Principles of X-ray Crystallography

___________________________________

Advisor: Raymond Kwok, Ph.D.

___________________________________

Coadvisor: Sotoudeh Hamedi-Hagh, Ph.D.

___________________________________

Committee Member: Masoud Mostafavi, Ph.D.

___________________________________

Supervisor: Tri Caohuu, Ph.D.

Luke Snow

58 Mount Hermon Rd

Scotts Valley, CA 95066

(831) 440-9170

LukeSnow@Gmail.com

Abstract

• The goal of this paper is to verify the

principles of crystallography at radio-

frequencies, and then use the principles to

design an antenna.

Outline

• Motivation and Introduction

• Verify Principles

– Bragg’s Law

– Scherrer Law

• Experimental Verification

• Switched beam design

• Conclusion and Further Work

The Classic Experimental Setup

Diff

ract

ed B

eam

Det

ecto

r

X-ray source Incident Beam2θ

Sample

Some Typical Data…

Some Typical Data…

X-ray Diffraction Powder Pattern

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20 40 60 80 100 120 140

2θ

Inte

ns

ity

Principles of X-ray Diffraction

• Bragg’s Law

• The Scherrer Equation

• The Reciprocal Lattice

• The Ewald Sphere

• The Scattering Factor

Motivation and Methodology

• To apply the concepts

verified to design an

antenna.

• To verify the concepts,

the flow chart at right

was used.

Hand Calculations.

Simulate each component of design.

Simulate the whole structure

Model Verified?

Success

No Yes

Motivation and Methodology

• The concepts verified

were employed to

design an antenna,

shown in the flow

chart. Fabricate the antenna

(Metal shop)

Verify Bragg's Law

Hand Calculations.

Simulate each component of design

Simulate the whole structure

Model Verified?

Success

No Yes

Background

• X-ray Crystallography is a well established

field.

• Born with the Discovery of Bragg’s Law, in

1912.

• Basic principles are used to determine

crystal structure, size, and defects.

Photonic Crystals

• Pioneered by E. Yablonovitch in 1987.

• Most applications employ the band stop and

band pass properties of photonic crystals

– Beam focusing antenna substrate

– Tunable 4-port switch

– Band pass or band block filters

Antenna

• The design can be thought of as an antenna

array.

• The design presented, and the analysis

behind it, appear to be unique.

Direction of Main Lobe

• The direction of the main lobe of the

antenna is determined by Bragg’s Law:

λ =2d sin θ

A

B

C

D

Incident Beam

Diff

ract

ed B

eam

θ

θ

θ

d sin θ

d

d sin

θ

Sample Level View

θ

2θ

k Crystal

k'

Bragg’s Law Verified

Experimental Setup

Top ViewSkewed View

Results

34 Degree Incidence

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120 140 160 180

Angle

Norm

alized E

-Fie

ld M

agnitude

22 Degree Incidence

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120 140 160 180

Angle

Norm

alized E

-Fie

ld M

agnitude

45 Degree Incidence

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120 140 160 180

Angle

Norm

alized E

-Fie

ld M

agnitude

Summary – Peak LocationsPredicted Observed

θ (φ -90)/2 ∆

22 22 0.00%

24 24.25 0.40%

26 25.75 0.40%

28 27.75 0.30%

30 30 0.00%

32 32.25 0.30%

34 34.5 0.60%

36 37.25 1.50%

38 39 1.20%

40 41 1.20%

42 44.25 2.60%

44 44.5 0.60%

45 46 1.10%

46 46.75 0.80%

Avg: 0.80%

RSD 0.87

Beam Width

• The beam width (FWHM) is given by the Scherrer Law:B(2θ) = Kλ / (Na cos θ)

• K-shape factor

• N - size of the crystal in unit cells

• a - unit cell length for a square crystal

• λ -Wavelength

• θ - Bragg angle

Verification

• The Scherrer law is verified in two ways

– By varying N, holding all other quantities

constant. Expect a 1/N dependence, and values

of K on the order of unity.

– Vary θ and a together; Use Bragg’s Law to

substitute for a in the Scherrer equation:

B(2θ) = 2K tan θ / N

Results

K = 1.02 Gave Best Fit

3 5 7 9 11 13

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

FWHM vs Crystal Size

HFSS Data Best Fit

Crystal Size - N x N [integer]

FW

HM

[d

eg

ree

s]

Results

K = 0.90 Gave Best Fit

Effect of Angle On Peak Width

0

0.05

0.1

0.15

0.2

0.25

22 27 32 37 42

Angle of Incidence

Peak W

idth

HFSS Data Best Fit

Experimental

• An antenna was constructed to verify

Bragg’s law.

