description of changes in crystal orientations by the...

Research ArticleDescription of Changes in Crystal Orientationsby the Elements of Logarithm of a Rotation Matrix

Susumu Onaka and Kunio Hayashi

Department of Materials Science and Engineering Tokyo Institute of Technology 4259-J2-63 Nagatsuta Yokohama 226-8502 Japan

Correspondence should be addressed to Susumu Onaka onakasaamtitechacjp

Received 29 July 2016 Accepted 13 October 2016 Published 11 January 2017

Academic Editor Jose Alvarez

Copyright copy 2017 S Onaka and K Hayashi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The logarithm lnR of rotation matrix R is a skew symmetric tensor consisting of three independent elements of real numbers Inaddition to the Euler angles and the axisangle pair the elements of lnR called the log angles are also the set of three parameters ofR In this paper we will show that the concept of the log angles is also useful to discuss changes in crystal orientationsThe changesin R as a function of the position are given by the changes in the log angles As an example orientation changes caused by arrays ofdislocations in a plastically deformed Cu single crystal are discussed

1 Introduction

Electron backscatter diffraction analysis with scanning elec-tron microscopy (SEMEBSD) is a powerful technique toanalyze orientations of crystalline materials Using this tech-nique we can measure variations of crystal orientations inmaterials with high accuracyThen there is a chance to assessdefect structures in grains such as dislocation structures thatcan vary crystal orientations around the defects [1] but itis necessary to express the variations in crystal orientationsreasonably

We can describe certain crystal orientation using arotation matrix R which is the 3 times 3 orthogonal matrixhaving nine elements [2] However since the number ofindependent elements of R is three sets of three parametersinstead of R are more convenient to understand the rotationTo discuss variations R in crystalline materials obtained byresults of SEMEBSDmeasurements appropriate parametersshould be selected to show the relationship between changesin crystal orientations and microstructures formed in thematerials

The logarithm lnR of R has been considered to discussthe rotation mathematically in the framework of the grouptheory [3ndash5] The logarithm lnR is a skew symmetric tensorconsisting of three independent elements of real numbers

In addition to the Euler angles and the axisangle pair [2]the elements of lnR are also the set of three parameters ofR [6] In this paper we will show that elements of lnR calledthe log angles [7 8] are useful parameters to discuss changesin crystal orientations As an example orientation changescaused by arrays of dislocations in a plastically deformed Cusingle crystal are discussed

2 Changes in Crystal Orientation

21 Logarithm of Rotation Matrix When the axisangle pairof R is given by a unit vector n = (ℎ 119896 119897) and a rotation angleΦ using the set of log angles (1199081 1199082 1199083) forR lnR is writtenas [4]

lnR = ( 0 minus1199083 11990821199083 0 minus1199081minus1199082 1199081 0 ) = ( 0 minus119897Φ 119896Φ119897Φ 0 minusℎΦminus119896Φ ℎΦ 0 ) (1)

On the other hand the relationship between R and lnR iswritten as [5]

R = lim119901rarrinfin

(E + lnR119901 )119901 (2)

HindawiScanningVolume 2017 Article ID 4893956 7 pageshttpsdoiorg10115520174893956

2 Scanning

yz

x

Crystal

x998400 y998400

z998400

Num

ber o

f sph

eric

al u

nits

N≫

1

(a)

R

yz

x

Crystal

x998400

y998400

z998400

Num

ber o

f120575R

N≫

1

120575R

120575R

120575R

120575R

w1N

w2N

w3N

(b)

Figure 1 Graphical or mechanical representations of R given by (3) Spherical units corresponding to 120575R given by (4) are stacked 119873 times(a) and (b) are the models for before and after the operation of R respectively In (b) R determines the primed coordinate axes of a crystal onthe top of the model with respect to the reference unprimed frame

x0Reference

R(x2)

R(x1)

x1 x2

ΔR

Figure 2 A model showing changes in crystal orientations for linescan of SEMEBSD measurements

where E is the unit matrix Hence when 119873 is a sufficientlylarge positive integer we have

