Method and apparatus for evaluating strains in crystals

The strains in crystals are evaluated by quantifying the sharpness of an electron channeling pattern and determining changes in the quantified sharpness of the electron channeling pattern. There is such a close correlation between the sharpness of the electron channeling pattern and the strains in crystals that the latter can be evaluated in terms of changes in the former. An apparatus for evaluating strains in crystals comprises a scanning electron microscope having a function to form an electron channeling pattern and an image analyzer having a function to quantify the sharpness of an electron channeling pattern.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates to a method and apparatus for evaluating the strain 
in steel, silicon and other crystalline substances. 
2. Description of the Prior Art 
The strain in steel and other crystalline substances has conventionally 
been evaluated commonly by determining the spreading of diffracted beams 
of X-rays passed therethrough or determining their Vickers hardness. 
But X-ray diffraction analysis is unfit for the evaluation of strain in a 
very small area because beams of X-rays are difficult to focus to such a 
small area as under 10 .mu.m across. Another shortcoming is low 
sensitivity of the spreading of diffracted X-ray beam to the strain. 
Evaluation by Vickers hardness, on the other hand, takes advantage of a 
phenomenon that the hardness of a material varies with its dislocation 
density (or strain). But this method also does not provide very accurate 
evaluation because solid solutions and precipitates in crystals also 
affect the hardness of the material being examined. 
Electron Channeling Pattern (hereinafter abbreviated ECP) is a phenomenon 
discovered by D. G. Coates (D. G. Coates: Phil. Mag., 16 (1967), p. 1179). 
When electron beams are irradiated on a specimen of crystalline substance 
not too thin, part of the electrons having entered the specimen is 
elastically scattered while maintaining the incident energy because of the 
interaction with the constituent atoms of the crystal, with the rest being 
inelastically scattered losing the incident energy. Part of the incidence 
energy lost is used for the excitation of the electrons in the atoms 
making up the crystalline substance. Of the excited electrons, those 
emitted from the surface of the specimen are called secondary electrons. 
Of the inelastically scattered electrons, those emitted from the surface 
of the specimen are called back-scattered electrons. When the surface of 
the specimen is scanned with electron beams irradiated at varying angles, 
the intensity of the secondary and back-scattered electrons changes 
greatly because of the diffraction at the crystal plane in the vicinity of 
the Bragg angle .theta..sub.B at which the incident angle .theta. of the 
electron beams with respect to the crystal plane satisfies Bragg's law 
n.lambda.=2d sin .theta., wherein n is the order of reflection, .lambda. 
the wavelength of electron beams, and d the interval between crystal 
planes. On detecting the intensity of the secondary or back-scattered 
electrons and inputting the signal representing the detected intensity to 
a CRT display or other recording device synchronously with the scanning 
signal of the electron beams, an image with varying light and shade 
appears at and near the Bragg angle .theta..sub.B. Scanning electron 
microscopes are widely used to obtain ECP's. 
The ECP is known to provide information about the orientation and 
perfectness of crystals. Regarding the perfectness of crystals, it is 
known that the ECP blurs when crystals are strained (D. E. Newbury and H. 
Yakowitz: Practical Scanning Electron Microscopy, ed, by J. I. Goldstein 
and H. Yakowitz (1975), p. 149 [Plenum Press]). But the intricateness of 
the ECP and other factors have so far hampered the quantitative evaluation 
of strains. Though it is known that strains can be quantified using the 
contrast of specific pseudo-Kikuchi lines in the ECP as a parameter (Wear, 
1976, 40, p. 59), this method is inapplicable to the quantification of 
strains in an arbitrarily chosen crystal orientation. 
Evaluation of the nonuniform distribution of work-induced strain, the 
amount of strain accumulated in different crystal orientations and the 
residual strain resulting from softening is extremely important for the 
development of recrystallization and texture control technologies. As 
such, there is a pressing need for the development of a reliable 
micro-strain evaluation method based on the conventionally established 
principles. 
Noting a phenomenon that the ECP suddenly blurs when a micro-strain is 
applied on crystals, the inventors made an extensive analysis of ECP's 
using an image analyzer. The analysis led to a new discovery that image 
analysis of the ECP is very effective for the evaluation of the 
microstrain in such crystalline substances as steel. 
SUMMARY OF THE INVENTION 
The object of this invention is to provide a method and apparatus for 
evaluating crystal strains that overcome the conventional difficulty in 
the quantitative evaluation of micro-strains in crystals. 
