- Email: [email protected]
Measurement of magnetostriction coefficient λs by speckle photography
Optics Communications 282 (2009) 635–639
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Measurement of magnetostriction coefficient ks by speckle photography F. Salazar a,*, A. Bayón a, J.M. Chicharro b a b
Departamento de Física Aplicada a los Recursos Naturales, E.T.S.I. Minas, Universidad Politécnica de Madrid, c/Ríos Rosas 21, 28003 Madrid, Spain Dept. de Mecánica Aplicada e Ingeniería de Proyectos, E.T.S.I. Industriales, Universidad de Castilla-La Mancha, Avd. Camilo José Cela s/n, 13071 Ciudad Real, Spain
a r t i c l e
i n f o
Article history: Received 14 April 2008 Accepted 27 October 2008
PACS: 75.80.+q 42.30.Ms 95.75.Kk
a b s t r a c t In this paper we present a general procedure based on the speckle photography to calculate both the longitudinal magnetostriction ksk and the transversal magnetostriction ks? in saturation, in only one experiment. For this purpose, we propose a model in which not only the deformation of the sample, but also its rigid movements are taken into consideration. To test the model, experimental results for nickel are reported, and a discussion of the uncertainty of the method is shown. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Magnetostriction Nickel Interferometry Speckle Uncertainty
1. Introduction As it is well known magnetostriction is a phenomenon that may be explained as the deformation of a material due to an external magnetic field H. This phenomenon has played an important role in technology because of its applications. For instance, the discovery of the giant magnetostriction has opened the door to new applications for industry and science. The application of these materials for industrial use has been important in areas such as robotics, automobiles, or scientific instrumentation. Another important group of magnetostrictive materials are the metallic glasses. These amorphous solids are very sensitive to mechanical stress, and this property can be used for applications in sensor technology. In this sense, the magnetostrictive characterization of materials is important for new applications as described above. One of the most important parameters of a magnetic material is the saturation magnetostriction coefficient ks . For isotropic materials the saturation magnetostriction at an angle h to the direction of the magnetic field H may be defined as follows [1]
ks ðhÞ ¼
3 1 : ks cos2 h 2 3
* Corresponding author. Tel.: +34 1 3364179; fax: +34 1 3366952. E-mail address: [email protected] (F. Salazar). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.10.052
ð1Þ
This equation shows that a magnetic field produces not only a longitudinal magnetostriction (ksk ) in the direction of H, but also a transversal deformation (ks? ). Substituting h ¼ 0 and h ¼ 90 into Eq. (1), gives
ks ¼
2 ðksk ks? Þ; 3
ð2Þ
which means that the saturation magnetostriction ks may be obtained from the difference between the saturation magnetostrictions measured parallel ðksk Þ and perpendicular ðks? Þ to the magnetic field H applied. The value of ðks Þ given by Eq. (2) remains independent of the demagnetizated state [2] while determining ðks Þ from only the measurement of ðksk Þ, according to Eq. (1), is sensitive to the presence of anisotropies in the demagnetized state. Direct and indirect methods have been employed to measure ks (a review of such techniques may be found in [3]). Among them, techniques based on optical interferometry have revealed to be suitable for measuring the magnetostriction of thin films [4–6], steel sheets [7], and ribbon samples [8]. The objective of this work is to obtain the saturation magnetostriction coefficient ks by measuring simultaneously the coefficients ksk and ks? in only one experiment. For this purpose we use a well known optical technique: the speckle photography. This method has been used to solve problems of thermal deformation [9], and strain analysis [10–12], among others. However, to our knowledge, this technique has not been applied to measurement
636
F. Salazar et al. / Optics Communications 282 (2009) 635–639
of ks . The proposed technique is a non-interacting method, precise, and its range of measurement is relatively easy to adjust. Furthermore, focused speckle photography allows the analysis of the inplane displacement in a strain experiment, being insensitive to out-of-plane movements. Our analysis deals with the measurement of strain when the external load is a magnetic field. Rigid movements of the sample during the experiment are undesirable because they lead to incorrect values of the physical unknowns. By means of the speckle photography technique, as we will show later, from the total displacement measured, it is possible to separate the rigid movements of the sample from the deformation suffered by the material as a consequence of the external magnetic field. All these characteristics make the proposed method adequate to measure ks .
2. Description of the technique The speckle photography method consists in taking two photographs of the sample on the same photographic plate, before and after the perturbation is applied (in our case the magnetic field), using a monochromatic light source. Once the plate is developed, we can determine the displacements on the surface of the material by means of the diffraction pattern obtained when a point on the specklegram is illuminated with a narrow laser beam [10,13] (Pointwise filtering technique) as shown in Fig. 1. The mentioned pattern is characterized for maxima and minima of intensity similar to those obtained in the Young’s experiment of double slit, but modulated by the so-called diffraction halo, which depends on the diameter D of the lens used to take the photograph, and on the distance z0 from this lens to the photographic plate (sin hh zD0 ) (Fig. 1). The characteristics of the lens are important because they are directly related with the sensibility of the meth-
od. So, the minimum displacement we can measure by this technique is of the order of the average speckle size, i.e. rs 1:2kðzD0 Þ ¼ 1:2kF 1 þ m1 , where F is the f-number of the lens ðf =]Þ and m the demagnification factor, which relates the dimensions of the object to the image recorded on the photographic emulsion. Focused speckle photography technique allows us to measure the in-plane displacements being not sensitive to out-of-plane displacements. The basic equation that relates the modulus of the inplane movements of the points on the sample plane with the directions of the intensity maxima is:
jDr j sin hn ¼ nkm;
ð3Þ
where jDr j is the modulus of the projection of the total displacement over the plane of the specimen, hn is the angle between the n-order maximum and the central maximum, and k is the wavelength of the laser. In this paper, the coefficients ksk and ks? are determined from measurement of displacements ðDx; DyÞ suffered by several points on the surface of the material when the specimen is perturbed by an homogeneous magnetic field. All measurements are carried out in a single test.
