- Email: [email protected]
Stress and microstructure in YBaCuO thin films on MgO and SrTiO3 substrates studied by X-ray diffraction and bending tests
Jourael of
A ~
AND ~ M ~ D S ELSEVIER
Journal of Alloys and Compounds 251 (1997) 37-40
Stress and microstructure in YBaCuO thin films on MgO and S r T i O 3 substrates studied by X-ray diffraction and bending tests S, A u z a r y , F. B a d a w i , L. Bimbault, J. Rabier, R.J. G a b o r i a u d * Universitd de Poitiers. Laboratoire de M~talhtrgie Physique. SP2MI. Bd 3, t~ldport 2, BPi79. 86960 Futuroscope cedex, France
Abstract
Stress-strain tensors and stress-free lattice parameters related to the microstmcture is studied in YBCO thin films deposited by pulsed-laser ablation on MgO and SrTiO~ substrates by means of X-ray diffraction and bending test measurements. The results show that the lattice of the thin film deposited on SrTiO~ is under compression in the (a,b) plane and the stress-free lattice parameters of this film are always lower than the parameters of the bulk material. Keywords: Stress; Microstructure;YBaCuO thin lilms; Lattice parameters
1. Introduction
It is well established that thin films of YBaCuO show very good superconducting properties with a high critical current depending upon the quality of the deposited layer. This so-called crystalline quality depends drastically on a large number of parameters involved in the deposition and growth process, Among those factors is the oxygen concentration which is of prime importance to obtain the relevant crystalline structure and therefore the supercon° ducting properties. The relationship between lattice paramo eters and the crystal stoichiometry has been widely used as a measurement of the oxygen concentration. Structural investigations of YBaCuO thin films deposited on different substrates mainly MgO and SrTiOa have been carried out by means of various analytical techniques. Nevertheless, further investigations seem necessary in the intricate field of internal stresses which are inherently present within a thin layer having some crystalline mismatch with the substrate. Physical or/and chemical causes of such stress, which promote strained crystal lattices, are still controversial and need complementary analysis. This study deals with structural determinations of the strain-stress state of thin films by X-ray diffraction and analysis of the results using the fundamental metric tensor
*Corresponding author. Fax: (33) 49 49 66 92, tel: (33) 49 49 66 62, e-mail: [email protected]. 0925-8388/971517.00 © 1997Elsevier Science S.A. All rights reserved Pll S0925-8388f96~02767-3
method generally used in the field of continuum mechanics.
I.I. ?Taeoretical basis
The analysis I'~r superficial residual stress measurement in polycrystals by X-ray diffraction 11-51 was used in this work. This method turned out to be very attractive fi~r the stress investigation of large grains considered as a single crystal. In our work this method has to be adapted to the very particular case of a thin layer (100-200 nm) which exhibits a rather good epitaxy relationship with a substrate and which can be therefore considered as a pseudo~single crystalline material. This method is based on the measurement of the variations of the lattice plane distances by X-ray diffraction measurements. The theoretical basis proposed by Ortner 16,7l is adapted to the orthorhombic (or tetragonal) structure of YBaCuO. For the sake of clarity only the guide line of the theoretical development used in this work is described. The details of this theory have ben developed elsewhere 141. On the basis of the crystalline lattice e~, e 2, e 3 one write a vector u=F,x~.e~ and JuJZ=go'x~'x/ where go is the fundamental metric tensor of the basis et(g,j =el %). In the so-called small deformation theory of continuum mechanics [8], the fundamental measure of deformation is
38
S. Anzu~:vel ai. / Journal of Alloys and COml)Omlds 251 (1997)37-40
