Diffraction Theory Introduction

Zhao Wang

January 31, 2013

Part I

Crystallography

Crystallography is the subject which studies the geometrical arrangement of

atoms in the crystal. Before crystallography was incorporated in the field of

solid state physics, it has been intensively studied through various means for

hundreds of years. However, Most of the studies had focused on the macro-

scopic vision of crystal until the famous physicists W.L Bragg discovered

Braggs Law, studying the X-ray diffracted pattern to reveal microscopic or

atomic structure of the crystal. Therefore, crystallography has been brought

into a new era and often was called X-ray crystallography. X-ray diffraction

technique have made big contribution to a variety of scientific research such

as physics,chemistry, material science, biology, mineralogy and so on. Newer

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technologies such as electron and neutron diffraction is also based on X-ray

diffraction theory. Therefore, in this report, we are going to first review the

basic knowledge of crystallography and explain why X-ray is a unparalleled

way to study the structure of crystal. Then, modern techniques LEED and

RHEED will be briefly discussed.

1 Bravais Lattice

Bravais lattice is an important concept in crystalline solid. It forms a ge-

ometrical picture in which a identical unit is so constructed that it repeats

itself into infinite 3-D space. This identical unit could be an single atom or

a cluster of atoms. This unit is defined as basis. Two definitions of Bravais

lattice could be made, one of which is more related to its geometry and the

other of which is more explicit in mathematics. They are stated as [1]:

1. A Bravais lattice is constructed in a way that the geometrical arrange-

ment and symmetry of atoms are exactly identical no matter from

which lattice point is viewed.

2. All the points in a (3-D)Bravais lattice can be reached by the vector

translational operation expressed as ~R = u1a1 + u2

a2 + u3a3 , provided

that u1, u2, u3 are integers.

2

Figure 1: Primitive vectors in a 2-D Bravais lattice

a1 ,a2 ,

a3 are called primitive vectors in lattice and are not all in the same

plane. A primitive cell that is constructed by primitive vectors will repeat

itself in an infinite space neither overlapping with neighbor cells nor leaving

a void between neighbor cells. Therefore, a primitive cell is a minimum-

volume cell and serves as the smallest building block. Equivalently speaking,

one can say that a primitive cell contains only one lattice point. It is worth

mentioning that the choices of primitive vectors is not unique, some choices

are preferred due to the simpler symmetry. See an example [3] in Fig.1 where

we have many choices of primitive vectors for a 2-D Bravais lattice excepted

the vectors in the red circle . The Bravais lattice point can be invariant with

respect to other points by translation or point symmetrical operation such

as rotations about certain axis or reflection about a certain plane. Some

rotations by 2pi, 22pi, 2

3pi, 2

4pi, 2

6pi are noted as one-, two-, three-, four- and six-

fold rotation. One would ask if there is other rotational symmetry found

in natural crystal such as fivefold rotation[3]. The answer is no. Imagine

we have a pentagon shape as a primitive cell in 2-D space. It is impossible

3

Figure 2: An example of non-existed fivefold symmetry of crystal lattice

for us to construct a Bravais lattice with such a primitive cell because these

primitive cells can not fill in all the space, as shown in Fig.2 [3]. However,

not only can the primitive cell repeat itself in space without leaving a void

or overlapping but also some non-primitive cells can do so as well although

these non-primitive cells are not minimum-volume cells. These non-primitive

cell are often preferred due to advantage in point symmetry (rotation and

reflection) and they are called as unit cell. There are only 14 lattices based on

the distinctive unit cell in 3-D for the point symmetry group, shown in Fig.3

Keep in mind that only simple cubic is primitive cell while the other unit cells

are non-primitive cells, such as face centered cubic (fcc) and volume-centered

or body-centered cubic (bcc). However, the primitive cell can be found by

vector translation . Take a bcc unit cell as an example[3] shown in Fig.4 .The

primitive vectors can be contructed through vector operation

a = 12a(x +y z ),

b = 1

2a(x +y +z ),c = 1

2a(x y +z )

with a being the lattice constant in x, y, z direcctions in a rectangular coor-

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Figure 3: 14 distinctive lattices in three dimensional

Figure 4: Primitive vectors of the volume-centered cubic cell being con-structed

5

dinate system . For the other lattices, the primitive cells can be constructed

in a similar fashion.