3. Equations for obtaining ksk and ks? simultaneously Let us suppose a prismatic bar whose dimensions are L1 , L2 and L3 along the coordinate system OXYZ, respectively (Fig. 2). Let us consider a point Pi ðxi ; yi ; 0Þ on the surface S1 defined by L1 and L2 ,
Fig. 2. Sample of dimensions L1 , L2 and L3 . Surface S1 is used to take the photograph.
Fig. 1. Experimental procedure of the pointwise filtering technique (PWFT). LB laser beam; PP photographic plate; S screen; YF Young’s fringes; DH diffraction halo; hn diffraction angle; b angle between the direction of displacement d, and the OX axis.
Fig. 3. View of the specimen. S1 surface; ri , position vector of P i ; bi , angle between the projection of displacement of point P i on S1 and the OX axis; Dri , displacement of P i due to the magnetic field.
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F. Salazar et al. / Optics Communications 282 (2009) 635–639
0
n1 km
cos b1
1
whose position vector with respect to the origin O is ri (Fig. 3). If we apply a magnetic field along the direction OX of the sample, it produces a displacement of the point P i to another position P 0i ðx0i ; y0i ; z0i Þ. As a consequence the sample suffers, in general, not only a deformation but also a translation and a rotation. In our case rigid motions perpendicular to the sample plane are impeded by the experimental constraints. This means in the presented model no tilt is allowed. If we choose the origin O as the reduction center and we suppose small continuous homogeneous deformations, when the applied field H is large enough to saturate the specimen, the projection of the displacement on the surface S1 suffered by Pi may be expressed as
0
Dxi ¼ ksk xi þ t x Xz yi ; Dyi ¼ ks? yi þ ty þ Xz xi ;
With these data and by applying the system (7), the values of the unknowns ksk , ks? , t x t y and Xz , may be obtained.
ð4Þ
where t x and t y are the components of translation, Xz is a small rotation around the OZ axis perpendicular to the sample surface, and ksk and ks? are the aforesaid saturation magnetostriction coefficients along the direction parallel and perpendicular to the applied field H, respectively. By introducing Eq. (3) in Eq. (4), we obtain the following system of equations
ni km cos bi ¼ ksk xi þ tx Xz yi ; sin hni ni km Dyi ¼ sin bi ¼ ks? yi þ t y þ Xz xi ; sin hni
Dxi ¼
ð5Þ
where bi is the angle between the direction of the in-plane displacement on S1 and the axis OX (Fig. 3). These equations have five unknowns, i.e. ksk , ks? , tx , t y and Xz , which means we need to measure at least the displacements of three points of the sample to obtain such values. However, as we work with experimental data, it is possible that the results for the unknowns differ of the actual values if the measurement of the displacement of these three points are not very accurate. Moreover, it is necessary to separate these points as much as possible and the disposition must be not in a straight line. The reason is that, in this case, the systematic uncertainty of the calculation would be quite large. A possibility to avoid that problem is to measure the displacements of many points and to use the multiple linear regression model in order to fit these data.
4. Multiple linear regression
1 sin h D x1 B n1 B n km B Dx2 C B sin2 hn 2 C B B C B B B C B C B n km B B Dxj C B sinj hn j C B B C¼B B B Dy1 C B n1 km C B sin hn1 B B Dy C B B 2 C B n2 km C B B @ A B sin hn2 B @ n km Dy j
j
sin hnj
0 x1 C cos b2 C C B x2 C B C B C B C B cos bj C B x C B j C¼B B0 sin b1 C C B C B0 B sin b2 C C B C @ C A 0 sin bj
1
0
1 0 y1
0
1 0 y2 C C 0k 1 C sk C B C B ks? C C C 1 0 yj C C B C: t CB x C 0 1 x1 C B C C B t A @ y C 0 1 x2 C Xz C A
0 y1 y2 yj
0 1
xj ð8Þ
5. Experimental arrangement The experimental setup employed is described in Fig. 4. A laser beam (LB) (He–Ne, 37 mW) is directed through a series of mirrors (M1 and M2) to a pin-hole (PH) to expand the beam (LB). The expanded beam is then directed to the sample plane (S). The rough specimen scatters the beam (producing speckles) and the resulting scattered light passes through the lens (L) reaching the photographic plate (PP). This experiment corresponds to the first exposure and the sample is demagnetised. Afterwards, a magnetic field is applied along the longitudinal direction of the sample and we take a second photograph for this new state of the object on the same photographic plate. This means that two photographs are superimposed on the same photographic emulsion. By developing the plate, we are able to measure the displacements suffered by any point of the sample plane through the diffraction pattern (Young’s fringes) described in Section 2. In order to avoid great rigid displacements and tilt of the sample between exposures, a face of the specimen was glued to an iron pole between the poles of the electromagnet (Figs. 3 and 4). To generate the magnetic field we used an electromagnet refrigerated by water (Newport Instruments) which may produce a magnetic field up to 1 T. This electromagnet has been previously calibrated by a magnetometer (Walker Scientific MG-4D) and with a transversal sound type of HP 14S. The applied optical system to register the displacement of the sample was corrected by aberrations. The lens has an f-number of 2.8 and the demagnification factor for all experiments was m = 1. With this set-up, the minimum displacement we can detect
Let us suppose we have obtained the components of the displacement of a number j of points, so that the number of equations of type (5) is greater than the number of unknowns. In this case, the resulting system of equations may be written in a matrix form as follows:
D ¼ X K;
ð6Þ
where D represents a vector in n-dimensional space with components corresponding to the projections of the infinitesimal displacements of each point analyzed, X is the matrix obtained by ordering the known coefficients of the system of equations, and K is the vector whose components are the unknowns of our problem. It may be demonstrated [14] by applying the multiple linear regression method, that best value of D which solves the system (6) satisfies
K ¼ ðX t XÞ1 X t D; t
1
ð7Þ
where X and X represent the transposed and inverse matrix of X, respectively. By setting the specific values of Eq. (5) in Eq. (6), for the points P1 , P 2 , . . ., P j , it may be obtained
Fig. 4. Experimental setup for the simultaneous measurement of ksk and ks? . L, laser; SH, shutter; M1 mirror 1; LB, laser beam; M2 mirror 2; PH pin-hole; S, sample of nickel; L, lens; PP, photographic plate; EM, electromagnet; DC, current generator.