R asymp (120575R)119873 (3)

where

120575R=E + lnR119873 = ( 1 minus1199083119873 11990821198731199083119873 1 minus1199081119873minus1199082119873 1199081119873 1 ) (4)

Equation (4) shows that the 119873(≫ 1) times successiveoperations of 120575R are equivalent to R Hence we have Figures1(a) and 1(b) as graphical or mechanical representations [7 8]of R given by (3) and (4) Spherical units corresponding to120575R with infinitesimal rotation angles are stacked 119873 timesSince 120575R is the rotationmatrix with infinitesimal off-diagonal

components this can be shown as a product of three basicrotations in a spherical unit [7 8]These figures show that thelog angles (1199081 1199082 1199083) are the sums of the divided rotationangles around the coordinate axes of the reference frame andinterpreted as the components of the rotation angles of R[7 8] This can be said for any R and is not limited to smallangles

Depending on selection of rotation axes and their ordersvarious sets of the Euler angles based on products of threebasic rotations are defined for certain R [7] The reasonof the many sets is that the products of rotations are notcommutative [7] Different from the various sets of the Eulerangles a set of the log angles is uniquely determined forcertain R [7] Using a concept similar to that of the Eulerangles components of the rotation angles of R have beenproposed by considering simple products of rotations [9]However as well as the Euler angles the components givenby considering the simple products of rotations may notbe uniquely determined for certain R The concept of thelog angles can be applied to other characteristic values ofrotations such as average orientations treated in previousstudies [6 10 11]

22 Log Angles for Small-Angle Rotation As a model for linescan of SEMEBSDmeasurements here we consider changesin crystal orientations as shown by Figure 2 The crystalorientations at the positions 1199091 and 1199092 with respect to thatat the origin O are given by R(1199091) and R(1199092) respectively

Scanning 3

y

z

x

Crystal

x998400 y998400

z998400

R(x1)

ΔR

(a)

y

z

x

Crystal

x998400 y998400

z998400

R(x2)

(b)

Figure 3 Graphical representations corresponding to the (a) right- and (b) left-hand sides of (5) where the difference ΔR in the crystalorientations between those at 1199091 and 1199092 is expressed by using the reference frame at the origin

ND asymp [111]

RD asymp [112]

TD asymp [110]

(a)

101001

111ND

A

O

ND

RD

200120583m

(b)

111

ND

TD RD

010

100

001121

110

011

211

101

11 2

(c)

Figure 4Cold rolling ofCu single crystal (a) Schematic illustration showing a shape and crystallographic orientations before cold rollingTheregion analyzed by SEMEBSD technique after rolling is also shown in (a) (b)The inverse pole figure (IPF) map showing the crystallographicorientation of ND on the TD plane after rolling to 15 reduction in thickness (c)The stereographic projection showing the orientation at theorigin O after the rolling Changes of crystal orientations along the line OA are also shown by 111 pole figures in (c)

Here we defineΔR as the difference in the crystal orientationsbetween those at 1199091 and 1199092 expressed by using the referenceframe at the origin Then the relation among the threerotation matrices is given by

R (1199092) = ΔRR (1199091) (5)

Figures 3(a) and 3(b) are graphical representations corre-sponding to the right- and left-hand sides of (5) respectivelyAlthough ΔR is an additional rotation from R(1199091) to R(1199092)the stacking of the spherical units for the right-hand side of

(5) becomes that as shown in Figure 3(a) IfΔR is the rotationdefined by the rotated reference frame at 1199091 the order of theproduct in the right-hand side of (5) and the stacking of thespherical units in Figure 3(a) are changed

Here we assume that ΔR is a small-angle rotation and therotation angleΔΦ of the axisangle pair forΔR satisfy |ΔΦ| ≪1 Then (1) for ΔR is written from (2) to (4) as

ΔR asymp E + lnΔR (6)

4 Scanning

minus005

0

005

010

015

minus010

200 300 400 5000 100minus015

Distance x (120583m)

TD co

mpo

nent

ofΦ

wTD

(rad

)

(a)

minus001

0

001

002

003

minus002

minus003

200 300 400 5000 100Distance x (120583m)

RD co

mpo

nent

ofΦ

wRD

(rad

)