2 Miller Indices

A notation system should be first established in order to study crystal struc-

ture more functionally. In a crystal structure, a specific plane will intercept

with three axes at position u1, u2, u3. Thus the coordinates of intercepts

(u1u2u3) can be used to specify a plane. However, it is more often interesting

to know the orientation of a plane for the structural analysis. Consequently,

Miller indices has been used. To form a Miller indices, one needs to do the

following steps:[3]

Find the intercepts (u1u2u3)

Take the reciprocals of (u1u2u3) and then it gives (1u1

1u2

1u3)

Let (hjk) = ( 1u1

1u2

1u3), find then convert (hjk) to three smallest integers

which have the same ratio. Thus, Miller indices is founded.

For example, in a cubic lattice, if a plane intercepts with three axes at (213),

the reciprocal values would be 12,1,1

3and then multiply these three values

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by 6. The smallest integers thus can be obtained as (hjk) = (362). It

is noted that Miller indices can indicate a family of parallel planes or a

single plane. For instance, if two parallel planes intercept with three axes

at (213) and (426), respectively. The Miller indices of both plane is (326).

Besides, (200) plane is parallel to (100) plane but cutting x axis at u12. If

the intercept position is in the negative side of origin, a little bar above the

number indicates negative like (hjk). If under certain condition of symmetry

a set of planes are equivalent, then {hkj}are used. For instance, in cubic

lattice, (100), (010), (001), (100), (010), (001) planes are all identical so that

the set of all these planes are denoted as {100}. Since we are generally

interested in a set of parallel planes, the inter-planar spacing for different sets

of parallel planes is a very important parameter. It can be indeed obtained

from the analysis of geometry which we will give full detail in this report.

In general, the inter-planar spacing formula of simple lattices is given in the

Table.1. a, b, c are the lattice constants on x, y, z axes which has occurred in

Fig.2 as a1, a2, a3 and h, j, k are Miller indices.

3 Reciprocal Lattice

At beginning, one may think why we would even bother defining a reciprocal

space if the crystal structure is perfectly defined in space domain. In compar-

7

Table 1: Inter-planar spacing of simple lattices

ison, electrical engineer often look at problem in frequency domain besides

time domain. It is natural to analyze problem in its Fourier domain as the

reciprocal lattice plays an important role in study of periodic structures such

as Bravais lattice. In fact, it also leads us to study the theory of diffraction

in crystal on which we focus.

A plane wave can form in the Bravais lattice, defined as eiK r . Generally

speaking, the wave vectorK could be any value. Most of plane waves which

have random wave vectors would destructively interfere among each other

and vanish eventually but some plane waves will constructively interfere as

long as its wave vector fulfill the requirement of periodicity in Bravais lattice,

defined as:[1]

eiK (r +

R ) = ei

K r (1)

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R is the periodicity of Bravais lattice, defined as a primitive vector in the

first section. we can say that such a wave vectorK belongs to the reciprocal

lattice of such a Bravais lattice. Factoring out the common term eiK r , we

can get [1]

eiK R = 1 (2)

Therefore, for each Bravais lattice system indicated byR , there is a cor-

respondent reciprocal lattice system indicated byK . Let

b1 ,

b2 ,

b3 be the

primitive vectors in the reciprocal lattice and a1 ,a2 ,

a3 be the primitive vec-

tors in the corresponding Bravais lattice. We getb1 ,

b2 ,

b3 as

b1 = 2pi

a2a3a1 (

a2a3 )

b2 = 2pi

a3a1a1 (

a2a3 )

b3 = 2pi

a1a2a1 (a2a3 )

(3)

As a result, the reciprocal lattice translational vectorK = v1

b +v2

b2 +v3

b3 .

v1, v2, v3 are nothing but Miller indices h, j, k. Therefore, it can be rewritten

asK = h

b + j

b2 + k

b3 .

4 Braggs Law

The crystal structure in atomic level is studied through the diffraction of

X-ray, electrons and neutrons. W.L. Bragg first presented X-ray diffraction

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Figure 5: Bragg equation derivation

theory which gives the constructive diffraction condition for a beam of X-ray

diffracted by crystal, show in Fig.5 . When X-ray beam is incident on a set of

parallel lattice planes which has an interval d, it will be elastically scattered

by the electron of the atoms. The zero order scattering beam that is the also

the direct reflection with respect to the planes will take up most of the power

and the higher order parts are not considered in this case. Its diffracted angle

is thus equal to incident angle. Therefore, the optical path difference (OPD)

of diffracted beams can be calculated. In Fig.5, we can tell that the OPD

between diffracted beam 1 and 2 is ABC = 2dsin and the OPD between

diffracted beam 1 and 3 is ABC = 4dsin = 2ABC. We can imagine

that the OPD between diffracted beam 1 and n+1 is n ABC. In order

to constructively interference between diffracted beam 1 and all the other

diffracted beams. the OPD between diffracted beam 1 and 2 ABC must be

multiple numbers of wavelength and consequently the OPD between other

beams are also multiple numbers of wavelength . The equation is given as[1]