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F. Salazar et al. / Optics Communications 282 (2009) 635–639
is about 4.2 lm (the average diameter of speckle). In our experiments a sample of nickel was investigated. 6. Experimental results To test the validity of the proposed procedure, we have used samples of nickel (99.75%) in the form of a prismatic bar 125 mm in length and square section of 40 mm height. The mechanical treatment followed was cold drawn and straight length and for the heat treatment the sample was annealed at 900 °C in continuous furnace. The magnetic field was applied in the longitudinal direction of the bar (direction of axis OX, Fig. 3). We have seen that only three points are needed to obtain the five unknowns. However, it is recommendable to employ a high number of points and use Eq. (7) referred to the multiple lineal regression method. With this aim we have measured the displacement of ten points which result in 20 equations and 5 unknowns. The specific form of equation 6 in the considered case is
0
1
20000
0
1000000
0
15000
B 54000 B B B 76000 7:22 B C B B 3:95 C B 75000 B C B B B 4:54 C B 58500 C B B C B B B 1:39 C B 66000 C B B B 1:74 C B 25000 C B B C B B B 0:79 C B 55000 C B B B 3:62 C B 61000 C B B C B B B 8:77 C B 28500 C B B B 8:18 C B 55000 C B B C B B B 7:06 C B 61000 C B B B 8:94 C ¼ B 0 C B B C B B B 15:12 C B 0 C B B B 23:07 C B 0 C B B C B B B 17:13 C B 0 C B B B 15:55 C B 0 C B B C B B B 17:17 C B 0 C B B C B B B 9:83 C B 0 C B B B 11:31 C B 0 C B B C B B @ 14:05 A B 0 B B 0 16:26 B B @ 0
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
0
0 0
1000000
0
1000000
0
0
1000000
0
15000 C C C 14000 C C C 28500 C C 29000 C C C 29000 C C 30000 C C C C 0 C C 0 C C C 0 C 30000 C C C C 0 C 20000 C C C 76000 C C 75000 C C C 54000 C C 76000 C C C 75000 C C C 58500 C C 66000 C C C 25000 C C 28500 C C C 55000 A
0
1
0
0 ksk
1
15000
0
1000000
19000
0
1000000
33000
0
1000000
15000
0
1000000
14000
0
1000000
28500 29000
0 0
1000000 1000000
29000
0
1000000
30000
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
C B B ks? C C B C B B tx C: C B @ ty A
61000
ð9Þ
Xz With these data, by applying Eq. (7) the values of ksk and ks? are simultaneously obtained, and then the saturation magnetostriction coefficient ks . The solutions for the system are ksk ¼ 54:7 106 , ks? ¼ 25:6 106 , t x ¼ 10:7 lm, ty ¼ 5:3 lm and Xz ¼ 192:9 lrad. With these results we obtain the following value for ks ,
ks ¼
2 ðksk ks? Þ ¼ 53:5 106 3
ð10Þ
This experimental result for ks is similar to that obtained by other authors [15]. The value of the magnetostriction ks for nickel depends on the chemical composition, and thermal and mechanical treatment. Since the sample used is finite, the form effect should
be taken into consideration [16–18]. The value estimated for our paralepipedic sample results to be of the order of 0:3 106 , which may be neglected as compared with the magnetostriction coefficient obtained. Taking into account the uncertainty of the method (see Section 7), we may write
ksk ¼ ð54:7 3:5Þ 106 ;
ð11Þ
and
ks? ¼ ð25:6 1:7Þ 106 ;
ð12Þ
respectively.
