ASTM E81-96(2024)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E81 − 96 (Reapproved 2024)
Standard Test Method for
Preparing Quantitative Pole Figures
This standard is issued under the fixed designation E81; the number immediately following the designation indicates the year of original
adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A superscript
epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 1.8 This standard does not purport to address all of the
safety concerns, if any, associated with its use. It is the
1.1 This test method covers the use of the X-ray diffracto-
responsibility of the user of this standard to establish appro-
meter to prepare quantitative pole figures.
priate safety, health, and environmental practices and deter-
1.2 The test method consists of several experimental proce-
mine the applicability of regulatory limitations prior to use.
dures. Some of the procedures (1-5) permit preparation of a
1.9 This international standard was developed in accor-
complete pole figure. Others must be used in combination to
dance with internationally recognized principles on standard-
produce a complete pole figure.
ization established in the Decision on Principles for the
1.3 Pole figures (6) and inverse pole figures (7-10) are two Development of International Standards, Guides and Recom-
mendations issued by the World Trade Organization Technical
dimensional averages of the three-dimensional crystallite ori-
entation distribution. Pole figures may be used to construct Barriers to Trade (TBT) Committee.
either inverse pole figures (11-13) or the crystallite orientation
2. Summary of Test Method
distribution (14-21). Development of series expansions of the
2.1 The test method consists of characterizing the distribu-
crystallite orientation distribution from reflection pole figures
tion of orientations of selected lattice planes with respect to
(22, 23) makes it possible to obtain a series expansion of a
sample-fixed coordinates (6). The distribution will usually be
complete pole figure from several incomplete pole figures. Pole
obtained by measurement of the intensity of X rays diffracted
figures or inverse pole figures derived by such methods shall be
by the sample. In such measurements the detector and associ-
termed calculated. These techniques will not be described
ated limiting slits are fixed at twice the appropriate Bragg
herein.
angle, and the diffracted intensity is recorded as the orientation
1.4 Provided the orientation is homogeneous through the
of the sample is changed (1-6, 25, 26, 27). After the measured
thickness of the sheet, certain procedures (1-3) may be used to
data have been corrected, as necessary, for background,
obtain a complete pole figure.
defocusing, and absorption, and normalized to have an average
1.5 Provided the orientation has mirror symmetry with
value of unity, the results may be plotted in stereographic or
respect to planes perpendicular to the rolling, transverse, and
equal-area projection.
normal directions, certain procedures (4, 5, 24) may be used to
2.2 The geometry of the Schulz (25) reflection method is
obtain a complete pole figure.
illustrated in Fig. 1. Goniometers employing this geometry are
1.6 The test method emphasizes the Schulz reflection tech-
commercially available. The source of X rays is indicated by L.
nique (25). Other techniques (3, 4, 5, 24) may be considered
Slit S1 limits divergence of the incident beam in the plane of
variants of the Schulz technique and are cited as options, but
projection. Slit S2 limits divergence perpendicular to the plane
not described herein.
of projection. The sample, indicated by crosshatching, may be
tilted about the axis FF', which is perpendicular to the
1.7 The test method also includes a description of the
diffractometer axis and lies in the plane of the sample. The tilt
transmission technique of Decker, et al (26), which may be
angle was denoted φ by Schulz (25). The sample position
used in conjunction with the Schulz reflection technique to
shown in Fig. 1 corresponds to φ = 0 deg, for which approxi-
obtain a complete pole figure.
mate parafocusing conditions exist at the detector slit, S3. With
the application of a defocusing correction, this method is useful
over a range of colatitude φ from 0 deg to approximately 75
This test method is under the jurisdiction of ASTM Committee E04 on
deg.
Metallography and is the direct responsibility of Subcommittee E04.11 on X-Ray
2.2.1 Tilting the sample about FF ', so as to reduce the
and Electron Metallography.
Current edition approved April 1, 2024. Published April 2024. Originally
distance between L and points in the sample surface above the
approved in 1949. Last previous edition approved in 2017 as E81 – 96 (2017). DOI:
plane of projection, causes X rays diffracted from these points
10.1520/E0081-96R24.
to be displaced to the left of the center of S3, while X rays
The boldface numbers in parentheses refer to the list of references at the end of
this test method. diffracted from points in the sample surface below the plane of
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E81 − 96 (2024)
μt and θ. If, for example, I /I is restricted to values ≥ 0.5, one
α 0
arrives at the series of curves shown in Fig. 3.