• The antenna consisted of a waveguide,

horn, and a parallel plate/crystal section.

• The antenna was designed to operate in the

6GHz region.

Experimental• Data was taken in a Compact Antenna Test Range

(CATR).

• A VNA with 0-40GHz capability was used to take data.

• A WR137 waveguide to coax adapter was used for the detector.

• Two WR137 waveguides were used for a reference.

• Far Field for this design was 12 ft.

• Data was taken at approximately 14ft, for an angular resolution of 0.5 deg.

Waveguide sectionParameter Value

OD 1.5" x 1.0"

ID 1.25" x 0.75"

fc104.72 GHz

fc119.17 GHz

Length 12"

Horn Section

Top View

b1

b1

ρ1

Front View, from the x'-direction

x'

a

y'Parameter Value

b1 12"

a 1"

ρ1 18"

Parallel Plate Section

d 1"

fc00 0 GHz

fc10 6 GHz

Crystal Section12 in

12 in

2 in

1 in

30deg

Post

Diam

1/8"

θ 30o

λ 2"

Results

• Return Loss was better than –20dB at

5.9GHz, and was about –10dB at 6.223GHz

• 5.9GHz corresponds to a wavelength of 2in,

but the best performance was obtained at

6.223GHz, with a gain over WR137 of 8dB

ResultsTransmission, 5.9GHz

-80

-75

-70

-65

-60

-55

-50

-45

-40

0 20 40 60 80 100 120 140 160 180

Angle (deg)

S21 (dB)

HFSS Data, 6GHz

-15

-10

-5

0

5

10

15

20

25

30

35

0 20 40 60 80 100 120 140 160 180

Angle (deg)

S21 (dB)

The best radiation pattern

obtainedrETotal [V] - Freq='24GHz' L='25cm' Theta='90deg'

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300 350 400

30 degree incidence

A polar plot

20

40

60

80

30

210

60

240

90

270

120

300

150

330

180 0

ResultsTransmission, 6.223 GHz

Frequency of Max transmission at 150 deg.

-75

-70

-65

-60

-55

-50

-45

-40

0 20 40 60 80 100 120 140 160 180

Angle (deg)

S21 (

dB

)

Best Performance

Conclusions and Observations

• Design could be improved with:

– Better grounding

– Higher quality plane wave.

– Larger diameter posts

– Longer interaction length

Switched Beam Antenna

• Each Crystal has an associated “reciprocal space” – a

lattice of points related to those of the direct space crystal.

• The units of this space are inverse length.

• For a direct space rectangular lattice of dimensions a and b,

the reciprocal lattice is of rectangular, of length 1/a, 1/b.

• The “Ewald Circle” may be drawn in reciprocal space to

describe an X-ray diffraction experiment, the circle having

radius 1/λ

• When the circle intersects two or more reciprocal lattice

points, one or more reflections are created.

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

1/λ

2θ

For the given

diagram, there are

two 45 degree

reflections. If a = b

= 0.5in, then λ =

0.707in

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

The reciprocal

lattice has been

altered by doubling

the length of the

basis vector in the

vertical direction,

corresponding to

halving the direct-

space lattice basis

vector

The two models

The two radiation patterns

0.00 125.00 250.00 375.00Phi [deg]

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

rET

ota

l [V

]

Ansoft LLC 45degDirection24GHzRectCryst_2LobesRectCryst2Lobes ANSOFT

m3

m2m1

Curve Info

rETotalSetup1 : LastAdaptive_lambda='1.25cm' Freq='24GHz' Theta='90deg'

Name X Y

m1 182.0000 33.6675

m2 358.0000 33.3849

m3 126.0000 0.7572

0.00 125.00 250.00 375.00Phi [deg]

0.00

10.00

20.00

30.00

40.00

50.00

60.00

rET

ota

l [V

]

Ansoft LLC 45degDirection24GHzRectCryst_1LobezRectCryst1Lobe ANSOFT

m1

m2

Curve Info

rETotalSetup1 : LastAdaptive_lambda='1.25cm' Freq='24GHz' Theta='90deg'

Name X Y

m1 190.5000 59.4786

m2 126.0000 33.7840

Conclusion

• Various concepts of crystallography have been verified.