7. Evaluation of the uncertainties of the method To this end, we must analyze the expression (5). In these equations, the appearance of translations and rotations correspond to rigid movements of the sample. However, if rotations and translations are eliminated, the order of magnitude for the uncertainty of the technique may be estimated easier. For the case of ksk , Eq. (5) gives
ksk ¼
Dxi ni km ¼ cos bi ; xi sin hni xi
ð13Þ
and its uncertainty can be expressed as [19]
us ðksk Þ ¼
X oksk o q us ðqi Þ;
ð14Þ
i
where qi represent the variables in Eq. (13), and us ðqi Þ the systematic uncertainty of these parameters. By expanding the summation and dividing by ksk we obtain
us ðksk Þ us ðkÞ us ðxi Þ þ ¼ þ jtan bi us ðbi Þj þ us ðhn Þ: k k x tan h sk i n
ð15Þ
The angle hn refers to the intensity maxima of the diffraction pattern. To measure hn the highest order of fringes n may be always chosen (see Fig. 1). It means we can measure the fringe on the border of halo with the aim to reduce the uncertainty. Taking into account that the limit of diffraction halo verifies sin hh ¼ ðzD0 Þ ¼ ð1þ1 1 Þ, m
and in our case m ¼ 1, this expression converts in sin hh ¼ 2fD . By identifying hn with hh , Eq. (15) yields
us ðksk Þ us ðkÞ us ðxi Þ 2f k ¼ k þ x þ jtan bi us ðbi Þj þ D us ðhn Þ: sk
ð16Þ
i
If instead of the angle hh formed by the direction of the intensity maximum on the border of halo and the central maximum, the angle 2hh is measured between the maxima of order n and n, the last term of Eq. (16) is divide by 2, obtaining
us ðksk Þ us ðkÞ us ðxi Þ ¼ þ jtan bi us ðbi Þj þ f us ðhn Þ þ k k x D sk i us ðkÞ us ðxi Þ þ jtan bi us ðbi Þj þ jF jus ðhh Þ; þ ¼ k xi
ð17Þ
where F is the f-number of the lens. Assuming, k ¼ 632 nm, bi 45 , f =2:8, m ¼ 1, uðkÞ ¼ 1 1010 m, xi ¼ 0:10 m, uðxi Þ ¼ 0:01 mm, and uðbi Þ ¼ uðhh Þ ¼ 0:017 rad, Eq. (17) gives
us ðksk Þ k ¼ 0:00001 þ 0:0001 þ 0:017 þ 0:047 ¼ 0:064; sk
ð18Þ
that is, about 6.4%. In the case of ks? the expression is
ks? ¼
Dy i ni km ¼ sin bi ; yi sin hni yi
ð19Þ
F. Salazar et al. / Optics Communications 282 (2009) 635–639
and by applying Eq. (14) the uncertainty holds
poles of a magnet. Different sources of uncertainty are evaluated and the estimated relative uncertainty is of the order of 6.4%.
us ðks? Þ us ðkÞ us ðyi Þ us ðbi Þ ¼ þ þ ks? k y tan b þ jF jus ðhh Þ i i ¼ 0:00001 þ 0:005 þ 0:017 þ 0:047 ¼ 0:065;
References
ð20Þ
that is, 6.5%. The main contribution to the uncertainty of ksk and ks? is the term corresponding to the diffraction angle hh . This means that the uncertainty could be minimized by choosing a lens with smaller f-number. Once the uncertainties of ksk and ks? are known, the value of uðks Þ may be calculated using Eqs. (2) and (14), resulting
us ðks Þ ¼
2 uðksk Þ þ juðks? Þj ¼ 3:4 106 ; 3
ð21Þ
and the relative uncertainty,
us ðks Þ ¼ 0:064 ¼ 6:4%: ks
639
ð22Þ
8. Conclusions An optical method based on speckle photography is applied to measure saturation magnetostriction of a nickel sample. The method provides direct measurement of magnetostritive strain. A model is proposed to simultaneously determine the magnetostriction coefficients measured in directions parallel and perpendicular to the magnetic field. A non-contact technique is used to measure such coefficients, minimal stress is therefore imposed on the sample. Only a single test is required. The model proposed allows us to distinguish between magnetostriction strain and spurious displacements, which can avoid errors in the measurement. The resolution of the method ranges from 1:7 lm and 70 lm, depending on the optical components used (lens and laser). The method can be used for measuring relatively large magnetostriction of magnetic samples with different geometrical forms placed between the
[1] B.D. Cullity, Introduction to Magnetic Materials, Addison-Wesley, Massachusetts, 1972. [2] E. Trémolet, Magnetostriction: Theory and Applications of Magnetoelasticity, CRC Press, Florida, 1993 (Chapter 3). [3] P.T. Squire, Magnetomechanical measurements of magnetically soft amorphous materials, Mess. Sci. Technol. 5 (1994) 67. [4] A.C. Tam, H. Schroeder, New high-precision optical technique to measure magnetostriction of a thin magnetic film deposited on a substrate, IEEE Trans. Magn. 25 (1989) 2629. [5] G.H. Bellesis, P.S. Harlle III, A. Renema, D.N. Lamberth, Anisotropy and magnetostriction measurement by interferometry, IEEE Trans. Magn. 29 (1993) 2989. [6] P.S. Harlle III, G.H. Bellesis, D.N. Lamberth, Anisotropy and magnetostriction measurement by interferometry, J. Appl. Phys. 75 (1994) 6884. [7] T. Nakata, N. Takahashi, M. Nakano, K. Muramatsu, M. Miyake, Magnetostriction measurements with a Laser Doppler velocimeter, IEEE Trans. Magn. 30 (1994) 4563. [8] L. Gates, G. Stroink, Magnetostriction measurements of ribbon samples using a fiber-optic interferometer, Rev. Sci. Instrum. 63 (1992) 2017. [9] F. Gascón, F. Salazar, Thermal expansion tensor measurement by speckle interferometry, Rev. Sci. Instrum. 64 (1993) 2241. [10] R.P. Khetan, F.P. Chiang, Strain analysis by one-beam laser speckle interferometry. 1: Single aperture method, Appl. Opt. 15 (1976) 2205. [11] F.P. Chiang, J. Adachi, R. Anastasi, J. Beatty, Subjective laser speckle method and its application to solid mechanic problems, Opt. Eng. 21 (1982) 379. [12] F. Gascón, F. Salazar, A procedure for calculating through laser speckle interferometry the elastic constants of isotropic materials, Opt. Commun. 