2dsin = n (4)

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The radiation of diffracted beam is a collection of reflection from all the par-

allel planes. Each plane reflects about 103 to 105 of the incident radiation

(give a citation). Despite of straightforwardness of this expression, X-ray

diffraction has become the most powerful tool for studying crystal structure.

Since the typical unit cell parameter (it is d in this case ) of most crystal is

on the order of several angstroms (A). In the above equation , the maximum

value of sin is 1 and n is integer. It is apparent that in order to make

diffraction occur, must be on the order of around 10 A or below. Conse-

quently, the X-ray is chosen or other particle source like electron beam or

neutron beam as their wavelengths are even smaller. Besides, this is also the

reason that we can not observe the diffraction from visible light. However,

one must notice that the X-ray diffraction does not give the information of

basis such as which kinds of atom a basis has.

5 Laue Condition and Ewald Sphere

Braggs law basically states a condition under which the diffracted beams by

a set of parallel planes will constructively interfere. A set of parallel planes

needs to be singled out so as to observe the diffracted radiation. Max Von

Laues method differs from Braggs but eventually approaches the same goal

which is to find the condition for constructive interference. Instead of form-

ing parallel planes, the crystal is constructed by identical microscopic objects

such as atoms and irons attached to the Bravais lattice points . Each micro-

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Figure 6: Illustration model of Laue condition

scopic object can been seen as scatterer. The scattering by these microscopic

objects is illustrated in Fig.6. An incoming parallel beam with wave vector

k in is incident on the objects and changes direction as outgoing parallel

beam with wave vectork out. The relation |

k in| = |

k out| =

2piholds due to

elastically scattering characteristic. Constructive interference will occur only

if[1]

OPD = dcos () + dcos () = m (5)

Convert it into the phase difference between incoming beam and outgoing

beam so that we can obtain

d2pi

cos () + d

2pi

cos () = 2pim (6)

From Fig.6, we can easily impose a vector operation a b = |a||b|cos on

Eq.8. Moreover, notice that the angle betweend and

k in is and the angle

betweend and

k out is 180

o so that we can get

d

k in

d

k out = 2pim (7)

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d is translation vector between two points in Bravais lattice. To consider all

the scattering points in space,d could be replaced by Bravais lattice vector

R . Therefore, the above equation can be rearranged to

R (

k in

k out) = 2pim (8)

This equation can be re-written in an equivalent form as

eiR (

k in

k out) = 1 (9)

Comparing the Eq.11 with Eq.4 which we have obtained the previous chap-

ter,we get

(k in

k out) =

K (10)

Therefore, the Laue condition states that constructive interference only occur

if (k in

k out) is a vector in the reciprocal plane.

We have already come up with the truth that Braggs Law and Laue condi-

tion must be equivalent criterions for forming a constructive interference in

the lattice. To prove such an equivalency[3], we can make some redundant

operations on the Laue condition (k in

k out) =

K . Due to elastically

scattering, we can obtain

|k in| = |

k out| = |

k in

K | (11)

13

Squaring both sides, we get

|k in|

2 = |k in| 2

k in

K + |

K |2 (12)

Rearrange the equation and we can get

k in

K =

1

2|K|2 (13)

We assume we are dealing with cubic lattice and the similar analysis can

be applied to the other cubic systems but with more complicated mathe-

matic deduction (citation here). The reciprocal lattice of a cubic lattice is

still a cubic lattice. WithK = h

b1 + j

b2 + k

b3 , the spacing of a certain

set of parallel planes dhjk =2pi|K|

andk in

K=2pi

|K|cos. Seeing the geo-

metrical arrangement of Bragg condition and Laue condition in Fig.5 and

Fig.6 respectively, we shall notice that is the angle between incident beam

and reciprocal vectord which is normal to crystalline planes while is the

angle between incident beam and crystalline planes. Thus, = pi2 and

cos = sin. Plug all terms into Eq. 13, we obtain

2pi

2pi

dhjksin =

1

2

(2pi)2

d2hjk(14)

dhjk may have contain a common factor n so that dhjk =dnfor a cubic lattice

. Thus rearranging the above function ,we can obtain

2dsin = n (15)

14

Figure 7: Ewald construction

Therefore, the equivalency is proved in an oversimplified but applicable way.

