General paper
by Raman Microspectroscopy
2-4-1 Mutsuno, Atsuta-ku Nagoya 456-8587, Japan ** Department of Engineering
, Nagoya University,
, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
Abstract: A new system of Raman microspectroscopy was developed to measure the residual stress in a local area with
about 2ƒÊm diameter in ceramics. The system was applied to a single crystal sapphire in order to evaluate the capability
of the system. The shift of two Raman spectra, 645 and 418cm-1, of A1g mode was proportional to the lattice strain, ƒÃ33,
along the c-axis of trigonal crystals. The proportional constant was larger for 418cm-1 than for 645cm-1 spectrum. The
distributions of loading strains due to four-point bending were measured across the minimum ligament of single-edge-
notched specimens by scanning the focused Ar+ laser beam. The strain concentration near the notch root was
successfully detected by Raman microspectroscopy. The measured distributions of strains agreed very well with the
results of a finite element analysis. The developed system of Raman microspectroscopy was applicable to the strain
measurement in the area having a steep gradient as observed near the crack tip in ceramics.
Key words: Local stress measurement, Raman microspectroscopy, Sapphire, Finite element method
1. INTRODUCTION
sensitive to the local residual and loading stresses in the
vicinity of the crack tip. The residual stress varies depend-
ing not only on geometrical singularities, but also on the
grain structure of ceramics [1]. It is important to develop
the method to measure the residual stress in a local area
with a few micrometer dimensions in ceramics. The X-ray
diffraction method is now widely used for the measure-
ment of residual stresses in ceramics. However, the spatial
resolution of the method is very limited. The minimum
size of the irradiation area of the conventional X-ray
equipment is about 100ƒÊm [2].
Laser Raman microspectroscopy is a novel nondestru-
ctive technique to measure the stress in materials such as
semiconductors [3] and ceramics [4-6]. The advantage of
this technique is its high spatial resolution of about 2ƒÊm,
which is easily accomplished by a combination of the
optical microscope and Raman spectroscopy equipment.
This high spatial resolution makes the method one of the
most powerful tools for estimating the stress in a local
area [7, 8].
In the present study, a new system of Raman micro-
spectroscopy was developed to measure the residual and
loading stresses in a local area with a few micrometer
diameter in ceramics. The system was applied to a single
crystal sapphire in order to evaluate the capability of the
system. The strain constants were first determined by
applying the known strain to smooth specimens, and then
the calibrated strain constants were applied to measure a
steep stress gradient in notched specimens subjected to
four-point bending. The measured distributions of strains
were compared with those calculated by a finite element
method.
The material used in the present study was a single
crystal sapphire, ƒ¿-Al2O3. Table 1 summarizes elastic
stiffnesses, cij, of this material [9]. Four types of specimen
were used for experiments as shown in Fig. 1. The one
was a smooth bend-bar specimen, Specimen A, with a
width of 4mm, a thickness of 3mm, and a length of 40
mm. This specimen was used for determining the strain
constants. The others were single-edge V-notched beam
(SEVNB) specimens [10] with a width of 3mm, a
thickness of 4mm, and a length of 40mm, which was
used for stress measurement. They are designated
Specimens B, C and D as shown in Fig. 1. These
specimens were cut from the bulk sapphire which was
identified the crystallographic directions, a1, a2 and c-
axes. For SEVNB specimens, V-notches were machined
by grinding using a V-shaped diamond wheel [10]. The
depth of each notch was about 1.18mm and the radius of
the notch root curvature was about 20ƒÊm as shown in Fig.
2. The angle of V-notch was about 20 degrees.
Table 1. Stiffness of single crystal sapphire [9].
Received March 24, 2000 Accepted September 25, 2000295
Yoshihisa SAKAIDA, Keisuke TANAKA and Kaori SHIRAKIHARA
Fig. 1. Specimens A, B, C and D. The tensile-surface of each specimen is shown with the gray-colored-face.
Fig. 2. Micrograph near the V-notch tip of Specimen C.
Fig. 3. Schema of Raman microspectroscopy.
2.2. Stress Measurement by Raman Microspectroscopy
The Raman spectra were measured at room temper-
ature in the backscattering configuration using a micro-
scope as shown in Figs. 3 and 4. An Ar+ laser with a
wavelength of 515 nm was used for excitation. The diam-
eter of the focused laser beam was about 2ƒÊm.
Figure 5 shows the unpolarized Raman spectrum
pattern of a single crystal sapphire. Since the crystal
structure of ƒ¿-Al2O3 is the trigonal system with D3d
(a) (b)
Fig. 4. Coordinate systems for determining the strain constants, (a), and stress measurement, (b).
ki and ks are wave number vectors of the incident and scattered beams, respectively.