• Fruitful parallels between X-ray diffraction and photonic crystals exist, with potential to illuminate ideas in both fields.

• More work to be done before the design is admitted to practical application.

– Additional Measurements with the improved model

– Switched beam measurement

Pattern after Improvement

0.00 20.00 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00Phi [deg]

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

rETo

tal [V

]

Ansoft LLC QuarterInchPostsXY Plot 2 ANSOFT

m1

m2

Name X Y

m1 97.0000 25.1515

m2 151.0000 34.9966

References• D. Pozar, Microwave engineering, Hoboken NJ: Wiley, 2005, pp. 105,113.

• C.A. Balanis, Antenna Theory: Analysis and Design, 3rd Edition, Wiley-Interscience, 2005, pp. 740,742,756.

• B. Warren, X-ray diffraction, Reading Mass.: Addison-Wesley Pub. Co., 1969, pp. 18,31,251-254.

• M. Woolfson, in An introduction to X-ray crystallography, 2nd ed., Cambridge: Cambridge Univ. Press, 1997, p. 108.

• Data Taken in Upper Division Physics Lab, University of California, Santa Cruz, 2001

• W. L. Bragg, “The diffraction of short electromagnetic waves by a crystal,” in Proceedings of the Cambridge Philosophical Society, vol. 17,

pp. 43–57, 1913.

• A. L. Patterson, “The Scherrer formula for X-ray particle size determination,” Physical Review, vol. 56, no. 10, pp. 978–982, 1939.

• P. Scherrer, “Göttinger Nachrichten,” Math. Phys, vol. 98, 1918.

• F. Molinet, “Geomatrical Theory of Diffraction(GTD) Part I: Foundation of the theory,” IEEE, vol. 29, no. 4, pp. 6-17, Aug. 1987.

• F. Molinet, “Geometrical Theory of Diffraction(GTD) Part II: Extensions and future trends of the theory,” IEEE, vol. 29, no. 5, pp. 5-16, Oct.

1987.

• J. B. KELLER, “Geometrical Theory of Diffraction,” Journal of the Optical Society of America, vol. 52, no. 2, pp. 116-130, Feb. 1962. See

figs. 18-20

• S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Physical Review Letters, vol. 58, no. 23, pp. 2486–

2489, 1987.

• E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Physical Review Letters, vol. 58, no. 20, pp.

2059–2062, 1987.

• J.M. J. Danglot, O. Vanbesien, D. Lippens, et al, “Toward Controllable Photonic Crystals for Centimeter and Millimeter Wave Devices,”

Journal of Lightwave Technology, Vol. 17, No. 11, November 1999

• E.R. Brown, C.D. Parker, E. Yablonovitch, “Radiation Properties of a Planar Antenna on a Photonic Crystal Substrate,” J. Opt. Soc. Am B10,

404-407,1993

• O. Vanbesien, J. Danglot, D. Lippens, “A Smart KBand 4-Port Resonant Switch Based on Photonic Band Gap Engineering.” 29th European

Microwave Conference, Munich 1999.

• J. Carbonell, O. Vanbesien, D. Lippens, “Electric field patterns in finite two-dimen-sional wire photonic lattices,” Superlattices and

Microstructures, Vol. 22, No. 4, 1997

Acknowledgements

• My advisor, Dr. Ray Kwok

• Bill Shull of Zygo corporation, for helping with the construction of the antenna, and use of his Machine Shop,

• My Supervisor and co-workers at Space Systems Loral, for providing facilities and assistance to make the measurements.

• My Wife and family, for their patience with my seemingly endless project.

Questions?

The effect of Post Diameter

Effect of Post Size

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2 0.25

Post Diameter (cm)

Inte

ns

ity

of

Pe

ak (

mV

)

Diffracted Beam Peak Straight Through Beam

Model Specifications

Parameter Value

Plate Thickness - Top 0.1cm

Plate Thickness - Bottom 0.1cm

Plate Spacing 1.25cm

Crystal Size 8x8

Post Spacing λ/(2 sinθ)

Post Radius 0.1cm

Angle of Incidence 22o-46o, 2o steps; 45o

Solution Frequency 24GHz

Max. ∆S 0.01