Song Li

Yudong Zhang

Gaowu Qin

Jean-Sébastien Lecomte

Claude Esling
email: claude.esling@univ-metz.fr

Xiang Zhao

Liang Zuo

Determination of surface crystallography of faceted nanoparticles under TEM imaging and diffraction mode

A general method to determine surface crystallography and therefore shape of faceted nanoparticles by TEM is proposed.

Introduction

Recently, much attention has been focused on faceted nanoparticles with various shapes due to their unique properties and potential applications in catalysis (Narayanan & El-Sayed, 2005;Xiong et al., 2007), surface-enhanced Raman scattering (SERS) (Orendorff et al., 2005) and optoelectronics (Maillard et al., 2003). It has been demonstrated that many shape-related properties of nanoparticles can be attributed to their surface crystallographic structure. For instance, tetrahexahedral Pt nanoparticles with high-index {730} plane exhibit enhanced catalytic activity for the electro-oxidation as compared with that of spherical particles (Tian et al., 2007). Also the surface crystal structure plays a critical role in the anisotropic growth of nanocrystals. The selective binding of capping molecules on different crystal planes may alter their surface energies and thus change their growth capacities [START_REF] Lee | [END_REF][START_REF] Stekolnikov | [END_REF]. Therefore, the characterization of surface crystallographic features of nanoparticles has become an important issue for the study of their surface related properties and the investigation of their shape evolution mechanism in growth process.

The application of TEM techniques to crystallographic analysis has been suggested for several years (Zaefferer, 2002). In the present case of a nanoparticle, the surface plane indices are conventionally indexed through comparing its diffraction pattern with its projection in image after bringing the plane to the edge on position in TEM. Sometimes, high-resolution TEM (HRTEM) images that record the atomic spacing of the on edge planes are used to verify the determination of the surface plane index (Xiang et al., 2006;Lee et al., 2002). The method is particularly suitable for nanoparticles with conventional shapes (Xiong et al., 2007), i.e., the particles are enclosed by symmetric low-index facets, as in most cases the low-indexed planes can easily satisfy the diffraction condition and generate diffraction spots. However, for particles with unconventional shapes (Xiong et al., 2007), i.e. covered with high-index facets, the method shows limitation. In such cases, it is often difficult to obtain the corresponding diffraction spots for those highindex planes. To solve this problem, an attempt has been made by measuring the angles between the surface plane projections when the planes are simultaneously on edge-on position and comparing them with the theoretical interfacial angles to conclude the miller

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3 indices of the facet (Tian et al., 2007). Obviously, the method requires great effort in searching for the edge-on positions and sometimes it is even impossible to obtain such position for some planes when the shape of the particle is not favorable. Moreover, the study of unconventional shaped nanoparticles has recently become intensive due to their unique catalytic properties resulting from the high index surface facet. Naturally, the correct indexing of such surface planes constitutes one of the important bases for the further property and growth kinetics investigation. Therefore, a general and accurate determination method appears to be in great need. In response to such a requirement, we developed a general calculation method that can be applied to determine any faceted surface plane indices of nanoparticles without requiring the corresponding diffraction spots, provided that the edges of the facets are visible under TEM imaging mode. This technique will facilitate the surface related properties research in nanoscience.

Methodology

The basic principle of this method is to determine the edge vectors that define the surface facets of a nanoparticle in the TEM screen coordinate system and then obtain the surface plane normal by cross product and further calculate their Miller indices in the crystal coordinate system through coordinate transformation. The full method will be detailed as follows.

Setting of coordinate systems

Considering the characteristics of TEM for sample loading and imaging, it is convenient to introduce three Cartesian coordinate systems in addition to the lattice base of the crystalline and its reciprocal base. All the coordinate systems are of the same handedness and the Cartesian coordinate systems are orthonormal. In the present work, we choose the right-hand set. Of the three Cartesian coordinate systems, one is referenced to the screen of the TEM that records the image of the particle. Its Z-axis is set in the inverse direction of the incident electron beam, one coordinate system is set to the sample holder also with its Z axis in the inverse direction of the incident electron beam when the holder is in non-tilt position and the third one to the crystal under the convention described in the "International Table of Crystallography" [START_REF] Hahn | International Tables for Crystallography[END_REF]. The orientation

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4 relationships between the Cartesian coordinate systems are defined as a set of rotations (Euler angles in Bunge notation (Bunge et al., 1981)) transforming one system into the other.