(b)

minus001

0

001

002

003

minus002

minus003

200 300 400 5000 100Distance x (120583m)

ND

com

pone

nt o

fΦw

ND

(rad

)

(c)

Figure 5 The variations of the log angles (119908TD 119908RD 119908ND) along the line OA in Figure 4(b) as a function of the distance from the origin OThe log angles show the orientation changes from the orientation at the origin O (a) (b) and (c) are for the components of 119908TD 119908RD and119908ND respectively

The log angles (Δ1199081 Δ1199082 Δ1199083) of ΔR written as

lnΔR = ( 0 minusΔ1199083 Δ1199082Δ1199083 0 minusΔ1199081minusΔ1199082 Δ1199081 0 ) (7)

satisfy |Δ1199081| |Δ1199082| |Δ1199083| ≪ 1 since |ΔΦ| ≪ 1 Using thisequation (5) is rewritten as

R (1199092) asymp ( 1 minusΔ1199083 Δ1199082Δ1199083 1 minusΔ1199081minusΔ1199082 Δ1199081 1 )R (1199091) asymp (1 0 00 1 minusΔ11990810 Δ1199081 1 )( 1 0 Δ11990820 1 0minusΔ1199082 0 1 )( 1 minusΔ1199083 0Δ1199083 1 00 0 1)R (1199091) (8)

Equation (8) shows that the infinitesimal log angles (Δ1199081Δ1199082 Δ1199083) of ΔR can be interpreted as the components ofrotation angles giving the difference in the crystal orienta-tions between R(1199091) at 1199091 and R(1199092) at 119909223 Position Dependence of Change in Crystal OrientationHere we assume that the difference Δ119909 in 1199091 and 1199092 is small

When R(119909) is differentiable with respect to 119909 the differencebetween R(1199091) at 1199091 and R(1199092) at 1199092 is written as[R (1199092) minus R (1199091)] = [R (119909 + Δ119909) minus R (119909)]asymp 119889R (119909)119889119909 Δ119909 (9)

Scanning 5

minus010 minus005 000 005 010

minus002

000

002

ΔwTD (rad)

Δw

RD(r

ad)

(a)

minus010 minus005 000 005 010

minus002

000

002

ΔwTD (rad)

Δw

ND

(rad

)

(b)

minus003 minus002 minus001 000 001 002 003minus003

minus002

minus001

000

001

002

003

ΔwRD (rad)

Δw

ND

(rad

)

(c)

Figure 6 The relations among Δ119908TD Δ119908RD and Δ119908ND at the same locations on the line OA in Figure 4(b) where Δ119908TD Δ119908RD and Δ119908NDare the changes in the log angles between neighboring two measuring points See text for an inclined broken line in (c)

From this equation and (5) to (7) we have

((

0 minusΔ1199083Δ119909 Δ1199082Δ119909Δ1199083Δ119909 0 minusΔ1199081Δ119909minusΔ1199082Δ119909 Δ1199081Δ119909 0 ))

asymp [119889R119889119909 ]Rminus1 (10)

As shown by this equation the changes in the rotationmatrixR(119909) as a function of the position 119909 are given by the changesin the log angles Δ1199081 Δ1199082 and Δ1199083

If all R(119909) for any 119909 are small-angle rotations products ofmatrices relating to R(119909) become commutative and lnΔR iswritten from (5) as

lnΔR = ln [R (1199091)minus1 R (1199092)] asymp lnR (1199092) minus lnR (1199091) (11)

Then a differential form for the small-angle rotations is givenby 119889 lnR119889119909 asymp [119889R119889119909 ]Rminus1 asymp Rminus1 [119889R119889119909 ] (12)

3 Changes in Crystal Orientations ofCold-Rolled Cu Single Crystal

31 Cold Rolling of Cu Single Crystal and Crystal Orientationsafter Cold Rolling As an example of applications of thepresent analysis we consider changes in crystal orientationsof a Cu single crystal caused by plastic deformation A plate-like Cu single crystal (999 mass purity) with a 2mmthickness and 20mm width and length was cold rolledFigure 4(a) shows a schematic of the Cu single crystal beforecold rolling Directions TD RD and ND shown in thisfigure are the transverse rolling and normal directions ofthe applied rolling respectively TD RD and ND constructan orthogonal coordinate system and their crystallographicdirections are also indicated in Figure 4(a)

