laue monochromator performance calculations for a future canadian light source diffraction beamline

$: laue monochromator performance calculations for a future canadian light source diffraction beamline$
LAUE MONOCHROMATOR PERFORMANCE CALCULATIONS

FOR A FUTURE CANADIAN LIGHT SOURCE DIFFRACTION

BEAMLINE

by

GABRIEL DINA

A Thesis

presented to

The University of Guelph

In partial fulfilment of requirements

for the degree of

Master of Science

in

Physics

Guelph, Ontario, Canada

© Gabriel Dina, October, 2011

ABSTRACT

LAUE MONOCHROMATOR PERFORMANCE CALCULATIONS

FOR A FUTURE CANADIAN LIGHT SOUCE DIFFRACTION

BEAMLINE

Gabriel Dina Advisor:

University of Guelph, 2011 Professor S. W. Kycia

The computational investigation of perfect and bent crystals both cylindrically

and sagittaly, have led to the development of sets of optimized parameters to be used for

the high energy wiggler beamline monochromator being built at the CLS. Using both Si

and Ge in Bragg and Laue geometries, the developed algorithms examine parameter

space for most photon flux at the crystal. Using programs in XOP, the calculation

analysis for a single incident beam revealed that for symmetric flat crystals the reflection

(1,1,1) in the Bragg geometry is most preferable for producing the most throughput at

energies below 24keV. For cylindrically bent crystals at energies higher than 24keV, a

Laue geometry is more preferred as a result of an increase in the rocking curve width and

throughput. Development of a program that calculates the diffracted intensity and energy

resolution of a saddle bent crystal with varying asymmetry angles are presented here.

iii

Acknowledgments

I would like to thank Dr. Kycia for the guidance and knowledge that he shared

with me. The explanations and ideas that he taught me helped me appreciate this field of

research, giving me more incentive to continue understanding and exploring new

monochromator conceptual designs. I am also grateful for the help Dr. Gomez provided

and for his suggestions on the parameters that needed to be determined for building the

monochormator at the CLS. I would also like to extend my thanks to Marcus Miranda

and all the other graduate students that have contributed to my learning and experience

here in Guelph.

I am thankful for the support and comfort that my family has given me over the

years and for the help God has given me as I pursued my studies. It is through these

years of learning that I learned to appreciate their love, care and patience.

iv

Contents

1 Introduction 1

The Synchrotron Source .............................................................................................1

The Wiggler and Undulator Insertion device ..............................................................3

Flat Bragg and Laue Monochromators .......................................................................6

The Cylindrically bent Laue Monochromator ............................................................9

The Saddle Bent Laue Monochromator ....................................................................12

2 Flat Symmetric crystals in Bragg and Laue geometry 16

2.1 Theoretical Perfect Bragg Crystals .................................................................17

2.1.1 X-ray Definitions ...................................................................................17

2.1.2 Analysis of Perfect Bragg Crystals using program XCRYSTAL ..........18

2.1.3 Analysis of Perfect Laue Crystals using program XCRYSTAL ...........24

3 Cylindrically Bent Laue crystals with comparisons to perfect Bragg crystals 29

3.1 Theoretical Cylindrically Bent and Perfect Laue Crystals .............................31

3.1.1 Algorithm for optimizing integrated reflectivity ..................................31

3.1.2 Si (1,1,1) cylindrically bent and perfect Bragg at 8 keV .....................34

3.1.3 Si (1,1,1) cylindrically bent and perfect Bragg at 17 keV ....................36

v

3.1.4 Si (1,1,1) cylindrically bent and perfect Bragg at 24 keV .....................37

3.1.5 Si (1,1,1) cylindrically bent and perfect Bragg at 50 keV ....................37

4 A single crystal saddle bent Laue 39

4.1 Theoretical single and double saddle bent Laue .............................................43

4.1.1 Fundamental saddle bent equations ....................................................43

4.1.2 Parameters fs, Rs, and Rm ...................................................................43

4.1.3 Parameter C ........................................................................................45

4.1.4 Parameter C Roland ............................................................................47

4.1.5 Elastic compliance coefficients ..........................................................47

4.1.6 Parameters ΔθBragg (T), Δθrot (T), and Δθ (T) ......................................50

4.1.7 Rocking curve width, ω ......................................................................53

4.1.8 Variation of angle of incidence, Δθ and energy resolution ................54

4.1.9 Parameters Q, B, and A .......................................................................55

4.1.10 Integrated reflectivity and mass attenuation coefficients ..................57

4.1.11 Peak reflectivity .................................................................................59

4.1.12 Diffracted intensity at the crystal ......................................................60

4.2 Analysis of a single saddle bent Laue with an infinitely large radius ..............62

4.2.1 Parameters constrained for flat symmetric Laue crystals ...................63

4.2.2 Beam size at monochromator and sample ..........................................64

vi

4.2.3 Energy resolution and diffracted intensity ..........................................66

4.2.4 Analysis of diffracted intensities and energy resolutions at sample ....69

4.3 Analysis of a single saddle bent Laue in the inverse-Cauchois geometry .......75

4.3.1 Parameters constrained for a single saddle bent Laue crystal ............76

4.3.2 Calculation of axes with an orientation ..............................................77

4.3.3 Energy resolution and diffracted intensity .........................................78

4.3.4 Analysis of optimized intensities and energy resolutions at

monochromator ...................................................................................79

5 Conclusion 93

5.1 Future Work .....................................................................................................97

Appendices 100

vii

List of Tables

3.1 Optimized throughput using XCRYSTAL_BENT at 8 keV . . . . . . . . . . . . . . 36

3.2 Optimized throughput using XCRYSTAL_BENT at 17 keV . . . . . . . . . . . . . 36



4.1 Literature values Rs and Rm for Si (1,1,1) on (1,0,0) . . . . . . . . . . . . . . . . . . 45

4.2 Literature values of CS'23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Calculated elastic compliance coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 Calculated ΔθBragg, Δθrot, Δθ (T), and ω for Si (1,1,1) . . . . . . . . . . . . . . . . . . 53

4.5 Calculated ΔθBragg, Δθrot, Δθ (T), and ω for Si (-1,1,1) . . . . . . . . . . . . . . . . . 53

4.6 Equations used for calculating mass attenuation coefficients . . . . . . . . . . . . 58

4.7 Calculated normalized thickness, rocking curve width and reflectivity . . . . 59

4.8 Energy range and orientation for flat symmetric Laue crystals . . . . . . . . . . . 63

4.9 Energy range of reflections for a single saddle bent Laue crystal . . . . . . . . 77

4.10 Optimized diffracted intensities and energy resolutions at a single bent Laue

crystal using low order reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83


crystal using higher order reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.12 Diffracted intensities and energy resolutions at a single bent Laue crystal

keeping T, F2, χ, and C fixed and varying energy for low order reflections . . . 89


keeping T, F2, χ, and C fixed and varying energy for higher order reflections . . 91

viii

List of Figures

1.1 An undulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Two Bragg crystal monochromator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Symmetric and asymmetric Bragg geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Crystal beam footprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Symmetric Laue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Cylindrically bent Laue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 Sagittally bent Laue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.8 The Roland condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Symmetric Bragg rocking curve using XCRYSTAL . . . . . . . . . . . . . . . . . . . . 21

2.2 Maximum throughputs for Si Bragg reflections . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Maximum throughputs for Ge Bragg reflections . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Symmetric Laue rocking curve using XCRYSTAL . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Maximum throughputs for Ge Laue reflections . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Maximum throughputs for Si Laue reflections . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Optimized throughputs for cylindrically bent Laue at 8keV . . . . . . . . . . . . . . . . . 35

4.1 Roland circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Single saddle bent Laue geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Change in d-spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

ix

4.4 Lattice plane rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Mass attenuation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.6 Incident flux for a super conducting wiggler . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Vertical beam size at sample for a flat symmetric Laue crystal . . . . . . . . . . . . . 65

4.8 Horizontal beam size at sample for a flat symmetric Laue crystal . . . . . . . . . . . 66

4.9 Pendellӧsung fringes of Si (5,1,1) symmetric Laue crystal . . . . . . . . . . . . . . . . 68

4.10 Fringes convoluted using a Gaussian profile for Si (5,1,1) . . . . . . . . . . . . . . . . 69

4.11 Flat symmetric Laue diffracted intensities and energy resolutions for the full

beam size at the sample and using 2nd harmonic reflections . . . . . . . . . . . . . . . 70


beam size at the sample and using multiple harmonics . . . . . . . . . . . . . . . . . . . 71


beam size at the sample and using all other low order reflections . . . . . . . . . . . 72

4.14 Flat symmetric Laue diffracted intensities and energy resolutions for a sample

size of 1mm (V) x 1mm (H) and using all other low order reflections . . . . . . . 73

4.15 Flat symmetric Laue diffracted intensities and energy resolutions for a sample

size of 0.1mm (V) x 0.1mm (H) and using all other low order reflections . . . . 74


crystal using low order reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


crystal using higher order reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84


keeping T, F2, χ, and C fixed and varying energy for low order reflections . . . 88


keeping T, F2, χ, and C fixed and varying energy for higher order reflections . . 90

1

Chapter 1

Introduction

The Synchrotron Source

Today, there are dozens of extremely sophisticated synchrotrons, world-wide,

dedicated to synchrotron x-ray applications. The energy of the charged particles directly

influences the photon spectrum produced by these machines. These machines can be

divided into three categories, low energy (0.5GeVto 2GeV), medium energy (2GeV to

4GeV) and high energy (4GeV to 8GeV). There are only a handful of high energy

synchrotrons in the world, the largest most powerful one being the 8 GEV, SPRING-8

synchrotron in Japan. In the past, low energy synchrotron were constrained to work in

the ultra violet to soft x-ray regime, medium energy synchrotron could work up to the

hard x-ray regime (<30keV) while the high energy machines could manage significant

flux of x-rays up to ~80keV. Canada has a medium energy (2.9GeV) electron storage

ring called the Canadian Light Source (CLS) in Saskatoon.

The theory of how synchrotron radiation is produced can be explained by the idea that

accelerating charges emit electromagnetic radiation. The path of a charged particle such

as an electron or positron begins in an injector. The particles are initially accelerated to

relativistic speeds by a linear accelerator. Upon reaching these high energies (~50-

2

100MeV), the particles are typically sent to a small booster ring which is an accelerator

that brings the electrons to their maximum energy (the energy of the ring). At this point

the electrons are injected into the storage ring. This process continues until the storage

ring is filled to its starting current (which may be 100mA to 500mA in general). With

time, the current in the storage ring decays in an exponential decay with a time constant

of hours. When the current drops below a certain threshold, the machine scientists refill

the storage ring with electrons. Some new machines can actually conduct continuous

injection throughout the day, called “top-up mode”. This has the desirable feature of

constant electron current and no filling interruptions.

The storage ring, which is an evacuated ring shaped pipe, is composed of straight

sections and bends. The electrons trajectories are kept approximately circular, as a result

of electromagnets and the placements of bending magnets throughout the storage ring.

Charged particles undergo centripetal acceleration and a change in direction as they pass

through a bending magnet, resulting in the emission of electromagnetic radiation. In the

storage ring, electrons lose energy as they circulate due to this emission of radiation, and

are replenished in energy by a radio-frequency accelerating cavity. In the past 20 years,

enhanced electromagnetic synchrotron radiation has been produced due to the

introduction of insertion devices. These are a sequence of magnets causing the electrons

in the straight section of the storage ring to follow a “slalom-like” trajectory. Two such

devices are a wiggler and an undulator. The electromagnetic radiation created from an

insertion device or a bending magnet is what we call synchrotron radiation.

Experimenters have introduced high magnetic field wigglers into storage rings to shift the

energy spectrum of synchrotron radiation to high energies. In this way the medium

3

energy CLS operating with a high field super conducting magnet wiggler can produce

intense x-ray flux up to even 100keV.

Synchrotron radiation is usually highly collimated in the vertical direction, and broad

in energy spectrum, extending from far infrared to hard x-rays [1]. The light produced is

horizontally polarized in the plane of the storage ring (electric field, E , is parallel to the

plane of the orbit) and becomes elliptically polarized as the direction is changed from this

plane. This radiation is a regularly pulsed flash that occurs every few or tens of few

nanoseconds, and only lasts less than 1ns as a result of the reenergized electrons being

bunched in groups inside the storage ring from the radiofrequency system [2]. The

source size is very small, about 1mm horizontal and 20 µm in the vertical [3] for the case

of the Canadian Light Source diffraction beamline. The combination of a very low

emittance (the amount of spread of the beam as it travels), large intensities, spectral

tunability, high polarization, and high collimation makes synchrotron radiation an

important tool for many types of condensed matter structure studies. In the case of the

Brockhouse sector at CLS, several uses of synchrotron radiation includes high resolution

powder diffraction, microcrystal crystallography, high resolution pair distribution

function analysis, and reciprocal space mapping.

The Wiggler and Undulator Insertion device

Radiation that enters the beamline is usually produced by an undulator or wiggler, and

is more intense than the electromagnetic radiation produced by the bending magnets. It is

for the purpose of gaining several orders of magnitude in flux, compared to the radiation

produced by bending magnets, that such insertion devices are used. The magnitude of the

4

flux at the source for bending magnets is on the order of 1012

to 1014

photons s-1

mrad-2

mm-2

(0.1% δE/E) and for undulators 1019

to 1021

photons s-1

mrad-2

mm-2

(0.1% δE/E)

[2]. An undulator consists of dipole magnets with alternating polarity where the

magnetic field alternates with wavelength, λu. As the electron beam passes through this

structure, electrons experience the periodic magnetic force that creates an oscillating

trajectory where each time the electrons change direction, they emit intense radiation in a

narrow energy band. A dimensionless parameter that categorizes the nature of the

electron motion is [4]:

𝐾 = 0.934𝐵𝑜[𝑇]𝜆𝑢[𝑐𝑚] (1.0.1)

where Bo is the peak magnetic field. For K ≤ 1, the insertion device is called an undulator

and the amplitude of the oscillation is small. The total divergence of the beam emitted by

the device is actually smaller than the divergence of the radiation emitted by a single

period of the undulator. This is a result of the interference pattern of photons emitted by

the total sum of undulator periods. For undulators with N periods, the brightness

(flux/(unit source size, unit solid angle)) increases as N2. In the limit that K >> 1, the

insertion device is called a wiggler. In this case, the oscillation amplitude is large, and

there is no coherent interference of radiation from the poles. This makes the total

divergence of the beam emitted by wigglers relatively large compared to that of

undulators. Being an incoherent sum of the radiation produced by each period, causes the

wiggler spectrum to be broad, flat and featureless. Intensity increases as 2N for a

wiggler, where N is the number of periods.

A more novel insertion device that may be used for the Brockhouse sector beamline is

a superconducting wiggler. This type of device makes use of high B-field

5

superconducting magnets which are electromagnets that are cooled to very low

temperatures, where electricity conduction can occur with zero resistance, increasing

current density. Advantages of using such a wiggler are higher magnetic fields which

increases the power of the beam, higher photon energy, and more photon flux incident on

the sample.

Figure 1.1: A schematic diagram of an undulator with the electron beam entering

the device, ID, from the top left.

6

Flat Bragg and Laue Monochromators

An important component of any synchrotron beamline is the monochromator. It

functions to select a certain band of x-ray energies from the incident beam by means of

diffraction. It is optimized to output a desired intensity, angular distribution, beam size

and energy resolution to a hutch where the sample and detectors are located. The beam

may encounter slits and other optical filters, such as thin plates of aluminium or graphite

during its travel, in order to reduce the heat load on the monochromator crystal. Most

synchrotrons today have the monochromator consisting of several crystals in the Bragg

geometry (figure 1.2). The Bragg diffraction geometry refers the case when the

diffracted x-rays are reflected from the same crystalline surface on which the incident x-

rays are impingent. This type of geometry works best at low energies where the Bragg

angle, the angle between the incident beam and the crystal planes, is large causing the

beam footprint at the crystal to be relatively small. The crystal may have an asymmetric

cut angle, α, which may be defined as the angle between the Bragg planes and the surface

and for the case of symmetric Bragg, α is 0° (figure 1.3). A brief investigation using this

type of symmetric Bragg geometry for the case of Ge and Si is performed for reflections

[2,2,0], [4,4,0], [1,1,1], and [3,3,3] and was conducted mainly to serve as a reference for

further Laue geometry studies.

7

Figure 1.2: Schematic of a typical synchrotron monochromator using two Bragg

crystals. The incident beam diffracts off the first flat Bragg crystal and

hits a second perfect Bragg crystal causing the beam to continue in the

horizontal direction.

(a) Symmetric Bragg crystal geometry (b) Asymmetric Bragg crystal geometry

Figure 1.3: X-rays diffracting in Bragg (reflecting) geometry for both symmetric, (a),

and asymmetric, (b), crystal cut case.

The future high energy wiggler beamline at the CLS is intended to take advantage of

8

the beneficial properties of high energy x-rays in the energy range from 30keV to

100keV. A consequence of using higher energies is smaller Bragg angles. This leads to a

disadvantage for the Bragg geometry resulting from the beam footprint becoming too

large, as seen in figure 1.4. The relationship between the Bragg angle and the size of the

crystal beam footprint is:

𝐶𝑟𝑦𝑠𝑡𝑎𝑙 𝑏𝑒𝑎𝑚 𝑓𝑜𝑜𝑡𝑝𝑟𝑖𝑛𝑡 ≅ 𝐵𝑒𝑎𝑚 𝑤𝑖𝑑𝑡 𝑕

𝑠𝑖𝑛𝜃𝐵 (1.0.2)

Figure 1.4: For high energies the Bragg angle is small, θB small, resulting in a large

crystal footprint as denoted with the dotted lines. For the same beam width

at smaller energies and larger Bragg angles, θB large, the footprint is

decreased.

To solve the problem of needing larger crystals when diffracting high energy x-

rays, a Laue (transmission) type monochromator may be used, as seen in figure 1.5. For

symmetric Laue crystals, the asymmetry angle, α, is 90° as it is the angle between the

planes and the surface. As the crystal is positioned to θ = θB, the diffracted beam exits

the crystal from the opposite side at an angle of 2θB from the incident beam. As a

preliminary step of this thesis work, the integrated reflectivity of the diffracted beam was

investigated for Si and Ge reflections [2,2,0], [4,4,0], [1,1,1], and [3,3,3] in the flat Laue

geometry. Comparisons of results for both Bragg and Laue geometries were made and

θB large θB small

Crystal Footprint

Width

9

the crystal and reflection were chosen that best optimize the total throughput for the

energy of 8keV.

Figure 1.5: The diffracted beam transverses the symmetric Laue crystal and exits on the

opposite side of the crystal at an angle 2θB from the direction of the incident

beam. Also shown are the forward diffracted beam, known as “anomalous

transmitted beam”, and the transmitted beam.

The cylindrically bent Laue monochromator

Upon investigating both symmetric Bragg and Laue geometries for the case that the

crystal is perfectly flat, the reflection chosen for further study was Si (1,1,1). This

reflection proved to be the most promising in producing more throughput for a single

incident low energy x-ray beam. Analysis of this reflection was performed to determine

if an increase in throughput occurs for a crystal strained as a result of bending. Variation

of throughput for the case of a cylindrically bent Laue crystal geometry was analyzed

10

using the program XCRYSTAL_BENT from XOP [5], as depicted in figure 1.6. In

addition, energy was varied from 8 keV to 50 keV to determine the preferred energy

range for which the cylindrically bent Laue geometry out performs the perfect Bragg

geometry in terms of throughput.

Cylindrically bending a crystal causes lattice distortion which has a significant effect

on the rocking curve width. As the bending radius is decreased, the crystal becomes

more bent causing increased distortion of the atomic planes. The atomic d-spacing in the

region close to the convex side of the bent crystal is larger than the d-spacings of atoms

near the concave side. In XCRYSTAL_BENT the crystal is approximated by a lamellar

model, such that the crystal is subdivided into a set of perfect crystal layers, each

individually dynamically diffracting with their specified lattice constant. The orientation

of each of these lamella, and consequently, the Bragg planes within, vary with bend

radius and asymmetry angle. The potential benefits of using a cylindrically bent crystal

includes gains in intensity at the sample position due to focusing and an increase in

throughput.