123 (1996) 734. [13] A.E. Ennos, Speckle interferometry, in: J.C. Dainty (Ed.), Laser Speckle and Related Phenomena, Topics in Applied Physics, vol. 9, 1985. [14] R.H. Myers, Classical and modern regression with applications, Prindle, Weber and Schmidt, Boston, MA, 1990. [15] Y. Chen, B.K. Kriegermeier-Sutton, J.E. Snyder, K.W. Dennis, R.W. McCallum, D.C. Jiles, Magnetomechanical effects under torsional strain in iron, cobalt and nickel, J. Magn. Magn. Matter. 236 (2001) 131. [16] R.I. Joseph, E. Schlömann, Demagnetizating field in nonellipsoidal bodies, J. Appl. Phys. 36 (1965) 1579. [17] E.W. Lee, Magnetostriction and magnetomechanical effects, Repts. Progr. Phys. 18 (1955) 184. [18] Du-Xing Chen, Enric Pardo, Alvaro Sánchez, Desmagnetizing factors of rectangular prims and ellipsoids, IEEE Trans. Magn. 38 (2002) 1742. [19] G.L. Squires, Practical Physics, Cambridge University Press, 1989.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Measurement of magnetostriction coefficient ks by speckle photography F. Salazar a,*, A. Bayón a, J.M. Chicharro b a b
Departamento de Física Aplicada a los Recursos Naturales, E.T.S.I. Minas, Universidad Politécnica de Madrid, c/Ríos Rosas 21, 28003 Madrid, Spain Dept. de Mecánica Aplicada e Ingeniería de Proyectos, E.T.S.I. Industriales, Universidad de Castilla-La Mancha, Avd. Camilo José Cela s/n, 13071 Ciudad Real, Spain
a r t i c l e
i n f o
Article history: Received 14 April 2008 Accepted 27 October 2008
PACS: 75.80.+q 42.30.Ms 95.75.Kk
a b s t r a c t In this paper we present a general procedure based on the speckle photography to calculate both the longitudinal magnetostriction ksk and the transversal magnetostriction ks? in saturation, in only one experiment. For this purpose, we propose a model in which not only the deformation of the sample, but also its rigid movements are taken into consideration. To test the model, experimental results for nickel are reported, and a discussion of the uncertainty of the method is shown. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Magnetostriction Nickel Interferometry Speckle Uncertainty
1. Introduction As it is well known magnetostriction is a phenomenon that may be explained as the deformation of a material due to an external magnetic field H. This phenomenon has played an important role in technology because of its applications. For instance, the discovery of the giant magnetostriction has opened the door to new applications for industry and science. The application of these materials for industrial use has been important in areas such as robotics, automobiles, or scientific instrumentation. Another important group of magnetostrictive materials are the metallic glasses. These amorphous solids are very sensitive to mechanical stress, and this property can be used for applications in sensor technology. In this sense, the magnetostrictive characterization of materials is important for new applications as described above. One of the most important parameters of a magnetic material is the saturation magnetostriction coefficient ks . For isotropic materials the saturation magnetostriction at an angle h to the direction of the magnetic field H may be defined as follows [1]
ks ðhÞ ¼
3 1 : ks cos2 h 2 3
* Corresponding author. Tel.: +34 1 3364179; fax: +34 1 3366952. E-mail address: [email protected] (F. Salazar). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.10.052
ð1Þ
This equation shows that a magnetic field produces not only a longitudinal magnetostriction (ksk ) in the direction of H, but also a transversal deformation (ks? ). Substituting h ¼ 0 and h ¼ 90 into Eq. (1), gives
ks ¼
2 ðksk ks? Þ; 3
ð2Þ
which means that the saturation magnetostriction ks may be obtained from the difference between the saturation magnetostrictions measured parallel ðksk Þ and perpendicular ðks? Þ to the magnetic field H applied. The value of ðks Þ given by Eq. (2) remains independent of the demagnetizated state [2] while determining ðks Þ from only the measurement of ðksk Þ, according to Eq. (1), is sensitive to the presence of anisotropies in the demagnetized state. Direct and indirect methods have been employed to measure ks (a review of such techniques may be found in [3]). Among them, techniques based on optical interferometry have revealed to be suitable for measuring the magnetostriction of thin films [4–6], steel sheets [7], and ribbon samples [8]. The objective of this work is to obtain the saturation magnetostriction coefficient ks by measuring simultaneously the coefficients ksk and ks? in only one experiment. For this purpose we use a well known optical technique: the speckle photography. This method has been used to solve problems of thermal deformation [9], and strain analysis [10–12], among others. However, to our knowledge, this technique has not been applied to measurement
636
F. Salazar et al. / Optics Communications 282 (2009) 635–639
of ks . The proposed technique is a non-interacting method, precise, and its range of measurement is relatively easy to adjust. Furthermore, focused speckle photography allows the analysis of the inplane displacement in a strain experiment, being insensitive to out-of-plane movements. Our analysis deals with the measurement of strain when the external load is a magnetic field. Rigid movements of the sample during the experiment are undesirable because they lead to incorrect values of the physical unknowns. By means of the speckle photography technique, as we will show later, from the total displacement measured, it is possible to separate the rigid movements of the sample from the deformation suffered by the material as a consequence of the external magnetic field. All these characteristics make the proposed method adequate to measure ks .