Determination of the edge vectors in the screen coordinate system

To determine the coordinates of a edge vector in the screen coordinate system, two sample positions are required. Assume that the unit edge vector in the sample holder coordinate system is v ( x , y , z ) and let the equivalent vectors in the screen coordinate system with respect to the two sample holder positions be 1 v ( 1 x , 1 y , 1 z ) (position 1) and 2 v ( 2 x , 2 y , 2 z ) (position 2). Under TEM imaging mode, let the orientation relationship between the screen coordinate system and the first sample position described by rotation

R ( 2 1 ϕ Φ ϕ , ,
) and the rotation from the first sample position to the second sample

position R ∆ ( 2 1 ϕ ∆ ∆Φ ϕ ∆ , ,
), as shown in Fig. 1. For simplicity, the first sample position could be taken without tilt operation. R is then characterized by a single rotation around the Z axis of the screen coordinate system caused by the magnetic rotation of the electron beam which is magnification dependent. Thus, the following relations between the unit vectors v , 1 v and 2 v hold for

v v ⋅ = 1 1 M (1a) v v ⋅ ⋅ = 2 1 2 M M (1b)
where 1 M and 2 M are the corresponding rotation matrices of R and R ∆ , and are known.

Then the relation between the

1 v ( 1 x , 1 y , 1 z ) and 2 v ( 2 x , 2
y , 2 z ) can be deduced as:

2 1 1 1 2 1 1 M M M v v ⋅ ⋅ ⋅ = - - (2)
where 1 1 M -and 1 2 M -are the inverse matrices of 1 M and 2 M . According to the imaging principle of the TEM, an image of a nanoparticle is the rotated (around the Z axis of the screen coordinate system and dependent on the magnification) and amplified projection of the particle. As the Z axis of the screen coordinate system is set parallel to the incident beam, i.e., the Z axis is parallel to the projection direction, the X and Y coordinates of

R E V I E W D O C U M E N T 5 1
v and 2 v in the screen coordinate system can be measured. Only 1 z and 2 z are unknown. With the three linear equations offered by Eq. ( 2), the two unknown 1 z and 2 z can be resolved. Thus the coordinates of the edge vector in the screen coordinate system are determined.

Coordinate transformation matrices

The coordinate transformation from the screen coordinate system to the crystal bases can be achieved through acquiring and indexing the electron diffraction pattern of the crystal. The diffraction pattern can either be in spot pattern or in Kikuchi line pattern. For simple operation, one of the two sample positions used to determine the edge vectors in the screen coordinate system could be dedicated to obtain a diffraction pattern of a low index zone axis. There is no automatic program available to directly determine the orientation of the crystallite with respect to either the sample holder or the screen of the microscope by indexing the spot pattern. Such software to determine the orientation by indexing the Kikuchi pattern is available. Thus we will treat the spot pattern and the Kikuchi pattern separately in this work.

According to the diffraction geometry, the TEM diffraction spot pattern of a crystallite is the amplified 2-dimentional reciprocal plane of the crystallite, which is perpendicular to the incident beam or the Z-axis of the screen coordinate system. Each spot in this reciprocal plane represents one set of diffracting planes expressed with Miller indices (h k l). The reciprocal vector with components (h, k, l) in the reciprocal base is perpendicular to the plane with Miller indices (h k l) in the direct lattice space. Thus with the correctly indexed diffraction pattern, it is easy to find two vectors r 1 and r 2 that are coincident with two reciprocal lattice vectors (h 1 k 1 l 1 ) and (h 2 k 2 l 2 ) in the XOY plane of the screen coordinate system. By vector cross product, one can easily obtain the 3 rd vector r 3 in the Z direction as shown in Fig. 2. The three vectors should be referred to the same basis, either to the reciprocal space or to the direct space. It should be noted that r 3 is not necessarily a reciprocal lattice vector. The coordinate transformation of the vectors between the direct space and the reciprocal space can be easily realized with the direct metric tensor or the reciprocal metric tensor (Eqs. (A1) and (A2) in Appendix 1).