11

Figure 1.6 : Cylindrically bent crystal of radius ρ using the lamellar model. The x-ray

beam is incident on a set of locally flat planes, called „lamella‟. The Bragg

condition is satisfied for a distribution of angles due to the spread of

lamella orientation and d-spacing. Top right: the effect of strain on the

rocking curve.

12

The Saddle Bent Laue Monochromator

By taking the concept of bending a crystal one step further, one realizes that the

bending of the crystal about a first axis naturally results in a bend about a second axis

perpendicular to the first. This effect is called anticlastic bending, as depicted in figure

1.7 in the meridional plane (x-y plane). For a Laue geometry with the beam diffracting in

the x-y plane, the crystal may be bent about the y axis, called the sagittal axis, resulting

in a primary curvature in the x-z plane. For focusing Laue crystals, the incident beam

must hit the convex side and exit the concave side of the crystal. The anticlastic effect

results in the crystal having a slight curvature in the x-y plane that can be viewed as a

bend about the z axis. In this case the z axis is called the meridional axis. The incident

beam hits the concave side of the meridional curvature and exits on the convex side of the

meridional curvature. The shape of a crystal bent in such a way would look sandal

shaped. Bending a crystal sagittally in this manner allows focusing of x-rays

perpendicular to the scattering plane.

13

Figure 1.7 : A saddle bent crystal with its y-direction as the sagittal axis, z-direction as

the meridional axis and x-direction as the surface normal axis. For the

CLS, the beam is diffracting horizontally in the x-y plane.

Sagittal bending of a crystal in the Bragg geometry has been previously used for

the energy range of 5-30keV[6]. For low energies the Bragg angle is large and thus a

small beam footprint is produced. For higher energies, on the other hand, the beam

Quadrant 1 Quadrant 4


14

footprint increases in size due to the smaller Bragg angle and thus larger crystals need to

be used. Shallow Bragg angles also requires tighter curvature for focusing resulting in

detrimental anticlastic curvature. As a result, the Bragg geometry becomes unfavorable

as we go to larger energies, and above 30keV a Laue crystal is preferred [6].

Throughout the world, there now exist hundreds of monochromators at

synchrotrons based on varying crystal arrangements and designs. In the case of double-

crystal monochromators, one common solution consists of a flat Bragg crystal and a

second sagittally bent Bragg crystal. This design allows for horizontal focusing of the

source beam and increases the total flux at the sample. The second crystal that is bent

sagittally is also bent anticlastically which is a problem because the Bragg planes

between the first and second crystal lose their parallelism. To fix this problem slotted

crystals or ribbed crystals, which are crystals with periodic slots, have been used. The

disadvantage of using such types of crystals is that they act like polygonal segmented

crystals rather than cylinders, compromising the minimum focus size [7].

Quintana et. al. [8] minimize anitclastic bending by using an unribbed rectangular

crystal with a length to width ratio of 2.42 and a four point bender. This special ratio,

that is used to minimize anticlastic bending, produces a local curvature which differs less

than 0.001% from a cylinder [8].

A novel approach for crystals bent in the Laue geometry is to sagittally focus high

energy x-rays, as suggested by Zhong Zhong et. al. [6]. The length of the beam‟s

footprint is relatively small and insensitive to energy. In the single or double Laue crystal

geometry, the anticlastic curvature is actually desirable. This is because the anticlastic

curvature can be tuned to satisfy the “inverse-Cauchois geometry”. This geometry is also

15

known as the Roland condition, where a diverging incident beam hits the crystal and

maintains its divergence upon exit, as depicted in figure 1.8. This has the beneficial

feature of minimizing the energy bandwidth of the resulting monochromatic beam.

Figure 1.8 : The Roland condition for a bent Laue geometry. A set of incident rays

diverging from the source each diffract off of planes at the same angle and

exit the crystal, conserving the incident divergence.

Source

16

Chapter 2

Flat Symmetric crystals in Bragg and Laue geometry using a single

incident beam

With the goal of designing a monochromator that will pass as much monochromatic

light as possible without sacrificing too much flux, we start with a computational study

performed for flat symmetric crystals. Programs nBeam [9] and XOP XCRYSTAL [5]

were used in order to determine the parameters that best optimize the throughput for flat

symmetric Ge and Si crystals in both Bragg (reflection) and Laue (transmission)

geometry. Program nBeam was written mainly for Laue geometries for a Unix type

platform, and XOP XCRYSTAL is an x-ray oriented widget based program for calculating

diffraction profiles of flat crystals using dynamical theory of diffraction developed by

Zachariasen [10].

Dynamical theory was described as early as 1914 independently by Ewald and

Darwin, and is a theory that explains how the diffracted beam is calculated by both

constructive and destructive interference of the waves as the incident beam passes

through the atomic planes of a perfect crystal. The elaborate dynamical theory refined by

Zachariasen describes the coupling between incident and diffracted waves by requiring

the internal wave field to satisfy the electromagnetic field equations and the surface

boundary conditions.

Zachariasen‟s equations used by XCRYSTAL involve calculating the reflected

17

power, the ratio of the diffracted intensity over the incident intensity as [5]:

𝑅𝐵𝑟𝑎𝑔𝑔 𝛥 = 1

|𝑏|

𝐼𝐻

𝐼𝑜=

1

|𝑏|

𝑥1𝑥2(𝑐1−𝑐2)

𝑐2𝑥2−𝑐1𝑥1

2

(2.0.1)

𝑅𝐿𝑎𝑢𝑒 𝛥 = 1

|𝑏|

𝐼𝐻

𝐼𝑜=

1

|𝑏|

𝑥1𝑥2(𝑐1−𝑐2)

𝑥2−𝑥1

2

(2.0.2)

where Δ is the deviation from the Bragg angle, IH is the external diffracted intensity, Io is

the incident diffracted intensity, b = γo

γ𝑕

where γo is the incident direction cosine and γh the

diffracted direction cosine, 𝑐1 = e−iυ1t where the crystal thickness is defined as t,

𝑐2 = e−iυ2t, υ1

= 2πkδo

′

γo

, and υ2

= 2πkδo

′′

γo

. The remaining variables are defined as [10]:

𝛿𝑜

′

𝛿𝑜′′ =

1

2(𝛹𝑜 − 𝑧 ± 𝑞𝑃2 + 𝑧2) (2.0.3)

𝑥1

𝑥2 =

−𝑧 ± 𝑞𝑃2+ 𝑧2

𝑃𝛹𝐻

(2.0.4)

𝑧 = 1−𝑏

2 𝛹𝑜 +

𝑏

2τ (2.0.5)

where 𝑘 = 1

λ, 𝛹𝐻 = −

4𝜋𝑒2𝐹𝐻

𝑚𝑤𝑜2𝑉

is the Fourier component of electric susceptibility Ψo and

FH is the structure factor, e is the charge of an electron, m is the mass of an electron, V is

volume of a unit cell, P is the polarization factor, 𝑞 = 𝑏𝛹𝐻𝛹𝐻 and τ ≈ 2Δsin(2θB).

2.1 Theoretical Perfect Bragg Crystals

2.1.1 X-ray Definitions

As a single incident beam enters a perfect crystal in the Bragg (reflecting) geometry,

the incident beam diffracts through planes of atoms, and if the Bragg condition is

18

satisfied, the wavelength λ is the defined as:

λ = 2dsinθB (2.1.1)

and constructive interference results. If the path length of each beam that traverses a

layer of atomic planes is an integer multiple of the wavelength, the Bragg condition is

satisfied. The Bragg angle, θB, is the angle between the incident beam and the planes of

the crystal, and for the symmetric Bragg case this angle is the same as the angle that the

diffracted beam makes with the surface. The lattice spacing d of a cubic crystal is

defined as:

𝑑 = 𝑙𝑎𝑡𝑡𝑖𝑐𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑕2+𝑘2+𝑙2 (2.1.2)

where h, k, and l are the indices of the reflection. The energy of the beam is related to the

wavelength as:

𝐸 (𝑒𝑉) =𝑕𝑐

𝜆 =

12398 .562 Å𝑒𝑉

𝜆 (2.1.3)

2.1.2 Analysis of Perfect Bragg Crystals using program XOP XCRYSTAL 2.1

Using program XCRYSTAL, the total throughputs were computed for both Ge and Si

symmetric crystals at an energy of 8.048keV, crystals whose planes are parallel to the

surface, for reflections (2,2,0), (4,4,0), (1,1,1) and (3,3,3). This low energy was chosen

because it is an energy that is attainable in laboratories, and for the most part it provides

an initial idea for selecting a reflection for making further comparisons to the Laue

geometry.

For each crystal and reflection, a table of reflectivity and rocking curve angles were

produced for varying crystal thickness. In all cases the incident beam was defined as σ

19

polarized, meaning the electric field, 𝐸𝜍 , is defined as being perpendicular to the plane

defined by the incident and scattered x-ray vectors. This plane is called the scattering

plane. The asymmetry angle, α, for the case of symmetric Bragg was set to 0°, and the

thickness was varied from 0.001 mm to just over the thickness that resulted in the

maximum integrated reflectivity in increments of 0.001 mm. From observation, at a very

thin crystal thickness the throughput was small and as the thickness was increased, an

optimum thickness would exist that results in the throughput being maximized. For larger

thicknesses the throughput is slightly lower than the maximum for each reflection. This

may be due to an interference effect of the two surfaces for very thin crystals.

Initially the program nBeam was used with a rocking curve composed of 2399

scanning points (rocking curve range) and a normalized width of 8 but due to the

programs limited capability of correctly calculating the throughput (for all thicknesses the

throughput remains constant as the program may not be properly taking account affects

like absorption) these calculations were reproduced using program XCRYSTAL. The

same rocking curve width as in nBeam with 2399 scanning points were again used. By

referring to the plots produced using program nBeam, the total scanned angular range

between the smallest and largest rocking curve angle was determined. This width value

was divided by two to get the max and min rocking curve scan value used in program

XCRYSTAL. For each reflection the step size or the width of each point was calculated

by taking the scanning width and dividing this by the 2399 scanning points. For the Si

reflections (1,1,1), (2,2,0), (3,3,3) and (4,4,0) the calculated widths of each rocking curve

point are 6.669x10-6

, 5.002x10-6

, 2.084x10-6

and 2.709x10-6

degrees respectively. For the

Ge reflections (1,1,1), (2,2,0), (3,3,3) and (4,4,0) the calculated widths of each rocking

20

curve point are 1.459 x10-5

, 1.146x10-5

, 4.168x10-6

and 5.836x10-6

degrees respectively.

The throughput was calculated by summing the reflectivity values of all the points under

a rocking curve and multiplying this by the calculated width of each point.

The distinct shape seen in the diffraction profile of a Bragg crystal may be attributed

to the fact that there exist two standing wave patterns inside the crystal. One standing

wave has its nodes coinciding with the atomic planes, and the other has its nodes between

the atomic planes. This results in the second standing wave experiencing more

photoelectric absorption as a result of it „seeing‟ more electrons [11]. As shown in figure

2.1, peak reflectivity exists at a Bragg angle of 14.2216° at 8048eV for Si (1,1,1) and

there exists constructive interference within the crystal for angles very close to the Bragg

angle which correspond to the size of the Darwin width. The Darwin width is intrinsic to

the crystal and is dependent on the crystal type, reflection and energy. There exists a

shift in the scattering angle to a slightly higher angles near the Lorentz point, where the

incident and diffracted wave vectors are shorter and have longer wavelengths.

21

Figure 2.1: Rocking curve of Si (1,1,1) flat symmetric Bragg at 8.048 keV using XOP

XCRYSTAL 1.2 for curves with maximum throughput using parameters:

min scan angle -0.008°, max scan angle = 0.008°, α = 0°, 2399 scan

points, σ and π polarizations.

When comparing among Ge Bragg crystals, using program XCRYSTAL, Ge (1,1,1) has

the largest integrated reflectivity with an area of 0.00558° at 2μm followed by Ge (2,2,0)

with an area of 0.00456° at 2μm for σ polarization, as seen in figure 2.3. Among Si

Bragg crystals for the case of σ polarization, Si (1,1,1) has the most throughput with an

area of 0.00259° at 4μm, and followed by Si (2,2,0) with an area of 0.00202° at 5μm, as

shown in figure 2.2.

Peak Reflectivity ~ 94%

Darwin width

= 0.002⁰ ± 0.001⁰

22

Figure 2.2: Maximum throughput and corresponding thickness for varying Si

reflections in flat symmetric Bragg geometry at 8.048 keV, for the σ

polarization using program XCRYSTAL.

23

Figure 2.3: Maximum throughput and corresponding thickness for varying Ge

reflections in flat symmetric Bragg geometry at 8.048 keV, for the σ

polarization using program XCRYSTAL.

As in figure 2.1 it was observed that the full width at half maximum, FWHM, of

rocking curve plots agrees with the calculated Darwin width from CHESS calculations

[12]. The Darwin width of Ge crystals for each reflection are much larger than that of Si

crystals which agrees with theory. A Ge atom has 32 electrons as compared to Si which

has 14 electrons. This results in Ge having a larger structure factor and consequently a

larger Darwin width.

Using the program, XCRYSTAL, it has become apparent that a Ge crystal has the most

throughput at 8.048 keV when comparing Ge and Si crystals. The preferred reflection for

optimizing throughput is the (1,1,1) followed by the (2,2,0).

24

2.1.3 Analysis of Perfect Laue Crystals using program XOP XCRYSTAL 2.1

Calculations of throughput for Ge and Si with flat symmetric Laue geometry at 8.048

keV was performed using program XCRYSTAL. The number of points scanned for each

rocking curve was 2399, as depicted in figure 2.4. Using the symmetric Laue geometry

plots produced using program nBeam for each crystal, and reflection, the scanning width

of the rocking curve plot was measured and divided by two to get the min and max

rocking curve scan angle. For each reflection the step size or the width of each point was

calculated by taking the scanning width and dividing this value by the 2399 scanning

points. For the Si reflections (1,1,1), (2,2,0), (3,3,3) and (4,4,0) the calculated widths of

each rocking curve point are 6.669x10-6

, 5.002x10-6

, 2.084x10-6

and 2.709x10-6

degrees

respectively. For the Ge reflections (1,1,1), (2,2,0), (3,3,3) and (4,4,0) the calculated

widths of each rocking curve point are 1.459 x10-5

, 1.146x10-5

, 4.168x10-6

and 5.836x10-

6 degrees respectively. The throughput was calculated by multiplying the sum of the

reflectivity values of all the points under a rocking curve by the calculated width of each

point. Thickness was increased from 0.001mm in increments of 0.001mm and the

asymmetry angle used in the program was 90°.

25

Figure 2.4: Rocking curve of Si (1,1,1) flat symmetric Laue at 8.048keV using XOP

XCRYSTAL for curves with maximum throughput using parameters: min

scan angle = -0.008°, max scan angle = 0.008°, α = 90°, 2399 scan

points, σ and π polarizations.

It was observed that for very small thicknesses the integrated reflectivity was small

and as thickness was increased, the throughput increased up to a maximum at a certain

thickness and then it dropped off. For very thin crystals where the beam is diffracting in

Laue (transmission) geometry, the beam exits mainly as the transmitted beam as a result

of too few atoms and little constructive interference. For very thick crystals, absorption

has a predominant role of limiting the intensity of the diffracted beam, as it transverses

thickness of the crystal. Between these two limits, there exists an optimum thickness

with a balance of having sufficient number of diffracting atoms while maintaining the

effects of absorption low.

26

Analysis of computational results using program XCRYSTAL for flat symmetric Laue

crystals reveals that among Ge crystals, Ge (1,1,1) produces the most throughput at 3μm

with an area of 0.00429° followed by Ge (2,2,0) for σ polarization, as shown in figure

2.5. Among Si crystals for the case of σ polarization, Si (1,1,1) at 7 μm and an area of

0.00196° produces the most throughput at the crystal, followed by Si (2,2,0) at 5μm with

an area of 0.00152°, as shown in figure 2.6. The oscillations seen in figures 2.5 and 2.6

are a result of the Pendellӧusng fringes, and are caused by an interference effect of the

wavefields.

Figure 2.5: Maximum throughput and corresponding thickness for varying Ge

reflections in symmetric Laue geometry, for the σ polarization using

program XCRYSTAL.

27

Figure 2.6: Maximum throughput and corresponding thickness for varying Si

reflections in symmetric Laue geometry, for the σ polarization using

program XCRYSTAL.

For a symmetric flat Laue crystal at 8.048keV, the reflection Ge (1,1,1) at 3μm would

be the preferred candidate for exhibiting the most throughput. The results of optimized

throughput using program XCRYSTAL reveals that the reflection (1,1,1) followed by the

reflection (2,2,0) should be chosen for flat symmetric Bragg and Laue crystals. The

crystal type that best optimizes throughput for both crystal geometries is Ge. In theory,

Ge does have a larger Darwin width which contributes to the increased throughputs.

Although throughputs were optimized at crystal thickness in the range of a few μm it

would be not practical to create such crystals, as the crystal would be very fragile and

hard to handle. This study provided us confidence in the calculation software and an idea

28

of how to pick a reflection that best optimizes the throughput while varying parameters.

For larger crystal thicknesses this ordering of reflections based on throughput is still

maintained as the throughputs remain constant and independent of thickness at large

thicknesses in the case of crystals with the Bragg geometry. For the case of crystals in the

Laue geometry the throughputs continue their trends of decay as the beam gets attenuated

due to absorption.

29

Chapter 3

Cylindrically bent Laue crystals versus flat Bragg crystals using a single

incident beam

Our goal is to determine the effects that geometry, reflection, and other parameters

such as asymmetry angle, thickness and bend radius have on the rocking curve and

consequently the throughput. Presented here is a study showing how a cylindrically bent

Laue monochromator crystal has higher throughputs, compared to a flat Bragg crystal, for

energies greater than ~20keV. In this chapter we treat the incident beam as a

monochromatic ray with infinitesimal angular divergence and spatial dimension at the

monochromator. The modelling of the throughput of a curved Laue monochromator for a

diverging white x-ray beam from a wiggler source at the CLS will be presented in the

next chapter.

An incident beam hitting a flat thick crystal in the Laue geometry will diffract

according to dynamical theory, which takes into account all the wave interactions. If the

crystal is bent cylindrically, the crystal can be approximated by several layers called

lamella, which each dynamically diffract the x-rays. The reflectivity of each lamella is

summed to give the total reflectivity. The rocking curve width can be relatively large

being a convolution of the, Darwin width, d-spacing variation term and lattice plane

orientation term that are experienced by the infinitesimal x-ray beam that traverses the

crystal.

30

The Penning-Polder method [5] was primarily used for the calculations performed in

XCRYSTAL_BENT. This method assumes the x-ray wave field propagates, undeflected,

through the distorted crystal, a composite of perfect crystal parts. In this method the

interference of two wavefields is neglected as it is assumed that a wavefield that hits a

layer of a lamella remains preserved upon passing to the next lamella layer [5]. A

weakness of this method is that it only works for slightly deformed crystals where the

bend radius, ρ, satisfies the constraint [6]:

𝜌 >2𝛬𝑠𝑖𝑛𝜃

|𝛹𝐻 | (3.0.1)

where 𝛹𝐻 = −4𝜋𝑒2𝐹𝐻

𝑚𝑤𝑜2𝑉

and is the Fourier component of electric susceptibility Ψo, and Λ is

the Pendelӧsung length. The Pendelӧsung length is defined as [2]:

𝛬 = 𝜋𝑉𝑐 𝛾𝑜 |𝛾𝑕 |

𝑟𝑒𝜆𝐾 |𝐹𝐻 | (3.02)

where Vc is the volume of a unit cell, γo and γh are the incident and diffracted direction

cosine, re is the radius of an electron, K = 1 for σ polarization or |cos2θ| for π

polarization, and FH is the structure factor dependent on the type of crystal and reflection.

