} can be used in accordance with Eqn. (6) toyield the general result for a cubic lattice:

a = d

h2 + k2 + l2 (25)

The above equation was then used to find the corresponding lattice parameter a for eachpeak given the candidate values {h,k,l}. Upon subsequent calculations, the results indicatedan approximately constant value of the lattice parameter for the candidate values of

{h,k,l

}for the FCC lattice, indicating that the surface was indeed an FCC structure. A tablesummarizing calculated values of the interplanar spacing, lattice parameter is shown in Table5.1 along with their Miller indexing assignments. The comparison to the values found forthe corresponding BCC values for {h,k,l} are attached in the Appendix. Only the first fivecandidate indices of {h,k,l} of Table 5.1 were given; however, using the criteria prescribedthe Theory Section 3.4 relating to the structure factor, the remainder of the {h,k,l} valueswere able to be deduced based whether the determined indices yielded a constant latticeparameter. All calculated values are displayed in Table 5.1.

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K peak centroid (A1) Interplanar spacing d (A) Lattice parameter a (A) (hkl)

2.9923 2.0998 3.6367 1113.4593 1.8163 3.6326 2004.9053 1.2809 3.6229 2205.7538 1.0920 3.6218 311

6.0114 1.0452 3.6207 2226.9406 0.9053 3.6212 4007.7649 0.8092 3.6189 3317.8058 0.8049 3.5990 420

Table 5.1 - K peaks identified by their centroid s, and the corresponding calculated values of theinterplanar spacings, lattice parameter and Miller index assignments. The values of {h,k,l} imply an FCC

crystal structure for the surface of the penny.

With the peaks assigned, the diffraction pattern was labeled, and is shown in Figure 5.2.

Figure 5.2 - Labelled diffraction pattern for the surface of the penny plotted with respect to s.

All of the K peaks were identifiable. Approximately half of the calculated K peakswere not resolvable above the background noise of the spectrum. This information is givenin Table 9.1.

From the lattice parameters of Table 5.2, an arithmetic mean of the lattice parameter wascomputed to arrive at an average lattice parameter a. A unit cell volume was computed, andthe number and mass densities were calculated as per Eqns. (20)-(22), using the atomic massA = 63.546 g mol1 [Ref. 5]. The results are given in Table 5.2, along with a comparisonto literature values for pure Cu.

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Average lattice parameter a Number density N (cm3) Density (g cmSurface 3.622 8.419 1022 8.884

Pure Cu 3.610 8.502 1022 8.900Percent Difference 0.3269 0.9762 0.1798

Table 5.2 - Experimental values for the lattice parameter, and the number and mass densities arecompared to literature values at STP. [Ref. x, lennentech, webelements, nrl]

All the experimental values all hold less than 1% error in regards to the literature values.These differences are minute, and could be a reflection on an inherent innaccuracy of theX-ray Diffractometer; however, more likely is just the presence of defects within the material.Since the coin used in lab has necessarily aged and been weathered, also due to the coin beingmade not entirely of pure copper reflects the possibility of vacancies, voids, susbtitutionalimpurities, etc. within the material that could alter the densities. The defects can createlocalized regions of stress giving rise to a small change in the lattice parameter.

5.1.2 Depth of Penetration

The depth of penetration was plotted as per Theory Section 3.7. The CRC Handbook ofChemistry and Physics was consulted for the mass attenuation coefficient values (given interms of mass attenuation coefficient per density of incident material). The energy calcula-tion using Eqn. (9) yielded an incident photon energy of 17.516 keV. The values listed inthe CRC Handbook of Chemistry and Physics did not list a value for this specific energy.Four values were taken in the vicinity of the this energy, and fitted to a power function.The function was used to find the mass attenuation coefficient corresponding to the incidentX-ray photon energy. The fitted plot is given in Figure 5.3.

Figure 5.3 - Mass attenuation coefficient per density of incident material is plotted versus photon energy,and fit to a function in order to interpolate for the proper coefficient corresponding to an energy of 17.516

keV.

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Using the density of copper (Ref. x), a mass attenuation coefficient was found, and acorresponding mean free path to be 2.071 103 cm. Using this value, a plot was made ofthe depth of penetration into the material versus the angle 2.

Figure 5.4 - Mass attenuation coefficient per density of incident material is plotted versus photon energy,and fit to a function in order to interpolate for the proper coefficient corresponding to an energy of 17.516

keV.

The plot intuitively exhibits sinusoidal character. The lesser degree to which this ispresent is due to the limited range of angles in the plot. The depth of penetration is shownto increase with increasing angle, and can be expected to reach a maximum when the photonis normally indicident on a material.