6 Ewald Construction

Ewald construction is a simple geometric illustration of Laue condition to

understand the occurrence of the diffraction. It also gives the guideline for a

practical X-ray diffraction experiment. The general picture is given by Fig.7

[1] . The incident beam and outgoing beam wave vectors arek and

k

,

respectively. The Ewald construction is to draw a sphere, centered on the

tip ofk with radius |

k | in the reciprocal lattice. Since we know from Eq.11

that |k | = |

k

|, the only possible positions whichK can point to are on

the surface of such a sphere. Fig.7 only gives a cross section of 3-D sphere

but one can imagine a Ewald sphere is formed in 3-D space. Apparently, any

point which lies on such a surface will satisfy the Laue condition in which

case there will be a Bragg diffraction from a certain set of parallel planes

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correspondent to this reciprocal lattice point. This sphere is called Ewald

sphere.

Ewald sphere is the most useful tool to guide a practical experiment.It can

be seen that generally there would be no Bragg diffraction peak observed if

the wave vector of the incident beam is not wisely picked up so that there

is no reciprocal lattice points intersecting with the surface of Ewald sphere.

To ensure that the Bragg diffraction peak can be observed during the ex-

periment, several techniques are proposed based on the Ewald construction.

There are 3 general schemes to make use of Ewald construction, which will

be given below.

The Laue Method

The crystal under test is fixed in a certain axis and instead of a monochro-

matic X-ray beam, a range X-ray beam (1 0) is used in order to make

sure the occurrence of diffraction. The Ewald construction is shown in Fig.8

[1]. It is noticed that for the reciprocal lattice points which locate in the

shadow region between Ewald sphere of radiusk 0 and Ewald sphere of ra-

diusk 1 , they will satisfy Laue condition and thus there will be Bragg

diffraction for them. The Laue method is suitable to study one single bulk

sample with a known structure. For example, if the direction of the incident

beam is same as the symmetry axis of the crystal, the diffraction pattern will

have the same symmetry.

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Figure 8: Ewald construction for the Laue method

The Rotating-Crystal Method

For this method, a monochromatic X-ray beam is used while the bulk crystal

under test is rotated around the rotation axis and the orientation of rotation

axis is varied several times. When the crystal is rotating, its correspondent

reciprocal lattice will rotate about the same angle as in the space domain.

Therefore, it could be imagined that the Ewald sphere is fixed in the recip-

rocal lattice space while the entire reciprocal lattice points will rotate about

the same rotation axis. If any reciprocal lattice point hits the surface of the

Ewald sphere, there will be a consequent Bragg diffraction observed. This

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Figure 9: Ewald construction for the rotating-crystal method

method is illustrated in the Fig.9 . [1]

The Debye-Scherrer Method

The Debye-Scherrer Method is often referred as the powder method. The

crystal will be first made into powder. As a result, the crystal axes of the

powder elements will be randomly oriented. When the powder sample is

rotating, it means rotating about all possible rotation axes. It is similar to

the rotating-crystal method whose orientation of rotation axis is only varied

several times. The resulted diffraction pattern of the Debye-Scherrer method

is the the accumulated diffraction from all possible orientations of a single

bulk crystal. When a reciprocal point is rotating about all possible axes, it

will construct a 3-D sphere with radiusK . When it extends to all the lattice

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Figure 10: Ewald construction for the Debye-Scherrer method. (a) 3-D model

indicating that the possible reciprocal vectorK for the occurrence of Bragg

diffraction. (b) a plane section of (a) and is the diffracted angle.

points, the results will be a series of concentric spheres with different radii

defined in the reciprocal lattice. The Ewald construction is shown in Fig.10

(a) [1]. The smaller sphere is Ewald sphere with radiusk while the larger

sphere is rotating reciprocal lattice plane with radiusK . These two spheres

will intersect in a ellipse, provided that |K | is less than 2|

k |. Otherwise, the

Ewald sphere will totally immerse inside the bigger sphere and thus there

wont be intersection. It is obvious to see that each reciprocal lattice vector

of length less than 2|k | will generate a cone of Bragg diffraction at an angle

with respect to the forward direction. The relation amongK ,k and is

illustrated in Fig.10 (b) when a cross section is singled out. The triangle is

isosceles and applying the sine rules in triangle, we get

|K |

sin=

|k |

sin(pi2

)(16)