Fig. 5. Raman spectrum pattern of a single crystal
sapphire. The 418 and 645cm-1 spectra are
classified into A18 mode. The 378, 432, 451, 578
and 751cm-1 spectra are classified into E8 mode.
symmetry, a single crystal sapphire has seven Raman-
active modes spectra [5, 6]: two A18 and five doubly
degenerate E8. In the present study, two Raman spectra,
645 and 418 cm-1, classified into A18 mode were selected
for stress measurement.
bending load in our Raman spectroscopy equipment [11],
as shown in Fig. 4 (a). The inner and outer spans were 20
and 36mm, respectively. The shifts of two Raman spectra
due to bending loads were detected by a charge coupled
device camera, and then strain constants were determined.
Next, a constant bending load of about 100N was applied
to each SEVNB specimen, and the shifts of Raman spectra
were measured by scanning the Ar+ laser beam from the
tip of the notch to the end of the ligament of the specimen,
as shown in Fig. 4 (b). From the measured Raman shifts,
the distributions of three strain components, ƒÃx, ƒÃy, and ƒÃz,
were determined, where ƒÃx and ƒÃy, are the strains parallel
and perpendicular to the notch face, and ƒÃz is the strain
parallel to the notch front.
296
3. EXPERIMENTAL RESULTS AND DISCUSSION
3.1. Determination of Strain Constants for Two Raman
Spectra of A18 Mode
According to the theory of Raman spectroscopy [4-6],
the shifts of two Raman spectra, 645 and 418cm-1, of A18
mode, ĢąA18, are expressed asĢąA
18 ƒÖA18-ƒÖ0,A18=C1, ƒÖ(ƒÃ11+ƒÃ22)+C2,ƒÖƒÃ33, (1)
where ƒÃ11 and ƒÃ22 are the normal strains on the basal plane
of trigonal crystal and ƒÃ33 is that in the direction of c-axis. ƒÖ0,A
18 is a wave number from the stress-free crystal, and
C1 ,ƒÖ and C2,ƒÖ, are strain constants. In the case of A18 mode spectrum, either red or blue shift is caused by the applied
strain.
The strain constants, C1 ,ƒÖ and C2,ƒÖ, of each Raman
spectrum were first determined by measuring the Raman
shift as a function of the applied strain. When a bending
load is applied to Specimen A, the ratio of (ƒÃ11+ƒÃ22) to ƒÃ33 is
theoretically -0.34. Substituting this relation into Eq. (1),
the Raman shift can be expressed as
ƒ¢ƒÖA18 (-0.34C1,ƒÖ+C2,ƒÖ)ƒÃ33, (2)
for A18 mode spectrum.
Raman shift, ƒ¢ƒÖA18, and the applied strain component, ƒÃ33.
Red shifts of both Raman spectra increase linearly with
increasing applied strain along the c-axis of the crystal.
The strain constants were determined by a multiple
regression analysis of experimental data. Table 2
summarizes optimized strain constants C1,ƒÖ and C2,ƒÖ for
645 and 418cm-1 spectra. Under the same applied strain,
the shift of 418cm-1 spectrum is about ten times larger
than that of 645cm-1 spectrum. The 418cm-1 spectrum is
highly sensitive to the strain component, ƒÃ33.
Fig. 6. Relationship between Raman shift, AƒÖA18, and the
applied strains, ƒÃ33. ƒ¢ƒÖ645 and ƒ¢ƒÖ418 are Raman
shifts of 645 and 418cm-1 spectra, respectively.
Table 2. Optimized strain constants C1,645, C2,645, C1,418, and C2 ,418 for 645 and 418cm-1 spectra.
3.2. Stress Measurement of Bent SEVNB Specimens by
Raman Microspectroscopy
The distributions of two Raman shifts, Ģą6a5 and
Ģą418, due to four-point bending were measured across the minimum ligament of SEVNB specimens by scanning the
laser beam from the notch tip to the end of the ligament
along x-axis (see Fig 4 (b)).