3. Significance and Use
3.1 Pole figures are two-dimensional graphic
representations, on polar coordinate paper, of the average
distribution of crystallite orientations in three dimensions. Data
for constructing pole figures are obtained with X-ray
diffractometers, using reflection and transmission techniques.
3.2 Several alternative procedures may be used. Some
produce complete pole figures. Others yield partial pole
figures, which may be combined to produce a complete figure.
4. Apparatus
FIG. 1 Geometry of Reflection Method.
4.1 Source of X Rays—A beam of characteristic X rays of
substantially constant intensity is required. Characteristic Ka-
lpha radiation of chromium, iron, cobalt, nickel, copper,
projection are displaced to the right of the center of S3. The
molybdenum, and silver have all been used successfully,
displacement is equal to 2D tan φ cos θ, where D is the distance
depending on the chemical composition of the specimen.
above or below the plane of projection. The integrated, or total,
Insofar as possible, the radiation selected shall provide suffi-
diffracted intensity is influenced only slightly by tilting the
cient angular dispersion to permit the resolution of peaks to be
sample (28). Insofar as possible, the detector slit shall be of
measured, and shall not produce excessive fluorescence in the
sufficient width to include the defocused line profile corre-
sample. Linear absorption coefficients (29) for selected ele-
sponding to the maximum sample tilt for which measurements
ments are given in Table 2. Lower energy radiation (Cr, Fe, Co,
are to be made. Because of interferences from neighboring
Ni, Cu) is generally preferred for reflection pole figure mea-
diffraction peaks and physical limitations on sample size and
surements as it provides greater angular dispersion. Higher
detector slit width, it is necessary to limit vertical divergence of
energy radiation (Mo, Ag) is generally preferred for transmis-
the incident beam. A widely used pole figure goniometer with
sion measurements.
a focal spot to the center of the sample distance of 172 mm
employs a 0.5-mm slit located 30 mm from the center of the
4.2 Slits—Suitable slits shall be provided to limit horizon-tal
sample for this purpose. Measured intensities may be corrected
(in the plane of projection of Figs. 1 and 2) and vertical
for defocusing by comparison with intensities diffracted by a
(perpendicular to the plane of projection of Figs. 1 and 2)
randomly oriented specimen of similar material, or byemploy-
divergence of the incident beam. Horizontal divergences of 1 to
ing the theoretically calculated corrections (28).
3 deg for reflection and 0.5 deg for transmission are typical.
Vertical divergences of 0.2 deg for reflection and 1 deg for
2.3 The geometry of the transmission technique of Decker,
transmission are typical. Insofar as possible, the receiving slit
et al (26) is shown in Fig. 2. In contrast to the reflection
shall be of sufficient width to include the diffracted peak.
method, X rays diffracted from different points in the sample
Receiving slits corresponding to 1 deg 2−theta are typical.
diverge, making the resolution of adjacent peaks more difficult.
The ratio of the diffracted intensity at α = −5, −10,··· , −70 deg,
4.3 Specimen Holder—Reflection Method:
to the diffracted intensity at α = 0 deg, calculated in accordance
4.3.1 The specimen holder for the reflection method shall
with the expression given by Decker, et al (26) for linear
preferably employ the Schulz reflection geometry illustrated in
absorption thickness product, μt, = 1.0, 1.4, ···, 3.0, and, for
Fig. 1 and described in 2.2. It is desirable that the specimen
θ = 5, 10,··· , 25 deg is given in Table 1. These data may be
holder be equipped with a means for oscillating the sample in
used as a guide to determine the useful range of α for a given
the plane of its surface without changing the orientation of the
sample. It is also desirable that the magnitude of the oscillation
be variable. The specimen holder shall preferably be provided
with automatic means for changing colatitude and longitude of
the sample.
4.3.2 Alternative reflection geometries include those of
Bakarian (1), Field and Marchant (27), and Jetter and Borie (2).