The incident and diffracted direction cosines are the cosine angles between the crystal

surface normal and the incident and diffracted directions respectively.

An important parameter that needs to be inputted prior to doing calculations is y, the

deviation of the incident angle from the Bragg angle. This parameter is dependent on the

depth of the crystal since the crystal is composed of layers called lamella, that change in

orientation depending on the bending curvature, as shown in figure 1.6. At the crystal‟s

entrance surface y is defined as:

31

𝑦𝑜 = 𝑧

𝑏 𝑃|𝛹𝐻 | (3.0.3)

where z is defined in equation (2.0.5), P is the polarization factor, and b = γo

γ𝑕

is the

asymmetry factor. For crystals bent cylindrically, y = yo + cA where A is the reduced

thickness and c is a dimensionless variable defined as [5]:

𝐴 = 𝜋𝑃 |𝛹𝑜 |

𝜆 | 𝑠𝑖𝑛 𝜃𝐵−𝛼 𝑠𝑖𝑛(𝜃𝐵 +𝛼)|𝑡 (3.0.4)

𝑐 = 𝑠𝑖𝑛 𝜃𝐵−𝛼 (𝑏−1)

𝜋 |𝛹𝐻 |2𝑏𝜌[1 + 𝑏 1 + 𝜐 𝑠𝑖𝑛2 𝜃𝐵 + 𝛼 ] (3.0.5)

where t is the crystal thickness, λ is the wavelength, α is the angle between diffraction

plane and surface, and ρ is the bending radius of the crystal.

3.1 Cylindrically Bent and Perfect Laue Crystals

3.1.1 Algorithm for optimizing integrated reflectivity

In XCRYSTAL_BENT, the parameters that were kept fixed for the Laue geometry were

the Si (1,1,1) reflection, the Penning-Polder method, a Poission ratio of 0.2202, σ-

polarization and 2000 scanning points. Parameters min and max y, which are the

normalized scanning angles for the diffraction profile, were chosen by making sure the

scanning range of the rocking curve plots were large enough to cover the entire curve.

The asymmetry angles 90°, 60°, and 30° (angles between crystal plane and surface) were

initially used, and angles in increments of 5° and 1° were used to narrow down the angle

α that optimized throughput. The bending radius was varied from the minimum bending

radius defined by equation (3.0.1) to 100,000 m which simulates a flat Laue, in increasing

32

increments of 5 m or 10 m. The thickness was varied until the bandwidth of the rocking

curve plots, Δθ or the FWHM, matched the desired bandwidth calculated by constraining

%ΔE/E = 0.2%. Analysis of optimized throughputs for energies 8 keV, 17 keV, 24 keV

and 50 keV were performed and comparisons were made with the Bragg geometry using

the Zachariasen method in XCRYSTAL_BENT. This was done to determine at what

energy the optimized throughput of bent Laue geometry would be preferred over a

perfect Bragg geometry.

The Zachariasen method in XCRYSTAL_BENT allows for calculations involving flat

Bragg crystals, ignoring parameters such as the bending radius and the Poission ratio.

The Zachariasen method was used to make comparisons to the cylindrically bent results

as the x-axis of a rocking curve plot is already in units of this normalized y, unlike the

program XCRYSTAL which has the scanning angle in units of degrees. Both Penning-

Polder and Zachariasen methods have their scanning angles in terms of y and are required

as input parameters for the calculations.

An additional constraint was imposed to ensure the crystal was not bent more than it is

realistically possible for its thickness. This condition was defined as:

𝑇𝑕𝑖𝑐𝑘𝑛𝑒𝑠𝑠

𝐵𝑒𝑛𝑑𝑖𝑛𝑔 𝑟𝑎𝑑𝑖𝑢𝑠= ≤ 10−3 =

0.5𝑚𝑚

500𝑚𝑚 (3.1.1).

The algorithm used for optimizing the maximum throughput at the crystal for a

cylindrically bent Laue can be summarized using the following steps:

1. Choose the Si (1,1,1) reflection, energy and desired energy bandwidth ΔE = 0.2%E.

2. Pick an asymmetry angle, such as 90°.

3. Calculate the minimum bending radius, 𝜌 ≅2𝛬𝑠𝑖𝑛𝜃

|𝛹𝐻 | and using equation (3.0.2) to

33

obtain Λ and 𝛹𝐻 = −4𝜋𝑒2𝐹𝐻

𝑚𝑤𝑜2𝑉

.

4. Vary the bending radius in increasing increments of 5 m or 10 m from the minimum

bending radius.

5. Calculate the desired bandwidth 𝛥𝜃 = 𝛥𝐸

𝐸𝑐𝑜𝑡 𝜃𝐵, for a desired energy resolution.

6. For each bending radius, thickness is varied until FWHM, Δθ, produced from

rocking curve plots matches the desired bandwidth Δθ.

7. Check to see that for all bending radii the constraint 𝑇𝑕𝑖𝑐𝑘𝑛𝑒𝑠𝑠

𝐵𝑒𝑛𝑑𝑖𝑛𝑔 𝑟𝑎𝑑𝑖𝑢𝑠 ≤ 10−3 is

satisfied.

8. Get integrated area from rocking curve.

9. Repeat steps 3 through 7 for asymmetry angles 60° and 30°

10. Determine which asymmetry angle has the most integrated area and pick

asymmetry angles 5° above and below the asymmetry angle that has the most

throughput and repeat steps 3 through 8.

11. Pick asymmetry angles with the same similar procedure as in step 10, in 1°

increments in order to arrive at the asymmetry angle that produces the most

throughput.

The throughput for the flat symmetric Bragg geometry was calculated for thicknesses

greater than 1mm in order to simulate an infinitely large Bragg crystal, as crystal

thicknesses of 0.5 mm or greater are typically used. Comparisons of throughput were

made between the optimized cylindrically bent Laue crystal and an infinitely thick

symmetric Bragg crystal. The reason for choosing an asymmetry angle before

calculating the minimum bending radius is because the Pendelӧsung length, Λ, is a

34

function of the direction cosines which is dependent on the asymmetry angle and the

Bragg angle.

Using XCRYSTAL_BENT program, it was observed that the throughput increased with

decreasing thickness and bending radii. For crystals with a smaller thickness there is less

attenuation of the diffracted intensity due to absorption as less atomic planes exist for the

beam to transverse through. Crystals bent to smaller bending radii experience more strain

resulting in increased rocking curve widths. Rocking curve widths takes into account an

additional term besides the Darwin Width, involving the change in d-spacing and lattice

plane orientation. It is for this reason that a minimum bending radius was first

investigated for each chosen asymmetry angle.

3.1.2 Analysis of Si (1,1,1) cylindrically bent and perfect Bragg crystals at 8 keV

Since the energy of 8.048 keV was previously investigated for both flat symmetric

Laue and Bragg, it was appropriate to initially analyze cylindrically bent Laue at the

same energy. A desired energy resolution of 0.2 % would require the energy bandwidth,

ΔE, to be ~16 eV, however a bandwidth of ~10 eV or an energy resolution of ~1.2 % was

used in the calculations. Using the algorithm described in section 3.1.1 the thickness was

varied until the FWHM of the rocking curve corresponded to the desired bandwidth using

the equation described in step 5 where the energy resolution (3.1.2) was rearranged for

the bandwidth, Δθ.

𝛥𝐸

𝐸= 𝑐𝑜𝑡𝜃𝐵𝛥𝜃 (3.1.2)

At 8keV for the case of Si (1,1,1) sigma polarized, optimized parameters for

cylindrically bent Laue are α=34° (angle between plane and surface), bending radius 1.1

35

m and thickness 0.21 mm with ΔE=10 eV. At this energy Bragg is the better choice of

geometry than that of the optimized cylindrically bent Laue crystal by a factor of ~365 in

throughput (row 1 to row 2, table 3.1). The asymmetry angle affects the throughput by

up to a factor of 5 as the asymmetry angle was decreased from 60° to 34° as shown in

figure 3.1.

Figure 3.1: Optimized throughput results from program XCRYSTAL_BENT at 8 keV

for a cylindrically bent Laue as asymmetry angle is varied and ΔE = 10 eV

is kept fixed.

36

Table 3.1: Throughput results from program XCRYSTAL_BENT at 8 keV for a

cylindrically bent Laue optimized for throughput and an infinitely thick flat

symmetric Bragg crystal.

Penning-Polder method for Laue Crystals

Row

# α

Min

Y

Max

Y ρ (m)

Thickness

(mm) Area

∆θ

(mRad)

∆E

(eV)

Peak

Reflectivity

1 34° -500 20 1.1 0.21 0.000006595 0.3272 10.39 0.03%

Zachariasen method for Bragg Crystals

Row

# α

Min

Y

Max

Y ρ (m)

Thickness

(mm) Area

∆θ

(mRad)

∆E

(eV)

Peak

Reflectivity

2 0° -25 25 ∞ ∞ 0.00240679 0.0369 1.17 94.27%


To analyze the effects that energy has on the throughput the energy 17 keV was

chosen. The energy resolution desired for this energy was 0.2% and the same procedure

was implemented as in the 8 keV case. Optimized parameters for bent Laue are α=40°,

bending radius of 6.46 m and thickness of 1.108 mm with ΔE=34 eV (0.2% energy

resolution). For this energy of 17 keV, Bragg is the better choice of geometry by a factor

of ~1.44 in throughput (row 1 to row 2, table 3.2) than that of a cylindrically bent Laue

crystal.





Row

# α

Min

Y

Max

Y ρ (m)

Thickness

(mm) Area

∆θ

(mRad)

∆E

(eV)

Peak

Reflectivity

1 40° -10000 10 6.46 1.108 0.000807 0.2347 34.07 6.10%


Row α Min Max ρ (m) Thickness Area ∆θ ∆E Peak

37

# Y Y (mm) (mRad) (eV) Reflectivity

2 0° -25 25 ∞ ∞ 0.001165 0.0167 2.423 98.63%


An energy of 24 keV was next chosen to determine if a cylindrically bent Laue would

produce better throughputs in comparison to the flat symmetric Bragg geometry. By

sampling small increments of energy in parameter space, one may determine the energy

boundary where a bent Laue is preferred over a perfect Bragg crystal. Optimized

parameters for bent Laue are α = 36°, bending radius of 11.97 m and thickness of 1.465

mm with ΔE = 48 eV (0.2% energy resolution). Bending the crystal cylindrically causes

the strained Laue to be the better choice of geometry at 24 keV by a factor of ~3.2 when

comparing throughputs to an infinitely thick symmetric flat Bragg crystal, as shown in

table 3.3.





Row

# α Min

Y Max

Y ρ (m) Thickness

(mm) Area ∆θ

(mRad) ∆E

(eV) Peak

Reflectivity

1 36° -10000 10 11.97 1.465 0.002644 0.16539 48.02 28.00%


Row

# α Min

Y Max

Y ρ (m) Thickness

(mm) Area ∆θ

(mRad) ∆E

(eV) Peak

Reflectivity

2 0° -25 25 ∞ ∞ 0.000829 0.01174 3.408 99.32%


38

To see whether this trend continues with having a bent Laue crystal giving off more

throughput than a symmetric Bragg crystal at energies higher than 24 keV, an energy of

50 keV was analyzed. Optimized parameters for bent Laue are α=13°, bending radius of

19.94 m and thickness of 1.275 mm with ΔE=100 eV (0.2% energy resolution). A

cylindrically bent Laue is the better choice of geometry for producing maximum

throughput by a factor of ~11 when comparing areas to an infinitely thick symmetric flat

Bragg crystal, as shown in table 3.4.





Row

# α Min

Y Max

Y ρ (m) Thickness

(mm) Area ∆θ

(mRad) ∆E

(eV) Peak

Reflectivity

1 13° -10000 10 19.94 1.275 0.004406 0.0790736 100.2 97.80%


Row

# α Min

Y Max

Y ρ (m) Thickness

(mm) Area ∆θ

(mRad) ∆E

(eV) Peak

Reflectivity

2 0° -25 25 ∞ ∞ 0.000399 0.0057373 7.272 99.86%

39

Chapter 4

A single crystal saddle bent Laue

A computational investigation of flat Ge and Si crystals using programs nBeam and

XCRYSTAL was performed in order to obtain parameters that maximize throughput. In

addition, cylindrically bent Si was investigated using program XCRYSTAL_BENT.

Calculations were then performed for single-crystal and double-crystal Laue

monochromators where in both cases, the crystals are bent into saddle shapes.

The high energy beamline of the Brockhouse Sector has been designed to work with a

single horizontal bounce monochromator rather than the more common double crystal

option based on the work of Zhong Zhong et al [6, 13-15]. The advantage of using a

double crystal monochromators is that the sample can be kept fixed in position as energy

is varied. The exit beam is fixed at the sample and only the monochromator crystals need

to be rotated and translated to satisfy the Bragg condition. A disadvantage of double bent

Laue monochromators is the challenge of maintaining parallelism between Bragg planes

while varying both crystal curvatures to produce a quality focus at the sample. The high

energy x-ray diffraction beamline will be located adjacent to an undulator beamline. For

this reason, the single bounce Laue option has the important advantage of shifting the

position of the end station (diffractometer) to the side and away from the undulator

beamline.

40

The advantages of using just one saddle bent Laue is its simplicity. Concerns of how

the beam couples between the crystals are eliminated, and only one crystal needs to be

rotated to match the correct Bragg angle for a desired energy. In addition, if the crystal is

bent to satisfy the inverse-Cauchois geometry, where the incident divergence is

conserved, the geometry using a single saddle bent crystal requires the source and virtual

image be located on the Roland circle, as show in figure 4.1. Disadvantages of using just

one saddle bent Laue is that as energy is varied, the diffracted beam swings so that the

sample needs to be moved to follow the beam focal point. In addition, because only one

crystal is responsible for focusing, the crystal needs to be bent tighter than a double

saddle bent Laue monochromator.

Figure 4.1: Roland Circle where an incident diverging beam starting on the sphere of a

41

circle hits the crystal and diffracts, coming into focus on the surface of the

sphere.

The geometry of a single saddle bent Laue is depicted in figure 4.2. For the CLS

beamline, the diffracted beam exits the monochromator crystal in the horizontal plane. In

the figure the horizontal diffraction plane is the x-y plane. The sagittal axis, y, has a

curvature perpendicular to the diffraction plane with a radius called sagittal radius, Rs. A

positive Rs implies the incident x-ray are impingent on a convex sagittal curvature. If the

crystal has no asymmetry angle then the sagittal radius will have no focussing effect. If

an asymmetry angle is introduced, the sagittal curvature can cause focusing for a Laue

diffracted beam passing from the fourth quadrant of the figure 4.2. As the asymmetry

angle increases for a fixed bending radius the diffracted beam comes into vertical focus at

shorter focal lengths. Perpendicular to the sagittal curvature is the ever present anticlastic

curvature resulting in the saddle shape of the monochromator. The anticlastic curvature

is in the horizontal plane which is also the diffraction plane. For this reason the

anticlastic curvature corresponds to a meridional curvature of radius Rm. The bending

axis of the anticlastic curvature is the meridional axis, z. When Rm is positive, the

incident beam is impingent on a concave meridional curvature. We take advantage of the

saddle shape of the monochromator to obtain: sagittal (vertical) focussing, an increased

flux, and reduced energy resolutions because of a properly chosen Rm.

The calculations in Zhong Zhong et al. [6, 13-15] are appropriate for a double crystal

monochromator working in the vertical diffraction plane in contrast to our single crystal

monochromator working in the horizontal diffraction plane. In the reference frame of the

crystal, we chose to keep the x-y-z axes and all conventions for the asymmetric angle, Rs,

42

and Rm identical to Zhong Zhong et al. In the reference frame of the lab, the z-axis for

the calculations happens to be the vertical direction while for Zhong Zhong et al. it is in

the horizontal and perpendicular to the incident beam direction. The equations derived

by Zhong Zhong et al. are therefore valid for the CLS design monochromator, and are re-

examined here with the necessary modifications made to account for the change of the

plane of diffraction.

Figure 4.2: A single saddle bent Laue geometry, where the incident beam hits the

crystal between the crystal planes and the surface normal. The diffraction

vector may be in the first or second quadrants for focusing of x-rays.



43

4.1 Theoretical single and double saddle bent Laue

4.1.1 Fundamental Equations

In order to gain confidence in our calculations we first reproduced the calculations

presented in the literature [6, 13-15]. Eventually our results agreed within a very small

margin of error. The parameters investigated in the next few sections are: 𝜃𝐵 , fs, Rs, Rm,

C, C Roland, elastic compliance coefficients, ΔθBragg (T), Δθrot (T), Δθ (T), rocking

curve width, variation of angle of incidence, Energy Resolution, Q, B, A, integrated

reflectivity, mass attenuation coefficients, peak reflectivity and the diffracted intensity at

the crystal.

4.1.1 Parameter 𝜽𝑩: Bragg angle

Using these several fundamental equations:

𝐸 (𝑒𝑉) =𝑕𝑐

𝜆 =

12398 .562 Å𝑒𝑉

𝜆 (4.1.1)

λ = 2dsinθB (4.1.2)

𝑑 Å = 5.431 (𝑓𝑜𝑟 𝑠𝑖𝑙𝑖𝑐𝑜𝑛 )

𝑕2+𝑘2+𝑙2 (4.1.3)

we rearrange these equations to get the Bragg angle:

𝜃𝐵 = 𝑠𝑖𝑛−1 𝜆

2𝑑 = 𝑠𝑖𝑛−1

12398.562

2𝑑𝐸 (4.1.4)

where h, k and l are the indices of the primary reflection.

4.1.2 Parameters fs, Rs and Rm

From optical theory the focal length, fs, for a single crystal is defined as the distance

which initially collimated rays are brought to a focus, and can be determined by the

44

equation:

1

𝑓𝑠=

1

𝐹1+

1

𝐹2 (4.1.5)

where F1 is the distance from the source to the crystal and F2 is the distance from the

crystal to the sample. For the CLS beamline F1 is fixed at 23 m, and F2 has a preferred

distance of 8 m due to layout constraints resulting in a focal length of 5.94 m. This focal

length value is used in calculating the sagittal radius, Rs, perpendicular to the diffraction

plane.

Radius sagittal, Rs, as defined by Zhong Zhong et al. [6], for the case of one and two

saddle bent Laue crystals are:

Rs (single crystal) = 2sinθBsinχfs (4.1.6)

Rs (double crystal) = 4sinθBsinχfs (4.1.7)

Equation 4.1.6 was used as the appropriate equation to obtain the focusing of x-rays in

the vertical for the CLS beamline. The asymmetry angle, χ (90- α), is the angle between

the Bragg planes and the surface normal. The asymmetry angle was calculated by taking

the dot product of the reflection and the surface normal to get angle α, the angle between

the plane and the surface, and subtracting this angle from 90° to get χ. Radius

meridional, Rm, for both single and double saddle bent Laue cases is defined as:

𝑅𝑚 = 𝑅𝑠

𝐶𝑆′23 (4.1.8)

where C is a unitless constant (discussed in the next section) and S‟23 is the Poisson ratio.

Calculations for two saddle bent Laue crystals were performed in Excel. Once the

focal length and CS'23 values were determined for each case (reflection, energy, and

orientation) in the literature we were able to obtain the exact same values for Rs and Rm

45

as were tabulated in the references (table 4.1).

Table 4.1: Table of Si (1,1,1) with a surface normal (1,0,0) and thickness 0.4 mm

from reference [16] for a double saddle bent Laue.