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Figure 11: Experiment setup on Debye-Scherrer method

Manipulate it with sin = 2cossin, we can rearrange the equation as

|K | = 2|

k |sin

1

2 (17)

The above equation tells that when the Bragg diffraction is observed at

diffracted angle . the corresponding reciprocal lattice vectorK can be

calculated with monochromatic X-ray wave vector |k |. This analysis indeed

provides a guideline for the experimental setup. Fig.11 gives a model of X-ray

diffraction setup based on the Debye-Scherrer method . In the Figure, the

red and green cones represent the Bragg diffraction radiation with different

reciprocal lattice vectors. The yellow cone represents the backscattered radi-

ation. All the radiations are recorded on a circular film so that the diffraction

angles can be obtained.

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7 Some Important Parameters

There are some interesting parameters which are particularly helpful to un-

derstand the structure of crystal. to summarize them, they include atomic

scattering factor, structure factor and shape factor.

Structure Factor and Forbidden Reflection

First , we are assuming that a monochromatic X-ray wave,traveling inside

the crystal with monoatomic lattice, has a complex amplitude electric field

defined as[1]

E in = A0e

ik inr (18)

Similarly,the scattered wave is defined as

E out = A0e

ik outr (19)

As we have concluded that the scattered wave will only constructively inter-

fere when it satisfies Eq.12. Insert Eq.12 into the above Equation we can

get

E out = A0e

i(k in

K )r (20)

Rearrange the equation to get

E out =

E ine

iK r (21)

It can be seen that the complex amplitude of electric field between outgoing

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wave and incident wave has a ratio eiK r . The crystal is consist of many

identical atoms so that we give j to index each scatterer. A little tweak is

made on above equation and we can get

E

j

out =E

j

ineiK rj (22)

If there are n atoms in the lattice. Accordingly, the electric field of outgoing

beam will be the sum of the electric fields of all the scattered wave from n

scatterers. Then we can get the electric field of outgoing beam

E out =

n

j=1

E

j

ineiK rj (23)

E

j

in is a constant complex number because of identical atoms so that it can

be factored out. Then we can get

E out =

E in

n

j=1

eiK rj (24)

The structure factor SK is then defined as

SK =n

j=1

eiK rj (25)

What information does the structure factor tell us? It is better to answer this

question by looking some examples. We are going to calculate the structures

factors of a bcc lattice to see the importance of the structure factor.[3]

Structure factor of the bcc lattice

The bcc cubic cell has identical atoms at position r 1 = 0x +0y +0z

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Table 2: Forbidden reflection in Miller indices

and r 2 =12x + 1

2y + 1

2z . insert them into Eq.27 and it becomes

SK = 1 + exp[ipi(h + j + k)] (26)

Provided thatK = h

b + j

b2 + k

b3 . Furthermore, we could get

SK = 0 when h + j + k = odd number

SK = 2 when h+ j + k = even number

Notice that when S = 0, the electrical field of outgoing beam is zero.

This means there is no diffraction on plane (hjk) if h + j + k =

odd number. This is called forbidden reflection condition.

Generally, to apply such a analysis to the other lattice types. We can get the

forbidden reflections for them. They are included in Table.2 The forbidden

reflection is very useful for analyzing the crystal structure as it can tell the

crystal type from the X-ray diffraction experimental results

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Atomic Form Factor

In the previous section , one of the important assumption we made is that

the crystal has a monochromatic lattice with identical points everywhere. If

the lattice points are not identical with different atoms attached, then the

electric field of the diffracted wave will be different for different point due to

different scattering power. Therefore, the structure factor has to be modified

as[3]

SK =n

j=1

fj(K )ei

K rj (27)

fj is called the atomic form factor. Without detailed prove, It can be calcu-

lated through

fj(K) =

1

e

dr ei

K r j(r ) (28)

The atomic form factor is said to be proportional to the Fourier transform

of the charge distribution of the corresponding ion (the basis attached to the

lattice point). The X-ray diffraction experimental results (give a citation)

between powder KCl (Potassium Chloride) and KBr (Potassium Bromide)

give the best visual illustration to explain the structure factor and the atomic

form factor. First of all, KCl and KBr both have the same crystal structure,

which is given in Fig. 12 [3]. In the picture, the iron with larger mass is

assigned to a larger atom. Thus, for KCl, the green atom is K+ and the blue

one is Cl-while for KBr, such an assignment is just opposite. Now, if we

discard the blue atoms, then the rest green atoms form a perfect fcc lattice.