Here

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6 we referred the three vectors to the reciprocal space. Therefore, the coordinate transformation matrix from the screen coordinate system to the r 1 -r 2 -r 3 reference base

g I M → is: ⎥ ⎥ ⎥ ⎥ ⎦ ⎥ ⎢ ⎢ ⎢ ⎢ ⎣ ⎢ θ + η ⋅ η ⋅ θ + η ⋅ η ⋅ = → 3 2 1 2 1 0 0 0 ) cos( sin 0 ) cos( cos r r r r r M g I (3)
where η is the angle between X-axis and r 1 , and θ is the angle between r 1 and r 2 . The coordinate transformation matrix from the r 1 -r 2 -r 3 reference base to the reciprocal space, r g M → is:

1 3 * 3 1 3 * 2 1 3 * 2 1 - → ⎥ ⎥ ⎥ ⎦ ⎥ ⎢ ⎢ ⎢ ⎣ ⎢ = c b a r l l r k k r h h r I M (4)
M M M → → → ⋅ = (5)
and

* G M M R I l I ⋅ = → → (6)
in which G* is the metric tensor of the reciprocal basis [START_REF] Shmueli | International Tables for Crystallography[END_REF]

(Eq. (A2) in Appendix 1).
The orientation of the crystallite with respect to the sample coordinate system can be directly determined by indexing the Kikuchi line pattern using the commercial software Euclid's Phantasies (EP) (Morawiec, 1999;Morawiec et al., 2002). The orientation of the crystalline is expressed with three Euler angles (

2 1 ϕ ϕ , , Φ ) (in Bunge notation) rotating R E V I E W D O C U M E N T 7
the sample holder coordinate system to the Cartesian crystal coordinate system. The orientation relationship between the Cartesian crystal coordinate system and the Bravais lattice basis is set under the convention given by the "International Table of Crystallography" [START_REF] Hahn | International Tables for Crystallography[END_REF] described in Appendix B. Thus the screen-to-reciprocal lattice base transformation matrix R i→

M

and the screen-to-crystal coordinate system transformation matrix

l i → M
will be:

R C EP i R i → → ⋅ = M M M M (7) and * G R i l i ⋅ = → → M M (8)
in which i M is the rotation matrix from the screen to one of the sample holder positions where the Kikuchi pattern is acquired, EP M is the transformation matrix from the sample holder coordinate system to the Cartesian crystal coordinate system and R C →

M

is the transformation matrix from the Cartesian crystal coordinate system to the reciprocal space.

Therefore, for any vector obtained in the screen coordinate system ) , , (

i i i i z y x v that
corresponds to the sample position acquiring the diffraction pattern, its components in the lattice space can be calculated as:

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ → i i i l i z y x 1 - M (9)
and the intercepts of the plane that is normal to ) , , (

i i i i z y x v
can be obtained as:

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ → i i i R i z y x 1 - M (10)

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In this way, the Miller indices of the edges and surface planes of the crystalline can be determined.

Application

Hereafter, we present one example to illustrate the application of the above method.

The nanoparticles used in this example are hematite (α - ∆R are given in Table 1. The X and Y components of vector i v corresponding to the two sample positions in the screen coordinate system were measured and their Z components were calculated. The results are displayed in Table 2. One Kikuchi pattern (the inset of Fig. 3(a)) was taken at sample position 1 and indexed using the Euclid's Phantasies (EP) (Moraviec et al., 2002). The set of Euler angles that denote the rotation from the sample coordinate system to the Cartesian crystal coordinate system is (182.63°, 80.11°, 33.99°).

One diffraction spot pattern (the inset of Fig. 3(b)) was taken after the sample was tilted (position 2) where the [010] zone axis is on edge. The two reference vectors r 1 and r 2 coincident with the ( 1 10 ) and ( 1 0 1 ) spots, respectively. The third vector r 3 that is in Z direction and calculated by vector product has the components (0.333, 0.584, 0.333) in

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9 the reciprocal space. The angle between the X-axis of the screen coordinate system and r 1 is 50.986° and that between r 1 and r 2 is 90°.

The Miller indices of the surface planes and the edges of the particle were calculated separately by indexing the diffraction spot pattern and the Kikuchi pattern, as given in Table 3. The results showed that hematite particles were enclosed by high-index facets.

Conclusions

In conclusion, a new method for determining the crystalline planes and directions of the surface facets and edges of nanoparticles has been proposed in this paper. By determining the edge vectors and then the plane normal vector of a surface facet in the TEM screen coordinate mode, their Miller indices can be calculated through coordinate transformation. The method offers a general and a systematical way to characterize the surface crystallographic features of a nanoparticle and will facilitate the related studies.