We observed how themicrostructure evolved after rollingto 15 reduction in thickness by SEMEBSD techniqueA JSM-7001F (JEOL) controlled by a program OIM DataCollection ver 701 (TSL) was used The acceleration voltageand step size were 15 kV and 10 120583m respectively The cross-section normal to the TD (TD plane) was observed as shownby the rectangular region in Figure 4(a) Figure 4(b) is the

6 Scanning

ND asymp [111]

RD asymp [112]

TD asymp [110]

111

10111 2

01 1

(a)

110

111

11 2

(b)

121 110

111

211

10111 2

01 1

(c)

Figure 7 Schematic illustrations showing dislocation structures in the rolled single crystal See text for the details

inverse pole figure (IPF) map showing the crystallographicorientation of ND on the TD plane of the rolled specimenThe color code for the IPFmap is shown in a standard triangleat the top-right The directions of RD and ND on the TDplane are also indicated

As shown in Figure 4(b) almost parallel blue and greenbands showing orientation splitting appeared after the rollingInclination of the bands is from 056 to 065 rad (32 to 37 deg)to RDWe discuss orientation changes along the line from theorigin O to A shown in this IPF map Orientation measure-ments along the line were made at intervals of 24 120583m Thestereographic projection in Figure 4(c) shows the orientationat the origin O Changes of crystal orientations along the lineOA are shown by 111 pole figures in Figure 4(c)These showthat the orientation changes caused by the rolling are mainlyrotations around TD

32 Changes in Log Angles Figure 5 shows the variationsof the log angles (119908TD 119908RD 119908ND) along the line OA as afunction of the distance from the origin O The log anglesshow the orientation changes from the orientation at theorigin O These show that the changes in 119908TD are aboutplusmn014 rad and the largest among the three components Theperiodic changes in 119908TD show that the interfaces betweenthe blue and green bands in Figure 4(b) are small-angletilt boundaries causing clockwise and anticlockwise rotationsaround TD Parallel directions of the traces of interfacesshown in Figure 4(b) are almost parallel to a red brokenline from the center of the projection to 111 in Figure 4(c)This red broken line is perpendicular to the dotted lineshowing directions perpendicular to [111] in the same figure

These show that the small-angle tilt boundaries are composedof arrays of edge dislocations on parallel (111) slip planesas indicated by dislocation marks at the middle-right ofFigure 4(b) Considering the initial orientation of the Cusingle crystal shown in Figure 4(a) we can say that this (111)is the slip plane operated during the rolling [12]

Figures 6(a)ndash6(c) show the relations amongΔ119908TDΔ119908RDandΔ119908ND at the same locations on the lineOA in Figure 4(b)where Δ119908TD Δ119908RD and Δ119908ND are the changes in the logangles between neighboring two measuring points 24120583mapart In the Δ119908TD minus Δ119908RD and Δ119908TD minus Δ119908ND relationsshown in Figures 6(a) and 6(b) the dispersion of data seemsto be symmetric and random However in the Δ119908ND minusΔ119908RD relation shown in Figure 6(c) the data dispersessystematically near an inclined broken line in this figure Theslope of the inclined broken line is discussed later

Figure 7(a) schematically shows the small-angle tiltboundaries observed on the TDplane Possible slip directions[01 1] and [101] on (111) are also indicated in Figure 7(a)During rolling macroscopic shear direction on (111)may be[11 2] since this causes the plane-strain plastic deformationon the ND-RD plane Such deformation is achieved bythe double slip of [01 1] and [101] on (111) When sheardeformation on (111) to [11 2] occurs crystal rotationsaround TD [110] occur as shown in Figure 7(b) Howeveras shown in Figure 7(c) this deformation is the combinationof the two shears (111) [01 1] and (111) [101] which causecrystal rotations around [211] and [121] respectively