4.1.3 Parameter C

The relationship between the sagittal radius, Rs, and the meridional (anticlastic)

radius, Rm, is dependent on the aspect ratio of the monochromator crystal. This can be

reduced to a variable called C. This variable C is a unitless value, typically ranging

between 0.2 and 1 [15]. C is calculated using the equation:

𝐶 = 𝑅𝑠

𝑅𝑚 𝜈 (4.1.9)

where ν, also defined as S'23, is the calculated Poisson ratio. Values of C depend on the

shape of the crystal and bending mechanism. The values of C for different orientations

may be similar for crystals with the same size, thickness, and similar type of bender used.

Zhong Zhong et al. uses crystal sizes of 40 mm high by 100 mm wide, and thickness of

0.4 mm -0.7 mm with a 4 bar bender for measuring the rocking-curve widths of saddle

bent crystals [15].

It is possible to experimentally determine this value of C by measuring the sagittal and

meridional radius for a fixed orientation and energy which gives the value of Cν, and

dividing this by the calculated Poission ratio, S'23 [13]. Experimentally, radius sagittal is

46

set manually to a certain radius by adjusting the four-bar bender. This is done by having

a divergent laser beam hitting the polished surface side of the crystal and having it

focused in the plane of bending. Radius meridional is determined by measuring the

rocking curves at different positions of the crystal using the equation [13]:

𝑅𝑚 = 𝛥𝑕

𝛥𝜃 𝑐𝑜𝑠 𝜒−𝜃𝐵 (4.1.10)

where Δh is the vertical translation of the crystal and Δθ is the angular difference between

center positions of two rocking curve measurements. Table 4.2 shows measured values

by Zhong Zhong et al. for Cν, or alternatively CS'23 for different orientations of saddle

shaped Si crystals with an aspect ratio of 40 mm by 100 mm.

Table 4.2: Literature values of measured CS'23 for a single saddle bent Laue with

different orientations using a silicon wafer [15].

Crystal Thickness

Crystal Orientations Cν (mm)

x'[100] y'[011] z'[0-11] CS'23=0.04 0.67

x'[511] y'[-255] z'[0-11] CS'23=0.058 0.67

x'[111] y'[1-10] z'[11-2] CS'23=-0.19 0.67

x'[111] y'[-211] z'[0-11] CS'23=-0.19 0.67

The value of C is required to be determined if a crystal has a different thickness and

aspect ratios than those used in literature. At present, ANSYS simulations at the CLS are

being performed to determine how changing the aspect ratio of a crystal affects the value

of C and what constrains C usually between 0.2 and 1 in terms of crystal dimensions. It

is only known that the thickness, the shape of a crystal and the type of bending

mechanism directly affects the magnitude of C.

47

4.1.4 Parameter C Roland

If the saddle bent Laue is meridionally bent in the inverse-Cauchois geometry, the

incident divergence is conserved. For the case when the source and the centre of

meridional bending are on different sides of the lattice planes, the following equation

holds true for the inverse-Cauchois geometry[6]:

𝐹1 = 𝑅𝑚 𝑐𝑜𝑠 𝜒 + 𝜃𝐵 = 𝑅𝑠

𝐶𝜈 𝑐𝑜𝑠 𝜒 + 𝜃𝐵 (4.1.11)

and rearranging for C Roland the equation becomes:

𝐶 𝑅𝑜𝑙𝑎𝑛𝑑 = 𝑅𝑠

𝐹1𝜈 𝑐𝑜𝑠 𝜒 + 𝜃𝐵 (4.1.12)

Using this equation the calculated sagittal radius, Poisson ratio, χ and θB values may be

substituted in order to define the variable C, which would satisify the inverse-Cauchois

geometry as parameters are varied in the program SagBentApp of the appendix. Further

investigation using ANSYS simulations will perhaps define the corresponding aspect ratio

for each C value.

4.1.5 Elastic Compliance Coefficients

Elastic compliance coefficients are defined as the proportionality constants between

stress and strain according to the generalized Hook's law, and are a function of the

direction cosines. Elastic coefficients need to be calculated as it is common that the

orientation of the crystal used is not described by the cubic axes coordinate system, but

rather by an arbitrary rectangular coordinate system. Elastic compliance coefficient

expressions most commonly used in calculated parameters are S'23, the Poission ratio, and

48

S'13, found in the paper by Wortman and Evans [17]. The coefficient S'63 had to be

derived from the general relation for deriving compliance coefficients using [18] and

some knowledge of the rules when changing subscripts from matrix notation to tensor

notation (S'63=2S'1233 from matrix to tensor notation).

The crystal cubic axes are defined as: x=[1,0,0], y=[0,1,0] and z=[0,0,1]. The

orientation of the crystal is described with primes and indicates an arbitrary rectangular

coordinate system. The surface normal is defined as x'=[h1',k2',l3'], the sagittal bending

axis as y'=[h4',k5',l6'], and the meridional bending axis as z'=[h7',k8',l9'].

The general relation for deriving the elastic compliance coefficients is given by the

equation [18]:

S‟ijkl (cm²/10¹²dyn) = δijδklS12 + ¼ (δikδjl + δilδjk)S44 +aimajmakmalmSo (4.1.13)

where δ are Kronecker deltas, art are the direction cosines, S12 = -0.214 cm²/10¹²dyn, S44

= 1.26 cm²/10¹²dyn, S11 = 0.768 cm²/10¹²dyn, and So = S11 – S12 – ½ S44 [17]. The

following equations for the compliance coefficients are of most interest:

S‟13 (cm²/10¹²dyn) = S12 + So(l1²l3²+m1²m3²+n1²n3²) (4.1.14)

S‟23 (cm²/10¹²dyn) = S12 + So(l2²l3²+m2²m3²+n2²n3²) (4.1.15)

S‟63 (cm²/10¹²dyn) = 2So(l1l2l3²+m1m2m3²+n1n2n3²) (4.1.16)

The direction cosines used in calculating these elastic compliance coefficients in the

spreadsheets and program are as follows:

l1= cosθ1 = (x‟[h1‟,k2‟,l3‟]•x[1,0,0]) /(√(h1‟²+k2‟²+l3‟²)*√(1²)) (4.1.17)

m1= cosθ2 = (x‟[h1‟,k2‟,l3‟]•y[0,1,0]) /(√(h1‟²+k2‟²+l3‟²)*√(1²)) (4.1.18)

n1= cosθ3 = (x‟[h1‟,k2‟,l3‟]•z[0,0,1]) /(√(h1‟²+k2‟²+l3‟²)*√(1²)) (4.1.19)

l2= cosθ4 = (y‟[h4‟,k5‟,l6‟]•x[1,0,0]) /(√(h4‟²+k5‟²+l6‟²)*√(1²)) (4.1.20)

49

m2= cosθ5 = (y‟[h4‟,k5‟,l6‟]•y[0,1,0]) /(√(h4‟²+k5‟²+l6‟²)*√(1²)) (4.1.21)

n2= cosθ6 = (y‟[h4‟,k5‟,l6‟]•z[0,0,1]) /(√(h4‟²+k5‟²+l6‟²)*√(1²)) (4.1.22)

l3= cosθ7 = (z‟[h7‟,k8‟,l9‟]•x[1,0,0]) /(√(h7‟²+k8‟²+l9‟²)*√(1²)) (4.1.23)

m3= cosθ8 = (z‟[h7‟,k8‟,l9‟]•y[0,1,0]) /(√(h7‟²+k8‟²+l9‟²)*√(1²)) (4.1.24)

n3= cosθ9 = (z‟[h7‟,k8‟,l9‟]•z[0,0,1]) /(√(h7‟²+k8‟²+l9‟²)*√(1²)) (4.1.25)

where each of the above direction cosines is calculated for the angle between the cubic

axis and the crystal orientation axis. Where it is understood that the direction cosine

between x' and x is taken as x'[h1',k2',l3']•x[1,0,0] = 𝑕1′² + 𝑘2′² + 𝑙3′² 1 cosθ as

described by Mason [19].

Using these direction cosines, the elastic compliance coefficients were calculated in

units of cm²/10¹² dyn. In order to make these elastic compliance coefficients unitless,

Zhong Zhong et al. [14] defines S‟ij as Sij/S33 (unitless), where Sij are for a specific

orientation. Similarly, the calculated elastic compliance coefficients may be normalized

by a factor, F = 1

𝑆33′ (𝑐𝑚 ²/10¹² 𝑑𝑦𝑛 ,𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 )

, found by equating:

S‟33 (cm²/10¹² dyn , calculated) x F = S‟33 (unitless, Z.Z.) = 𝑆33

𝑆33 = 1 (4.1.26)

where F = 1.689 in units of 10¹² dyn/ cm² for all crystal orientations of interest. The

calculated elastic compliance coefficients agree with those obtained in literature, as in

table 4.3.

Table 4.3: Elastic compliance coefficients calculated and those obtained by Zhong

Zhong et al., Z.Z. [15].

x'[1,0,0]

x'[5,1,1]

x'[1,1,1]

x'[1,1,1]

y'[0,1,1]

y'[-2,5,5]

y'[1,-1,0]

y'[-2,1,1]

z'[0,-1,1]

z'[0,-1,1]

z'[1,1,-2]

z'[0,-1,1]

50

Z.Z.

(Unitless)

Calculated

values

(unitless)

Z.Z.

(Unitless)

Calculated

values

(unitless)

Z.Z.

(Unitless)

Calculated

values

(unitless)

Z.Z.

(Unitless)

Calculated

values

(unitless)

S'13 -0.36 -0.361

-0.34 -0.339

-0.16 -0.163

-0.16 -0.163

S'23 -0.061 -0.0642

-0.083 -0.086

-0.26 -0.262

-0.26 -0.262

S'63 0 0

0.16 0.156

0 0

0.28 0.280

It is evident from table 4.3 that the calculated elastic compliance coefficients agree

with those obtained in the literature. In addition, S'23 also known as the Poisson ratio, ν,

is the ratio of the resulting transverse contraction strain and longitudinal extension strain

that occur when a crystal is stretched. This ratio changes for different orientations

because crystalline silicon is anisotropic [6].

4.1.6 Parameters ΔθBragg (T), Δθrot (T) and Δθ (T)

An important parameter that contributes to the energy resolution is Δθ (T), the angle in

which the crystal needs to rotate to satisfy the Bragg condition for a lamella at depth T.

The term Δθ (T) is defined as [14]:

Δθ (T) = -Δθrot (T) + ΔθB (T) (4.1.27)

where ΔθB (T), as seen in figure 4.3, is the change in Bragg angle due to the change in d-

spacing at the depth T in the crystal. ΔθB (T) is usually of the order of a few tens of µrads

[14]. The term Δθrot (T) is defined as the change in the angle between lattice planes and

the incident beam as seen in figure 4.4. These equations are defined as:

ΔθB (T) = - 𝑇

𝑅𝑠 tanθB[S‟13sin²χ + CS‟23cos²χ + S‟63sinχcosχ] (4.1.28)

Δθrot (T) = - 𝑇

𝑅𝑠 [(S‟13 - CS‟23)sinχcosχ – CS‟23tan(χ-θB) + S‟63cos²χ] (4.1.29)

where thickness is denoted as T. The elastic compliance coefficients are defined by

51

equations (4.1.14) through (4.1.16).

Figure 4.3: The change in d-spacing that contributes to ΔθB (T)

52

Figure 4.4: Lattice plane rotation that contributes to Δθrot (T)

Substitution of (4.1.28) and (4.1.29) into (4.1.27) results in the following equation:

Δθ (T) = 𝑇

𝑅𝑠 [(S‟13 - CS‟23)sinχcosχ – CS‟23tan(χ-θB) + S‟63cos²χ] (4.1.30)

- 𝑇

𝑅𝑠 tanθB[S‟13sin²χ + CS‟23cos²χ + S‟63sinχcosχ]

Using the parameter CS'23 from [14, table 1] for the case of a single saddle bent Laue

at 18 keV, and keeping Rs fixed at 0.76 m, calculations were performed and comparisons

were made to values obtained in literature as shown in tables 4.4 and 4.5. The calculated

results agree very well with those obtained in the journal [14], confirming that the

parameters were calculated correctly.

Lattice Planes

53

Table 4.4: Si (1,1,1) calculations of ∆θB (T), Δθrot (T), ∆θ (T), and ω, the rocking

curve width, for a single saddle bent Laue with 0.67 mm thickness.

Included are calculated values of these parameters in literature [14].

Si (1,1,1) x' [1,0,0] y' [0,1,1] z' [0,-1,1]

Z.Z.

Values

Our

Calculated

Values

%

Difference

∆θB (T) (µrad) 14 14.63 4.50

∆θrot (T) (µrad) 114 113.13 0.76

∆θ (T) (µrad) -100 -98.50 1.50

ω, theo. (µrad) 101 100.20 0.79

Table 4.5: Si (-1,1,1) calculations of ∆θB (T), Δθrot (T), ∆θ (T), and ω, the rocking

curve width for a single saddle bent Laue with 0.67 mm thickness.

Si (-1,1,1) x' [1,0,0] y' [0,1,1] z' [0,-1,1]

Z.Z.

Values

Our

Calculated

Values

%

Difference

∆θB (T) (µrad) 14 14.30 2.14

∆θrot (T) (µrad) -102 -102.35 0.34

∆θ (T) (µrad) 117 116.64 0.31

ω, theo. (µrad) 118 117.13 0.74

4.1.7 Rocking curve width, ω

The rocking curve width was previously defined as the Darwin width for crystals in

flat Bragg geometry. The rocking curve width, ω, for the case of strained Laue crystals is

54

the convolution of the Darwin width, and ∆θ (T), a term that takes into account both the

change in the d-spacing and the change in the lattice plane orientation. It is defined as

[14]:

ω ≈ 𝛥𝜃²(𝑇) + 𝑊𝑎 ² (4.1.31)

where Δθ (T) is defined using (4.1.30), the Darwin Width, Wa, and the rest of the

variables are defined using [20] and [21] as:

𝑊𝑎 = 2𝜆

𝜋𝛬𝑠 𝑠𝑖𝑛 2𝜃

𝛾𝐻

𝛾𝑜 (4.1.32)

𝛬𝑠 = 𝑉𝑐

𝑟𝑒𝜆𝐾 𝐹𝐻 (4.1.33)

ɣo = cos(χ-θB) (4.1.34)

ɣH = cos(χ+θB) (4.1.35)

where Λs is the extinction length for a symmetrical reflection, K is the polarization, FH is

the structure factor calculated [12], Vc is the volume of a units cell, re is the radius of an

electron, ɣo is the incident direction cosine and ɣH is the reflected direction cosine.

Looking at tables 4.4 and 4.5 above, we can see that the rocking curve width calculated is

off by less than 1 µrad. This confirms that the calculations are working properly.

4.1.8 Variation of angle of incidence, Δθ, and Energy Resolution, 𝜟𝑬

𝑬

Another important term that contributes to the energy resolution is the variation of

angle of incidence along the surface of the crystal, Δθ. This term is defined as [6]:

𝛥𝜃 = 𝜑𝑕 1 −𝐹1

𝑅𝑚 𝑐𝑜𝑠 𝜒+𝜃𝐵 (4.1.36)

where F1is the distance from the source to the crystal and υh is the horizontal divergence

55

of the incident beam if the diffraction plane is in the horizontal, otherwise it is the vertical

divergence for a beam diffracting in the vertical. This term is independent of the crystal

thickness and if the crystal is bent to satisfy the inverse-Cauchois geometry, this term

becomes zero and the following holds true:

𝐹1 = 𝑅𝑚 𝑐𝑜𝑠 𝜒 + 𝜃𝐵 (4.1.37)

The energy bandwidth of the diffracted X-rays can be defined as [6]:

𝛥𝐸

𝐸=

𝛥𝜃² + 𝛥𝜃 (𝑇)² + 𝜍𝑠𝐹1

2

𝑡𝑎𝑛 𝜃𝐵 (4.1.38)

where Δθ (T) is defined previously in (4.1.30), Δθ is defined using (4.1.36) and σs/F1 is

the horizontal size of the source divided by the source to crystal distance (for a beam

diffracting horizontally). In the spreadsheet calculations and in the program

SagBentApp, the horizontal divergence of 0.3 mrads was used and the horizontal source

size, σs, of 1 mm was used for the CLS beamline.

An energy resolution of ~0.1% or better is desired for the CLS diffraction beamline as

the diffracted intensity is optimized, and parameters such as reflection, energy, F2, χ and

thickness are varied. Good energy resolutions is preferred in order to get better pair

distribution function analysis, PDF's, which is the Fourier transform of the entire

diffraction pattern of a sample. The advantage of using PDF‟s to analyze samples is that

it probes the bulk average positions of all atoms [22]. As the energy resolutions are

enhanced, the peaks of a PDF corresponding to large r are better resolved.

4.1.9 Parameters Q, B and A

In order to calculate the remaining last few parameters such as the integrated

56

reflectivity and the reflectivity, Q, the integrated kinematical reflectivity per unit length

needs to be calculated and it is defined here [15]:

𝑄 = 𝑟𝑒2𝐾2

𝐹𝐻2𝜆3

𝑠𝑖𝑛 2𝜃𝐵 (4.1.39)

where K = 1 for normal polarization or K = |cos2θB|, the radius of an electron, re, and FH

is the structure factor. The problem with (4.1.39) is that the units are m5 and not m-1 as

suggested in literature [13]. Erola et al. defines Q as [20]:

𝑄 = 𝜆

𝛬𝑠2 𝑠𝑖𝑛 2𝜃

(4.1.40)

where Λs was defined using (4.1.33). Using this equation (4.1.40) results in Q being in

the correct units of m-1. Combining equations (4.1.40) and (4.1.33) results in the Q that

was used in the calculations:

𝑄 = 𝑟𝑒2𝐾2

𝐹𝐻2𝜆3

𝑉𝑐2𝑠𝑖𝑛 2𝜃𝐵

(4.1.41)

This derived Q has units of m5/m6 or m-1, and accounts for 1

𝑉𝑐2 that was previously

missing.

The other few parameters that needs to be calculated are B(θB,χ,S‟ij), a unitless

constant, and A, the normalized thickness. Equation (4.1.30) can also be defined using

B(θB,χ,S‟ij) as follows [15]:

Δθ (T) = -Δθrot (T) + ΔθB (T)

= 𝑇

𝑅𝑠ɣ𝑜𝐵(𝜃𝐵 ,𝜒 ,𝑆𝑖𝑗′ )

(4.1.42)

where T is the crystal thickness and ɣo is defined using (4.1.34). We get B by rearranging

57

(4.1.42) as:

𝐵 𝜃𝐵 , 𝜒, 𝑆𝑖𝑗′ =

𝑇

𝑅𝑠ɣ𝑜 𝛥𝜃(𝑇) (4.1.43)

where the absolute value of Δθ (T) was used, in order that normalized thickness, A, would

have a positive extinction length.

The normalized thickness, A, is defined as [15]:

A = RsBQ (4.1.44)

where B, a unitless constant is defined using (4.1.43), and Q, with units of m-1, is defined

using (4.1.41) to give A in units of m/m or dimensionless.

4.1.10 Integrated reflectivity, I, and the mass attenuation coefficient, 𝝁

𝝆

The integrated reflectivity, I, is defined as the area under the rocking curve or the

throughput and is defined as [15]:

I ≈ (𝑡𝑎𝑛 𝑕 𝐴

𝐴)(

𝑄

𝑎µ)𝑒

−𝜇𝑇

𝛾𝐻 [1 - 𝑒−

𝑎𝜇𝑇

𝛾𝑜 ] (4.1.45)

where A, Q, ɣo, ɣH are defined using (4.1.44), (4.1.41), (4.1.34) and (4.1.35) respectively.

The linear attenuation coefficient, µ, and a are defined as:

µ = µ

𝜌 (cm²/g) x ρ (g/cm³) (4.1.46)

a = 1 - ɣo/ɣH (4.1.47)

where the values used for µ/ρ, the mass attenuation coefficients, are found in table 4.6

and were calculated using the equation of a line for the two nearest literature coefficient

values and their corresponding energy values [23]. For coefficient values that are close

in energy, the curve of linear attenuation coefficients versus energy may be approximated

58

by straight lines connecting these coefficients. In the limit that there exist many linear

attenuation coefficients for smaller steps in energy, the straight lines connecting them

appears to be approximating a smooth decaying curve.