24

Figure 12: Crystal structure of KCL and KBr

Then, if we discard the green atoms, then the red atoms form a fcc lattice as

well. Assume the lattice constant of fcc it is a. Then, the structure can be

considered as two fcc lattice interlacing with each other by a translation of

a2. The X-ray experimental result is show in Fig.13 [3]. Two comments can

be made by this comparison of the result.

In KCl, the density of electrons of K+ and Cl-are equal and the atomic

form factor fK+ fCl. Therefore, it looks like a sc lattice of lattice

constant a2which has identical atom everywhere. Therefore, the X-

ray diffraction result is similar to the result from a sc lattice while only

even integers appears because the actual lattice constant is a.

On the viewpoint of forbidden reflection given in Table.2, Bragg reflec-

25

Figure 13: Comparison of X-ray reflections from KCl and KBr

tion can not arise in a fcc lattice if the indices are mixed with odd and

even number. In KBr, the atomic factors of K+ and Br- are quit differ-

ent. The result shows that the indices only contain all odd numbers or

even numbers, implying that this is actually a fcc lattice.

Therefore, the comparison of experiments qualitatively explains the structure

factor and atomic form factor. Basically, for the material with unknown crys-

tal structure, its lattice type can also be told by matching the experimental

X-ray to the experimental results of the typical lattice types.

26

Shape Factor

Ideally, the Bragg diffraction should reflect as a delta peak in the experiment

result only of the crystal is infinitely large. However, in realistic experiment

based on Debye-Scherrer method, the Bragg diffraction peak is broadened.

This is mainly due to the smaller crystal size. Therefore, the Bragg equation

must be modified in order to deal with this finite size effects. Therefore,the

Scherrer Formula is given as a modified formula in X-ray science:[7]

Dhjk =W

Bhjkcos(hjk)(29)

Dimensionless W is the shape factor , B is the line broadening at half the

maximum intensity (FWHM ) in radian and is the Bragg angle. Therefore,

we get a mean size of crystalline distance D. In the original paper written by

Schnerrer, the shape factorW is approximately equal to 0.93. The Schnerrer

equation also has it limitation. Mainly, they are

Schnerrer equation is only applicable for the crystallite size short than

0.1um.

It does not include the other effects which may broaden the line-width

of the peak as well,such as inhomogeneous strain or instrument effects.

27

Part II

Electron Diffraction Technique

Apart from the traditional X-ray diffraction, the particle such as electron

can also exhibit wave-like behavior, so called particle-wave duality. The

application of the electron diffraction was first sparked by the famous the

Debrogie hypothesis:

=h

p(30)

where h is the Planck constant and p is linear momentum. Therefore, when

comes to crystallography, the electron diffraction theory should be very close

to the X-ray diffraction theory and this means that the general rules such as

Bragg law, Laue condition can apply for both of techniques. However, it is

also worth mentioning the differences between them in order to understand

how the electron diffraction technique evolved from the X-ray diffraction one.

These differences can be concluded as:

The wavelength of electrons (e.g. 2pm for 300keV electron, obtained

by the above equation)is much shorter than that of X-rays (typical

value around 100pm). Therefore, the electron diffraction Technique

can complement X-ray diffraction for studying even smaller objects

28

The X-ray is scattered by the electrons in the material under test to

yield the information of atomic structure. On the other hand, the

electron is primarily scattered by the nuclei, which carry the direct

information of atom location. The interaction strength of the latter

technique is dramatically larger (> 105) than that of the former one.

Therefore, the diffracted electron beam has much higher intensity than

diffracted X-rays which allow us observe better experimental results.

However, one should also notice that not only the stronger elastic scat-

tering presents in the electron diffraction but also does the inelastic

scattering due to phonon, plasmon excitation as well as Coulomb in-

teraction. The deeper the electron beam travels through the mate-

rial, the more inelastic scattering it produces. Therefore, The electron

diffraction is very used to study the structure of the surface, another

advantage over X-ray diffraction because X-ray will most of time pen-

etrate through a thin layer without strong interaction. The inelastic

process in the electron diffraction can be quantitatively described by

the penetration depth equation which is given by

I(d) = I0exp(d

(E)) (31)

I is the intensity with respect to the propagation distance d.(E) is

the penetration depth, defined as the distance an electron can travel

before its intensity has decreased by the factor 1e.