The axis (ℎ 119896 119897) of rotation is related to the componentsof the log angles as shown by (1)The rotation between neigh-boring two measuring points along the line OA is caused by

Scanning 7[01 1] and [101] dislocations between themThe componentsof the log angles are proportional to the components of therotation axis The rotation axis may change between [211]and [121] on (111) depending on the ratio of numbers ofthe [01 1] and [101] dislocations Hence the change in therotation axes is expressed by the Δ119908ND minus Δ119908RD relation asshown by the broken line in Figure 6(c) We have consideredthe changes in crystal orientations in the rolled Cu singlecrystal with the changes in the log angles The systematicdispersion of the Δ119908ND minusΔ119908RD relation shown in Figure 6(c)is thus explained by the dislocation structures shown inFigure 7(c)

4 Conclusions

The logarithm lnR of rotation matrix R is a skew symmetrictensor consisting of three independent elements of realnumbersThe log angles are the elements of lnR and the set ofthree parameters ofR We have shown that the concept of thelog angles is useful to discuss changes in crystal orientationsThe changes in crystal orientations as a function of theposition are given by the changes in the log angles As anexample orientation changes caused by arrays of dislocationsin a cold-rolled Cu single crystal are discussed

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank Motoki Osada and Yuki Kurosufor their contributions to the work on Cu single crystalsThis work was supported by JSPS KAKENHI Grant no16K06703

References

[1] A Yoshida YMiyajima and S Onaka ldquoEvolution of the spreadof crystal orientation with plastic deformation in a cold-rolledCu single crystalrdquo Journal of Materials Science vol 49 no 5 pp2013ndash2017 2014

[2] D A Varshalovich A N Moskalev and V K KhersonskiiQuantum Theory of Angular Momentum World ScientificSingapore 1988

[3] D H Sattinger and O L Weaver Lie Groups and Algebras withApplications to Physics Geometry andMechanics Springer NewYork NY USA 1986

[4] J E Marsden and T S Ratiu Introduction to Mechanics andSymmetry Springer New York NY USA 2nd edition 1999

[5] B C Hall Lie Groups Lie Algebras and Representations AnElementary Introduction vol 222 of Graduate Texts in Mathe-matics Springer New York NY USA 2015

[6] N R Barton and P R Dawson ldquoOn the spatial arrangementof lattice orientations in hot-rolled multiphase titaniumrdquoMod-elling and Simulation in Materials Science and Engineering vol9 no 5 pp 433ndash463 2001

[7] K Hayashi M Osada Y Kurosu Y Miyajima and S OnakaldquoLog angles characteristic angles of crystal orientation given bythe logarithm of rotation matrixrdquo Materials Transactions vol57 no 4-5 pp 507ndash512 2016

[8] S Onaka and K Hayashi ldquoInterpretation of elements of thelogarithm of a rotation matrix as rotation components aroundcoordinate axes of a reference framerdquo Journal of MathematicalChemistry vol 54 no 8 pp 1686ndash1695 2016

[9] J A Wert Q Liu and N Hansen ldquoDislocation boundaryformation in a cold-rolled cube-oriented Al single crystalrdquoActaMaterialia vol 45 no 6 pp 2565ndash2576 1997

[10] R Quey J H Driver and P R Dawson ldquoIntra-grain orientationdistributions in hot-deformed aluminium orientation depen-dence and relation to deformation mechanismsrdquo Journal of theMechanics and Physics of Solids vol 84 pp 506ndash527 2015

[11] J C Glez and J Driver ldquoOrientation distribution analysis indeformed grainsrdquo Journal of Applied Crystallography vol 34 no3 pp 280ndash288 2001

[12] Y Ajisaka Y Miyajima and S Onaka ldquoSEMEBSD analysis ofchanges in crystal orientation of 111 lt112gt Cu single crystalcaused by the cold-rollingrdquo Journal of the Japan Institute ofMetals vol 80 pp 553ndash561 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of

2 Scanning

yz

x

Crystal

x998400 y998400

z998400

Num

ber o

f sph

eric

al u

nits

N≫

1

(a)