Figure 4.5: Mass attenuation coefficients at varying energies using literature values

[23] and connected by straight lines.

Table 4.6: Equations of a line used for calculating mass attenuation coefficients for

energies 10-150 keV.

Energies (keV) Equation used where y is in units of (cm²/g) and x in (keV)

10-15 y=-4.710x+80.99

15-20 y=-1.1752x+27.968

59

20-30 y=-0.3024x+10.512

30-40 y=-0.0739x+3.657

40-50 y=-0.0262x+1.749

50-60 y=-0.0118x+1.029

60-80 y=-0.0049x+0.615

80-100 y=-0.00192x+0.379

100-150 y=-0.000774x+0.2609

4.1.11 Peak reflectivity

Reflectivity, R, is calculated using (4.1.31) and (4.1.45), and is defined in literature as

[15]:

𝑅 ≅ 𝐼

𝜔=

𝐼

𝛥𝜃²(𝑇) + 𝑊𝑎 ² (4.1.48)

where I is the integrated reflectivity and ω is the rocking curve width. For 67 keV with a

fixed sagittal radius of 0.76 m and 0.95 m, and thickness = 0.67 mm, the values of I and

R that Zhong Zhong et al. obtains were reproduced with reasonable agreement, as shown

in table 4.7.

Table 4.7: Calculations of the normalized thickness, A, Reflectivity and Integrated

reflectivity, I, with comparisons made between Zhong Zhong values [15]

and calculated values for varying reflections and orientations.

Si (1,1,1) reflection, Rs = 0.76m, CS'23 = -0.040, x = [1,0,0], y = [0,1,1], z = [0,-1,1]

Z.Z.

Calculation Z.Z.

Measured Our

Calculation

% Difference

between

Calculations

A (unitless) 0.89 0.869 2.36

ω (µrad) 107 118 106.430 0.53

Reflectivity 0.68 0.56 0.663 2.50

Integrated ref. (µrad) 72 68 70.59 1.96

Si (-3,1,1) reflection, Rs = 0.76m, CS'23 = -0.040, x = [1,0,0], y = [0,1,1], z = [0,-1,1]

60

Z.Z.

Calculation Z.Z.

Measured Our

Calculation

% Difference

between

Calculations

A (unitless) 1.4 1.787 27.64

ω (µrad) 37 31 37.2 0.54

Reflectivity 0.79 0.76 0.849 7.47


Si (-1,1,1) reflection, Rs = 0.95m, CS'23 = -0.058, x = [5,1,1], y = [-2,5,5], z = [0,-1,1]

Z.Z.

Calculation Z.Z.

Measured Our

Calculation

% Difference

between

Calculations

A (unitless) 0.53 0.562 6.04

ω (µrad) 150 145 147.1 1.93

Reflectivity 0.46 0.43 0.49 6.52


For the reflections shown in table 4.7, there exists good agreement between calculated

values and those obtained by the calculations produced in literature [15]. These

calculated values differ from experimentally measured values by a few micro radians.

4.1.12 Diffracted Intensity at the crystal

The diffracted intensity may be defined as:

Diffracted Intensity at crystal (ph/s) = Incident flux (ph/s/0.1%bw) x (4.1.49)

0.6 x Reflectivity x (EcotθB 𝛥𝜃²(𝑇) + 𝑊𝑎 ² )/0.1%bw

where the incident flux values that were calculated was for a super conducting wiggler

using XOP „WS‟ algorithm, for a 2.9 GeV, 500 mA electron beam, and for wiggler

parameters: ky = 11.08, period=3.65 cm, B=3.25 T, and 27.5 periods. The incident flux

values are calculated for an incident beam divergence of 0.3 mrad horizontal and 0.3

mrad vertical [3]. A figure of the incident flux for energies up to 100 keV is shown in

61

figure 4.6. This incident flux was multiplied by a factor of 0.6 to reduce the heat load and

prevent melting of the crystal optics and since electron beam current is ~250mA. This

may be accomplished by adding a filter, such as a thin layer of aluminum in front of the

beam in order to reduce the intensity going through the rest of the monochromator. The

reflectivity value was obtained using equation (4.1.48), and Δθ (T) and Wa are defined in

equations (4.1.30) and (4.1.32) respectively. In addition, 0.1%bw is defined as 0.001

multiplied by the energy.

Figure 4.6: Incident flux using XOP 'WS' algorithm for a 2.9 GeV, 500 mA electron

beam and a super conducting wiggler with paramters: ky = 11.08,

period=3.65 cm, B=3.25 T, and 27.5 periods.

62

4.2 Analysis of a single saddle bent symmetric Laue with an infinitely

large bending radius

Presented here is the flat symmetric Laue monochromator crystal case that was used as

a reference for comparison and to gain an idea of the intensities, energy ranges and

energy resolutions for all attainable silicon reflections. A flat symmetric Laue crystal is a

special case of the saddle bent Laue where the sagittal bend radius goes to infinity. As a

result of the very large bend radius there is no anticlastic curvature. Instead of getting

vertical focusing for a beam diffracting in the horizontal, the focal length goes to infinity

as the asymmetry angle, χ goes to 0°. Rearranging equation (4.1.6) for the focal length,

fs, confirms that the focal length does indeed become large:

𝑓𝑠 = 𝑅𝑠

2 𝑠𝑖𝑛 𝜃𝐵 𝑠𝑖𝑛 𝜒 ≈ ∞ (4.2.1)

The Si reflections that were used in the calculations were chosen according to the

selection rules for a fcc diamond structure, where either all indices are odd or even, and

h+k+l are divisible by 4 if all indices are even. The reflections chosen can be divided

into two groups, those with a 2nd

harmonic and those without. We first present

calculations for the (1,1,1) and its harmonics: (3,3,3), (4,4,4) and (5,5,5). Next,

calculations were performed for the (0,2,2) and the 2nd

harmonic (0,4,4). Calculations

were performed for the (4,0,0) which is a reflection with a 2nd

harmonic. Finally

calculations were performed for the low order reflections which do not exhibit a 2nd

harmonic: (3,1,1), (1,3,3), (5,1,1), (5,3,3), (1,5,5), (7,1,1) and (3,5,5). These reflections

were chosen out of many others in order to narrow our search in parameter space, as

63

these reflections cover the energy range of 18 keV to 100 keV, but also because they are

all perpendicular to the (0,-1,1). This final feature makes introducing an asymmetry angle

as a variable convenient if the meridional axis is aligned with the (0,-1,1) for all cases.

4.2.1 Parameters constrained for a flat symmetric Laue

Due to the space allocated for the High Energy beamline at the CLS several

parameters were constrained. The distance from the source to the crystal, F1, was fixed at

23 m. The distance between the sample and the crystal used in the calculations was 8 m.

We defined the incident horizontal and vertical divergences used in the calculations as

0.3 mrads, and for the calculations in this section, it was assumed that the incident beam

was diverging from a point source as opposed to a finite sized source. In addition,

investigation of a 0.5 mm thick crystal was initially chosen. Bragg angles of 3.625° to

6.25° were used for each reflection and the corresponding energy range is tabulated in

table 4.8, using the equation:

𝐸 = 𝑕2+𝑘2+𝑙2

2 𝑠𝑖𝑛 𝜃𝐵

12398 .562 𝑒𝑉Å

5.43102 Å (4.2.2)

Table 4.8: Energies for the Bragg angles 6.25° and 3.625° for all the reflections

used. Included are the calculated surface normal axis from the cross

product of sagittal and meridional axes. The meridional axis [0,-1,1] was

chosen as it was perpendicular to the reflections used in the calculations.

Reflection, meridional

axis z' Surface

normal x'

θB = 6.25° θB = 3.625°

Sagittal Minimum Maximum

axis y' Energy (keV) Energy (keV)

[1,1,1] [0,-1,1] [2,-1,-1] 18.160 31.269

[3,3,3] [0,-1,1] [6,-3,-3] 54.481 93.809

[4,4,4] [0,-1,1] [8,-4,-4] 72.641 125.079

[5,5,5] [0,-1,1] [10,-5,-5] 90.801 156.349

64

[0,2,2] [0,-1,1] [4,0,0] 29.656 51.063

[0,4.4] [0,-1,1] [8,0,0] 59.312 102.127

[4,0,0] [0,-1,1] [0,-4,-4] 41.940 72.214

[4,2,2] [0,-1,1] [4,-4,-4] 51.365 88.444

[3,1,1] [0,-1,1] [2,-3,-3] 34.774 59.877

[1,3,3] [0,-1,1] [6,-1,-1] 45.703 78.694

[5,1,1] [0,-1,1] [2,-5,-5] 54.481 93.809

[5,3,3] [0,-1,1] [6,-5,-5] 68.754 118.386

[1,5,5] [0,-1,1] [10,-1,-1] 74.877 128.929

[7,1,1] [0,-1,1] [2,-7,-7] 74.877 128.929

[3,5,5] [0,-1,1] [10,-3,-3] 80.536 138.673

4.2.2 Beam size at the monochromator and sample positions

The horizontal and vertical beam size at the monochromator is 6.9 mm for a 0.3 mrad

(V) x 0.3 mrad (H) diverging beam from a source 23 m upstream of the monochromator.

This beam size on the crystal was derived using the formula:

𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑏𝑒𝑎𝑚 𝑠𝑖𝑧𝑒 𝑜𝑛 𝑐𝑟𝑦𝑠𝑡𝑎𝑙 𝑚 = 𝐹1 𝑡𝑎𝑛 0.3 ∗ 10−3 (4.2.3)

𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑏𝑒𝑎𝑚 𝑠𝑖𝑧𝑒 𝑜𝑛 𝑐𝑟𝑦𝑠𝑡𝑎𝑙 𝑚 = 𝐹1 (𝑡𝑎𝑛 0.3 ∗ 10−3 ) (4.2.4)

As there is no sagittal bend about the sagittal axis, y', the beam continues to diverge

vertically, as shown in figure 4.7. Using geometry, the vertical beam height at the sample

was calculated as:

𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑏𝑒𝑎𝑚 𝑠𝑖𝑧𝑒 𝑎𝑡 𝑚𝑜𝑛𝑜𝑐𝑕𝑟𝑜𝑚𝑎𝑡𝑜𝑟

𝐹1=

𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑏𝑒𝑎𝑚 𝑠𝑖𝑧𝑒 𝑎𝑡 𝑠𝑎𝑚𝑝𝑙𝑒

𝐹1 + 𝐹2

𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑏𝑒𝑎𝑚 𝑠𝑖𝑧𝑒 𝑎𝑡 𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑚 = 23+8

23 6.9 𝑚𝑚 = 9.3 𝑚𝑚 (4.2.5)

65

Figure 4.7: Beam diffracting in the vertical, perpendicular to the diffraction plane,

continues to diverge if χ = 0°. Largest vertical beam size at the sample is

9.3 mm.

In the horizontal for a symmetric flat Laue crystal there exists 1:1 focusing. Upon

diffraction, the horizontal beam with a F1 of 23 m will have its rays converge at a sample

approximately 23 m away, as shown in figure 4.8. A finite sized sample placed at 8 m

from the crystal may only have a fraction of the converging beam strike the surface.

Using geometry, the horizontal beam height at the sample was calculated as:

𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑏𝑒𝑎𝑚 𝑠𝑖𝑧𝑒 𝑎𝑡 𝑚𝑜𝑛𝑜𝑐𝑕𝑟𝑜𝑚𝑎𝑡𝑜𝑟

𝐹1=

𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑏𝑒𝑎𝑚 𝑠𝑖𝑧𝑒 𝑎𝑡 𝑠𝑎𝑚𝑝𝑙𝑒

𝐹1 − 𝐹2

66

𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑏𝑒𝑎𝑚 𝑠𝑖𝑧𝑒 𝑎𝑡 𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑚 = 23−8

23 6.9 𝑚𝑚 = 4.5 𝑚𝑚 (4.1.6)

Therefore the total size of the beam at the sample position, 8 m from the monochromator,

is 9.3 mm high and 4.5 mm wide.

Figure 4.8: Top view of a beam diffracting in the horizontal, parallel to the diffraction

plane, and coming to a focus at the distance F1, 23 m away. Largest

horizontal beam size at the sample is 4.5mm.

4.2.3 Energy resolution and diffracted intensity for a flat symmetric Laue

In the energy resolution equation, 𝛥𝐸

𝐸=


2

𝑡𝑎𝑛 𝜃𝐵, the term 𝛥𝜃 is defined as

𝛥𝜃 = 𝜑𝑕 1 −𝐹1

𝑅𝑚 𝑐𝑜𝑠 𝜒+𝜃𝐵 and since Rm is infinity the term 𝛥𝜃 simply becomes 𝜑𝑕 , the

horizontal divergence. The term 𝛥𝜃 (𝑇) in equation (4.1.30) goes to zero as the bending

radius increases, as it is proportional to the inverse of the sagittal radius. Therefore, the

67

energy resolution at the sample becomes:

𝛥𝐸

𝐸=

𝜑𝑕 𝑕𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒 𝑚𝑚

4.5𝑚𝑚 ² +

𝜍𝑠𝐹1

2

𝑡𝑎𝑛 𝜃𝐵 (4.1.7)

where the horizontal beam size of the source, σs, is 1 mm.

The diffracted intensity at the sample using the general equation (4.1.49) becomes:

Diffracted Intensity at sample (ph/s) = Incident flux (ph/s/0.1%bw) x (4.1.8)

0.6 x 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒 𝑚𝑚

9.3𝑚𝑚 x

𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒 𝑚𝑚

4.5𝑚𝑚 x Reflectivity

x (EcotθB 𝑊𝑎 ² )/0.1%bw

where the ratio of the sample size to the beam size at the sample position is now included.

In addition, 𝛥𝜃 (𝑇) in the rocking curve width equation goes to zero with very large bend

radii. Incident flux was calculated using XOP „WS‟ algorithm and the parameters defined

in section 4.1.12. Reflectivity's were calculated using the XOP „XCRYSTAL‟ algorithm,

and the Pendellӧusng fringes were convoluted with a Gaussian profile as shown in figure

4.10, resulting in a FWHM within 20% of the Darwin Width. The Pendellӧusng fringes

observed in figure 4.9 are a result of interference of the forward and diffracted waves,

which periodically exchange their energy as the beam traverses the crystal.

68

Figure 4.9: Rocking curve plot of Si (5,1,1) symmetric Laue with Pendellӧusng fringes

at 55 keV and 0.5 mm thickness. The y-axis is the reflectivity and the x-

axis is a scan of the Bragg angles. Input parameters used are shown using

program XCRYSTAL.

69

Figure 4.10: Rocking curve plot of figure 4.9 convoluted with a Gaussian profile for Si

(5,1,1) symmetric Laue at 55 keV and 0.5 mm thickness. The y-axis is

reflectivity and the x-axis is a scan of the Bragg angles. Input parameters

used are shown using program XCRYSTAL.

4.2.4 Analysis of diffracted intensity and energy resolution at the sample

For each reflection, the meridional axis chosen was the [0,-1,1], as this axis was

perpendicular to all chosen reflections. In addition, for a symmetric Laue crystal the

primary reflection is parallel to the bending axis, y'. The surface normal was determined

by calculating the cross product between the bending axis and the [0,-1,1] axis. For each

reflection a crystal thickness of 0.5 mm was chosen, and the energy was varied in

increments of 5 keV within the Bragg angle range of 3.625° and 6.25°. Diffracted

70

intensity and the energy resolution were calculated for sample sizes of the full beam at

the sample (9.3 mm high x 4.5 mm wide), 1 mm x 1 mm and 100 μm x 100 μm. The

incident flux values were calculated from 'WS' algorithm for a super conducting wiggler,

with an incident divergence of 0.3mrads x 0.3mrads. Results were plotted, and the

reflections were separated into three categories: 2nd

harmonics, multiple harmonics, and

all other low order reflections. Figures 4.11 through 4.13 shows the results of a full beam

at the sample for all three categories of reflections, and figures 4.14 and 4.15 shows all

other low order reflections for sample sizes (1 mm)2 and (0.1 mm)

2 respectively.

Diffracted intensity is denoted with a circle symbol and energy resolution points are

diamond shaped.

71

Figure 4.11: Diffracted intensities and energy resolutions on a 9.3Vmm x 4.5Hmm

sample, using flat symmetric Si Laue crystals and 2nd harmonic

reflections. Crystal thickness used was 0.5 mm, and the incident

horizontal and vertical divergence was 0.3 mrads. Calculated

reflectivity's have a margin of error of ± 20% resulting in a ± 20%

diffracted intensity error.

Figure 4.12: Diffracted intensities and energy resolutions on a 9.3 Vmm x 4.5 Hmm

sample, using flat symmetric Si Laue crystals and multiple harmonic





Reflections that have 2nd

harmonics such as (0,2,2) which has the (0,4,4) as a

second harmonic, should be avoided from being selected as monochromator

reflections since the second harmonic contamination is as much as 4% of the primary

intensity at only double the energy of the primary (figure 4.11). In contrast, for the

72

case of the Si (1,1,1) primary reflection, the first higher harmonic is the Si (3,3,3)

(figure 4.12) which is still on the order of 1% contamination but at least it‟s energy is

three times higher than that of the Si (1,1,1).

Figure 4.13: Diffracted intensities and energy resolutions on a 9.3 mm(V) x 4.5 mm(H)

sample, using flat symmetric Si Laue crystals and all other low order





73

Figure 4.14: Diffracted intensities and energy resolutions on a 1 mm(V) x 1 mm (H)






74

Figure 4.15: Diffracted intensities and energy resolutions on a 0.1 Vmm x 0.1 Hmm






Changing the sample size smaller than the full beam size, results in decreased

intensities. For the case of Si (1,1,1) at 20 keV, intensity is reduced by a factor of 50

(from ~ 1013

ph/s to 2x1011

ph/s) for a sample size of (1 mm)2 and ~ 5000 for a sample

size of (100 μm)2 (from ~ 10

13 ph/s

to 2x10

9 ph/s). Better energy resolutions are obtained

for smaller sample sizes, and for Si (1,1,1) at 20 keV the energy resolution drops from

~0.3% to ~0.08% for a (1 mm)2

sample and to ~0.044% for a (100 μm)2 sample.

For all the reflections there exists more intensity for a larger 9.3x4.5 mm2 sample size:

4x109-2x10

13 ph/s (9.3x4.5 mm

2 sample), and higher energy resolutions: ~0.28%-0.48%

75

(figure 13). This is because for larger sample sizes, more of the incident rays, which are

vertically diverging and horizontally converging, are impingent on the sample. For very

small sample sizes such as (100 μm)2, the intensity at the sample is 1x10

6 - 4x10

9 ph/s

and energy resolution decreases to ~0.04%-0.07%. For the case that the sample is small

fewer rays from the monochromator will strike the sample, resulting in better energy

resolutions since less of the horizontal divergence from the source is reaching the

sample.

For energies between 20 keV to 100 keV and acceptable Bragg angles between 3.625°

and 6.25°, four single bounce Laue crystals can be used to satisfy these conditions. We

choose the (1,1,1) , (3,1,1), (5,1,1) and (5,3,3) reflections.

4.3 Analysis of single saddle bent Laue in the inverse-Cauchois

geometry

Although the energy resolutions produced by a single flat Laue are good and within

the constraint of ≤ 0.2% for small sample sizes, the diffracted intensities are not as large

as desired. To better optimize for flux while still maintaining good energy resolution, a

saddle shaped bent monochromator crystal was investigated using the inverse-Cauchois

geometry. Presented here is the analysis of the low order silicon reflections: (1,1,1),

(3,1,1), (1,3,3), (5,1,1), (5,3,3), (1,5,5), (7,1,1) and (3,5,5), covering the range of 20 keV

to 100 keV.