defined by dl'-dl~=2~, d~',.dx i' with dl.=g,(.r) (I.~',.d~'j and dl ~'=gkt(x) d.vk'dx~ (dl. and dl are elementary distances before and after deformation). Rotating from the i, j, k axis of the sample to the et, e., e~ basis of the crystal one can - " o where E o is the strain tensor in the write: _~Eo-~o--g,i crystal axis system and go and g'/~ are the metric tensors related to the deformed and unreformed crystal respectively. The general case is rather complicated but cubic or orthorhombic cases are quite simple. In a cubic system, E.=aoe . (ao=latttce parameter)and g.=ao8 o then 2E,j= go-go gives go=a~(2~,s+8o) which is the metric tensor related to the deformed crystal basis, The angular position of the X-ray diffraction peaks gives dhk I which are directly related to the reciprocal lattice. Therefore it is more appropriate to write: dh~;=g IDhr where g" is the metric tensor related to the reciprocal basis of the crystal, g" is obtained from g,j by a matrix inversion and gives: g " - a . ~" (7t,j - 2 d ' )' which leads for the orthorhombie crystal to: ~,~ = -~(8 ~ 'J - a,~, ace g ') where a,, are the lattice parameters of the orthorhombic structure, The metric tensor of the delbnned basis is obtained from the X-ray diffraction measurements which therelbre requires at least six peaks in the general case. From the ltooke law. the stress tensor can be determined: o b = c , , . . e ; j where c,~ are the elastic moduli (compliance) of the single crystal 191, As seen above the determination of the strain tensor ~/',, requires the knowledge o1' the lattice parameters a., ( g . ) of the tmdeformed lattice (stressofree lattice) which are _generally not known, 'l'herefi~re a rather reasonable as= sumptkm is made: the outer ~urface of the sample is in equilibrium whicll gives ¢r~ lil(lrnlal)~-I). It is postuhiled ihiit no griidi~nt oo ,,~ cxi,,,ii,,,, and ihel'c~forc tr~ ~l) all along the coliXiS of the thin fihll, The stress tensor is then ¢lileulai~d rising the lattice piu'alllc~icr~ ot' ll'it~ bulk (or powdt~r) mater)ill t'Oulld in the Iiieraitu'e II01 following an algorithm where tr~ must reach zero vahld by an iterative calculation as a timction of new lattice parameters. When t r ~ = 0 the results give the stress (strain)tensor and the soocalled stress:free hittite parameters a.,. .
•
(
and in our laboratory. Up to seven diffraction peaks were registered per sample. As described latter on, bending tests were also carried out on the film deposited on SrTiO 3 sutlstrate using a Decktak apparatus. Young modulus of the SrTiO 3 substrate was furthermore determined by resonant frequency technique.
3. Results Experimental values of the lattice plane distarces obtained from the X-ray diffraction peaks introduced in the algorithm briefly described above lead to the determination of the stress and strain tensor and to the so-called stressfree lattice parameters of both YBaCuO thin films. Those results are summarized in the Figs. l and 2 which compare the as-deposited lattice parameters a
s,os
a
3,9
1 II
~
$,0S
~ 3,# 3,?l"
Bulk
11,75
r
~ Two thin lilms of YBaCuO prepared in-situ at 750°C by pulsedqaser deposition from a stoichiometric target were studied. The first layer deposited on a MgO substrate is I ~ nm thick. The second layer deposited on a Sfl'iO, substrate is 200 nm thick. Both thin films are cooriented and exhibit the usual epitaxy I~lationship with the sub= strate. The quality of this epitaxy as shown by X=ray pole figure allows us to consider the samples a~ pseudo single crystal material, X-ray diffraction measurements were I~rformed both on a 4-circle goniometer at LURE (Orsay)
MgO
(b) 1 1 , 8
_
2. Experimental
SrTIO~
11,7 11,6§
u 11,6 11,S5 11,5
Bnlk
SrTIO3
MIO
FiB, I. (a) XRD measurements of the lattice parameters of the thin lilms of YBaCuO oil MgO and st'rio, substrates a 0 bd (arrows). a, and b. are the stress-free lattice parameters obtained I'ronl the calculation described ill Section I,I. Also sllown are the values corresponding to the bulk material: a~ and h~,.(hi tdem us (a) but for the c-axis of the YBaCuO thin lilms.
S. Au:'.'.O el al. I Journal of Alloys and Compmmds 251 (1997) 37-40 (a)
3,9
XRD measurements for "b"
v///7/2ti 3,8z98|
3,85
!