2. Description of the technique The speckle photography method consists in taking two photographs of the sample on the same photographic plate, before and after the perturbation is applied (in our case the magnetic field), using a monochromatic light source. Once the plate is developed, we can determine the displacements on the surface of the material by means of the diffraction pattern obtained when a point on the specklegram is illuminated with a narrow laser beam [10,13] (Pointwise filtering technique) as shown in Fig. 1. The mentioned pattern is characterized for maxima and minima of intensity similar to those obtained in the Young’s experiment of double slit, but modulated by the so-called diffraction halo, which depends on the diameter D of the lens used to take the photograph, and on the distance z0 from this lens to the photographic plate (sin hh zD0 ) (Fig. 1). The characteristics of the lens are important because they are directly related with the sensibility of the meth-
od. So, the minimum displacement we can measure by this technique is of the order of the average speckle size, i.e. rs 1:2kðzD0 Þ ¼ 1:2kF 1 þ m1 , where F is the f-number of the lens ðf =]Þ and m the demagnification factor, which relates the dimensions of the object to the image recorded on the photographic emulsion. Focused speckle photography technique allows us to measure the in-plane displacements being not sensitive to out-of-plane displacements. The basic equation that relates the modulus of the inplane movements of the points on the sample plane with the directions of the intensity maxima is:
jDr j sin hn ¼ nkm;
ð3Þ
where jDr j is the modulus of the projection of the total displacement over the plane of the specimen, hn is the angle between the n-order maximum and the central maximum, and k is the wavelength of the laser. In this paper, the coefficients ksk and ks? are determined from measurement of displacements ðDx; DyÞ suffered by several points on the surface of the material when the specimen is perturbed by an homogeneous magnetic field. All measurements are carried out in a single test.
3. Equations for obtaining ksk and ks? simultaneously Let us suppose a prismatic bar whose dimensions are L1 , L2 and L3 along the coordinate system OXYZ, respectively (Fig. 2). Let us consider a point Pi ðxi ; yi ; 0Þ on the surface S1 defined by L1 and L2 ,
Fig. 2. Sample of dimensions L1 , L2 and L3 . Surface S1 is used to take the photograph.
Fig. 1. Experimental procedure of the pointwise filtering technique (PWFT). LB laser beam; PP photographic plate; S screen; YF Young’s fringes; DH diffraction halo; hn diffraction angle; b angle between the direction of displacement d, and the OX axis.
Fig. 3. View of the specimen. S1 surface; ri , position vector of P i ; bi , angle between the projection of displacement of point P i on S1 and the OX axis; Dri , displacement of P i due to the magnetic field.
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F. Salazar et al. / Optics Communications 282 (2009) 635–639
0
n1 km
cos b1
1
whose position vector with respect to the origin O is ri (Fig. 3). If we apply a magnetic field along the direction OX of the sample, it produces a displacement of the point P i to another position P 0i ðx0i ; y0i ; z0i Þ. As a consequence the sample suffers, in general, not only a deformation but also a translation and a rotation. In our case rigid motions perpendicular to the sample plane are impeded by the experimental constraints. This means in the presented model no tilt is allowed. If we choose the origin O as the reduction center and we suppose small continuous homogeneous deformations, when the applied field H is large enough to saturate the specimen, the projection of the displacement on the surface S1 suffered by Pi may be expressed as
0
Dxi ¼ ksk xi þ t x Xz yi ; Dyi ¼ ks? yi þ ty þ Xz xi ;
With these data and by applying the system (7), the values of the unknowns ksk , ks? , t x t y and Xz , may be obtained.
ð4Þ
where t x and t y are the components of translation, Xz is a small rotation around the OZ axis perpendicular to the sample surface, and ksk and ks? are the aforesaid saturation magnetostriction coefficients along the direction parallel and perpendicular to the applied field H, respectively. By introducing Eq. (3) in Eq. (4), we obtain the following system of equations
ni km cos bi ¼ ksk xi þ tx Xz yi ; sin hni ni km Dyi ¼ sin bi ¼ ks? yi þ t y þ Xz xi ; sin hni
Dxi ¼
ð5Þ
where bi is the angle between the direction of the in-plane displacement on S1 and the axis OX (Fig. 3). These equations have five unknowns, i.e. ksk , ks? , tx , t y and Xz , which means we need to measure at least the displacements of three points of the sample to obtain such values. However, as we work with experimental data, it is possible that the results for the unknowns differ of the actual values if the measurement of the displacement of these three points are not very accurate. Moreover, it is necessary to separate these points as much as possible and the disposition must be not in a straight line. The reason is that, in this case, the systematic uncertainty of the calculation would be quite large. A possibility to avoid that problem is to measure the displacements of many points and to use the multiple linear regression model in order to fit these data.
