Liangwei Chen, Yuyu Hui, Lan Yu, Jin Hu and Jianhong Yi

(School of Materials Science and Engineering, Kunming University of Science and Technology, Kunming 650093, Yunnan, China)

Abstract: The formulae of the relationships between Euler angles and texture are important for the orientation distribution functions of research materials.At present, it is found that the formulae for the orthogonal and hexagonal crystal structure proposed by Roe and Bunge are not in accord with the normal direction(ND)being perpendicular to the rolling direction(RD).In this paper, these formulae are deduced independently according to the notations proposed by Roe and Bunge.The results show that in addition to the coincidence with the formulae for the cubic crystal structure given by the Roe and Bunge, the other formulae are different from Roe’s and Bunge’s.The new formulae are in accord with ND, RD, and transverse direction(TD)at right angles.

Keywords: texture; Euler space; Bunge notation; Roe notation; orientation distribution function

0 Introduction

The orientation distribution function sections(ODFs)are analyzed by the formulae of the Euler angles and texture when the EBSD micro-texture and the X-ray macro-texture characterizations are carried out, and the textures in materials are obtained[1-4].Whether these formulae are correct or not depends on the accuracy and reliability of texture characterizations.The formulae for cubic, orthorhombic, and hexagonal crystal structures given by Roe are listed as Eqs.(1)-(6)[3-4]:

Cubic crystal structure:

H∶K∶L=(-sinθcosφ)∶(sinθsinφ)∶cosθ

(1)

U∶V∶W=(cosψcosθcosφ-sinψsinφ)∶

(-cosψcosθsinφ-sinψcosφ)∶(sinθcosψ)

(2)

Orthorhombic crystal structure:

H∶K∶L=(-asinθcosφ)∶(bsinθsinφ)∶(ccosθ)

(3)

U∶V∶W=[(cosψcosθcosφ-sinψsinφ)/a]∶

[(-cosψcosθsinφ-sinψcosφ)/b]∶

[sinθcosψ/c]

(4)

Hexagonal crystal structure:

(5)

(6)

The formulae for cubic, orthorhombic, and hexagonal crystal structures given by Bunge are listed as Eqs.(7)-(12)[3-4]:

Cubic crystal structure:

H∶K∶L=(sinθsinφ1)∶(-sinθcosφ1)∶cosθ

(7)

U∶V∶W=(cosφ2cosφ1-sinφ2sinφ1cosθ)∶

(cosφ2sinφ1+sinφ2cosφ1cosθ)∶(sinφ2sinθ)

(8)

Orthorhombic crystal structure:

H∶K∶L=(asinθsinφ1)∶(-bsinθcosφ1)∶

(ccosθ)

(9)

U∶V∶W=[(cosφ2cosφ1-sinφ2sinφ1cosθ)/a]∶

[(cosφ2sinφ1+sinφ2cosφ1cosθ)/b]∶

[(sinφ2sinθ)/c]

(10)

Hexagonal crystal structure:

(11)

(12)

Eqs.(1)and(2)for the cubic crystal structure given by Roe are calculated as follows:

HU+KV+LW=-sinθcosφ(cosψcosθcosφ-sinψsinφ)+sinθsinφ(-cosψcosθsinφ-sinψcosφ)+cosθsinθcosψ=0

This meets the condition that ND is perpendicular to RD.Let

hc=-sinθcosφ;kc=sinθsinφ;lc=cosθ

uc=cosψcosθcosφ-sinψsinφ

vc=-cosψcosθsinφ-sinψcosφ

wc=sinθcosψ

∴hcuc+kcvc+lcwc=0

The formula of the crystal plane angle in the orthorhombic crystal structure is written as Eq.(13):

(13)

Eqs.(3)and(4)for orthogonal crystal structure given by Roe are substituted into the numerator of the Eq.(13), as listed below:

The formula of the angle between the two crystal planes for hexagonal crystal structure is listed as Eq.(14):

(14)

Eqs.(5)and(6)for hexagonal crystal structure given by Roe are also substituted into the numerator of the Eq.(14), as listed below:

kcvc+lcwc)=0

It does not satisfy the definition that ND is perpendicular to RD in orthogonal and hexagonal crystal structure according to Roe’s formulae, neither does Bunge’s.