i
m
~ t o ,m~n.ewae, br "".~/'~ I
,,~ 3,8
F'////~
mounted in the deposition chamber. The film was deposited on one of the substrates but both underwent the same thermal treatment. The substrate without deposition gives the reference state for the radius of curvature. Measurements in two orthogonal directions ((100),(OlO)) give the stress by the Stoney formula:
el
o'f=
3,'/5
[nenmng test~ 3,7
(b)
bulk o free lattice o - - 0..5Gim o - ---0.85GI material e, jisotrop. (XRD) Y B a C u O / S r T i O 3 (lattice p a r a m e t e r s a ar, d b)
tt,'l
~ 1 !,6$
<
1i,6
I I,$$
It,$
~,aCeOtSr'riOs (latt~ p=rsmem" e)
Fig. 2. (a) XRD measurement of the lattice parameters of the YBaCuO film deposited on SrTiO~ (arrows). Bulk material values are sho~n (,, and h~,). Also in the figure are the stress-free lattice parameters calculated from the results of the bending test measurements. (b) idem as (a) but for the voaxis of the YBaCuO thin films,
39
E (l-v)
]
~2 1~ 1 ~'6h r RWhere
L., v and h, are the Young modulus, Poisson ratio and thickness of the substrate, h f is the thickness of the film and, R is the radius of curvature. The same values of R are obtained in orthogonal direction which lead to a bump-like shape of the thin film indicating, as expected from the X-ray diffraction measurement, a lattice under compression. The stress values are in the range - 0 . 5 to - 0 . 8 5 Gpa. If those values are introduced into the iteration calculations mentioned above, the stress-free lattice parameters a o, b o, c o shown in Fig. 2(a-b), are obtained. Those values are still lower than the bulk material values and it is worth mentioning that the values obtained from o-= - 0 . 8 5 Gpa are not very far from thc~se obtained with the hypothesis of an isotropic deformation of the orthorhombic lattice. A clear physical interpretation of the stress-free lattice parameters together with the difference observed with respect to the bulk material are underway.
5. Conclushm 4. Discussion Stress tensors calculated from the algorithm mentioned above indicate that tile lilm deposited on Sr FiO~ is under compression in the (a, b) plane. Furthemtore the values o1' the stress-free lattice paralneters are all lower than the values corresponding to the bulk material. The thin film deposited on MgO exhibits a quasitetragonal structure (o d ~bd). The lattice is expanded, with respect to the bulk lattice, for ad ~bo >ah and c d >cb. The stress-free parameters of this film show au, be and co larger than at,, bb and c b of the bulk material. It is worthy of mention that in the iteration process which leads to the determination of the stress and the strain tensor the hypothesis of homogeneous isotropic delormation is made and that the starting values of the lattice parameters are the values of the bulk material. In order to check the results o1" the calculation of stress tensor and stress-free lattice parameters, an attempt to measure the stress in the lilm was also performed by the bending test, i.e.. the measurement of the n~dius ol' curvature of the sample. This experiment was realized only with the film deposited on SrTiO~. Owing to :he high temperature in-situ deposition process two substrates wet~
h ) our knowledge the work reported here is the first attempt to determine the stress tensor and the soocalled
stress-free lattice parameters in a thin film of YBaCuO on MgO and SrTiO~. Those preliminary results obtained by both Xoray diffraction and bending tests measurements show a rather good coherency: lattice under compression in the (a, b) plane, stress°free lattice parameters lower than the bulk material lattice for SrTiO~ suhstrat¢ This technique of course needs to Ix imp~'oved pat'= ticularly concerning the bending tests measurements which give rough results mainly due to the thickness of the substrate. Experin~ents with thinner substrates are under progres,;.
Acknowledgments "File authors are indebted to A. Catherinot and C. Champeaux, University of Limoges for the deposition of thc: 'i' BaCuO film on MgO substrate, to D. Duhreuii and G. G~crry, Thomson LCR Orsay tbr the sample deposited on Sr'l'iO 3 and to S. Pautrot, ENSMA Poitiers tbr the measurem,3nt of the Young modulus of the SrTiO~ substrate.
40
S. Auzm~' et ai. I Jourmll of Alloys al~d Compounds 251 (1997) 37-40
References [11 G. Maeder, Chem. Scripta. 26A (1986l 23. {21 L. Castex, J.L. Lebrun. G. Maeder and J.M. Sprauel, Pub. Sci. Techn. ENSAM, Pads, 1981. [31 N. Dur,md, L. Bimbaud, F. Badawi and P. Goudeau, J. Phys. Ill France. 4 (1994) 1025. [41 P. Getgaud, Th6se ENSAM, Paris. 1992.
[5] M. Banal and J.M. Sprauel, in Macherauh and Hauk (eds.), Residual Stress in Science and Technology, 1987, p. 265. [6] B. Ortner, Eigenspanmmgen DGM. 2 (1983) 49. [7] B. Ortner, Adv. X-ray Anal.. 29 (1986). [8] G.E. Mase, Continuum rheas', Schaum's outlbw series, McGrawHill, 1970. [9] C.S. Menon. N.V. Eldhose, Solid State Commun., 81 (1992) 357. [lOl Card (JCPDS-ICDD[38-1433).