A method of evaluating crystal strains according to this invention 
comprises quantifying the sharpness of the ECP and determining the 
magnitude of crystal strain from a change in the quantified sharpness of 
the ECP. Here, the sharpness of the ECP means the contrast between the 
dark and bright areas in an overall image of the ECP quantified. Because 
of the close correlation between them, crystal strain can be evaluated by 
determining the sharpness of the ECP. 
The ECP can be obtained by use of, for example, a scanning electron 
microscope having an ECP forming function. 
The sharpness of the ECP can be quantified by several methods. One of them 
differentiates (by unidirectional differentiation, Sobel method, etc.) the 
light and shade across the ECP with respect to length (or distance) using 
an image analyzer and determines the ratio of an area showing higher 
derivatives to the entire area of the ECP. Another method uses the highest 
derivative as the criterion of sharpness. Still another method uses the 
difference between the brightest and darkest portions of the ECP as the 
criterion of sharpness. This invention can quantify the sharpness of the 
ECP by any of these conventional methods. 
The crystal strain may also be evaluated by determining the ratio of an 
area in which the light and shade varies greatly to the entire area of the 
ECP. The ECP is an extremely complex image in which waves diffracted by 
all planes of crystals are shown at a time, with a strain blurring 
individual diffracted waves differently. Thus, it is effective to 
determine the ratio of an area in which the light and shade changes beyond 
a given threshold limit and evaluate the average sharpness of the 
diffracted waves appeared. The higher the threshold limit, the smaller 
will be the chosen area (a sharp portion of the ECP). On the other hand, 
more noise will be involved if the threshold limit is lowered. Therefore, 
a threshold limit that permits a large enough sharp area to be chosen 
while keeping the noise level at a satisfactorily low level should be 
determined experimentally. The threshold limit must of course be chosen 
from within a proper range. A threshold limit chosen from within a proper 
range assures an accurate relative evaluation of strains. If the 
best-suited threshold limit is chosen for each individual application, the 
accuracy of crystal strain evaluation will be increased. Changes in the 
light and shade may be determined by any of the conventional methods such 
as unidirectional differentiation and the Sobel method with an image 
analyzer. 
A crystal strain evaluation apparatus of this invention comprises a 
scanning electron microscope with an ECP forming function and an image 
analyzer having a function to quantify the sharpness of an ECP. Any type 
of image analyzer may be used so far as it has the required ECP sharpness 
quantifying function. ECP images produced by a scanning electron 
microscope may be input into an image analyzer either by directly sending 
electric signals representing an ECP image from the scanning electron 
microscope to the image analyzer or inputting into the image analyzer 
image signals made by a television camera that picks up an image produced 
by the scanning electron microscope (either as an image on a CRT display 
or in the form of a photograph thereof). 
The method and apparatus of this invention permit evaluating micro-strains 
in crystals by quantifying the sharpness of ECP's. This feature permits 
accurate evaluation of residual strains in hot-rolled steel plates and 
those resulting from the softening processes (recovery and 
recrystallization). The ECP provides information about the perfectness of 
crystals in a surface region of not more than 500 .ANG. in depth and 
approximately 3 .mu.m across. Therefore, it can be used in the 
investigation and analysis of nonuniform deformation in working, 
difference in accumulated strains due to the crystal orientation and the 
nucleation site of recrystallization. As the strain can be construed as 
representing the amount of lattice defects, this invention is also 
applicable to the investigation of defects in semiconductors etc. 
ascribable to lattice defects and to the detection of strains in silicon 
and other semiconductor materials.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 1 shows a crystal strain evaluating apparatus of this invention. 
The crystal strain evaluating apparatus comprises a scanning electron 
microscope 11 and an image analyzer 41. The scanning electron microscope 
11 is an ordinary electron microscope having an ECP forming function. At 
the top of a microscope tube 12 is provided an electron gun 13 connected 
to an acceleration high-voltage power supply 16. A focusing lens 18 is 
provided directly under the anode of the electron gun 13, with an 
objective lens 20 at the bottom of the tube 12. The focusing lens 18 and 
objective lens 20 are connected to a lens power supply 22. The focusing 
lens controls the amount and spot size of electrons reaching a specimen 3. 