In this paper, according to the calculation conditions proposed by Roe and Bunge, the formulae of Euler angles and texture types are deduced independently.Some differences among these formulae will be found and explained.

1 Recalculation of the Formulae of the Euler Angles and the Texture According to the Roe Notations

In order to describe the spatial orientation of each grain in the plate texture specimen, it is necessary to specify two rectangular coordinate systems.A Cartesian coordinate systemO-ABCis used to represent the specimen coordinate system, andOA,OB, andOCare usually used to represent RD, TD, and ND of the rolling surface, respectively.The other Cartesian coordinate systemO-XYZis fixed on the grain, indicating the crystal coordinate system.In cube crystal structure,OX,OY, andOZcoincide with[100],[010], and[001]directions, respectively.In this way, the normal, rolling, and transverse orientations of the grain in the crystal space can be completely expressed by Euler angles.The Euler angles refer to three rotations which, when performed in the correct sequence, transform the specimen coordinate system into the crystal coordinate system-in other words, specify the orientation.There are several different conventions for expressing the Euler angles[5-6].The most commonly used are those formulated by Roe, as shown in Fig.1.The rotations are[5]

1)φabout the normal directionOC, transforming the transverse directionOBintoOB1and the rolling directionOAintoOA1;

2)θabout the axisOB1(in its new orientation), transforming the normal directionOCintoOC1and the rolling directionOA1intoOA2;

3)ψaboutOC1(in its new orientation), transforming the transverse directionOB1intoOB2and the rolling directionOA2intoOA3;

φ,θ, andψare the Euler angles(Roe notation).The effect of the operation sequence of these three rotations can be followed in Fig.1[4].

Fig.1 Relationships between O-ABC and O-XYZ coordinate system according to the Roe notations[4](a)Initial state;(b)-(d)State of different rotation angle;(e)Final state

According to Roe notation, the coordinates ofA1andB1inO-XYZcrystal coordinate system are shown as

A1=(cosφ,-sinφ, 0)

B1=(sinφ, cosφ, 0)

The coordinates ofA2andC1inO-A1B1Cspecimen coordinate system are shown as

A2=(cosθ, 0, sinθ)

C1=(-sinθ, 0, cosθ)

The coordinates ofA2andC1inO-XYZcrystal coordinate system are shown as

A2=(cosθcosφ,-cosθsinφ, sinθ)

C1=(-sinθcosφ, sinθsinφ, cosθ)

The coordinates ofA3andB2inO-A2B1C1specimen coordinate system are shown as

A3=(cosψ,-sinψ, 0)

B2=(sinψ, cosψ, 0)

The coordinates ofA3andB2inO-XYZcrystal coordinate system are shown as

A3x=cosψcosθcosφ-sinψsinφ

A3y=-cosψcosθsinφ-sinψcosφ

A3z= cosψsinθ

B2x= sinψcosθcosφ+cosψsinφ

B2y=-sinψcosθsinφ+cosψcosφ

B2z=sinψsinθ

BecauseOA3,OB2, andOC1indicate the rolling direction RD[UVW], transverse direction TD[RST], and normal direction ND[HKL]of the rolling surface, respectively, so the coordinates ofC1,A3, andB2inO-XYZcrystal coordinate system represent the indices[HKL],[UVW], and[RST]of ND, RD, and TD, respectively.The formulae of the texture and the Euler angles in the cubic crystal structure are listed as follows:

H=-sinθcosφ

K=sinθsinφ

L=cosθ

U=cosψcosθcosφ-sinψsinφ

V=-cosψcosθsinφ-sinψcosφ

W=sinθcosψ

R=sinψcosθcosφ+cosψsinφ

S=-sinψcosθsinφ+cosψcosφ

T=sinψsinθ

The formulae of ND, RD, and TD in the cubic crystal structure can be abbreviated as Eq.(15),(16), and(17), respectively.