By bending the crystal, the diffracted intensity will in fact increase compared to a flat

symmetric Laue as a result of vertical focusing of the beam. The bent crystal must have

an asymmetry angle between the surface normal and the Bragg planes in order to enable

76

focusing. The rocking curve equation used in calculating the diffracted intensity includes

the additional term Δθ (T), which accounts for the change in the d-spacing and lattice

plane rotation. Increasing the rocking curve width results in decreased reflectivity‟s

using equation (4.1.49), 𝑅 ≅ 𝐼

ω=

𝐼

Δθ²(T) + Wa ² , however, the total area under the rocking

curve increases.

4.3.1 Parameters constrained for a single saddle bent Laue

There exist many parameters in parameter space that may be varied for each silicon

reflection, but these are reduced in number if constraints such as the beamline layout and

the conditions for the inverse-Cauchois geometry are considered. Accordingly, for the

following single saddle bent Laue crystal we fix the following parameters: F1 = 23 m,

incident divergence of 0.3 mrads in both the vertical and horizontal, and Bragg angles

between 3.625° and 6.25°. This leads to the horizontal and vertical footprint size on the

crystal to both be 6.9 mm, as calculated in 4.2.2. The energy range for each reflection

varies as tabulated in table 4.9, corresponding to the narrow Bragg angle constraint. The

variable C which is dependent on the aspect ratio of the crystal and the bending

mechanism was constrained to values between 0.2 and 1, and for each set of varying

input parameters (reflection, energy, F2, χ, and thickness) the crystal was bent to satisfy

the inverse-Cauchois geometry, by setting the value of C = 𝐶 𝑅𝑜𝑙𝑎𝑛𝑑 = 𝑅𝑠

𝐹1𝜈 𝑐𝑜𝑠 𝜒 +

𝜃𝐵 using equation (4.1.12). In addition, energy resolution was constrained in

calculations to be ≤ 0.2%.

77

Table 4.9: Energies corresponding to the Bragg angles 6.25° and 3.625°, for a single

saddle bent Laue.

θB = 6.25° θB = 3.625°

Reflection Minimum Energy (keV) Maximum Energy (keV)

(1,1,1) 18.160 31.269 (3,1,1) 34.774 59.877 (1,3,3) 45.703 78.694 (5,1,1) 54.481 93.809 (5,3,3) 68.754 118.386 (1,5,5) 74.877 128.929 (7,1,1) 74.877 128.929

(3,5,5) 80.536 138.673

4.3.2 Calculation of axes with an orientation

For each reflection the meridional axis, z', was chosen as [0,-1,1], because this axis

was perpendicular to all the reflections. For crystals bent with an asymmetry angle of 0°

(symmetric Laue) the sagittal axis, y', is parallel to the reflection. The sagittal axis needs

to be calculated for each asymmetry angle, χ, between the Bragg planes and the surface

normal. This angle χ is also the angle the crystal reflection makes with the vertical, y‟

axis. The calculation of the sagittal axis involves rotating the reflection vector (x,y,z)

about the meridional axis z'(u,v,w) = z'[0,-1,1] by χ and is given as[24]:

𝑦′ = 𝑕′

𝑘′

𝑙′ = 4.3.1

𝑢 𝑢𝑥 + 𝑣𝑦 + 𝑤𝑧 (1 − 𝑐𝑜𝑠 𝜒) + 𝑢2 + 𝑣2 + 𝑤2 𝑥 𝑐𝑜𝑠 𝜒 + 𝑢2 + 𝑣2 + 𝑤2(−𝑤𝑦 + 𝑣𝑧) 𝑠𝑖𝑛 𝜒

𝑢2 + 𝑣2 + 𝑤2

𝑣 𝑢𝑥 + 𝑣𝑦 + 𝑤𝑧 (1 − 𝑐𝑜𝑠 𝜒) + 𝑢2 + 𝑣2 + 𝑤2 𝑦 𝑐𝑜𝑠 𝜒 + 𝑢2 + 𝑣2 + 𝑤2(𝑤𝑥 − 𝑢𝑧) 𝑠𝑖𝑛 𝜒

𝑢2 + 𝑣2 + 𝑤2

𝑤 𝑢𝑥 + 𝑣𝑦 + 𝑤𝑧 (1 − 𝑐𝑜𝑠 𝜒) + 𝑢2 + 𝑣2 + 𝑤2 𝑧 𝑐𝑜𝑠 𝜒 + 𝑢2 + 𝑣2 + 𝑤2(−𝑣𝑥 + 𝑢𝑦) 𝑠𝑖𝑛 𝜒

𝑢2 + 𝑣2 + 𝑤2

The surface normal, x', is orthogonal to the z' and y' axes and is therefore calculated by

78

doing the cross product of the sagittal axis with the meridional axis

(y'[h'calculated,k'calculated,l'calculated) x z'[0,-1,1]). Once these orientation axes are determined

the elastic compliance coefficients and the Poisson ratio, S'23, may be calculated.

4.3.3 Energy resolution and Diffracted intensity at the crystal for a saddle bent Laue

For a set of input parameters (reflection, energy, F2, χ, and thickness) satisfying the

inverse-Cauchois geometry, C is set equal to 𝐶 𝑅𝑜𝑙𝑎𝑛𝑑 = 𝑅𝑠

𝐹1𝜈 𝑐𝑜𝑠 𝜒 + 𝜃𝐵 , and the

general energy resolution equation (4.1.38), 𝛥𝐸

𝐸=


2

𝑡𝑎𝑛 𝜃𝐵, becomes:

ΔE

E=

Δθ (T)² + σsF1

2

tan 𝜃𝐵 (4.3.2)

where 𝛥𝜃 is equal to zero and equation (4.1.37) holds true: 𝐹1 = 𝑅𝑚 𝑐𝑜𝑠 𝜒 + 𝜃𝐵 . The

most appropriate values of C were determined by means of equation (4.1.12) to satisfy

the inverse-Cauchois geometry. In cases that the C value did not lead to the inverse-

Cauchois condition being satisfied, the general energy resolution equation (4.1.38) was

applied.

The diffracted intensity from the monochromator using the general equation (4.1.49)

becomes:

Diffracted Intensity (ph/s) = Incident flux (ph/s/0.1%bw) x (4.3.2)

0.6 x Reflectivity x (EcotθB 𝛥𝜃2 𝑇 + 𝑊𝑎 ² )/0.1%bw

where 𝛥𝜃 (𝑇) contributes to a larger rocking curve width and more throughput.

Reflectivity's were calculated using equation (4.1.48) from literature where 𝑅 ≅ 𝐼

ω=

79

𝐼

Δθ²(T) + Wa ² .

4.3.4 Analysis of the diffracted intensity, energy resolution and optimized intensity

at the monochromator for a saddle bent Laue satisfying the inverse-Cauchois

geometry

A routine was created using nested for-loops to calculate at the diffracted

monochromator intensities and energy resolutions, assuming the crystal is bent with an

aspect ratio corresponding to C (Cauchois). In addition, the parameters that optimize

diffracted intensity at the monochromator and the corresponding energy resolution was

calculated for each energy and reflection using several constraints (refer to step 5 below).

The steps of the algorithm that were implemented in the program SagBentApp are:

1. Pick a reflection, fix F1 = 23 m and set C = 𝐶 𝑅𝑜𝑙𝑎𝑛𝑑 = 𝑅𝑠

𝐹1𝜈 𝑐𝑜𝑠 𝜒 + 𝜃𝐵

2. Vary energy in increments of 5keV within the Bragg angle constraint of 3.625° to

6.25°

3. Vary F2 from 8 to 11.5 m in increments of 0.5 m

3. Vary thickness from 0.5 to 1 mm in increments of 0.1 mm

4. Vary χ from 5° to 85° in increments of 2°

5. If optimizing diffracted intensity set constraints:

a) 0.2 ≤ C ≤ 1

b) 𝛥𝐸

𝐸 ≤ 0.2%

c) 𝑅𝑠 ≥ 500 x 𝑡𝑕𝑖𝑐𝑘𝑛𝑒𝑠𝑠

80

d) Print parameter results that corresponds to optimized diffracted intensities

Initially, a variable max_intensity is set to zero at each energy in the program

SagBentApp. For every energy, when a larger intensity value is calculated in the nested

for-loops max_intensity is assigned the higher intensity value and printed out. In the

program, each time intensity is calculated an if-statement determines if the new

calculated intensity is larger than the max_intensity value calculated in the previous for-

loop step. If the intensity calculated at a step of a for-loop is greater than the intensity

previously defined as max_intensity, then max_intensity is assigned the higher intensity

value. To get the optimized intensity result for a specific energy and reflection, the print

line of the last row of each energy was picked out as it contains the parameters for the

maximum intensity.

Using this algorithm, the maximum diffracted monochromator intensities were

calculated for each reflection and are depicted in figures 4.16 and 4.17, with energy

resolutions better than 0.2% and having the crystal bent with an aspect ratio

corresponding to the inverse-Cauchois geometry. Diffracted intensity is denoted with a

circle symbol and the energy resolution points are diamond shaped. The bending radius

was constrained to ≥ 500 x thickness, as it would be nearly impossible to further bend a

crystal with the same thickness without it breaking. The calculated results for all eight

reflections are tabulated in table 4.10 for the low order reflections and table 4.11 for the

higher order reflections.

By performing these calculations, one can observe the importance that certain

parameters (thickness and asymmetry angle) have on optimizing diffracted intensity as

energy is varied. In addition, the calculations for each reflection provide an initial idea of

81

the diffracted intensity magnitude and the results of the parameters close to the Bragg

angle 4.9°, which corresponds to the middle of the Bragg angle constraint. This is

important, because the inverse-Cauchois condition can only be satisfied at a single energy

if χ and C are fixed for a reflection. For the CLS beamline, each crystal used will have a

fixed set of parameters (thickness, χ and C) and a set of crystals will be chosen to cover

the full energy range of 20 to 100 keV. Using equation (4.1.12) where

𝐶 = 𝑅𝑠

𝐹1𝜈 𝑐𝑜𝑠 𝜒 + 𝜃𝐵 , if F1, χ, C and the orientation remains fixed (ν is also fixed) for a

specific reflection, this equation will only hold true at one specific Bragg angle. Varying

the energy away from this specific Bragg angle results in worse energy resolutions,

because the incident horizontal divergence causes an „energy fan‟ when not in the

inverse-Cachois condition.

82

Figure 4.16: Diffracted intensities optimized at the monochromator with the

corresponding energy resolutions for Si reflections: (1,1,1), (3,1,1), (1,3,3)

and (5,1,1). Values were calculated using program SagBentApp where F2

was varied between 8-11.5 m, χ = 5°-85°, thickness = 0.5-1 mm and energy

was varied between Bragg angles 3.625°-6.25°. The constraints are: ∆E/E

≤ 0.2%, 0.2 ≤ C ≤1, Rs ≥ 500 x thickness and the crystal was bent to

satisfy inverse-Cauchois geometry at all energies.

83

Table 4.10: Diffracted intensities optimized at the monochromator for the points plotted

in figure 4.16 and Si reflections (1,1,1), (3,1,1), (1,3,3) and (5,1,1). The

surface normal is denoted [xh', xk', xl'], the sagittal axis [yh',yk',yl'], the

rocking curve width by w and reflectivity by Ref beside the energy resolution

column. The bandwidth terms Δθrot, ΔθB, Δθ (T) and σs/F1 are all in μrads.

84

Figure 4.17: Diffracted intensities optimized at the monochromator with the

corresponding energy resolutions for Si reflections: (5,3,3), (1,5,5), (7,1,1)

and (3,5,5). Values were calculated using program SagBentApp where F2

was varied between 8-11.5 m, χ = 5°-85°, thickness = 0.5-1 mm and energy

was varied between Bragg angles 3.625°-6.25°. The constraints are: ∆E/E

≤ 0.2%, 0.2 ≤ C ≤1, Rs ≥ 500 x thickness and the crystal was bent to

satisfy inverse-Cauchois geometry at all energies.

85

Table 4.11: Diffracted intensities optimized at the monochromator for the points plotted

in figure 4.17 and Si reflections (5,3,3), (1,5,5), (7,1,1) and (3,5,5). The

surface normal is denoted [xh', xk', xl'], the sagittal axis [yh',yk',yl'], the

rocking curve width by w and reflectivity by Ref beside the energy resolution

column. The bandwidth terms Δθrot, ΔθB, Δθ (T) and σs/F1 are all in μrads.

It is apparent that for a majority of the reflections when optimizing for the diffracted

intensity at the monochromator the parameters chosen are the larger crystal thicknesses of

1mm. This is a consequence of the term Δθ (T) being proportional to the thickness and

86

that having a larger crystal thickness results in this term becoming larger in the rocking

curve width and increasing the diffracted intensities. The term that contributes the most

to Δθ (T) is the change in lattice plane rotation, Δθrot, which is much larger than the

change due to the d-spacing, ΔθB. For Si reflections: (1,1,1), (3,1,1), and (1,3,3) a small

asymmetry angle is favoured for low energies and larger asymmetry angles at higher

energies. For higher order reflections the asymmetry angles decrease slightly with

increasing energies.

An important find from producing these calculations is that a low C value of 0.2 is

desired for all the energies and reflections. The sagittal bend at low energies is less tight

than at high energies where the bend radius is more severe. Having a large crystal bend

radius results in the lamella being nearly parallel. The term Δθrot is much smaller at

lower energies which corresponds with having the lamella nearly parallel for lower

energies. By bending the crystal sagittally about the y' axis, the meridional curvatures

obtained are much larger by up to a factor of 10 in some instances.

Low order reflections produce the most diffracted intensities and up to ~4x1014

ph/s

for the case of Si (1,1,1). Since the diffracted intensity is directly proportional to the

reflectivity, low order reflections are preferred for experimental use as they have high

reflectivity's. A set of reflections that would cover the energy range of interest with very

high diffracted intensities would be Si (1,1,1) from 18 keV to 31 keV, Si (3,1,1) from

34.7 keV to 59.9 keV, Si (5,1,1) from 54.5 keV to 93 keV and Si (5,3,3) from 68 keV to

118 keV.

For each reflection the thickness, F2, χ and C were chosen from the tabulated results of

optimized diffracted intensity, from the middle of the constrained Bragg angle range at an

87

angle closest to 4.9375°. Since the inverse-Cauchois geometry can only be satisfied at

only a single energy, the middle energy was chosen from table 4.9. When the crystal is

bent away from this inverse-Cauchois geometry the energy resolution increases. For

reflections that varies little in energy like Si (1,1,1) between 20 keV and 35 keV, the

incident bandwidth, Δθ, contributes little to the energy resolution at the extreme ends of

the energy range, but for reflections such as Si (3,5,5) with energies between 80 keV and

139 keV there is more contribution of this term, Δθ, to the energy resolution. These

parameters (thickness, F2, χ and C) were kept fixed for the Bragg angle 4.9° as energy

was again varied and diffracted intensity at the monochromator and energy resolution

was calculated. The calculated results are tabulated in tables 4.12 and 4.13. The low

order reflection calculation results are depicted in figure 4.18 and the higher order

reflections are illustrated in figure 4.19.

88

Figure 4.18: Diffracted intensities and the corresponding energy resolutions as energy

is varied for monochromator crystals with fixed thickness, F2, χ and C

close to the Bragg angle 4.9°. Silicon reflections calculated are: (1,1,1),

(3,1,1), (1,3,3) and (5,1,1).

89

Table 4.12: Table of diffracted intensities for monochromator crystals with fixed

thickness, F2, χ and C close to the Bragg angle 4.9° and silicon reflections:

(1,1,1), (3,1,1), (1,3,3), (5,1,1) and (5,3,3). The surface normal is denoted

[xh', xk', xl'], the sagittal axis [yh',yk',yl'], the rocking curve width by w and

reflectivity by Ref beside the energy resolution column. The bandwidth

terms Δθrot, ΔθB, Δθ (T) and σs/F1 are all in μrads.

90

Figure 4.19: Diffracted intensities and the corresponding energy resolutions as energy

is varied for monochromator crystals with fixed thickness, F2, χ and C

close to the Bragg angle 4.9°. Silicon reflections calculated are: (5,3,3),

(1,5,5), (7,1,1) and (3,5,5).

91

Table 4.13: Table of diffracted intensities for monochromator crystals with fixed

thickness, F2, χ and C close to the Bragg angle 4.9° and silicon reflections:

(1,5,5), (7,1,1) and (3,5,5). The surface normal is denoted [xh', xk', xl'], the

sagittal axis [yh',yk',yl'], the rocking curve width by w and reflectivity by

Ref beside the energy resolution column. The bandwidth terms Δθrot, ΔθB,

Δθ (T) and σs/F1 are all in μrads.

Fixing the parameters (thickness, F2, χ, and C) that optimizes the diffracted intensity

in the middle of the Bragg range for each reflection, was performed to calculate the

diffracted intensities and energy resolutions when the inverse-Cauchois condition was not

satisfied, and for energies away from the Bragg angle: ~4.9°. Using these results reveals

92

that only a set of ~4 crystals needs to be selected for the CLS beamline monochromator,

covering the entire energy range of 20 keV to 120 keV. The layout of the CLS beamline

would have to accommodate for distances of F2 up to 11.5 m..

The reflections that are best suited for optimizing the diffracted intensity at the crystal

and satisfy the constraint ∆E/E ≤ 0.2% are reflections: Si (1,1,1) for 18-31 keV with a

photon flux at the monchromator of 1.25-2.87x1014

ph/s, Si (3,1,1) for 34-59 keV with

6.92-3.54x1013

ph/s, Si (5,1,1) for 55-94 keV with 1.05x1013

-1.98x1012

ph/s and Si

(5,3,3) for 69-118 keV with 2.18x1012

-1.25*1011

ph/s. Although the energy resolutions

for Si (1,1,1) are 0.14-0.3% and for Si (3,1,1) are 0.12-0.37% (these are slightly larger

than our constraint of 0.2%), they are the only available reflections to cover these low

energy ranges of 18-45keV. All the other reflections have good energy resolutions while

still maintaining sufficiently large flux values.

Because the sagittal radius is dependent on the Bragg angle, for slightly deformed

crystals where the bend radius is large the term Δθ(T), which is inversely proportional to

Rs, is small resulting in smaller energy resolutions at low energies (large Bragg angles)

and larger energy resolutions at high energies. The diffracted intensities are proportional

to the incident flux (figure 4.6), and for low energies the incident flux is greater than at

higher energies resulting in increased diffracted intensities at lower energies.

93

Chapter 5

Conclusion

For a low energy of 8keV, research was performed to determine which crystal

geometry (symmetric Bragg or symmetric Laue), reflection and thickness produced the

most throughput using programs nBeam and XCRYSTAL. The reflections investigated

were: (1,1,1), (3,3,3), (2,2,0) and (4,4,0). These reflections were used to provide an

initial idea of the size of the throughputs and to make comparisons in order to select one

reflection. The reflection Si (1,1,1) was chosen for further analysis of when the crystal is

bent cylindrically.

For a flat Ge crystal with symmetric Bragg geometry and σ polarization, the reflection

(1,1,1) produces the most throughput at 2 μm among all the reflections, and followed by

the reflection (2,2,0) at 2 μm. For a flat Si crystal with symmetric Bragg geometry, the

reflection (1,1,1) produces the most throughput at 4 μm among all the reflections, and

followed by the reflection (2,2,0) at 5 μm. The optimized Ge crystal has ~ 2.15 times

more throughput than that of an optimized Si Bragg crystal.

For a flat Ge crystal with symmetric Laue geometry and σ polarization, the reflection

(1,1,1) produces the most throughput at 3 μm among all the reflections, and followed by

the reflection (2,2,0) at 2 μm. For a flat Si crystal with symmetric Laue geometry, the

reflection (1,1,1) produces the most throughput at 7 μm among all the reflections, and

94

followed by the reflection (2,2,0) at 5 μm. The optimized Ge crystal has ~ 2.19 times

more throughput that of an optimized Si Laue crystal.