R

yz

x

Crystal

x998400

y998400

z998400

Num

ber o

f120575R

N≫

1

120575R

120575R

120575R

120575R

w1N

w2N

w3N

(b)

Figure 1 Graphical or mechanical representations of R given by (3) Spherical units corresponding to 120575R given by (4) are stacked 119873 times(a) and (b) are the models for before and after the operation of R respectively In (b) R determines the primed coordinate axes of a crystal onthe top of the model with respect to the reference unprimed frame

x0Reference

R(x2)

R(x1)

x1 x2

ΔR

Figure 2 A model showing changes in crystal orientations for linescan of SEMEBSD measurements

where E is the unit matrix Hence when 119873 is a sufficientlylarge positive integer we have

R asymp (120575R)119873 (3)

where

120575R=E + lnR119873 = ( 1 minus1199083119873 11990821198731199083119873 1 minus1199081119873minus1199082119873 1199081119873 1 ) (4)

Equation (4) shows that the 119873(≫ 1) times successiveoperations of 120575R are equivalent to R Hence we have Figures1(a) and 1(b) as graphical or mechanical representations [7 8]of R given by (3) and (4) Spherical units corresponding to120575R with infinitesimal rotation angles are stacked 119873 timesSince 120575R is the rotationmatrix with infinitesimal off-diagonal

components this can be shown as a product of three basicrotations in a spherical unit [7 8]These figures show that thelog angles (1199081 1199082 1199083) are the sums of the divided rotationangles around the coordinate axes of the reference frame andinterpreted as the components of the rotation angles of R[7 8] This can be said for any R and is not limited to smallangles

Depending on selection of rotation axes and their ordersvarious sets of the Euler angles based on products of threebasic rotations are defined for certain R [7] The reasonof the many sets is that the products of rotations are notcommutative [7] Different from the various sets of the Eulerangles a set of the log angles is uniquely determined forcertain R [7] Using a concept similar to that of the Eulerangles components of the rotation angles of R have beenproposed by considering simple products of rotations [9]However as well as the Euler angles the components givenby considering the simple products of rotations may notbe uniquely determined for certain R The concept of thelog angles can be applied to other characteristic values ofrotations such as average orientations treated in previousstudies [6 10 11]

22 Log Angles for Small-Angle Rotation As a model for linescan of SEMEBSDmeasurements here we consider changesin crystal orientations as shown by Figure 2 The crystalorientations at the positions 1199091 and 1199092 with respect to thatat the origin O are given by R(1199091) and R(1199092) respectively

Scanning 3

y

z

x

Crystal

x998400 y998400

z998400

R(x1)

ΔR

(a)

y

z

x

Crystal

x998400 y998400

z998400

R(x2)

(b)


ND asymp [111]

RD asymp [112]

TD asymp [110]

(a)

101001

111ND

A

O

ND

RD

200120583m

(b)

111

ND

TD RD

010

100

001121

110

011

211

101

11 2

(c)



R (1199092) = ΔRR (1199091) (5)





4 Scanning

minus005

0

005

010

015

minus010

200 300 400 5000 100minus015


TD co

mpo

nent

ofΦ

wTD

(rad

)

(a)

minus001

0

001

002

003

minus002

minus003

200 300 400 5000 100Distance x (120583m)

RD co

mpo

nent

ofΦ

wRD

(rad

)

(b)

minus001

0

001

002

003

minus002

minus003

200 300 400 5000 100Distance x (120583m)

ND

com

pone

nt o

fΦw

ND

(rad

)

(c)








Scanning 5

minus010 minus005 000 005 010

minus002

000

002

ΔwTD (rad)

Δw

RD(r

ad)

(a)

minus010 minus005 000 005 010

minus002

000

002

ΔwTD (rad)

Δw

ND

(rad

)

(b)


minus002

minus001

000

001

002

003

ΔwRD (rad)

Δw

ND

(rad

)

(c)



((


asymp [119889R119889119909 ]Rminus1 (10)








6 Scanning

ND asymp [111]

RD asymp [112]

TD asymp [110]

111

10111 2

01 1

(a)

110

111

11 2

(b)

121 110

111

211

10111 2

01 1

(c)