H∶K∶L=(-sinθcosφ)∶(sinθsinφ)∶cosθ

(15)

U∶V∶W=(cosψcosθcosφ-sinψsinφ)∶

(-cosψcosθsinφ-sinψcosφ)∶

(sinθcosψ)

(16)

R∶S∶T=(sinψcosθcosφ+cosψsinφ)∶

(-sinψcosθsinφ+cosψcosφ)∶

(sinψsinθ)

(17)

Compared with the above Eqs.(1)and(2)given by Roe, it is found that Eqs.(15)and(16)are exactly the same as Eqs.(1)and(2).

ND and TD vector dot products are calculated as below:

HR+KS+LT=

(-sinθcosφ)(sinψcosθcosφ+cosψsinφ)+(sinθsinφ)(-sinψcosθsinφ+cosψcosφ)+cosθ(sinψsinθ)=-sinθcosθsinψcos2φ-sinθsinφcosφcosψ-sinθcosθsinψsin2φ+sinθsinφcosφcosψ+cosθsinψsinθ=

-sinθcosθsinψ(cos2φ+sin2φ)+

cosθsinψsinθ=0

∴HR+KS+LT=0

In the same way,

UR+VS+WT=0,HU+KV+LW=0

Therefore, the formulae of Euler angles and the texture in cubic crystal structure meet the conditions that ND, RD, and TD are perpendicular to each other.

The unit lengths ofX,Y, andZaxes in the tetragonal crystal structure area,a, andc, respectively.The miller indices of ND, RD, and TD in the tetragonal crystal structure are depended on ND(hc,kc,lc), RD(uc,vc,wc), and TD(rc,sc,tc)in cubic crystal structure.They are multiplied by the unit lengtha,a, andc, respectively.The miller indices of ND, RD, and TD in the tetragonal crystal structure are listed as ND(ahc,akc,clc), RD(auc,avc,cwc), and TD(arc,asc,ctc).The expressions of the Euler angles and texture type in the tetragonal crystal structure are shown in Eqs.(18)-(20).

H∶K∶L=(-asinθcosφ)∶(asinθsinφ)∶(ccosθ)

(18)

U∶V∶W=[a(cosψcosθcosφ-sinψsinφ)]∶

[a(-cosψcosθsinφ-sinψcosφ)]∶

(csinθcosψ)

(19)

R∶S∶T=[a(sinψcosθcosφ+cosψsinφ)]∶

[a(-sinψcosθsinφ+cosψcosφ)]∶

(csinψsinθ)

(20)

In the same way, the miller indices of ND, RD, and TD in the orthorhombic crystal structure are listed as(ahc,bkc,clc),(auc,bvc,cwc), and(arc,bsc,ctc), respectively.The expressions of the Euler angles and texture type in the orthorhombic crystal structure are shown in Eqs.(21)-(23):

H∶K∶L=(-asinθcosφ)∶(bsinθsinφ)∶(ccosθ)

(21)

U∶V∶W=[a(cosψcosθcosφ-sinψsinφ)]∶

[b(-cosψcosθsinφ-sinψcosφ)]∶

(csinθcosψ)

(22)

R∶S∶T=[a(sinψcosθcosφ+cosψsinφ)]∶

[b(-sinψcosθsinφ+cosψcosφ)]∶

(csinψsinθ)

(23)

There are some differences between Eqs.(21)-(22)and Eqs.(9)-(10)given by Roe.

The miller indices Eqs.(21)-(23)of ND, RD, and TD in the orthorhombic crystal structure are substituted into the numerator of Eq.(5), then the following equations are obtained:

c2(wctc)/c2=ucrc+vcsc+wctc=0

Therefore, the formulae of Euler angles and texture in orthorhombic crystal structure meet the conditions that ND, RD, and TD are perpendicular to each other, so does the tetragonal crystal structure.