Figure 7 shows the Raman shift distribution Ģą418 of
bent Specimen B. In this case, a constant bending load of
105.3 N is applied to Specimen B. A large red shift is
observed in the close vicinity of the notch tip. The red shift
of 418cm-1 spectrum reaches the maximum of about 0.85
cm-1 at the notch tip, and decreases drastically with an
increase in the distance from the notch tip. And then the
Raman shift changes to blue shift in the distance of about
1.0mm. From Eq. (1), it is impossible to separate the
elastic strain components, ƒÃ11, ƒÃ22 and ƒÃ33 from the single
Raman shift, ĢąA18, of A1g mode spectrum. In the present
study, the only strain component ƒÃ33 can be separated by
ƒÃ33= C1 ,645ƒ¢ƒÖ418-C1,418ƒ¢ƒÖ645/C1,645C2,418-C1,418C2,645
- 667Ģą418+90.3Ģą645,/435593(3)
where the unit of Raman shifts is cm-1. The separated strain of bent Specimen B corresponds to the strain com-
Fig. 7. Raman shift distribution Ģą418 of bent Specimen B.
297
Fig. 8. Strain distribution ƒÃy(x) of bent Specimen B
estimated from Ģą.
Fig. 9. Raman shift distribution Ģą418 of bent Specimen C.
ponent ƒÃy in the direction perpendicular to the notch face.
Figure 8 shows the strain distribution ƒÃy (x) estimated
by two Raman shifts of Specimen B. A large tensile strain
is observed in the close vicinity of the notch tip. Near the
notch tip, 0<x<0.4mm, the tensile strain shows a very
steep gradient. On the other hand, from 0.4mm to the
back side of the specimen, the strain ƒÃy changes linearly. It
can be concluded that the elastic strain distribution a in a
local area with about 2ƒÊm diameter can be estimated by
measuring two Raman shifts of A18 mode for Specimen B.
In a similar way, the distributions of remaining two elastic
strain components, ƒÃx and ƒÃz around the notch tip were
estimated from two measured Raman shifts, Ģą645 and
Ģą418, of Specimens C and D, respectively. Figures 9 and 10 indicate the Raman shift distribution
Ģą418 of bent Specimens C and D, respectively. In these
cases, a bending load of Specimen C is 132.1N and that of
Specimen D is 104.8N. Figure 11 shows the strain distri-
bution ƒÃx(x) estimated by two Raman shifts for Specimen
Fig. 10. Raman shift distribution Ģą418 of bent Specimen D.
Fig. 11. Strain distribution e (x) of bent Specimen C
estimated from Ģą.
estimated from Ģą.
Local Stress Measurement in Al2O3 by Raman Spectroscopy
C on the basis of Eq. (3). The estimated value of ƒÃx(x) is
very small at the notch root, where both 645 and 418cm-1
spectra hardly shift from the stress-free wave number,
ƒÖ0,A18. The tensile strain reaches the maximum of about
300•~10-6 at x=0.06mm and decreases monotonously to
zero with an increase in the distance from the notch tip.
On the other hand, Fig. 12 shows the strain distribution
ƒÃz(x) estimated for Specimen D. In the case of ƒÃz(x), the
compressive strain is observed near the notch tip and
almost zero elsewhere.
specimen were calculated by a three-dimensional finite
element analysis using a commercial program ANSYS.
Because of symmetry, one-fourth part of the SEVNB
specimen is modeled as shown in Fig. 13. The model is
divided with eight-node hexahedron and six-node trigonal
prism elements. The volume along the x-axis ahead of the
V-notch tip are subdivided to finer elements whose mesh
size is comparable to the laser beam size as shown in Fig.
14. In this model, the total number of elements and nodes
were 11920 and 15162, respectively. The 21 independent
constants of the elastic stiffness matrix [D] were substi-
tuted by anisotropic stiffness constants in accordance with
the crystallographic directions of each specimen. Table 3
summarizes the elastic stiffnesses used in the calculation
of each SEVNB specimen.
experimental data. The strain distributions in the SEVNB
specimen estimated by Raman microspectroscopy agree
very well with the results of the finite element analysis. In
particular, the complicated strain singularities near the
Fig. 13. Finite element model of one-fourth part of SEVNB specimen.
Fig. 14. Finer elements along the x-axis ahead of the V-notch tip.
notch tip were precisely detected by scanning the laser
beam with the diameter of 2ƒÊm. It is concluded that the
present system of Raman microspectroscopy is applicable
to the local stress measurement having a steep stress
gradient such as that in the stress field around the crack tip.
In the present study, the elastic strain fields around the
Table 3. Elastic stiffness, dij, of [D]-matrix
used in FEM analysis.
with the calculated strain distribution.
299
Fig. 16. Comparison of the estimated strains, ƒÃx(x) and
ƒÃz(x), with the calculated strain distributions.