4. Multiple linear regression
1 sin h D x1 B n1 B n km B Dx2 C B sin2 hn 2 C B B C B B B C B C B n km B B Dxj C B sinj hn j C B B C¼B B B Dy1 C B n1 km C B sin hn1 B B Dy C B B 2 C B n2 km C B B @ A B sin hn2 B @ n km Dy j
j
sin hnj
0 x1 C cos b2 C C B x2 C B C B C B C B cos bj C B x C B j C¼B B0 sin b1 C C B C B0 B sin b2 C C B C @ C A 0 sin bj
1
0
1 0 y1
0
1 0 y2 C C 0k 1 C sk C B C B ks? C C C 1 0 yj C C B C: t CB x C 0 1 x1 C B C C B t A @ y C 0 1 x2 C Xz C A
0 y1 y2 yj
0 1
xj ð8Þ
5. Experimental arrangement The experimental setup employed is described in Fig. 4. A laser beam (LB) (He–Ne, 37 mW) is directed through a series of mirrors (M1 and M2) to a pin-hole (PH) to expand the beam (LB). The expanded beam is then directed to the sample plane (S). The rough specimen scatters the beam (producing speckles) and the resulting scattered light passes through the lens (L) reaching the photographic plate (PP). This experiment corresponds to the first exposure and the sample is demagnetised. Afterwards, a magnetic field is applied along the longitudinal direction of the sample and we take a second photograph for this new state of the object on the same photographic plate. This means that two photographs are superimposed on the same photographic emulsion. By developing the plate, we are able to measure the displacements suffered by any point of the sample plane through the diffraction pattern (Young’s fringes) described in Section 2. In order to avoid great rigid displacements and tilt of the sample between exposures, a face of the specimen was glued to an iron pole between the poles of the electromagnet (Figs. 3 and 4). To generate the magnetic field we used an electromagnet refrigerated by water (Newport Instruments) which may produce a magnetic field up to 1 T. This electromagnet has been previously calibrated by a magnetometer (Walker Scientific MG-4D) and with a transversal sound type of HP 14S. The applied optical system to register the displacement of the sample was corrected by aberrations. The lens has an f-number of 2.8 and the demagnification factor for all experiments was m = 1. With this set-up, the minimum displacement we can detect
Let us suppose we have obtained the components of the displacement of a number j of points, so that the number of equations of type (5) is greater than the number of unknowns. In this case, the resulting system of equations may be written in a matrix form as follows:
D ¼ X K;
ð6Þ
where D represents a vector in n-dimensional space with components corresponding to the projections of the infinitesimal displacements of each point analyzed, X is the matrix obtained by ordering the known coefficients of the system of equations, and K is the vector whose components are the unknowns of our problem. It may be demonstrated [14] by applying the multiple linear regression method, that best value of D which solves the system (6) satisfies
K ¼ ðX t XÞ1 X t D; t
1
ð7Þ
where X and X represent the transposed and inverse matrix of X, respectively. By setting the specific values of Eq. (5) in Eq. (6), for the points P1 , P 2 , . . ., P j , it may be obtained
Fig. 4. Experimental setup for the simultaneous measurement of ksk and ks? . L, laser; SH, shutter; M1 mirror 1; LB, laser beam; M2 mirror 2; PH pin-hole; S, sample of nickel; L, lens; PP, photographic plate; EM, electromagnet; DC, current generator.
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F. Salazar et al. / Optics Communications 282 (2009) 635–639
is about 4.2 lm (the average diameter of speckle). In our experiments a sample of nickel was investigated. 6. Experimental results To test the validity of the proposed procedure, we have used samples of nickel (99.75%) in the form of a prismatic bar 125 mm in length and square section of 40 mm height. The mechanical treatment followed was cold drawn and straight length and for the heat treatment the sample was annealed at 900 °C in continuous furnace. The magnetic field was applied in the longitudinal direction of the bar (direction of axis OX, Fig. 3). We have seen that only three points are needed to obtain the five unknowns. However, it is recommendable to employ a high number of points and use Eq. (7) referred to the multiple lineal regression method. With this aim we have measured the displacement of ten points which result in 20 equations and 5 unknowns. The specific form of equation 6 in the considered case is
0
1
20000
0
1000000
0
15000
B 54000 B B B 76000 7:22 B C B B 3:95 C B 75000 B C B B B 4:54 C B 58500 C B B C B B B 1:39 C B 66000 C B B B 1:74 C B 25000 C B B C B B B 0:79 C B 55000 C B B B 3:62 C B 61000 C B B C B B B 8:77 C B 28500 C B B B 8:18 C B 55000 C B B C B B B 7:06 C B 61000 C B B B 8:94 C ¼ B 0 C B B C B B B 15:12 C B 0 C B B B 23:07 C B 0 C B B C B B B 17:13 C B 0 C B B B 15:55 C B 0 C B B C B B B 17:17 C B 0 C B B C B B B 9:83 C B 0 C B B B 11:31 C B 0 C B B C B B @ 14:05 A B 0 B B 0 16:26 B B @ 0
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
0
0 0
1000000
0
1000000
0
0
1000000
0
15000 C C C 14000 C C C 28500 C C 29000 C C C 29000 C C 30000 C C C C 0 C C 0 C C C 0 C 30000 C C C C 0 C 20000 C C C 76000 C C 75000 C C C 54000 C C 76000 C C C 75000 C C C 58500 C C 66000 C C C 25000 C C 28500 C C C 55000 A
0
1
0
0 ksk
1
15000
0
1000000
19000
0
1000000
33000
0
1000000
15000
0
1000000
14000
0
1000000
28500 29000
0 0
1000000 1000000
29000
0
1000000
30000
0
1000000
0
0
1000000
0
0
1000000
0
0
1000000
C B B ks? C C B C B B tx C: C B @ ty A
61000
ð9Þ
Xz With these data, by applying Eq. (7) the values of ksk and ks? are simultaneously obtained, and then the saturation magnetostriction coefficient ks . The solutions for the system are ksk ¼ 54:7 106 , ks? ¼ 25:6 106 , t x ¼ 10:7 lm, ty ¼ 5:3 lm and Xz ¼ 192:9 lrad. With these results we obtain the following value for ks ,
ks ¼
2 ðksk ks? Þ ¼ 53:5 106 3
ð10Þ
This experimental result for ks is similar to that obtained by other authors [15]. The value of the magnetostriction ks for nickel depends on the chemical composition, and thermal and mechanical treatment. Since the sample used is finite, the form effect should
be taken into consideration [16–18]. The value estimated for our paralepipedic sample results to be of the order of 0:3 106 , which may be neglected as compared with the magnetostriction coefficient obtained. Taking into account the uncertainty of the method (see Section 7), we may write
ksk ¼ ð54:7 3:5Þ 106 ;
ð11Þ
and
ks? ¼ ð25:6 1:7Þ 106 ;
ð12Þ
respectively.