A ~
AND ~ M ~ D S ELSEVIER
Journal of Alloys and Compounds 251 (1997) 37-40
Stress and microstructure in YBaCuO thin films on MgO and S r T i O 3 substrates studied by X-ray diffraction and bending tests S, A u z a r y , F. B a d a w i , L. Bimbault, J. Rabier, R.J. G a b o r i a u d * Universitd de Poitiers. Laboratoire de M~talhtrgie Physique. SP2MI. Bd 3, t~ldport 2, BPi79. 86960 Futuroscope cedex, France
Abstract
Stress-strain tensors and stress-free lattice parameters related to the microstmcture is studied in YBCO thin films deposited by pulsed-laser ablation on MgO and SrTiO~ substrates by means of X-ray diffraction and bending test measurements. The results show that the lattice of the thin film deposited on SrTiO~ is under compression in the (a,b) plane and the stress-free lattice parameters of this film are always lower than the parameters of the bulk material. Keywords: Stress; Microstructure;YBaCuO thin lilms; Lattice parameters
1. Introduction
It is well established that thin films of YBaCuO show very good superconducting properties with a high critical current depending upon the quality of the deposited layer. This so-called crystalline quality depends drastically on a large number of parameters involved in the deposition and growth process, Among those factors is the oxygen concentration which is of prime importance to obtain the relevant crystalline structure and therefore the supercon° ducting properties. The relationship between lattice paramo eters and the crystal stoichiometry has been widely used as a measurement of the oxygen concentration. Structural investigations of YBaCuO thin films deposited on different substrates mainly MgO and SrTiOa have been carried out by means of various analytical techniques. Nevertheless, further investigations seem necessary in the intricate field of internal stresses which are inherently present within a thin layer having some crystalline mismatch with the substrate. Physical or/and chemical causes of such stress, which promote strained crystal lattices, are still controversial and need complementary analysis. This study deals with structural determinations of the strain-stress state of thin films by X-ray diffraction and analysis of the results using the fundamental metric tensor
*Corresponding author. Fax: (33) 49 49 66 92, tel: (33) 49 49 66 62, e-mail: [email protected]. 0925-8388/971517.00 © 1997Elsevier Science S.A. All rights reserved Pll S0925-8388f96~02767-3
method generally used in the field of continuum mechanics.
I.I. ?Taeoretical basis
The analysis I'~r superficial residual stress measurement in polycrystals by X-ray diffraction 11-51 was used in this work. This method turned out to be very attractive fi~r the stress investigation of large grains considered as a single crystal. In our work this method has to be adapted to the very particular case of a thin layer (100-200 nm) which exhibits a rather good epitaxy relationship with a substrate and which can be therefore considered as a pseudo~single crystalline material. This method is based on the measurement of the variations of the lattice plane distances by X-ray diffraction measurements. The theoretical basis proposed by Ortner 16,7l is adapted to the orthorhombic (or tetragonal) structure of YBaCuO. For the sake of clarity only the guide line of the theoretical development used in this work is described. The details of this theory have ben developed elsewhere 141. On the basis of the crystalline lattice e~, e 2, e 3 one write a vector u=F,x~.e~ and JuJZ=go'x~'x/ where go is the fundamental metric tensor of the basis et(g,j =el %). In the so-called small deformation theory of continuum mechanics [8], the fundamental measure of deformation is
38
S. Anzu~:vel ai. / Journal of Alloys and COml)Omlds 251 (1997)37-40