Fig. 17. Change in the 432cm-1 Raman spectrum
with applied strain.
notch tip of a single crystal sapphire is estimated by Raman
microspectroscopy using two Raman shifts of A18 mode
spectra. On the other hand, each shift of remaining five
Raman spectra of E8 mode, ĢąEg, is expressed as [5, 6],
ƒ¢ƒÖ Eg=C3,ƒÖ(ƒÃ11+ƒÃ22)+C4,ƒÖƒÃ33
•}•ã{C5,ƒÖ(ƒÃ11-ƒÃ22)+C6,ƒÖƒÃ23}2+(2C5,ƒÖ12+C6 ,ƒÖƒÃ31)2, (4)
where ƒÃ12, ƒÃ23 and ƒÃ31 are shear strains of trigonal crystal,
and C5 ,ƒÖ, and C6,ƒÖ, are strain constants of each E8 mode spectrum. Figure 17 shows the transition of E8 mode
spectrum due to the applied strain. In the case of 432cm-1
spectrum, both the shift and splitting of Raman spectrum
are induced by the applied strain. It is seen from Eqs. (1)
and (4) that unknown stress and strain fields can be deter-
mined, in principle, by solving simultaneous equations
obtained for seven Raman spectra. However, it is difficult
to estimate experimentally the elastic stress or strain field,
ƒÐij or ƒÃij, using all the changes in seven Raman spectra,
because the wave shift and split of E8 mode spectrum can
not be precisely separated in the spectrum measured by a
charge coupled device camera. Future development of
local stress measurement using all seven Raman spectra is
required for a complete determination of unknown six
strain components.
4. CONCLUSION
(1) The shift of two Raman spectra, 645 and 418cm-1, of
A18 mode was proportional to the lattice strain,ƒÃ33, in the c-
axis of trigonal crystals. The proportional constant was
larger for 418cm-1 than for 645cm-1 spectrum.
(2) Under the application of four-point bending of single-
edge notched specimens, the distributions of loading
strains were measured across the minimum ligament of
the specimens by scanning the focused Ar+ laser beam. A
steep gradient of each strain near the notch root was
successfully detected by Raman microspectroscopy.
(3) The strain distributions of bent specimens were
calculated by a three-dimensional finite element analysis
with anisotropic elasticity. The distributions of strains
measured by Raman microspectroscopy agreed very well
with the calculated distributions. The developed system of
Raman microspectroscopy was applicable to the measure-
ment of a steep stress gradient near the crack tip in
ceramics.
Acknowledgment-This research sponsorship by AIST,
MITI, Japan, as a part of the Synergy Ceramics Project of
ISTF program is greatly appreciated.
REFERENCES
1. Y. Sakaida, A. Okada, K. Tanaka, Y. Yasutomi and H.
Kawamoto, Proc. 22nd Annual Conf. Composites, Ad-
vanced Ceramics, Materials, and Structures, Vol.B (ed. by
D. Bray), J. Am. Ceram. Soc., Cocoa Beach (1998) 169.
2. S. Tanaka, Proc. 8th JIM Int. Symposium Interface Science
and Materials Interconnection (ed. by Y. Ishida et al.), Japan
Inst. Met., Toyama (1996) 459.
3. G. Kolb, Th. Salbert, and G. Abstreitor, Fresenius J. Anal.
Chem., 341, (1991) 166.
4. V.J. Tekippe, A.K. Ramdas and S. Rodriguez, Phys. Rev.,
B 8-2 (1973) 706.
5. S.H. Shin, F.H. Pollak and P.M. Raccah, Proc. 3rd Int.
Conf. Light Scattering in Solids, Paris (1975) 401.
6. W. Jia and W.M. Yen, J. Raman Spectroscopy, 20 (1989)
785.
7. H. Sakata, G. Dresselhaus, M.S. Dresselhaus and M. Endo,
J. Appl. Phys., 63 (1988) 2769.
8. J.D. Belnap, J.F. Tsai and D.K. Shetty, J. Mater. Res., 9
(1994) 3183. 9. J.B. Wachtman, W.E. Tefft, D.G. Lam and R.P. Stinchfield,
J. Res. Natl. Bur. Std. -A. Phys & Chem., 64A (1960) 213.
10. H. Awaji and Y. Sakaida, J. Am. Ceram. Soc., 73 (1990) 3522.
11. Y. Takeda, N. Shibata, Y. Sakaida and A. Okada, Proc. 22nd
Annual Conf. Composites, Advanced Ceramics, Materials,
and Structures, Vol.A (ed. by D. Bray), J. Am. Ceram. Soc.,
Cocoa Beach (1998) 493.