(24)

(25)

W=cz

(26)

In hexagonal crystal system, the three indices(h,k,l)of crystal plane are not consistent with the normal direction indices[u,v,w]of the same crystal plane.However, the four indices(h,k,i,l)of a crystal plane are consistent with the normal direction indices[u,v,t,w].The three indices[U,V,W]can be changed into the four indices[u,v,t,w]as follows:

Here,xis replaced byax,ybyby, andzbycz.

(27)

(28)

(29)

w=W

(30)

Eqs.(24),(25), and(26)are substituted into Eqs.(27),(28),(29), and(30), respectively.The general expressions of ND, RD, and TD in hexagonal crystal structure are obtained as Eqs.(31)-(34):

(31)

v=ay

(32)

(33)

w=cz

(34)

(35)

The complete matrix is then transformed into

(36)

(37)

(38)

The direction indices of the ND and RD in Eqs.(36)and(37)are substituted into the numerator of the hexagonal crystal plane angle Eq.(14)as below:

Thus, ND is always perpendicular to RD in Eqs.(36)-(37).In the same way, it is easy to verify that ND, RD, and TD are perpendicular to each other in Eqs.(36)-(38).There are some differences between Eq.(37)and Eq.(12)provided by Roe.

2 Recalculation of Formulae between Euler Angles and the Texture According to Bunge Notations

Besides expressing the Euler angles according to Roe notations, there is another expression of the Euler angles according to Bunge notations, as shown in Fig.2.The rotations are[5-6]

1)φ1about the normal directionOC, transforming the transverse directionOBintoOB1and the rolling directionOAintoOA1;

2)θabout the axisOA1(in its new orientation), transforming the normal directionOCintoOC1and transverse directionOB1intoOB2;

3)φ2aboutOC1(in its new orientation), transforming the transverse directionOB2intoOB3and the rolling directionOA1intoOA2;

φ1,θ, andφ2are the Euler angles(Bunge notations).The effect of the operation sequence of these three rotations can be followed in Fig.2.

Fig.2 Relationship between O-ABC and O-XYZ coordinate system according to Bunge notations(a)Initial state;(b)-(d)States of different rotation angles;(e)Final state

The coordinates ofA1,B1, andCinO-XYZcrystal coordinate system are shown as

The coordinates ofA1,B2, andC1inO-A1B1Cspecimen coordinate system are shown as

The coordinates ofA1,B2, andC1inO-XYZcrystal coordinate system are shown as

A1x=cosφ1,A1y=sinφ1,A1z=0

B2x=-cosθsinφ1,B2y=cosθcosφ1,B2z=sinθ

C1x=sinθsinφ1,C1y=-sinθcosφ1,C1z=cosθ

The coordinates ofA2,B3, andC1inO-A1B2C1specimen coordinate system are shown as

The coordinates ofA2,B3, andC1inO-XYZcrystal coordinate system are shown as

A2x=cosφ2cosφ1-sinφ2sinφ1cosθ

A2y=cosφ2sinφ1+sinφ2cosφ1cosθ

A2z=sinφ2sinθ

B3x=-sinφ2cosφ1-cosφ2cosθsinφ1

B3y=-sinφ2sinφ1+cosφ2cosθcosφ1

B3z=cosφ2sinθ

C1x=sinθsinφ1,C1y=-sinθcosφ1,C1z=cosθ

In this case,OC1is ND of the specimen,OA2is RD, andOB3is TD.The formulae of the texture dependent on Euler angle(φ1,θ,φ2)is highly similar to these provided by Roe.It is quite easy to obtain all formulae for cubic, tetragonal, orthorhombic, hexagonal, and other crystal structures by Bunge notations.These formulae are shown as follows:

For cubic crystal structure,

ND:

H∶K∶L=(sinθsinφ1)∶(-sinθcosφ1)∶cosθ

(39)

RD:

U∶V∶W=(cosφ2cosφ1-sinφ2sinφ1cosθ)∶

(cosφ2sinφ1+sinφ2cosφ1cosθ)∶

(sinφ2sinθ)

(40)

TD:

R∶S∶T=(-sinφ2cosφ1-cosφ2cosθsinφ1)∶

(-sinφ2sinφ1+cosφ2cosθcosφ1)∶

(cosφ2sinθ)

(41)

For tetragonal crystal structure,

ND:

H∶K∶L=(asinθsinφ1)∶(-asinθcosφ1)∶(ccosθ)

(42)

RD:

U∶V∶W=[a(cosφ2cosφ1-sinφ2sinφ1cosθ)]∶[a(cosφ2sinφ1+sinφ2cosφ1cosθ)]∶

[c(sinφ2sinθ)]

(43)

TD:

R∶S∶T=[a(-sinφ2cosφ1-cosφ2cosθsinφ1)]∶[a(-sinφ2sinφ1+cosφ2cosθcosφ1)]∶

[c(cosφ2sinθ)]

(44)

For orthorhombic crystal structure,

ND:

H∶K∶L=(asinθsinφ1)∶(-bsinθcosφ1)∶(ccosθ)

(45)

RD:

U∶V∶W=[a(cosφ2cosφ1-sinφ2sinφ1cosθ)]∶[b(cosφ2sinφ1+sinφ2cosφ1cosθ)]∶

[c(sinφ2sinθ)]

(46)

TD:

R∶S∶T=[a(-sinφ2cosφ1-cosφ2cosθsinφ1)]∶

[b(-sinφ2sinφ1+cosφ2cosθcosφ1)]∶

[c(cosφ2sinθ)]

(47)

For hexagonal crystal structure,

(48)

(49)

(50)

It has been proved that ND, RD, and TD are perpendicular to each other in Eqs.(39)-(46)by the method used in Roe notation.There are some differences among the formulae in orthorhombic and hexagonal crystal structures compared with Eqs.(9)-(12)provided by Bunge.

3 Discussion

The relationships between the texture and Euler angles is actually the coordinate transformation between the crystallographic coordinate and the specimen appearance coordinate according to certain rules.Frankly speaking, this mathematical derivation is not complicated.For the cubic crystal system, the formulae deduced by this paper are exactly the same as those of the Roe and Bunge.For the orthorhombic and hexagonal crystal systems, the trigonometric function parts of the formulae deduced by the authors is exactly the same as those of the Roe and Bunge except for the coefficient parts in the formulae.At expression formulae of the texture deduced in this paper, ND, RD, and TD are uniformly expressed as crystal plane indices.When calculating the standard pole figures of the common textures of hexagonal crystal system materials, ND and RD are calculated according to the crystal plane indices[7-8].When designing the pole figure measurement software for the characterization of hexagonal crystal system texture, ND and RD are also used as the crystal plane indices.Nevertheless, the expression of ND and RD may be inconsistent in the textural formulae deduced by Roe and Bunge, where ND is used as the crystal plane indices, and RD may be used as the crystal direction indices.For cubic crystal system, the angle between crystal plane and crystal direction is the same, but for other crystal systems, the formula of angle between crystal plane index and crystal direction index has not been deduced.Therefore, for orthorhombic and hexagonal crystal systems, the traditional formulae deduced by Roe and Bunge may still be correct, but in the practice of material texture characterizations, it is easy to bring confusion to users.

4 Conclusions

The formulae deduced in this paper are inspired by the notations of Roe and Bunge.In the formulae of a texture, ND, RD, and TD are uniformly used as the crystal plane indices, so they are suitable for the corresponding formulae of crystal plane angles, which brings convenience to texture characterizations.

In the traditional formulae deduced by Roe and Bunge, ND is used as the crystal plane indices, and RD and TD may be used as the crystal direction indices, which may cause the differences among the formulae derived by the authors and by Roe and Bunge.

During the teaching of crystal orientation characterizations and the design of commercial software for material texture characterizations, users should pay attention to the differences among the formulae.