defined by dl'-dl~=2~, d~',.dx i' with dl.=g,(.r) (I.~',.d~'j and dl ~'=gkt(x) d.vk'dx~ (dl. and dl are elementary distances before and after deformation). Rotating from the i, j, k axis of the sample to the et, e., e~ basis of the crystal one can - " o where E o is the strain tensor in the write: _~Eo-~o--g,i crystal axis system and go and g'/~ are the metric tensors related to the deformed and unreformed crystal respectively. The general case is rather complicated but cubic or orthorhombic cases are quite simple. In a cubic system, E.=aoe . (ao=latttce parameter)and g.=ao8 o then 2E,j= go-go gives go=a~(2~,s+8o) which is the metric tensor related to the deformed crystal basis, The angular position of the X-ray diffraction peaks gives dhk I which are directly related to the reciprocal lattice. Therefore it is more appropriate to write: dh~;=g IDhr where g" is the metric tensor related to the reciprocal basis of the crystal, g" is obtained from g,j by a matrix inversion and gives: g " - a . ~" (7t,j - 2 d ' )' which leads for the orthorhombie crystal to: ~,~ = -~(8 ~ 'J - a,~, ace g ') where a,, are the lattice parameters of the orthorhombic structure, The metric tensor of the delbnned basis is obtained from the X-ray diffraction measurements which therelbre requires at least six peaks in the general case. From the ltooke law. the stress tensor can be determined: o b = c , , . . e ; j where c,~ are the elastic moduli (compliance) of the single crystal 191, As seen above the determination of the strain tensor ~/',, requires the knowledge o1' the lattice parameters a., ( g . ) of the tmdeformed lattice (stressofree lattice) which are _generally not known, 'l'herefi~re a rather reasonable as= sumptkm is made: the outer ~urface of the sample is in equilibrium whicll gives ¢r~ lil(lrnlal)~-I). It is postuhiled ihiit no griidi~nt oo ,,~ cxi,,,ii,,,, and ihel'c~forc tr~ ~l) all along the coliXiS of the thin fihll, The stress tensor is then ¢lileulai~d rising the lattice piu'alllc~icr~ ot' ll'it~ bulk (or powdt~r) mater)ill t'Oulld in the Iiieraitu'e II01 following an algorithm where tr~ must reach zero vahld by an iterative calculation as a timction of new lattice parameters. When t r ~ = 0 the results give the stress (strain)tensor and the soocalled stress:free hittite parameters a.,. .
•
(
and in our laboratory. Up to seven diffraction peaks were registered per sample. As described latter on, bending tests were also carried out on the film deposited on SrTiO 3 sutlstrate using a Decktak apparatus. Young modulus of the SrTiO 3 substrate was furthermore determined by resonant frequency technique.
3. Results Experimental values of the lattice plane distarces obtained from the X-ray diffraction peaks introduced in the algorithm briefly described above lead to the determination of the stress and strain tensor and to the so-called stressfree lattice parameters of both YBaCuO thin films. Those results are summarized in the Figs. l and 2 which compare the as-deposited lattice parameters a
s,os
a
3,9
1 II
~
$,0S
~ 3,# 3,?l"
Bulk
11,75
r
~ Two thin lilms of YBaCuO prepared in-situ at 750°C by pulsedqaser deposition from a stoichiometric target were studied. The first layer deposited on a MgO substrate is I ~ nm thick. The second layer deposited on a Sfl'iO, substrate is 200 nm thick. Both thin films are cooriented and exhibit the usual epitaxy I~lationship with the sub= strate. The quality of this epitaxy as shown by X=ray pole figure allows us to consider the samples a~ pseudo single crystal material, X-ray diffraction measurements were I~rformed both on a 4-circle goniometer at LURE (Orsay)
MgO
(b) 1 1 , 8
_
2. Experimental
SrTIO~
11,7 11,6§
u 11,6 11,S5 11,5
Bnlk
SrTIO3
MIO
FiB, I. (a) XRD measurements of the lattice parameters of the thin lilms of YBaCuO oil MgO and st'rio, substrates a 0 bd (arrows). a, and b. are the stress-free lattice parameters obtained I'ronl the calculation described ill Section I,I. Also sllown are the values corresponding to the bulk material: a~ and h~,.(hi tdem us (a) but for the c-axis of the YBaCuO thin lilms.
S. Au:'.'.O el al. I Journal of Alloys and Compmmds 251 (1997) 37-40 (a)
3,9
XRD measurements for "b"
v///7/2ti 3,8z98|
3,85
!