The objective lens 20 performs image focusing. Between the focusing lens 
18 and objective lens 20 is disposed a deflecting coil 24. To the 
deflecting coil 24 is connected a scanning signal generator 27 through a 
multiplier 26. Comprising an X-and Y-direction scanning coils, the 
deflecting coil 24 deflects the electron beams 1 on signals from the 
scanning signal generator 27, thus two-dimensionally moving the incident 
angle of the electron beams 1 reaching the surface 4 of the specimen. A 
blanking circuit 29 is connected to the scanning signal generator 27. The 
lower end of the tube 12 communicates with a specimen chamber 31. A 
specimen stand 33 is provided in the specimen chamber 31, with a detector 
35 disposed to face the specimen stand 33. The detector 35 detects the 
amount or intensity of secondary electrons emitted from the surface 4 of 
the specimen on irradiation of electron beams 1 and convert the intensity 
into an electrical signal. For detecting back-scattered electrons, the 
detector is disposed around a hole through which electron beams are sent 
into the specimen chamber 31. The detector 35 is connected to a CRT 
display 37 through an amplifier 36. The blanking circuit 29 is connected 
to the electron gun 38 of the CRT display 37 and the scanning signal 
generator 27 is connected to a deflecting coil 39. The signals resulting 
from the detection by the detector 35 are fed into the CRT display 37 in 
synchronism with the scanning of the surface 4 of the specimen by the 
electron beams 1. 
The image analyzer 41 comprises an A-D (analog-to-digital) converter 42 to 
convert the analog signals from the detector 35 of the scanning electron 
microscope 11 into digital signals and a central processing unit 46 
connected to the A-D converter 42 through an image memory 44. To the 
central processing unit 46 is connected a main memory 48, an image 
processor 50, a CRT display 52 and a printer 54. The scanning signal 
generator 27 of the scanning electron microscope 11 is connected to the 
A-D converter 42, image memory 44 and central processing unit 46. The 
image memory 44 serves as a buffer when feeding the image signal from the 
A-D converter 42 into the main memory 48. The image processor 50 performs 
noise elimination, edge detection and other image processing. 
The following paragraphs describe how the apparatus just described analyzes 
the strain in crystals. 
As schematically shown in FIG. 2, the surface 4 of a specimen 3 of a 
crystalline substance is two-dimensionally scanned with electron beams 1 
directed to a fixed point of impact P, with the angle of incidence .theta. 
rocked. While doing angular scanning by varying the incidence angle 
.theta. of the electron beams 1 in plane a, the detector detects secondary 
or back-scattered electrons. When the angular scanning in plane a is over, 
the same process is repeated in planes b, c and so on. The straight lines 
R in imaginary scanning plane Q correspond to the scanning angles in 
individual planes a, b, c and so on. Scanning is performed so that the 
incidence angle .theta. of the electron beams 1 with respect to the 
surface 4 of the specimen ranges from an angle smaller than Bragg angle 
.theta..sub.B to a larger one. The intensity of the secondary or 
back-scattered electrons detected by the detector is converted into an 
electric signal which is fed into the CRT display 37 in synchronism with 
the scanning signal of the electron beams 1. The intensity of the 
secondary or back-scattered electrons varies greatly within a narrow 
angular range on both sides of the Bragg angle .theta..sub.B. When the 
signal is reproduced on the CRT display 37, therefore, a pattern of bands 
and lines appear thereon. FIG. 3a schematically shows an ECP reproduced on 
the CRT display 37. For simplicity, linear patterns are omitted in FIG. 
3a. The ECP is made up of several differently oriented bands because the 
electron beams having entered crystals are reflected on many crystal 
planes whose orientations are different from the angle of incidence. 
Meanwhile, the A-D converter 42 converts the signals from the detector 35 
into digital signals that are then sent to the main memory 48 through the 
image memory 44. An instruction from the central processing unit 46 calls 
out an image data from the main memory 48 to the image processor 50 where 
image processing is carried out according to a flow chart in FIG. 4. 