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Table1

The rotation angles that characterize the coordinate transformation from the screen coordinate system to the first sample position R and that from the first sample position to the second position ∆R

R ( 1 ϕ ,Φ , 2 ϕ ) ∆R ( 1 ϕ ∆ , ∆Φ , 2 ϕ ∆ ) (165.84°, 0, 0) (0, 10.26°, -5.80°)
Table 2 Coordinates of the unit edge vectors i v with respect to the two sample holder positions (Positions 1 and 2) in the screen coordinate system calculated with their measured trace coordinates in the screen coordinate system.

1 x 1 y 1 z 2 x 2 y 2 z 1 v -0.

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12 Table 3 Calculated Miller indices of the edge vectors i v and the planes i P from indexing the corresponding diffraction spot pattern and Kikuchi line pattern.

[ uvw]

( hkl ) The corresponding reciprocal metric is:

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 * sin ) cos cos (cos ) cos cos (cos ) cos cos (cos sin ) cos cos (cos ) cos cos (cos ) cos cos (cos sin V b a V bc a V c ab V bc a V c a V abc V c ab V abc V c b G γ α γ β β γ α α γ β β γ β α β γ α γ β α α - - - - - - = (A2) in which ) cos cos cos cos cos cos 2 1 ( 2 2 2 2 2 2 2 γ β α γ β α - - - + = c b a V
. With Eq. ( A1) and (A2), the vector calculations and the coordinate transformation of the vectors between the two bases can be easily realized. of Crystallography" [START_REF] Hahn | International Tables for Crystallography[END_REF], the Cartesian system is set at the same origin of the Bravais lattice base with its X axis parallel to the base vector a and the base vector c is in the XOZ plane of the Relationship between the Cartesian coordinate system and the lattice base under the convention of "International Table of Crystallography" [START_REF] Hahn | International Tables for Crystallography[END_REF].

  components of r 3 referred to the reciprocal space. In this way, the coordinate transformation matrices from the screen coordinate system to

2 P

 2 belong to the trigonal crystal system (space group c 3 R ) with lattice parameters a=b=c=0.5419nm, α = β = γ =55.36°. Hereafter we express the miller indices in the trigonal base. The images and diffraction patterns of the hematite nanoparticle to be determined were recorded using PHILIPS CM 200 TEM at 200kV. Fig.3shows the TEM images of the hematite particle at the two sample positions and the corresponding diffraction patterns. The Kikuchi and spot diffraction images are inserted after correcting the additional rotation from the image of the particle. It can be seen that the particle has a polyhedron shape. The three distinct edges of the particle are denoted with the unit vector 1 v , 2 v , and 3 v and the three distinct surfaces are defined by the edge vectors are 1 P , , and 3 P , as shown in Fig.3. The rotation angles that characterize the orientation relationship between the screen coordinate system and the first sample position (position 1: no tilt) R and from the first sample position to the second position (position 2: tilted)

ACKNOWLEDGEMENT

  This work was supported by the Ministry of Education of China (under the projects No. 108039, No. B07015 and No. IRT0713) and by the CNRS (PICS No. 4164). Li Song gratefully acknowledges the Education Ministry of France for the Ph. D Grant.

  Figure 1Relative positions of the Screen coordinate system and the Sample coordinate system.

Appendix 2 .

 2 Coordinate transformation matrices between the Cartesian crystal coordinate system and the direct lattice space and its reciprocal spaceWith the automatic Kikuchi pattern indexing software packages, such as HKL's Channel software package, TSL's OIM software package and the Euclid's Phantasies (EP), the crystallographic orientation of a crystal can be determined by indexing the Kikuchi patterns of the crystallite. The orientation of the crystallite is always expressed by a set of rotations (Euler angles) from the sample coordinate system to the Cartesian to the crystal. Therefore, the coordinate transformation matrices from the Cartesian coordinate system to the Bravais lattice base and its reciprocal base are always in need. The derivation of such matrices would be helpful for the related calculations. According to the "International Table

  Figure A1Relationship between the Cartesian coordinate system and the lattice base under the convention of "International Table of Crystallography"[START_REF] Hahn | International Tables for Crystallography[END_REF].