At an energy of 8 keV, perfect Bragg geometry has ~26% larger throughput than a

Laue geometry when comparing the throughputs of Ge (1,1,1), and ~27% when

comparing the throughputs of Si (1,1,1). These optimized throughputs are obtained at

very thin crystal thicknesses of only a few µm. The symmetric reflection (1,1,1) in the

Bragg geometry produced the most favourable throughput and merits closer investigation

when the crystal is bent cylindrically.

Calculations involving the use of XCRYSTAL_BENT have thus far provided evidence

that there exists a critical energy between 17 keV and 24 keV, which defines the point

where both the Laue and Bragg geometries are equally favorable for maximizing the

throughput. The program XCRYSTAL_BENT has allowed us to narrow this range to

within a few keV of this unknown energy, but due to the constraints of the program

where the crystal is only slightly deformed it would be difficult to accurately define this

energy without going to smaller bending radii.

For energies 8 keV and 17 keV a flat symmetric Si Bragg crystal produces more

throughput than a cylindrically bent Si Laue crystal. At energies of 24 keV and 50 keV

the cylindrically bent Si Laue crystal proved to be the preferred choice of geometry for

producing larger throughputs. The rocking curve width for the cylindrically bent Laue

crystal at 24 keV increased by a factor of ~14 times that of a flat symmetric Bragg crystal

with a reduction in the peak reflectivity by a factor of ~3.5. This suggests that bending a

crystal is critical for improving throughput and increasing the rocking curve width.

When a crystal is strained, the expansion of the rocking curve width predominately

95

contributes to the increased throughput.

Bending a crystal cylindrically to optimize for throughput at low energies, results in

parameters that are small in bend radii and thickness. In the case of 8 keV, optimized

parameters include a bend radius of 1.1m and a thickness of 0.21 mm. At higher energies

of 50 keV the required bend radius is much larger, and is 19.9 m with a crystal thickness

of 1.3 mm. At low energies the strain on the crystal plays a significant role on increasing

the size of the rocking curve width, as there exists a larger change in the lattice plane

orientations and the d-spacing of atomic planes. The distance that a beam has to travel

along the atomic planes is longer for crystals with a tighter bend. To get sufficient

transmission when the crystal is very bent, the thickness is reduced, as seen in the 8 keV

case. But at higher energies where the bending radius is larger there exists enough

transmission through the crystal and as a result increased thicknesses are preferred for

optimizing throughput.

Analysis of a single horizontally diffracting flat Laue monochromator (no focusing),

revealed that large sample sizes at the sample lead to larger intensity values and less

favorable energy resolution values. Smaller sample sizes have less intensity but have

better energy resolution (within our constraint of ΔE/E ≤ 0.2%). In addition, four single

bounce Laue crystal can cover an energy range of 20-100 keV for this narrow Bragg

angle range (3.625° to 6.25°). A set of crystals that covers this entire energy range are

the Si reflections: (1,1,1), (3,1,1), (5,1,1) and (5,3,3). By analyzing a symmetric flat Laue

crystal, the magnitudes of the diffracted intensities for the full beam size at the sample

(same number of photons as those diffracted at the monochromator) could be used to

make comparisons with a saddle bent Laue, in order to determine if straining a crystal

96

increases the diffracted intensity.

When optimizing the diffracted intensity at the monochromator using a saddle bent

Laue crystal in the inverse-Cauchois geometry, it became apparent that a large crystal

thickness of 1mm was preferred. The term that contributes the most to the energy

resolution is 𝛥𝜃 (𝑇) followed by 𝜍𝑠

𝐹1 . The change in lattice plane rotation, 𝛥𝜃𝑟𝑜𝑡 , is much

larger than 𝛥𝜃𝐵 and contributes the most to the term to 𝛥𝜃 𝑇 , increasing the bandwidth

at higher energies. The other noticeable trends observed in the parameters that optimized

intensity are low C values of 0.2, and a tighter sagittal bend at higher energies. Low

order reflections do produce the largest optimized intensities and are preferred for their

high peak reflectivity's.

The results of a saddle bent Laue when fixing the optimized parameters at the Bragg

angle of 4.9° reveals that indeed the diffracted intensity at the monochromator is

increased in comparison to a flat Laue crystal, and the energy resolutions are much better

(from 0.3-0.45% for a flat Laue crystal to less than 0.2% for most reflections). The four

crystals that would perhaps be the best candidates for being used in the CLS beamline

monochromator are Si (1,1,1) for 18-31keV, Si (3,1,1) for 34-59keV, Si (5,1,1) for 55-

93keV and Si (5,3,3) for 69-118keV. Using the source specifications discussed in 4.1.12,

photon flux ranges from 1.25x1011

ph/s for Si (5,3,3) to 2.87x1014

ph/s for Si (1,1,1) for

the four selected saddle bent Laue crystals. These same four reflections for the case of a

flat symmetric Laue crystal produced flux values of ~1x1010

ph/s for Si (5,3,3) to ~1x1013

ph/s for Si (1,1,1). Bending a crystal into a saddle shape has its advantage of increasing

diffracted flux by at least a factor of 10, while reducing the energy resolution for energies

close to the angle 4.9° (angle whose parameters satisfy the inverse-Cauchois geometry) in

97

the middle of the Bragg range.

5.1 Future work

The next step for this research is to determine how changing the aspect ratio of a

crystal affects the value of C, and what constrains C usually between 0.2 and 1 in terms

of crystal dimensions. This depends on the ANSYS calculations currently being

performed at the CLS. The possible aspect ratios of a crystal still needs to be determined

in order to take the calculations performed in section 4.3 one step further, such as adding

a realistic constraint on the variable C. Upon further completion of these theoretical

calculations, these crystal parameters may be tried experimentally to determine the

reliability of the results. We will need to consider effects of aberrations on focusing and

the effects of monochromator thickness and asymmetry angle on potential horizontal

focusing size.

98

Bibliography

[1] Lalita Acharya. The Canadian Light Source - Canada's Synchrotron. Parliamentary

Research Branch, PRB 03-47E, 2004.

[2] Andre Authier. Dynamical Theory of X-Ray Diffraction. Oxford University

Press, Oxford, New York, 2001.

[3] Personal communication with Dr. Gomez.

[4] George Brown, Klaus Halback, John Harris, and Herman Winick. Wiggler and

Undulator Magnets - A Review. Nuclear Instruments and Methods, 208:65-77,

1983.

[5] M. Sánchez del Río, C. Ferrero and V. Mocella. Computer simulations of bent

perfect crystal diffraction profiles. SPIE proceedings, 3152, 1997.

[6] Z. Zhong, C. C. Kao, D.P. Siddons, and J.B. Hastings. Sagittal focusing of high-

energy synchrotron X-rays with asymmetric Laue crystals. I. Theoretical

considerations. Journal of Applied Crystallography, 34:504-509, 2001.

[7] V.I. Kushnir, J.P. Quintana, and P. Georgopoulos. On the sagittal focusing of

synchrotron radiation with a double crystal monochromator. Nuclear Instruments

and Methods in Physics Research, A328:588-591, 1993.

[8] J.P. Quintana, V.I. Kushnir, and G. Rosenbaum. Synchrotron experimental results

on an unribbed sagittally focusing crystal monochromator. Nuclear Instruments and

Methods in Physics Research, A362:592-594, (1995).

[9] Collaboration with Marcus Miranda on the use of program nBeam.

[10] William H. Zachariasen. Theory of X-ray Diffraction in Crystals. Dover

Publications, New York, 1945.

[11] Boris W. Batterman, and Henderson Cole. Dynamical Diffraction of X Rays by

Perfect Crystals. Reviews of Modern Physics, 36:681-717, 1964.

[12] http://www.chess.cornell.edu/oldchess/operatns/xrclcdwn.htm (Retrieved on

27/10/2011)

http://inside.mines.edu/~gmurray/ArbitraryAxisRotation/ArbitraryAxisRotation.pdf

http://www.chess.cornell.edu/oldchess/operatns/xrclcdwn.htm

99

[13] Z. Zhong, C.C. Kao, D.P. Siddons, and J.B. Hastings. Sagittal focusing of high-

energy synchrotron X-rays with asymmetric Laue crystals. II. Experimental studies.

Journal of Applied Crystallography, 34:646-653, 2001.

[14] Z. Zhong, C.C. Kao, D.P. Siddons, and J.B. Hastings. Rocking-curve width of

sagittally bent Laue crystals. Acta Cryst., A58:487-493, 2002.

[15] Z. Zhong, C.C. Kao, D.P. Siddons, H. Zhong, and J.B. Hastings. A lamellar model

for the X-ray rocking curves of sagittally bent Laue crystals. Acta Cryst., A59:1-6,

2003.

[16] Zhong Zhong. x7b_thoughts1. PowerPoint slides, 1-17.

[17] J.J. Wortman, and R.A. Evans. Young's Modulus, Shear Modulus, and Poisson's

Ratio in Silicon and Germanium. Journal of Applied Physics, 36:153-156, 1965.

[18] Jian-Min Zhang, Yan Zhang, Ke-Wei Xu, and Vincent Ji. , General compliance

transformation relation and applications for anisotropic cubic metals. Materials

Letters, 62: 1328-1332, 2008.

[19] Mason, and P. Warren. Physical Acoustics and the Properties of Solids. D. Van

Nostrand Company, Princeton, New Jersey, 1958.

[20] E. Erola, V. Etelaniemi, P. Suortti, P. Pattison, and W. Thomlinson. X-ray

Reflectivity of Bent Perfect Crystals in Bragg and Laue Geometry. Journal of

Applied Crystallography, 23:35-42, 1990.

[21] V. Etelaniemi, P. Suortti, and W. Thomlinson. Reflect - A computer Program for

the X-Ray Reflectivity of Bent Perfect Crystals. Brookhaven National Laboratory

Report, 43247:1-15, 1989.

[22] Bayden Pritchard. Optimized pair distribution function analysis of gold

nanoparticles. Master's thesis, University of Guelph, 2010.

[23] J. H. Hubbell, and S. M. Seltzer. Tables of X-Ray Mass Attenuation Coefficients

and Mass Energy-Absorption Coefficients. NIST, 1996.

http://physics.nist.gov/PhysRefData/XrayMassCoef/ElemTab/z14.html

(Retrieved on 27/10/2011)

[24] Glenn Murray. Rotation About an Arbitrary Axis in 3 Dimensions. July 2011.


(Retrieved on 27/10/2011)

mailto:[email protected]


100

Appendix: Java code for a single saddle bent Laue

A program was written in java using expressions typed previously in an Excel

spreadsheet for the single saddle bent Laue crystal. Parameter space can be analyzed

more efficiently using nested for-loops written in a program rather than manually

changing each set of parameters one step at a time. Presented here is a program written

in java containing codes SagBentApp and SagBent, both of which requires compiling for

the program to work. SagBentApp contains instructions for inputting and printing

parameters in both a terminal and in a graphical user interface calculation box. SagBent

contains all the methods, these are sections of the code where a parameter has its

expression defined.

SagBentApp Code:

/* SagBentApp.java

One Sagittally Bent Laue Calcualtions for CLS using Si

*/

import org.opensourcephysics.controls.*;

import org.opensourcephysics.frames.*;

public class SagBentApp extends AbstractCalculation {

public double initial;

public double initial2=0.0;

public double largest;

SagBent crystal = new SagBent();

// Creates plots of Intensity & Energy Resolution vs Energy

101

PlotFrame Intensity = new PlotFrame("Energy (keV)","Diffracted intensity

Sample","Diffracted Intensity Sample vs. Bragg angle");

PlotFrame Energyresolution = new PlotFrame("Energy (keV)","% dE/E at

sample","Energy Resolution at Sample vs. Energy");

PlotFrame Reflectivity = new PlotFrame("Energy (keV)","% Reflectivity","%

Reflectivity vs. Energy");

public void calculate() {

//Upon clicking calculate button program goes through Energy,F2,Thickness and Chi for-

loops

for(double e=control.getDouble("Min Energy (eV)");e<control.getDouble("Max

Energy (eV)")+5000.0;e=e+5000.0){

//increasing increments of 5000eV in energy

initial2=0.0;

for(double f=control.getDouble("Min F2 (m)");(f<control.getDouble("Max F2

(m)")) || (f==control.getDouble("Max F2 (m)"));f=f+0.5){

// increasing increments of 0.5m in F2

for(double i=control.getDouble("Min Thickness (m)");i<control.getDouble("Max

Thickness (m)")+0.0001; i=i+0.0001){

// increasing increments of 0.1mm in thickness

for (double a=control.getDouble("Min Asymmetry angle (degrees)

");a<control.getDouble("Max Asymmetry angle (degrees) ")+2.0;a=a+2.0){

// increasing increments of 2 degrees in chi

crystal.E = e;

crystal.Asym = a;

crystal.T = i;

crystal.F2=f;

// gets inputs of parameters from GUI

crystal.results = control.getInt("All results 1, max ph/s 2, results chosen C & chi 3, max

ph/s fix C and chi 4, all C & Rs constrained 5, max C & Rs & %dE/E constrained 6");

crystal.h = control.getInt("Reflection h");

crystal.k = control.getInt("Reflection k");

crystal.l = control.getInt("Reflection l");

crystal.zh = control.getInt("Meridional axes (z') h'");

crystal.zk = control.getInt("Meridional axes (z') k'");

crystal.zl = control.getInt("Meridional axes (z') l'");

crystal.F1 = control.getDouble("F1 (m)");

crystal.C = control.getDouble("C (unitless)");

crystal.horizdiv = control.getDouble("Horizontal Divergence (mrad)");

crystal.verticaldiv = control.getDouble("Vertical Divergence (mrad)");

crystal.horizsource = control.getDouble("Horizontal size of source (m)");

crystal.horizsample = control.getDouble("Horizontal sample size (mm)");

crystal.verticalsample = control.getDouble("Vertical sample size (mm)");

crystal.K = control.getDouble("K Polarization factor");

102

crystal.B = control.getDouble("Size of B(T) of source");

crystal.sourcetype = control.getInt("If SC Wiggler type 1, if Conv. Wiggler type

2");

// Prints parameter results in GUI box

control.println("Chi (degrees) = "+crystal.chi());

control.println("(x') h' Surface normal = "+crystal.xh());

control.println("(x') k' Surface normal = "+crystal.xk());

control.println("(x') l' Surface normal = "+crystal.xl());

control.println("(y') h' Sagittal axes= "+crystal.yh());

control.println("(y') k' Sagittal axes= "+crystal.yk());

control.println("(y') l' Sagittal axes= "+crystal.yl());

control.println("Bragg angle (degrees) = "+crystal.thetaB());

control.println("Poission Ratio or S'23 (unitless) = "+crystal.S23());

control.println("Vertical beam size at crystal(mm) =

"+crystal.verticalbeamcrystal());

control.println("Horizontal beam size at crystal(mm) =

"+crystal.horizontalbeamcrystal());

control.println("CS'23 (unitless) = "+crystal.CS23());

control.println("Rs (m) = "+crystal.Rs());

control.println("Rs Roland (m) = "+crystal.RsRoland());

control.println("Rm (m) = "+crystal.Rm());

control.println("Rm Roland (m) = "+crystal.RmRoland());

control.println("C Roland (unitless) = "+crystal.CRoland());

control.println("Thickness (m) = "+crystal.T);

control.println("dthetarot (microrad) = "+crystal.dthetarot());

control.println("dthetaB (microrad) = "+crystal.dthetaB());

control.println("dthetatotal(microrad) = "+crystal.dthetatotal());

control.println("Structure Factor (unitless) = "+crystal.Fh());

control.println("Lex extinction length (m) = "+crystal.Lex());

control.println("Darwin width (mircorad) = "+crystal.darwinWidth());

control.println("Rocking Curve Width (mircorad) =

"+crystal.rockingCurvewidth());

control.println("sigma/F1 (mircorad) = "+crystal.sigmaoverF1());

control.println("Q (m^-1) = "+crystal.Q());

control.println("B (unitless) = "+crystal.B());

control.println("A (unitless) = "+crystal.A());

control.println("Mass attenuation coeff (cm^2/g) = "+crystal.massatt());

control.println("Linear attenuation coeff (m^-1) = "+crystal.linearatt());

control.println("a (unitless) = "+crystal.a());

control.println("I Integrated reflecting power (microrads) = "+crystal.I());

control.println("Incident flux ph/s/0.1%bw = "+crystal.fluxIncident());

control.println("Reflectivity (unitless) = "+crystal.R());

control.println("% dE/E crystal = "+crystal.percentdEoverE());

103

control.println("Diffracted intensity (ph/s) = "+crystal.intensity());

control.println(" ");

if(((crystal.R()< 1.0) || (crystal.R()==1.0))){

//Make sure reflectivity <1

if(crystal.results == 1){ // Prints all results

crystal.C = crystal.CRoland(); // Defines C as the C Roland

System.out.format("%d%d%d,%.4f,%.2f,%.2f,%.2f,%.2f,%.2f,%.2f,%d%d%d,%.