First, noise elimination, linear conversion, contrast accentuation and 
other gray image processing are performed. The gray levels in linear 
conversion ranges from 0 to 255. Following this gray image processing, the 
light and shade of the image is differentiated in the direction of 
scanning. FIG. 3b schematically shows an image formed by outputting the 
absolute value of the derivative. The light and shade of the pattern 
varies greatly near the edge of the bands shown in FIG. 3a (because the 
intensity of the secondary or back-scattered electrons varies greatly 
within a narrow angular range on both sides of the Bragg angle 
.theta..sub.B). Therefore, paired parallel bright lines appear along the 
edges of each band. After making another gray image processing, 
segmentation is carried out. After eliminating isolated points and 
applying other necessary processing, the extraction width of each line on 
the binary image is made equal. Then the ratio of the area of the bright 
lines to the entire area of the ECP is determined, which is then used as 
the criterion for the evaluation of crystal strains. Assume, for example, 
that lines A and B in FIG. 3b are chosen as the bright portions as a 
result of segmentation. After the width of both lines is made equal, the 
sum of the product of the length of line A multiplied by its width and the 
product of the length of line B multiplied by its width is divided by the 
whole area of the ECP to determine the parameter for evaluating crystal 
strains. The parameter thus determined is output by the printer 54. The 
light and shade of an ECP corresponds to the intensity of the secondary or 
back-scattered electrons detected by the scanning electron microscope 11, 
the brightness of the ECP reproduced on the CRT display 37 or the light 
and shade on a photograph of the ECP. Anyway, the image analyzer 41 
processes the light and shade or sharpness of an ECP as electric signals. 
The mechanism by which crystal strains can be evaluated through the 
quantification of the sharpness of an ECP is not definitely clear. But the 
inventors' interpretation is as follows: The lines on an ECP represent the 
beams diffracted from various crystal planes. When crystals deform, 
therefore, the diffracted lines blur, reducing the sharp contrast between 
the light and shade on the ECP. Presumably, therefore, the strain in 
crystals can be evaluated by determining the area ratio of a portion in 
which the sharpness exceeds a given threshold limit. Now the results of 
some experiments conducted on the basis of this invention will be 
described in the following: 
The experiments were conducted by making image analysis according to the 
flow chart shown in FIG. 4 using a crystal strain evaluating apparatus 
comprising a scanning electron microscope (JSM-840) and an image analyzer 
(TOSPIX-II). 
EXPERIMENT I 
FIG. 5 shows the relationship among the true strain in cold rolling 
(.epsilon.), cold reduction ratio (CR), sharpness of an ECP (S/S.sub.O) 
and half-value width of the (200) diffracted X-rays (FWHV). In this 
experiment, 0.340 mm thick annealed steel sheets containing 3.25% of 
silicon (with crystal grain size ranging from 10 .mu.m to 50 .mu.m) were 
cold-rolled. Changes in the sharpness of ECP's resulting from cold rolling 
were quantified with the image analyzer. The acceleration voltage of the 
electron beams was 35 KeV, irradiation current 6.times.10.sup.-9 A, 
magnification 50, operating distance 8 mm and rocking angle .+-.8.degree.. 
Measurement was made at the center of the thickness of each sheet. The 
points plotted in FIG. 5 are the averages of the measurements made at 
fifty each points (i.e., five points in the direction of cold rolling 
times ten points in the direction perpendicular thereto, the individual 
points being spaced at intervals of 1 mm). S represents the ratio of an 
area in which the light and shade of an ECP varies greatly, whereas 
S.sub.O denotes the value of S under an unstrained condition. As is 
obvious from FIG. 5, the sharpness of an ECP (S/S.sub.O) is so sensitive 
to the presence of micro-strains that it can evaluate micro-strains more 
effecitively than the conventional methods (such as the measurement of the 
half-value width of diffracted beams). An experimental strain-ECP 
sharpness (S/S.sub.O) curve as shown in FIG. 5 must be prepared in 
advance. Then, the magnitude of the strain in a specimen under analysis 
can be determined by comparing the measured ratio S/S.sub.O of the 
specimen with that reference curve. 
EXPERIMENT II 
Annealed steel sheets having a thickness of 0.340 mm and containing 3.25% 
silicon were cold rolled with a cold reduction ratio of 0 to 15 percent. 
With each specimen thus prepared, the ratio of an area (S) in which the 
light and shade varies greatly to the whole area of the ECP was determined 
as shown in a histogram in FIG. 6. The ECP measurement was made at fifty 
points (five points in the direction of cold rolling times the points in 
the direction perpendicular thereto, the individual points being spaced at 
intervals of 1 mm) at the center of the thickness of each sheet. .epsilon. 
is the true strain. The acceleration voltage of the electron beams was 35 
KeV, irradiation current 6.times.10.sup.-9 A, magnification 50, operating 
distance 8 mm and rocking angle .+-.8.degree.. 
FIG. 6 shows that internal strains developed in the specimens as a result 
of cold rolling. 