B L Wu

Y D Wang

L Zuo

Z D Liang

J S Lecomte

C Esling

Development of the recrystallization texture in copper heated by laser beam

Introduction

The effects of the heating rate on recrystallization texture were investigated in some papers 1, 2, 3. Whereas the materials and the heating methods adopted in the previous investigations were different, the results obtained were not identical. Alexandrov and Davies [START_REF] Alexandrov | [END_REF] heated OFHC copper by means of transverse flux induction heating (TFIH) and found that the recrystallization texture consists mainly of {110}<112>, {112}<111> and cube components, which is obviously different from that observed in conventional annealing. Up to now, the process and physical mechanisms controlling the recrystallization textures during fast annealing remain unclear.

One of the significant characteristics of laser beam heating is that the heating rate is very high and the heating time may be controlled by a proper choice of some heating parameters. It is the aim of this paper to examine the development of the annealing texture in commercial pure copper heated by laser heating technique.

Experimental

Three pieces of commercial pure copper sheets of different thickness, with chemical composition given in Table 1 , were used in the present investigation. Two sheets with an average grain size of about 35μm were cold rolled to 70% reduction (Sample I) and 92% reduction (Sample II), respectively. Another sheet with a grain size of about 15μm was cold rolled to 88% reduction (Sample III). The final thickness of all the three samples was 0.30mm and each was cut into the size of 20 × 40mm for annealing by laser heating. In order to increase the efficiency of energy absorption, the upper surfaces of the samples were blackened. A 2kW CO2 laser generator was used and the light spot diameter was adjusted to d = 3.0 mm. The samples were then heated by means of laser beam scanning with the working power 600W. The distance between the scanning lines was 2.0mm and the heating time t was controlled by the scanning speed v (t = d/v). The Schulz back reflection technique [4] was adopted to measure {111}, {200} and {220} incomplete pole figures (χf = 70°) and the ODF analysis was performed by the two-step method [START_REF] Liang | Proceedings of ICOTOM[END_REF].

Results

The ODFs of cold rolled samples show the typical copper texture which consists of α and β tubes, i.e. mainly {112}<111> (C), {110}<112> (B) and {123}<634> (S). In the present work, the heating time during annealing process was chosen as t = 9.0, 11.2, 13.2, 15.0, 22.5, 45.0 (ms). As t increases, recrystallization occurs in Sample I, II and III. The optical microscopy observation showed that the recrystallization processes are the same as those found in conventional annealing. When t = 15.0ms the recrystallization completes in all the samples (see Fig. 2 ), but different texture components were formed. In Sample I the grain orientation distribution is almost random (Fig. 1(b)), whereas a strong cube component and a weak {122}<221> component are evident in Sample II (Fig. 1(c)). For Sample III the strong cube and relatively weak rolling components are seen (Fig. 1(d)). As t increases further, grains grow and the intensity of the cube component increases in all the three samples. The development of the C and cube components versus the heating time is shown in Fig. 3. It is noted that, for all the samples, the intensity of the rolling texture component {112}<111> (the B and S components as well) decreases considerably and there is no obvious increase of the cube component at the initial stage of heating. Indeed, no new grains could be observed by optical microscopy.

Discussion

It is evident that the texture development for Samples I and III is different from that reported by Liu et al. [START_REF] Liu | [END_REF], where the fast salt bath heating was used and almost the same process of texture development as in conventional annealing was found. Such a difference may be attributed to the different heating rates. In the present experiment the annealing rate is much higher than that of the salt bath heating. Now the question arises, what are the differences of texture formation mechanisms between conventional and ultrarapid annealing?

From Fig. 3 we can see that when the heating time t increases (but less than 13.2ms), the intensity of the cold rolling component C in all the samples decreases but there are no increases in the intensity of the cube component. The optical microscopy observation has confirmed that there were no microstructural differences between the annealed samples and the cold rolled samples at the time. This may indicate that at the early stage of recrystallization, there exist a larger number of nuclei with random orientation and the growth of these nuclei governs the further texture development process.

It is generally admitted that the recrystallization driving forces are mainly the stored energy, including the energy of defects such as vacancies, dislocations, grain or subgrain boundaries. In recrystallization, the grain boundary migration depends on both the difference of stored energy between two neighboring grains and the grain-boundary energy. A grain (or subgrain) which is in favour of the above two aspects would grow selectively. Recently, Vatne and Nes [7] cited Gibbs-Thomson relation for the critical radius RC of nuclei, (1)

(1)
where γ is the grain boundary energy, PD the driving pressure. All grains fulfilling Equation (1) will become successful in terms of further transformation with a growth rate G given by [7] (2) [START_REF] Lebrun | Proceedings of ICOTOM[END_REF] where M is the mobility of a grain boundary. The grain boundaries having 38.21°<111> (Σ = 7) relation are known to have lower energy γ111 than random grain boundaries. As the cube component has a 40°<111> relation with the S component, it is easier to nucleate according to Equation (1). On the other hand, if assuming M111 = Mrandom = M, one may deduce from Equation ( 2) that the growth rate of the cube component is higher than that of the others.