4 Conclusions


Competing Interests


Acknowledgments


References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Scanning 3

y

z

x

Crystal

x998400 y998400

z998400

R(x1)

ΔR

(a)

y

z

x

Crystal

x998400 y998400

z998400

R(x2)

(b)


ND asymp [111]

RD asymp [112]

TD asymp [110]

(a)

101001

111ND

A

O

ND

RD

200120583m

(b)

111

ND

TD RD

010

100

001121

110

011

211

101

11 2

(c)



R (1199092) = ΔRR (1199091) (5)





4 Scanning

minus005

0

005

010

015

minus010

200 300 400 5000 100minus015


TD co

mpo

nent

ofΦ

wTD

(rad

)

(a)

minus001

0

001

002

003

minus002

minus003

200 300 400 5000 100Distance x (120583m)

RD co

mpo

nent

ofΦ

wRD

(rad

)

(b)

minus001

0

001

002

003

minus002

minus003

200 300 400 5000 100Distance x (120583m)

ND

com

pone

nt o

fΦw

ND

(rad

)

(c)








Scanning 5

minus010 minus005 000 005 010

minus002

000

002

ΔwTD (rad)

Δw

RD(r

ad)

(a)

minus010 minus005 000 005 010

minus002

000

002

ΔwTD (rad)

Δw

ND

(rad

)

(b)


minus002

minus001

000

001

002

003

ΔwRD (rad)

Δw

ND

(rad

)

(c)



((


asymp [119889R119889119909 ]Rminus1 (10)








6 Scanning

ND asymp [111]

RD asymp [112]

TD asymp [110]

111

10111 2

01 1

(a)

110

111

11 2

(b)

121 110

111

211

10111 2

01 1

(c)










4 Conclusions


Competing Interests


Acknowledgments


References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



4 Scanning

minus005

0

005

010

015

minus010

200 300 400 5000 100minus015


TD co

mpo

nent

ofΦ

wTD

(rad

)

(a)

minus001

0

001

002

003

minus002

minus003

200 300 400 5000 100Distance x (120583m)

RD co

mpo

nent

ofΦ

wRD

(rad

)

(b)

minus001

0

001

002

003

minus002

minus003

200 300 400 5000 100Distance x (120583m)

ND

com

pone

nt o

fΦw

ND

(rad

)

(c)








Scanning 5

minus010 minus005 000 005 010

minus002

000

002

ΔwTD (rad)

Δw

RD(r

ad)

(a)

minus010 minus005 000 005 010

minus002

000

002

ΔwTD (rad)

Δw

ND

(rad

)

(b)


minus002

minus001

000

001

002

003

ΔwRD (rad)

Δw

ND

(rad

)

(c)



((


asymp [119889R119889119909 ]Rminus1 (10)








6 Scanning

ND asymp [111]

RD asymp [112]

TD asymp [110]

111

10111 2

01 1

(a)

110

111

11 2

(b)

121 110

111

211

10111 2

01 1

(c)










4 Conclusions


Competing Interests


Acknowledgments


References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Scanning 5

minus010 minus005 000 005 010

minus002

000

002

ΔwTD (rad)

Δw

RD(r

ad)

(a)

minus010 minus005 000 005 010

minus002

000

002

ΔwTD (rad)

Δw

ND

(rad

)

(b)


minus002

minus001

000

001

002

003

ΔwRD (rad)

Δw

ND

(rad

)

(c)



((


asymp [119889R119889119909 ]Rminus1 (10)








6 Scanning

ND asymp [111]

RD asymp [112]

TD asymp [110]

111

10111 2

01 1

(a)

110

111

11 2

(b)

121 110

111

211

10111 2

01 1

(c)










4 Conclusions


Competing Interests


Acknowledgments


References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



6 Scanning

ND asymp [111]

RD asymp [112]

TD asymp [110]

111

10111 2

01 1

(a)

110

111

11 2

(b)

121 110

111

211

10111 2

01 1

(c)










4 Conclusions


Competing Interests


Acknowledgments


References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




4 Conclusions


Competing Interests


Acknowledgments


References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



description of changes in crystal orientations by the...

Documents