EXPERIMENT III 
Hot-rolled silicon steel plates, 2.3 mm in thickness, containing 0.054% of 
carbon and 3.23% of silicon were treated under the following three 
different conditions: (1) Held at 1150.degree. C. for 30 seconds, slowly 
cooled to 900.degree. C. in 2 minutes, and then air-cooled; (2) Held at 
900.degree. C. for 4 minutes and then air-cooled; and (3) Without 
annealing. With each specimen thus prepared, ECP measurement was made (a) 
at the center of the thickness of the plate and (b) at a depth 1/4 of the 
thickness away from the surface of the plate. The ratio of the area S in 
which the light and shade varies greatly to the whole area of the ECP was 
determined as shown in a histogram in FIG. 7 (plotting the measurements at 
the center of the thickness) and FIG. 8 (plotting the measurements at 1/4 
of the thickness). The ECP measurement was made at fifty points (five 
points in the direction of cold rolling times ten points in the direction 
perpendicular thereto, the individual points being spaced at intervals of 
1 mm; but ECP measurement was not made at a point where carbide was found 
and the number of measuring points in the direction of cold rolling was 
increased so that the total number always remains 50). The acceleration 
voltage of the electron beams was 35 KeV, irradiation current 
6.times.10.sup.-9 A, magnification 50, operating distance 8 mm and rocking 
angle .+-.8.degree.. 
As is obvious form FIGS. 7 and 8, work strains in the rolled steels 
decreased as a result of annealing. 
EXPERIMENT IV 
An ingot of silicon single crystal having a composition shown in Table 1 
and growing in the &lt;100&gt; direction was sliced in the direction 
perpendicular to the direction of growth into 0.7 mm thick wafers. 
The wafers were subjected to donor-killing annealing to eliminate oxygen at 
650.degree. C. for 20 minutes and reduced to have an ultimate thickness of 
0.6 mm by mirror polishing the top surface and lapping the under surface. 
The mirror-finished surface was damaged by a sand-blasting applied under 
the conditions shown in Table 2. The acceleration voltage of the electron 
beans was 8 KeV, irradiation current 6.times.10.sup.-9 A, magnification 
50, operating distance 8 mm and rocking angle .+-.8.degree.. 
As is obvious from FIG. 9, the ratio of the area (S) in which the light and 
shade varies greatly to the whole area of the ECP corresponds to the 
intensity of the damage caused by sand-blasting, thus serving as a useful 
parameter in the evaluation of strains. 
TABLE 1 
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Oxygen Concentration 
Carbon Concentration 
______________________________________ 
9.5 .times. 10.sup.17 atoms/cm.sup.3 
1.5 .times. 10.sup.17 atoms/cm.sup.3 
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TABLE 2 
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Specimen 
Blasting Pressure (kgf/cm.sup.2) 
Particles Blasted 
______________________________________ 
0 SiO.sub.2 
2 1.4 3.4.about.3.8 .mu.m 
3 2.0 in diameter 
4 3.0 
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Involving all pseudo-Kikuchi lines, the method of this invention can be 
applied to the determination of strains in all crystallographic 
orientations. By contrast, the parameter employed by the conventional 
methods is the relative intensity of a pseudo-Kikuchi line corresponding 
to a specific crystal plane, such as, for example, the one in the (200) 
plane. Generally, not all pseudo-Kikuchi lines appear on a displayed ECP. 
The kind and number of pseudo-Kikuchi lines appearing vary with the 
crystallographic orientation of each measuring point. As such, the 
applicability of the conventional methods employing only specific 
pseudo-Kikuchi lines has been limited to certain crystallographic 
orientations. 
This invention is by no means limited to the specific embodiments 
described. In feeding the signals representing an ECP into the image 
analyzer 41 in the apparatus shown in FIG. 1, for example, the pattern on 
the CRT display 37 of the scanning electron microscope 11 may be recorded 
on a video tape. The resulting signals are then fed into the A-D converter 
42 of the image analyzer 41. In this case, the A-D converter 42, image 
memory 44 and control processing unit 46 are adapted to work in 
synchronism with the synchronizing signals from the videotape recorder. By 
the use of the SACP (selected area electron channeling pattern), crystal 
strains in as small an area as about 1 .mu.m can be determined, too. An 
SACP can be obtained by, for example, bringing the cross-over point of the 
scanning electron beams 1 close to the surface of the specimen by turning 
off the lower one of the two deflecting coils one placed over the other. 
The technology according to this invention evaluates the strains in 
crystals by quantifying the sharpness of the ECP. And technologies to 
quantify the sharpness of other similar improved patterns, such as the 
EBSP (electron back scattering pattern, J. A. Venables and C. J. Harland: 
Phil. Mag., 27 (1973), 1193), should be construed to come within the scope 
of this invention.