5.2 Bulk of Penny

The penny was sanded down so as to expose the bulk material in the X-ray diffratometryexperiment. The procedure was essentially the same as for the surface of the penny. TheK peaks were identified, and fitted to an equation of the form of Eqn. (x). This fit wasused to determine the centroid, which was then used to calculate interplanar spacings. Tworepresenatitive peaks are shown in Figure 5.5.

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Figure 5.5 - The first two peaks in the X-ray diffraction pattern of the bulk material along with theircorresponding fitted functions are shown.

Fitting parameters are attached in the Appendix. Because of the more complex structure,a direct methodology for determining the indices {h,k,l} was not utilized. Instead, a dataset consisting of labeled peaks with corresponding interplanar spacings dlit from literaturewas compared to the experimental calculations for the interplanar spacings dexp. In this

fashion, the indices were assigned to the experimental peaks. All peaks that were unableto be assigned were presumed to be due to substitutional impurities in the form of Copper.The analysis is summated in Table 5.3.

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Centroid s (A1) dexp (A) dlit (A) (hkl)

2.5474 2.467 2.473 0022.7171 2.313 2.307 1003.0027 2.093 3.7283 1.685 1.687 102

4.204 1.347 1.332 1105.0384 1.247 1.236 0045.3585 1.173 1.173 1125.4421 1.154 1.153 2005.5897 1.124 5.6242 1.117 5.7806 1.087 1.090 1046.0172 1.0442 1.045 2026.6517 0.9446 6.9408 0.9053 0.906 114

7.3112 0.8594 0.872 2108.1545 0.7705

Table 5.3 - Experimental values of the interplanar spacing were compared with pre-indexed K peakswith corresponding interplanar spacings. The Miller indexing was done for all peaks that matched the

data, all other peaks are taken to be due to substitutional impurities.

A plot of the Intensity as a function of the quantity s was labeled, and shown in Figure5.6.

Figure 5.6 - A plot is shown of the X-ray diffraction pattern of the bulk material in a penny with Millerindexing assignments.

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It is evident that the structure in the pennys bulk material is different than that of pureZn. To quantify this, the Cu concentration was assessed under the assumption that theexperimental atomic volume is a weighted average of the atomic volumes of Cu and Zn aspure elements. For the experimental data, the volume of the HCP lattice is computed viaEqn. (10). The height c is identified as 2d(002)exp, and the lattice parameter can be extracted

from the length d(110)exp, which is realized as the altitude of each of the basal triangles inthe structure. Simple trigonometric analysis yields the lattice parameter a = 2dtan6 . Thus,inputting these results into Eqn. (x), the volume of the HCP lattice was found. Becausethe actual density of the bulk material was not known, an approximation was enforced suchthat the number densities of the materials were directly proportional to the atomic volumes.Furthermore, it is noted that Copper is found as a pure element in a cubic lattice, andthus the approximation is only valid if the number density does not differ signifantly. Theexperiment indicated a bulk number density of 1.9347 1023 atoms/cm3. The number densityof copper was found from the previous section to be 8.419 1022 atoms/cm3, and literaturevalues indicated that the number density of Zn is 9.16 1022 atoms/cm3. Using these values,in accordance with Eqn. (x), it was found that there is approximately 3.729 atom percentcopper in the Bulk material.

6 Error Propagation

The standard deviation in the centroid s is listed in Table 9.4, and Table 9.5. Theerror the calculation of the interplanar spacing d was found via Eqn. (24), and yielded thefollowing standard deviations.

K peak centroid (A1) Interplanar spacing d (A) Standard deviation in d (A) x 103

2.9923 2.0998 39.283.4593 1.8163 37.454.9053 1.2809 6.2105.7538 1.0920 1.2066.0114 1.0452 1.9526.9406 0.9053 86.827.7649 0.8092 9.206

Table 6.1 - Error values for the calculations of the surface values for the interplanar spacing d.

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Centroid s (A1) dexp (A) Standard deviation in dexp (A) x 104

2.5474 2.467 3.934e-42.7171 2.313 5.746e-43.0027 2.093 2.238e-43.7283 1.685 6.317e-5

4.204 1.347 1.029e-45.3585 1.173 4.549e-55.4421 1.154 6.613e-55.5897 1.124 4.836e-55.6242 1.117 1.249e-45.7806 1.087 6.445e-56.0172 1.0442 3.099e-46.6517 0.9446 4.432e-56.9408 0.9053 5.800e-57.3112 0.8594 4.461e-5

8.1545 0.7705 6.136e-5Table 6.2 - Error values for the calculations of the bulk values for the interplanar spacing d.

In both the bulk material and the surface, the error was found to be small.