1f,",crystal.h,crystal.k,crystal.l

,crystal.T,crystal.xh(),crystal.xk(),crystal.xl(),crystal.yh(),crystal.yk(),crystal.yl(),crystal.z

h,crystal.zk,crystal.zl,crystal.F2);

System.out.format("%.1f,%.1f,%.1f,",crystal.chi(),crystal.E*Math.pow(10,-

3),crystal.thetaB());

System.out.format("%.3f,%.3f,%.3f,%.3f,%.1f,",crystal.C,crystal.CRoland(),cryst

al.Rs(),crystal.Rm(),crystal.RmRoland());

System.out.format("%.3f,",crystal.dthetarot());

System.out.format("%.3f,",crystal.dthetaB());

System.out.format("%.2f,",crystal.dthetatotal());

System.out.format("%.2f,", crystal.dthetasurface());

System.out.format("%.2f,", crystal.sigmaoverF1());

//System.out.format("%.1f ", crystal.I());

System.out.format("%.1f ", crystal.rockingCurvewidth());

System.out.format("%.2f,", crystal.R());

System.out.format("%.3f,", crystal.percentdEoverE());

System.out.format("%.3e,", crystal.intensity());

System.out.format("%.3f,", crystal.S13());



System.out.format("%.3f\n", crystal.S33());

}

// Finding results with max intensity and Rs>500*thinkness

if((crystal.results == 2) && ((crystal.Rs()>500*crystal.T) || (crystal.Rs() ==

500*crystal.T))){

crystal.C = crystal.CRoland();

if(crystal.intensity()>initial2){

// initial2 value set initially set to zero at beginning of energy for loop

largest = crystal.intensity();

}else if(crystal.intensity()<initial2){

largest = initial2;

}

// variable initial2 takes value largest and is used for next step of for-loop

initial2=largest;

104

if((crystal.intensity()==largest)){

// Each time a larger value of intensity is found the parameters are printed

//Display max intensity over max F2 range























}

}

// Code if C = default value from GUI, and prints all results for default input C

if(crystal.results == 3){













105











}

// Finding results with max intensity for fixed C using default GUI value

if(crystal.results == 4){

if(crystal.intensity()>initial2){



largest = initial2;

}

initial2=largest;

if((crystal.intensity()==largest)){ //Display max intensity over max F2 range























}

106

}

// All Results within boundaries of C>=0.2 and C<=1.0 and Rs>500*thinkness

if((crystal.results ==

5)&&((crystal.CRoland()>0.2)||(crystal.CRoland()==0.2))&&((crystal.CRoland()<1.0)||(c

rystal.CRoland()==1.0)) && ((crystal.Rs()>500*crystal.T) || (crystal.Rs() ==

500*crystal.T))){
























}

// Results within boundaries of C>=0.2 and C<=1.0 with max Intensity and

Rs>500*thickness

if((crystal.results ==

6)&&((crystal.CRoland()>0.2)||(crystal.CRoland()==0.2))&&((crystal.CRoland()<1.0)||(c

rystal.CRoland()==1.0)) && ((crystal.Rs()>500*crystal.T)||(crystal.Rs() ==

500*crystal.T))){


if((crystal.intensity()>initial2)&&(crystal.percentdEoverE()<0.2)){

107

/* If conditions of if loop above are true, print parameters in increasing order of intensity

for each energy for energy resolutions < 0.2%! To get Parameters that optimized for

intensity for each energy, just take the last row of results for each energy.*/



largest = initial2;

}

initial2=largest;

if((crystal.intensity()==largest)){ //Display max intensity over max F2 range























}

}

Intensity.append(0,crystal.E*Math.pow(10,-3),crystal.intensity());

Energyresolution.append(0,crystal.E*Math.pow(10,-3),crystal.percentdEoverE());

Reflectivity.append(0,crystal.E*Math.pow(10,-3),100*crystal.R());

Intensity.setAutoclear(false);

Energyresolution.setAutoclear(false);

Reflectivity.setAutoclear(false);

}

}

}

108

}

}

}

// Sets initial parameter values

public void reset() {

control.setValue("All results 1, max ph/s 2, results chosen C & chi 3, max ph/s fix

C and chi 4, all C & Rs constrained 5, max C & Rs & %dE/E constrained 6",6);

control.setValue("Reflection h",1);

control.setValue("Reflection k",3);

control.setValue("Reflection l",3);

control.setValue("Meridional axes (z') h'",0);

control.setValue("Meridional axes (z') k'",-1);

control.setValue("Meridional axes (z') l'",1);

control.setValue("Min Asymmetry angle (degrees) ",5);

control.setValue("Max Asymmetry angle (degrees) ",85);

control.setValue("Min Energy (eV)",30000);

control.setValue("Max Energy (eV)",95000);

control.setValue("F1 (m)", 23);

control.setValue("Min F2 (m)", 8);

control.setValue("Max F2 (m)", 11.5);

control.setValue("C (unitless)", 0.74);

control.setValue("Min Thickness (m)",0.0005);

control.setValue("Max Thickness (m)",0.001);

control.setValue("Horizontal Divergence (mrad)",0.3);

control.setValue("Vertical Divergence (mrad)",0.3);

control.setValue("Horizontal size of source (m)",0.001);

control.setValue("Horizontal sample size (mm)", 1);

control.setValue("Vertical sample size (mm)", 1);

control.setValue("K Polarization factor",1);

control.setValue("Size of B(T) of source",3.25);

control.setValue("If SC Wiggler type 1, if Conv. Wiggler type 2",1);

}

public static void main(String[] args) {

System.out.format("%s %s %s %s %s %s %s %s %s %s

","Ref","T(m)","xh'","xk'","xl'","yh'","yk'","yl'","z'","F2");

System.out.format("%s %s %s ","Chi","E(keV)","ThB");

System.out.format("%s %s %s ","C","CRol","Rs(m)");

System.out.format("%s %s %s %s %s

","Rm(m)","RmRol","dthrot","dthB","dth(t)");

System.out.format("%s ","dth");

System.out.format("%s %s ","sig/F1","w");

System.out.format("%s ","Ref");

109

System.out.format("%s ","%dE/E");

System.out.format("%s ","I(ph/s)");

System.out.format("%s %s %s %s\n","S'13","S'23","S'63","S'33");

CalculationControl.createApp(new SagBentApp());

}

}

SagBent Code:

/* SagBent.java

One Sagittally Bent Laue Calculations for CLS using Si

*/

import org.opensourcephysics.controls.*;

import org.opensourcephysics.frames.*;

public class SagBent {

public int h,k,l; // Reflection

public int zh,zk,zl; // Meridional axes

public double Asym;

public double E;

public double F1,F2;

public double C;

public int xhc = 1; // x cubic axes

public int xkc = 0;

public int xlc = 0;

public int yhc = 0; // y cubic axes

public int ykc = 1;

public int ylc = 0;

public int zhc = 0; // z cubic axes

public int zkc = 0;

public int zlc = 1;

public double s11 = 0.768; // cm^2/10^12 dyn

public double s12 = -0.214;// cm^2/10^12 dyn

public double s44 = 1.26; // cm^2/10^12 dyn

public double so = s11-s12-(0.5*s44); // cm^2/10^12 dyn

public double T; // thickness in m

public double Vc = Math.pow(5.4282*(Math.pow(10,-10)),3); // m^3

public double re = 2.81794*Math.pow(10,-15); // m

public double structfactor;

public double horizdiv; // horizontal divergence mrad

110

public double verticaldiv; // vertical divergence mrad

public double horizsource; // horizontal size of source m

public double horizsample; // horizontal sample size mm

public double verticalsample; // vertical sample size mm

public double K; // polarization factor=1 for normal polarization or

// k = abs(cos2theatB) for parallel Zachariasen pg. 123

public double att;

public double density = 2.33; // density of Si g/cm^3

public double B; // B field of source

public int sourcetype; // 1 for SCW and 2 for CW

public int results; // type of results in terminal

public double flux; // flux (ph/s/0.1%bw) incident

public double yh(){ // Sagittal axes

return ((zh*((zh*h)+(zk*k)+(zl*l))*(1-

Math.cos(Asym*Math.PI/180)))+(((zh*zh)+(zk*zk)+(zl*zl))*h*Math.cos(Asym*Math.P

I/180))+((Math.sqrt((zh*zh)+(zk*zk)+(zl*zl)))*(-

(zl*k)+(zk*l))*Math.sin(Asym*Math.PI/180)))/((zh*zh)+(zk*zk)+(zl*zl));

}

public double yk(){ // Sagittal axes

return ((zk*((zh*h)+(zk*k)+(zl*l))*(1-

Math.cos(Asym*Math.PI/180)))+(((zh*zh)+(zk*zk)+(zl*zl))*k*Math.cos(Asym*Math.P

I/180))+((Math.sqrt((zh*zh)+(zk*zk)+(zl*zl)))*((zl*h)-

(zh*l))*Math.sin(Asym*Math.PI/180)))/((zh*zh)+(zk*zk)+(zl*zl));

}

public double yl(){ // Sagittal axes

return ((zl*((zh*h)+(zk*k)+(zl*l))*(1-

Math.cos(Asym*Math.PI/180)))+(((zh*zh)+(zk*zk)+(zl*zl))*l*Math.cos(Asym*Math.PI

/180))+((Math.sqrt((zh*zh)+(zk*zk)+(zl*zl)))*(-

(zk*h)+(zh*k))*Math.sin(Asym*Math.PI/180)))/((zh*zh)+(zk*zk)+(zl*zl));

}

public double xh(){ // Surface normal

return ((yk()*zl)-(yl()*zk));

}

public double xk(){ // Surface normal

return ((yl()*zh)-(yh()*zl));

}

public double xl(){ // Surface normal

return ((yh()*zk)-(yk()*zh));

}

111

public double d() { // d spacing in ang

return 5.43102/(Math.sqrt((h*h)+(k*k)+(l*l)));

}

public double lambda() { // wavelength in ang

return 12398.562/E;

}

public double thetaB() { // Bragg angle in degrees

return Math.asin(lambda()/(2*d()))*180/Math.PI;

}

public double alpha() { // angle between BP and surface in degrees

return

Math.acos(((h*xh())+(k*xk())+(l*xl()))/((Math.sqrt((h*h)+(k*k)+(l*l)))*(Math.sqrt((xh()

*xh())+(xk()*xk())+(xl()*xl())))))*180/Math.PI;

}

public double chi() { // angle between BP and Surface Normal in degrees

return 90-alpha();

}

public double fs() { // focal length in m

return 1/((1/F1)+(1/F2));

}

public double CS23() { // CS'23 unitless absolute

return Math.abs(C*S23());

}

public double Rs() { // radius sagittal in m

return

fs()*2*(Math.sin(thetaB()*Math.PI/180))*(Math.abs(Math.sin(chi()*Math.PI/180)));

}

public double Rm() { // radius meridional in m

return Rs()/CS23();

}

public double RmRoland() { // radius meridional Roland Condition in m

return F1/(Math.cos((chi()+thetaB())*Math.PI/180));

}

public double RsRoland() { // radius sagittal Roland Condition in m

112

return RmRoland()*CS23();

}

public double CRoland() { // unitless usually between 0.2-1

return (Rs()*Math.cos((chi()+thetaB())*Math.PI/180))/(F1*Math.abs(S23()));

}

public double l1() { // direction cosine

return

((xh()*xhc)+(xk()*xkc)+(xl()*xlc))/(Math.sqrt(xhc*xhc+xkc*xkc+xlc*xlc)*Math.sqrt(xh

()*xh()+xk()*xk()+xl()*xl()));}

public double m1() { //direction cosine

return((xh()*yhc)+(xk()*ykc)+(xl()*ylc))/(Math.sqrt(yhc*yhc+ykc*ykc+ylc*ylc)

*Math.sqrt(xh()*xh()+xk()*xk()+xl()*xl()));}

public double n1() { //direction cosine

return((xh()*zhc)+(xk()*zkc)+(xl()*zlc))/(Math.sqrt(zhc*zhc+zkc*zkc+zlc*zlc)*

Math.sqrt(xh()*xh()+xk()*xk()+xl()*xl()));}


return

((yh()*xhc)+(yk()*xkc)+(yl()*xlc))/(Math.sqrt(xhc*xhc+xkc*xkc+xlc*xlc)*Math.sqrt(yh

()*yh()+yk()*yk()+yl()*yl()));}

public double m2() { //direction cosine

return((yh()*yhc)+(yk()*ykc)+(yl()*ylc))/(Math.sqrt(yhc*yhc+ykc*ykc+ylc*ylc)

*Math.sqrt(yh()*yh()+yk()*yk()+yl()*yl()));}

public double n2() { //direction cosine

return((yh()*zhc)+(yk()*zkc)+(yl()*zlc))/(Math.sqrt(zhc*zhc+zkc*zkc+zlc*zlc)*

Math.sqrt(yh()*yh()+yk()*yk()+yl()*yl()));}


return ((zh*xhc)+(zk*xkc)+(zl*xlc))/(Math.sqrt(xhc*xhc+xkc*xkc+xlc*xlc)*

Math.sqrt(zh*zh+zk*zk+zl*zl));}

public double m3() { // direction cosine

return((zh*yhc)+(zk*ykc)+(zl*ylc))/(Math.sqrt(yhc*yhc+ykc*ykc+ylc*ylc)*


public double n3() { // direction cosine

return((zh*zhc)+(zk*zkc)+(zl*zlc))/(Math.sqrt(zhc*zhc+zkc*zkc+zlc*zlc)*


public double S11() { // elastic compliance coefficient rotated axes

return (s12+(0.25*2*s44)+((l1()*l1()*l1()*l1())+(m1()*m1()*m1()*m1())+

(n1()*n1()*n1()*n1()))*so)/S33();}


113

return (s12+((l1()*l1()*l2()*l2())+(m1()*m1()*m2()*m2())+(n1()*n1()*n2()*

n2()))*so)/S33();}


return (s12+((l1()*l1()*l3()*l3())+(m1()*m1()*m3()*m3())+(n1()*n1()*n3()*

n3()))*so)/S33();}


return (2*so*(l1()*l1()*l2()*l3()+m1()*m1()*m2()*m3()+n1()*n1()*n2()*

n3()))/S33();}



n3()))/S33();}



n2()))/S33();}


return (s11-so*(1-(l2()*l2()*l2()*l2()+m2()*m2()*m2()*m2()+n2()*n2()*

n2()*n2())))/S33();}


return (s12+so*((l2()*l2()*l3()*l3())+(m2()*m2()*m3()*m3())+(n2()*n2()*

n3()*n3())))/S33();}



n3()))/S33();}



n3()))/S33();}



n2()))/S33();}

public double S33() { // elastic compliance coefficient rotated axes units ie. not

normalized

return (s11-so*(1-(l3()*l3()*l3()*l3()+m3()*m3()*m3()*m3()+n3()*n3()*

n3()*n3())));}



n2()))/S33();}



n1()))/S33();}



n2()))/S33();}


return (s44+4*so*(l2()*l2()*l3()*l3()+m2()*m2()*m3()*m3()+n2()*n2()*

n3()*n3()))/S33();}

114



n3()))/S33();}



n2()))/S33();}



n3()*n3()))/S33();}



n3()))/S33();}



n2()*n2()))/S33();}



n3()))/S33();

}

public double dthetarot() { // change in lattice rotation in microrads

return ((-T/Rs())*((S13()-(C*S23()))*Math.sin(chi()*Math.PI/180)*

Math.cos(chi()*Math.PI/180)-(C*S23())*Math.tan((chi()-thetaB())*

Math.PI/180)+S63()*Math.cos(chi()*Math.PI/180)*Math.cos(chi()*

Math.PI/180)))*1000000;

}

public double dthetaB() { // due to change in lattice spacing in microrads

return ((-T/Rs())*Math.tan(thetaB()*Math.PI/180)*(S13()*Math.sin(chi()*

Math.PI/180)*Math.sin(chi()*Math.PI/180)+C*S23()*Math.cos(chi()*

Math.PI/180)*Math.cos(chi()*Math.PI/180)+S63()*Math.sin(chi()*

Math.PI/180)*Math.cos(chi()*Math.PI/180)))*1000000;

}

public double dthetatotal() { // -dthetarot+dthetaB in microrads

return -dthetarot()+dthetaB();

}

public double gammah() { // cos(chi+thetaB) in rads

return Math.cos((chi()*Math.PI/180)+(thetaB()*Math.PI/180));

}

public double gammao() { // cos(chi-thetaB) in rads

return Math.cos((chi()*Math.PI/180)-(thetaB()*Math.PI/180));

115

}

public double sqrtgammahgammao() { // in rads

return Math.sqrt(Math.abs(gammah()*gammao()));

}

public double Fh() { // Structure Factor (unitless)

if(h==5 && k==1 && l==1) {

structfactor = 36.50203661;

}else if(h==0 && k==2 && l==2) {

structfactor = 69.3291351469501;

}else if(h==3 && k==3 && l==3) {

structfactor = 36.50203660659736;

}else if(h==1 && k==3 && l==3) {

structfactor = 40.75423502727092;

}else if(h==4 && k==4 && l==4) {

structfactor = 39.755302928564426;

}else if(h==0 && k==4 && l==4) {

structfactor = 48.340610030792114;

}else if(h==4 && k==0 && l==0) {

structfactor = 60.21195770580344;

}else if(h==4 && k==2 && l==2) {

structfactor = 53.7553947069231;

}else if(h==5 && k==5 && l==5) {

structfactor = 21.229648535500022;

}else if(h==1 && k==5 && l==5) {

structfactor = 27.16581633259372;

}else if(h==5 && k==3 && l==3) {

structfactor = 29.811695633763573;

}else if(h==3 && k==5 && l==5) {

structfactor = 24.890078160075063;

}else if(h==7 && k==1 && l==1) {

structfactor = 27.16581633259372;

}else if(h==3 && k==1 && l==1) {


}else if(h==3 && k==3 && l==1) {


}else if(h==1 && k==1 && l==1) {


}return structfactor;

}

public double Lex() { // extinction length (m)

return (Vc*sqrtgammahgammao())/(re*lambda()*Math.pow(10,-10)*Fh()*K);

}

116

public double gammas() { // in m

return Lex()/sqrtgammahgammao();

}

public double sqrtgammahovergammo() { // in rads

return Math.sqrt(Math.abs(gammah()/gammao()));

}

public double darwinWidth() { // Darwin width in microrads

return (2*lambda()*Math.pow(10,-

10)*sqrtgammahovergammo())/(Math.PI*gammas()*Math.sin((2*thetaB())*Math.PI/180

))*Math.pow(10,6);

}

public double rockingCurvewidth() { // microrads

return (Math.sqrt(Math.pow(dthetatotal(),2)+Math.pow(darwinWidth(),2)));

}

public double dthetasurface() { // along surface or dth(cauch) in microrads

return (horizdiv*Math.pow(10,-3)*(1-

(F1/(Rm()*Math.cos((chi()+thetaB())*Math.PI/180)))))*Math.pow(10,6);

}

public double sigmaoverF1() { // used in energy resolution microrads

return (horizsource/F1)*Math.pow(10,6);

}

public double percentdEoverE() { // percent energy resolution

return 100*(Math.sqrt(Math.pow((dthetatotal()*Math.pow(10,-

6)),2)+Math.pow((dthetasurface()*Math.pow(10,-

6)),2)+Math.pow((sigmaoverF1()*Math.pow(10,-

6)),2)))/(Math.tan(thetaB()*Math.PI/180));

}

public double Q() { // m^-1

return

(Math.pow(re,2)*Math.pow(K,2)*Math.pow(Fh(),2)*Math.pow(lambda()*Math.pow(10,

-10),3))/(Math.pow(Vc,2)*Math.sin((2*thetaB()*Math.PI/180)));

}

public double B() { // absolute B unitless

return (T/(Rs()*gammao()*Math.abs(dthetatotal()*Math.pow(10,-6))));

}

117

public double A() { // normalized thickness =RsBQ unitless

return (Rs()*B()*Q());

}

public double massatt() { // mass attenuation coeff cm^2/g

if(E == 10000){

att = 33.89;

}else if(E>10000 && E<15000){

att = -0.004710*E+80.99;

}else if(E == 15000){

att = 10.34;

}else if(E>15000 && E<20000){

att = -0.0011752*E+27.968;

}else if(E == 20000){

att = 4.464;

}else if(E>20000 && E<30000){

att = -0.000302*E+10.5;

}else if(E == 30000){

att = 1.436;

}else if(E>30000 && E<40000){

att = -0.0000739*E+3.657;

}else if(E == 40000){

att = 0.7012;

}else if(E>40000 && E<50000){

att = -0.0000262*E+1.749;

}else if(E == 50000){

att = 0.4385;

}else if(E>50000 && E<60000){

att = -0.0000118*E+1.029;

}else if(E == 60000){

att = 0.3207;

}else if(E>60000 && E<80000){

att = -0.0000049*E+0.615;

}else if(E == 80000){

att = 0.2228;

}else if(E>80000 && E<100000){

att = -0.00000192*E+0.379;

}else if(E == 100000){

att = 0.1835;

}else if(E>100000 && E<150000){

att = -0.000000774*E+0.2609;

}else if(E == 150000){

att = 0.1448;

}

return att;

118

}

public double linearatt() { // linear attenuation coeff m^-1

return massatt()*density*(1/Math.pow(10,-2));

}

public double a() { // 1-gammao/gammah

return 1-(gammao()/gammah());

}

public double I() { // Integrated reflecting Power mirorad

return ((Math.tanh(A())*Q())/(A()*a()*linearatt()))*Math.exp(-

linearatt()*T/gammah())*(1-(Math.exp(-a()*linearatt()*T/gammao())))*Math.pow(10,6);

}

public double R() { // Reflectivity unitless

return I()/rockingCurvewidth();

}

public double fluxIncident() { // flux ph/s/0.1%bw incident

if(B == 3.25 && sourcetype == 1 && horizdiv == 0.3 && verticaldiv == 0.3){

if(E == 10000){ flux = 3.59411*Math.pow(10,14);}

else if(E == 15000){ flux = 3.50861*Math.pow(10,14);}























119

















flux = flux*0.6; // multiplied by 0.6 to reduce heat load

}

return flux;

}

public double verticalbeamcrystal() { // vertical beam size at crystal mm

return F1*Math.tan(verticaldiv*Math.pow(10,-3))*Math.pow(10,3);

}

public double horizontalbeamcrystal() { // horizontal beam size at crystal mm

return F1*Math.tan(horizdiv*Math.pow(10,-3))*Math.pow(10,3);

}

public double intensity() { // intensity at crystal ph/s

return (fluxIncident()*R()*E*rockingCurvewidth()*Math.pow(10,-

6))/(Math.tan(thetaB()*Math.PI/180)*(0.001*E));

}

}

laue monochromator performance calculations for a future canadian light source diffraction beamline

Documents