7. Evaluation of the uncertainties of the method To this end, we must analyze the expression (5). In these equations, the appearance of translations and rotations correspond to rigid movements of the sample. However, if rotations and translations are eliminated, the order of magnitude for the uncertainty of the technique may be estimated easier. For the case of ksk , Eq. (5) gives
ksk ¼
Dxi ni km ¼ cos bi ; xi sin hni xi
ð13Þ
and its uncertainty can be expressed as [19]
us ðksk Þ ¼
X oksk o q us ðqi Þ;
ð14Þ
i
where qi represent the variables in Eq. (13), and us ðqi Þ the systematic uncertainty of these parameters. By expanding the summation and dividing by ksk we obtain
us ðksk Þ us ðkÞ us ðxi Þ þ ¼ þ jtan bi us ðbi Þj þ us ðhn Þ: k k x tan h sk i n
ð15Þ
The angle hn refers to the intensity maxima of the diffraction pattern. To measure hn the highest order of fringes n may be always chosen (see Fig. 1). It means we can measure the fringe on the border of halo with the aim to reduce the uncertainty. Taking into account that the limit of diffraction halo verifies sin hh ¼ ðzD0 Þ ¼ ð1þ1 1 Þ, m
and in our case m ¼ 1, this expression converts in sin hh ¼ 2fD . By identifying hn with hh , Eq. (15) yields
us ðksk Þ us ðkÞ us ðxi Þ 2f k ¼ k þ x þ jtan bi us ðbi Þj þ D us ðhn Þ: sk
ð16Þ
i
If instead of the angle hh formed by the direction of the intensity maximum on the border of halo and the central maximum, the angle 2hh is measured between the maxima of order n and n, the last term of Eq. (16) is divide by 2, obtaining
us ðksk Þ us ðkÞ us ðxi Þ ¼ þ jtan bi us ðbi Þj þ f us ðhn Þ þ k k x D sk i us ðkÞ us ðxi Þ þ jtan bi us ðbi Þj þ jF jus ðhh Þ; þ ¼ k xi
ð17Þ
where F is the f-number of the lens. Assuming, k ¼ 632 nm, bi 45 , f =2:8, m ¼ 1, uðkÞ ¼ 1 1010 m, xi ¼ 0:10 m, uðxi Þ ¼ 0:01 mm, and uðbi Þ ¼ uðhh Þ ¼ 0:017 rad, Eq. (17) gives
us ðksk Þ k ¼ 0:00001 þ 0:0001 þ 0:017 þ 0:047 ¼ 0:064; sk
ð18Þ
that is, about 6.4%. In the case of ks? the expression is
ks? ¼
Dy i ni km ¼ sin bi ; yi sin hni yi
ð19Þ
F. Salazar et al. / Optics Communications 282 (2009) 635–639
and by applying Eq. (14) the uncertainty holds
poles of a magnet. Different sources of uncertainty are evaluated and the estimated relative uncertainty is of the order of 6.4%.
us ðks? Þ us ðkÞ us ðyi Þ us ðbi Þ ¼ þ þ ks? k y tan b þ jF jus ðhh Þ i i ¼ 0:00001 þ 0:005 þ 0:017 þ 0:047 ¼ 0:065;
References
ð20Þ
that is, 6.5%. The main contribution to the uncertainty of ksk and ks? is the term corresponding to the diffraction angle hh . This means that the uncertainty could be minimized by choosing a lens with smaller f-number. Once the uncertainties of ksk and ks? are known, the value of uðks Þ may be calculated using Eqs. (2) and (14), resulting
us ðks Þ ¼
2 uðksk Þ þ juðks? Þj ¼ 3:4 106 ; 3
ð21Þ
and the relative uncertainty,
us ðks Þ ¼ 0:064 ¼ 6:4%: ks
639
ð22Þ
8. Conclusions An optical method based on speckle photography is applied to measure saturation magnetostriction of a nickel sample. The method provides direct measurement of magnetostritive strain. A model is proposed to simultaneously determine the magnetostriction coefficients measured in directions parallel and perpendicular to the magnetic field. A non-contact technique is used to measure such coefficients, minimal stress is therefore imposed on the sample. Only a single test is required. The model proposed allows us to distinguish between magnetostriction strain and spurious displacements, which can avoid errors in the measurement. The resolution of the method ranges from 1:7 lm and 70 lm, depending on the optical components used (lens and laser). The method can be used for measuring relatively large magnetostriction of magnetic samples with different geometrical forms placed between the
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