29

Figure 14: Bragg condition for electron beam diffraction

Mainly, there are two popular techniques associated with the electron diffrac-

tion. One is called low-energy electron diffraction (LEED) and the other one

is called reflection high-energy electron diffraction (RHEED). Before we in-

troduce them, a simple theoretical background of surface science is given.

The Bragg condition is basically no different from X-ray diffraction except

that we are more concerned about the surface structure instead of a set of

parallel planes. The Bragg condition is illustrated in Fig.14 [5].The only

change is that d is the interval of a set of points instead of parallel planes (a

citation here). Second, as we did not mention in the previous, the Bravais

lattice is not only defined in 3-D but also in 2-D. There are five distinctive

types of Bravais lattice, Listed in Table.3 [8].To name the planes, the Miller

indices (hjk) is still used,shown in Fig.15 [8]. In Fig.5, fcc(100) is the (100)

face of an fcc structure and fcc(111) is the (111) face. (11) is the primitive

mesh of the substrate layer (shown as green atoms) while (2 2) are the

30

Table 3: 2-D Bravais lattice

Figure 15: Notation of 2-D Bravais planes

31

primitive mesh of red atoms with doubled magnitude. c(2 2) is centered

mesh. R300denotes a rotation of 30 degree. A compact notation is written

as X(hjk)m nR.

8 Surface Reconstruction

In the surface science, one usually is involved with growing a single layer with

one type of lattice on top of a substrate layer with another type of lattice.

For example, Molecular beam epitaxy (MBE) is the method to deposit layer

by layer to make a semiconductor device. Surface reconstruction happens

when the newly deposited layer has a different structure as shown in Fig.15

where deposited layer (red atoms) has a different lattice compared to the

substrate (green atoms). Therefore, the rearrangement of the atoms will

introduce a new set of primitive vectors. Finding the new set of primitive

vectors with respect to the reference plane is called surface reconstruction.

Typically, the reference plane is the substrate. For example, if the primitive

mesh of substrate is a1 a2 and the primitive mesh of new layer on top is

b1 b2 shown in Fig.16[8], then we can find a matrix G satisfy

(a1

a2

) = (G11 G12

G21 G22

)(b1

b2

) (32)

Therefore, the surface reconstruction is described by matrix G.

32

Figure 16: Surface reconstruction

9 LEED

Indicated by its name, LEED is diffraction method to determine the sur-

face structure with low energy electrons (typically from 20 to 200eV). The

schematically experimental setup is illustrated in Fig.17 [2] .

1. The experiment is conducted in a ultra-high-vacuum (UHV) chamber

in order to avoid the unwanted absorption and diffraction by the atoms

in the air.

2. The sample must be well prepared,more specifically, to form a clean

surface. The containments on the surface can be removed by ion sput-

tering or by chemical process [2].

3. Electrons are emitted from a electron gun and then accelerated by

the external electrical field created by a series of electrodes. These

electrodes are not drawn in the picture. The energy of electrons can be

tuned by these electrodes. Thus, through a hole which has normally

33

Figure 17: Schematic of LEED

has the same size as the samples, the electron beam impinges on the

surface and the diffracted electron beam is detected by the florescent

screen for direct eye viewing. A demonstration of such a screen is given

in Fig.18.

4. Before collecting the diffracted beam, a LEED system normally con-

tains several hemispherical concentric grids. These grids play an im-

portant role on receiving clean image by blocking the inelastically scat-

tered electrons. The first grid, closest to the sample, is used to filter out

the non-spherical field from the retarding field. the second and third

grids(drawn as one grid as they are very close to each other), known

as suppressor grids is for blocking the low-energy electron in order to

34

Figure 18: An example of LEED pattern of Si(100)

make the diffracted electrons more homogeneous. The fourth grid(not

drawn in this picture as it is attached to the screen is for electronically

reading out the signal[2].

5. The screen directly reveals the result through the fluorescence effect.

The Ewald construction for LEED method is given in Fig.19 [10].