i
m
~ t o ,m~n.ewae, br "".~/'~ I
,,~ 3,8
F'////~
mounted in the deposition chamber. The film was deposited on one of the substrates but both underwent the same thermal treatment. The substrate without deposition gives the reference state for the radius of curvature. Measurements in two orthogonal directions ((100),(OlO)) give the stress by the Stoney formula:
el
o'f=
3,'/5
[nenmng test~ 3,7
(b)
bulk o free lattice o - - 0..5Gim o - ---0.85GI material e, jisotrop. (XRD) Y B a C u O / S r T i O 3 (lattice p a r a m e t e r s a ar, d b)
tt,'l
~ 1 !,6$
<
1i,6
I I,$$
It,$
~,aCeOtSr'riOs (latt~ p=rsmem" e)
Fig. 2. (a) XRD measurement of the lattice parameters of the YBaCuO film deposited on SrTiO~ (arrows). Bulk material values are sho~n (,, and h~,). Also in the figure are the stress-free lattice parameters calculated from the results of the bending test measurements. (b) idem as (a) but for the voaxis of the YBaCuO thin films,
39
E (l-v)
]
~2 1~ 1 ~'6h r RWhere
L., v and h, are the Young modulus, Poisson ratio and thickness of the substrate, h f is the thickness of the film and, R is the radius of curvature. The same values of R are obtained in orthogonal direction which lead to a bump-like shape of the thin film indicating, as expected from the X-ray diffraction measurement, a lattice under compression. The stress values are in the range - 0 . 5 to - 0 . 8 5 Gpa. If those values are introduced into the iteration calculations mentioned above, the stress-free lattice parameters a o, b o, c o shown in Fig. 2(a-b), are obtained. Those values are still lower than the bulk material values and it is worth mentioning that the values obtained from o-= - 0 . 8 5 Gpa are not very far from thc~se obtained with the hypothesis of an isotropic deformation of the orthorhombic lattice. A clear physical interpretation of the stress-free lattice parameters together with the difference observed with respect to the bulk material are underway.
5. Conclushm 4. Discussion Stress tensors calculated from the algorithm mentioned above indicate that tile lilm deposited on Sr FiO~ is under compression in the (a, b) plane. Furthemtore the values o1' the stress-free lattice paralneters are all lower than the values corresponding to the bulk material. The thin film deposited on MgO exhibits a quasitetragonal structure (o d ~bd). The lattice is expanded, with respect to the bulk lattice, for ad ~bo >ah and c d >cb. The stress-free parameters of this film show au, be and co larger than at,, bb and c b of the bulk material. It is worthy of mention that in the iteration process which leads to the determination of the stress and the strain tensor the hypothesis of homogeneous isotropic delormation is made and that the starting values of the lattice parameters are the values of the bulk material. In order to check the results o1" the calculation of stress tensor and stress-free lattice parameters, an attempt to measure the stress in the lilm was also performed by the bending test, i.e.. the measurement of the n~dius ol' curvature of the sample. This experiment was realized only with the film deposited on SrTiO~. Owing to :he high temperature in-situ deposition process two substrates wet~
h ) our knowledge the work reported here is the first attempt to determine the stress tensor and the soocalled
stress-free lattice parameters in a thin film of YBaCuO on MgO and SrTiO~. Those preliminary results obtained by both Xoray diffraction and bending tests measurements show a rather good coherency: lattice under compression in the (a, b) plane, stress°free lattice parameters lower than the bulk material lattice for SrTiO~ suhstrat¢ This technique of course needs to Ix imp~'oved pat'= ticularly concerning the bending tests measurements which give rough results mainly due to the thickness of the substrate. Experin~ents with thinner substrates are under progres,;.
Acknowledgments "File authors are indebted to A. Catherinot and C. Champeaux, University of Limoges for the deposition of thc: 'i' BaCuO film on MgO substrate, to D. Duhreuii and G. G~crry, Thomson LCR Orsay tbr the sample deposited on Sr'l'iO 3 and to S. Pautrot, ENSMA Poitiers tbr the measurem,3nt of the Young modulus of the SrTiO~ substrate.
40
S. Auzm~' et ai. I Jourmll of Alloys al~d Compounds 251 (1997) 37-40
References [11 G. Maeder, Chem. Scripta. 26A (1986l 23. {21 L. Castex, J.L. Lebrun. G. Maeder and J.M. Sprauel, Pub. Sci. Techn. ENSAM, Pads, 1981. [31 N. Dur,md, L. Bimbaud, F. Badawi and P. Goudeau, J. Phys. Ill France. 4 (1994) 1025. [41 P. Getgaud, Th6se ENSAM, Paris. 1992.
[5] M. Banal and J.M. Sprauel, in Macherauh and Hauk (eds.), Residual Stress in Science and Technology, 1987, p. 265. [6] B. Ortner, Eigenspanmmgen DGM. 2 (1983) 49. [7] B. Ortner, Adv. X-ray Anal.. 29 (1986). [8] G.E. Mase, Continuum rheas', Schaum's outlbw series, McGrawHill, 1970. [9] C.S. Menon. N.V. Eldhose, Solid State Commun., 81 (1992) 357. [lOl Card (JCPDS-ICDD[38-1433).