For ultrarapid annealing, because of the relative short time at the low temperature stage, the thermal kinetics process occurs mainly at high temperature. Thus the pressure PD is higher and the grain boundary energy γ is relatively lower than that in the conventional annealing. According to , , grains with different orientation would have more chance to nucleate and grow. Therefore, the cube formation is less favoured as compared to the conventional annealing case. A deduction is made that, when recrystallization just completes, the resulting textures may be quite different depending on the nucleation and growth rates of all components which are affected by the γ and PD. In fact, the smaller the grain size and the higher the rolling reduction, the higher the pressure PD. This makes the differences in the recrystallization texture development for Samples I, II and III. As for the increase in the density of the cube component with further grain growth after 15.0ms, this may be due to the fact that the cube grains have a relatively large size and they can swallow small grains of other orientations. Consequently, it leads to the cube texture strengthening.

Conclusions

1.The development of texture in commercial pure copper annealed by laser beam is affected by both the rolling reduction and the initial grain size prior to rolling. For the coarse grained sample I with 70% reduction, the recrystallization texture is nearly random when recrystallization just completes. However, the cube component is evident in the sample II with 92% reduction. For the fine grained sample with 88% reduction, the recrystallization texture consists of the cube component and the cold rolling components at the just full recrystallization state. As grains grow further, the cube component in all the samples increases and the cold rolling components disappear.

2.For the samples heated by laser beam with short time (<13.2ms), no new grains could be observed and the cold rolling components decrease considerably. There is no obvious increase of the cube component. It may be deduced that at the initial stage of recrystallization, there exist a larger number of nuclei of different orientations including the cube and cold rolling components. The further texture development are mainly controlled by the growth competition of all these components.

Fig. 1 (

 1 a) gives the constant ψ (Δψ = 10°) sections of the ODF of Sample III, where the orientation intensities of the C, B and S components are each 8 times random units (tru). The C component in Sample I and II has the peak value of 11 and 4 tru, respectively.

Figure 1 .

 1 Figure 1. Constant ψ sections (Δψ = 10°) of the ODFs of (a) as-cold rolled Sample III, (b) annealing Sample I (t = 15.0ms), (c) annealing Sample II (t = 15.0ms) and (d) annealing Sample III (t = 15.0ms).In the present work, the heating time during annealing process was chosen as t = 9.0, 11.2, 13.2, 15.0, 22.5, 45.0 (ms). As t increases, recrystallization occurs in Sample I, II and III. The optical microscopy observation showed that the recrystallization processes are the same as those found in conventional annealing. When t = 15.0ms the recrystallization completes in all the samples (see Fig.2), but different texture components were formed. In Sample I the grain orientation distribution is almost random (Fig.1(b)), whereas a strong cube component and a weak {122}<221> component are evident in Sample II (Fig.1(c)). For Sample III the strong cube and relatively weak rolling components are seen (Fig.1(d)). As t increases further, grains grow and the intensity of the cube component increases in all the three samples. The development of the C and cube components versus the heating time is shown in Fig.3. It is noted that, for all the samples, the intensity of the rolling texture component {112}<111> (the

Figure 2 .

 2 Figure 2. Microstructures of copper samples cold rolled and annealed by laser beam. (a) As-cold rolled Sample III (88%Re.), (b) Sample I (t = 15.0ms), (c) Sample II (t = 15.0ms), (d) Sample III (t = 15.0ms).

Figure 3 .

 3 Figure 3. Orientation intensity versus the laser beam heating time t. C1, C2 and C3 are the C components of Sample I, II and III, respectively; Cube1, Cube2 and Cube3 are the cube components of Sample I, II and III, respectively.

TABLE 1 .

 1 Composition of Commercial Pure Copper, % (impurities, not more than).

Acknowledgements L. Zuo acknowledges support from Trans-Century Training Programme Foundation for the Talents by the State Education Commission of China and from Northeastern University, China.