10 RHEED

Differing from LEED, RHEED typically operates with much higher-energy

electrons (>1keV) so that it can cause more inelastic reflection. It is nec-

essary to work in grazing incident angle to avoid vertical penetration. The

35

Figure 19: Ewald construction for LEED method in one dimension

experimental setup is shown in Fig.20 [9]. A electron gun shoots high-energy

electron beam in a grazing angle (about 1 degree). Also same as LEED, a

fluorescent screen is set up for recording the diffracted electron beam. Ide-

ally, if the sample has a perfect flat surface, the diffracted electron beams

should be reflected as an array of bright spots on the screen. However, due

to irregularity and defect on the surface, the received electron beams become

so-called streak pattern, an array of elongated spots. A real case of streak

pattern as an example is give in Fig.21 This streak can be also explained by

Ewald construction. An ideal Ewald construction is shown in Fig.22 . There-

fore, we can see that the reciprocal lattice vector intercept with Ewald sphere

about a point. However, for a rough surface, the Ewald sphere is broadened

36

Figure 20: Schematic experimental setup of RHEED

Figure 21: Real case streak pattern of TiO2

37

Figure 22: Ewald construction of RHEED method

so that the resulted interception becomes a elongated lines instead of points.

This is the reason for the formation of streak pattern.

One of the biggest breakthrough of RHEED is the capability of real-time

monitor for crystal layer growth in molecular beam expitaxy (MBE). It refers

to the oscillation of the specular beam intensity as a function of time during

MBE growth. The reason for the occurrence of the oscillation is given in

Fig.23 [4]. Ideally, we have a clean and smooth surface so that the we get

maximum intensity of the diffracted beam. Then the new layer starts growing

in MBE. When = 0.25, a few atoms land or become localized on the surface

so that the electron beam is defocused due to the imperfection of the layer.

Therefore, the intensity of the diffracted beam drops as shown in the figure.

When = 0.5, the new layer become half-complete and most imperfect so

that the intensity drops down to the valley. However, when the new layer

keep growing, the surface becomes more complete so that the electron beam

38

Figure 23: Formation of a single compete monolayer and intensity oscillationof the diffracted electon beam where is monolayer completion fraction

39

Figure 24: Typical RHEED oscillation during the deposition of GaAs on aGaAs(001) substrate

becomes more focused until the entire monolayer is finished and the intensity

recover back to the maximum. As a result, one can say that the oscillation

period (one cycle) is equal to the time required to deposit one monolayer. An

experimental result of a RHEED oscillation is given in Fig.24 [6]. One can

notice that a damping occurs to the oscillation. From the previous analysis,

we assume a perfect monolayer is formed. However, in Fig.24, When growing

GaAs, the newly deposited layer becomes rougher and rougher as the grows

continue. This is normally associated with the defect on the surface such as

40

a void, missing atoms, mismatch and so on.

Part III

Conclusion

In this report,we give an introduction to the diffraction theory and its appli-

cations: X-ray diffraction, LEED and RHEED. This review is by no means

complete. However, one should learn that the diffraction technique is a pow-

erful method to study the periodic structure. The geometrical structure of a

crystal may seem to be very complicated but its reciprocal space is straight-

forward when it deals with diffraction theory.Besides, lots of mathematical

prove in this report is oversimplified and lack of auxiliary diagrams due to

the scope of discussion. One should look up other literature about solid-state

physics in order to gain deeper insight on this topic. At last, thanks for Prof.

R. Lapierre for his kind help.

References

[1] Neil W. Ashcroft and N. David Mermin. Solid State Physics. BrooksCole, 1976.

[2] California Institute of Technology. Surface Science Laboratory : LowEnergy Electron Diffraction. 2009.

41

[3] Charles Kittel. Introduction to Solid State Physics. Wiley, 1995.

[4] Milton Ohring. Materials Science of Thin Films, Second Edition. Aca-demic Press, 2001.

[5] Dirk Rosenthal. Surface crystallography. 214(1991), 2008.

[6] S. Martini,Quivy A. A. In-situ determination of indium segregation inInGaAs/GaAs quantum wells grown by molecular beam epitaxy. Brazil-ian Journal of Physics, 2002.

[7] Detlef-M Smilgies. Scherrer grain-size analysis adapted to grazing-incidence scattering with area detectors. Journal of applied crystallog-raphy, 42(Pt 6):10301034, December 2009.

[8] Lund University. Surface crystallography.

[9] B. K. Vajn. Structure analysis by electron diffraction. Nuovo Cimento(Italy) Divided into Nuovo Cimento A . . . , 3(S4):773797, April 1956.

[10] MA VanHove, WH Weinberg, and CM Chan. Low-energy electrondiffraction, volume 214. 1986.

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