7 Summary and Conclusions

X-ray Diffraction was used to probe the crystal structure of a penny. The analysisallowed for an identification of the lattice type, assignment of Miller indices to the peaks inthe diffraction patternf, and calculation of the lattice parameter, and both the number and

mass densities. An analysis of the surface revealed a Copper Face-Centered Cubic latticestructure with an average lattice parameter of 3.622 A, a number density of 8.419 1022, anda mass density of 8.884 g cm3. All of these values were within 1% of literature values.

The bulk material of the penny was also analyzed. The K peaks in the diffractionpattern were assigned according to the Miller indexing scheme by matching correspondinginterplanar spacings with given values of {h,k,l}. Five of the peaks were unidentifiable inthe lattice, and this taken to be due to the presence of substitutional impurities in the formof Cu. The number density of the bulk material was found to be 1.9347 1022 atoms/cm3,and the concentration of Copper impurity in the material was calculated to be 3.729 atompercent.

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8 References

1. Atzmon, M. X-Ray Diffraction Lecture Notes, January 20072. Atzmon, M. Transparencies, January 20073. CRC. CRC Handbook of Chemistry and Physics, 87th Edition. 2003.

4. Hirose, Akira. Lonngren, Karl. Wave Phenomena. Kreiger Publishing Company.Malabar, Florida. 2003.

5. USGS Coastal and Marine Geology Program, http://pubs.usgs.gov/of/2001/of01-041/htmldocs/xrpd.htm, Accessed - February 16, 2007.

6. Wikipedia, X-ray Crystallography, http://en.wikipedia.org/wiki/X-raycrystallography,Accessed - February 16, 2007.

7. Web Elements. Zinc,Copper, http://www.webelements.com, Accessed - February 16,2007.

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9 Appendix

9.1 Figures

Figure 9.1 - A plot of the raw data for the surface of the penny acquired from the XRD experiment. Theintensity is plotted versus the angle 2.

Figure 9.1 - A plot of the raw data for the bulk material of the penny acquired from the XRDexperiment. The intensity is plotted versus the angle 2.

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Centroid si (A1) A B c s 2 R

2.5474 4.0619 104 1672.8 572.26 14403 .0078031 3.7347107 .990692.7171 6.7494 104 -5185.5 18892 4148.2 .0070848 1.3746107 0.960843.0027 3.211 104 -1117.2 8181 25960 .0068209 1.668108 0.985053.7283

1.3978

104 -959.09 6977.7 6357.1 0.0060571 3.972

106 0.99298

4.204 3.211 104 -1117.2 8181 25960 0.0068209 1.668108 0.985055.3585 2.0783 104 863.16 -1329.4 5259 0.0055909 3.0207106 0.990575.4421 3.1188 104 -1222.3 9794.9 463.57 0.0048169 11650 0.991695.5897 2.4051 104 -508.79 6139.3 2148.3 0.0051995 66304 0.997785.6242 6.2899 104 -20815 1.2032105 1074.6 0.0042943 1.071105 0.987225.7806 3.4272 104 -1599.9 12089 641.62 0.010186 15839 0.996086.0172 1.7861 103 -0.3278 2740.5 97.941 0.011606 1.938106 0.776246.6517 3.1211 104 387.97 50.867 985.31 0.0062698 1.7708105 0.98576.9408 4.4472 104 -588.47 6741.9 1204.2 0.012487 1.832105 0.992957.3112 3.7953 104 -1458.6 13387 1599.8 0.0059565 5.4451105 0.981498.1545 6.4937 10

4

365.85 -517.82 623.47 0.0085668 1.4725105

0.97295

Table 9.4 - Fitting parameters with associated error in the interplanar spacing d for the bulk material ofthe penny.

Centroid si (A1) A B c s 2 R

2.9923 2.7561 104 389.47 73.077 6823.2 0.015172 7.0319106 0.996743.4593 1.9662 104 -365.41 2443.1 3706.9 0.014853 5.9032105 0.0.99894.6914 1.3825 103 269.98 -657.73 149.46 0.011866 20873 0.958094.9053 3.6584 104 14.86 578.86 770.48 0.012472 1.0239105 0.994905.7538 2.2895 104 177.49 -332.43 1358.4 0.014487 5.6459105 0.995716.0114 3.3934 10

4

-135.62 1461.3 551.51 0.011659 38809 0.994366.9406 1.1324 104 -138.51 1531 207.34 0.015678 34138 0.97087.5685 8.9627 104 221.62 -1103.7 201.12 0.014212 21702 0.977257.7649 9.5937 104 -103.57 1357.5 178.4 0.015274 21996 0.97645

Table 9.5 - Fitting parameters with associated error in the interplanar spacing d for the surface materialof the penny.

X-Ray Diffraction - Final Version

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