The method of Bakarian requires machining a number of
cylindrical specimens whose axes are perpendicular to the
sheet normal direction. Each specimen provides intensity data
along one parallel of longitude. The method of Jetter and Borie
entails the preparation of a spherical specimen. In the methods
of Bakarian and of Jetter and Borie, the sample shall, insofar as
possible, be prepared from homogeneous material. These
methods have the advantage that intensity data need not be
FIG. 2 Geometry of Transmission Method. corrected for absorption or defocusing. They do not permit
E81 − 96 (2024)
TABLE 1 (I /I ) × 1000
α 0
−α
θ
μt 5 10 15 20 25 30 35 40 45 50 55 60 65 70
5 1.0 992 984 976 966 954 939 918 890 851 796 703 617 480 313
1.4 991 978 962 941 915 882 840 786 719 636 533 412 277 146
1.8 989 972 948 917 878 828 768 695 608 508 395 276 162 070
2.2 988 966 935 893 842 778 702 614 515 406 294 186 095 034
2.6 986 960 922 871 807 731 643 544 436 326 219 126 057 017
3.0 985 954 909 849 775 687 589 481 370 261 164 086 034 009
10 1.0 984 969 952 934 912 887 855 815 762 694 603 486 344 191
1.4 983 962 938 908 873 831 779 716 640 548 440 320 198 094
1.8 981 956 924 884 836 779 710 630 538 435 325 215 119 049
2.2 980 950 911 861 801 730 649 556 455 348 242 147 074 027
2.6 978 944 898 839 768 686 593 492 385 280 183 103 047 016
3.0 977 938 885 817 737 644 543 436 328 226 139 073 030 009
15 1.0 976 952 927 900 868 832 789 735 668 583 477 349 209 085
1.4 975 946 912 874 829 776 714 640 553 453 342 227 123 046
1.8 973 939 898 850 792 725 648 560 462 358 252 155 078 027
2.2 972 933 885 826 758 678 590 492 389 286 190 110 052 017
2.6 970 927 872 804 725 636 538 435 331 232 146 080 036 011
3.0 968 921 859 783 695 597 493 386 283 190 115 060 025 007
20 1.0 968 935 901 863 822 774 718 649 566 465 345 214 093 000
1.4 966 928 885 836 781 717 643 557 460 354 243 140 058 000
1.8 964 921 870 811 743 666 579 484 381 278 180 099 039 000
2.2 963 915 857 788 709 621 525 424 321 224 139 074 028 000
2.6 961 909 843 766 678 582 479 375 274 185 111 057 020 000
3.0 960 903 831 746 650 547 440 335 238 155 090 044 015 000
25 1.0 959 917 872 824 771 710 639 555 455 339 214 096 000
1.4 957 909 856 796 728 651 565 468 362 253 151 065 000
1.8 955 902 840 770 690 602 505 402 298 200 115 048 000
2.2 953 895 826 746 657 560 456 352 253 164 092 038 000
2.6 952 889 812 724 627 523 417 314 219 139 076 031 000
3.0 950 883 800 705 601 493 384 283 194 121 065 025 000
4.5 Detector—The detector shall preferably be of an energy-
dispersive type, for example, a solid state, proportional, or
scintillation counter, and used in conjunction with a pulse
height selector circuit to discriminate against X rays whose
energies differ markedly from that of the characteristic K-alpha
radiation being used. Reduction of the characteristic K-beta
radiation requires the use of a monochromator or appropriate
beta filter. Pd, Zr, Ni, Co, Fe, Mn, and V are appropriate beta
filters for Ag, Mo, Cu, Ni, Co, Fe, and Cr, respectively.
5. Test Specimens
5.1 For the reflection method, the sample shall be of
sufficient thickness that loss of intensity due to transmission
FIG. 3 α versus μt for I /I = 0.5, θ = 5, 10, ···, 25 deg.
α 0
through the sample may be ignored. If a maximum loss of 1 %
the incident beam is acceptable, the specimen must have a
linear absorption thickness product equal to or greater than 2.3
oscillation of the sample. Equipment is not currently commer-
sin θ. For an iron sample with molybdenum K-alpha radiation,
cially available for these methods.
this requires that μt be greater than 0.4, 0.6, and 0.7 for the
4.3.3 The method of Field and Marchant (27) requires an
(110), (200), and (211) reflections, respectively.
absorption correction. If this method is used in conjunction
5.1.1 Surface preparation is particularly important in the
with the transmission method of Decker, et al (26), it is
reflection method. Calculations due to Borie (30), who as-
necessary to use either different orders of reflection or different
sumed a sawtooth surface of spacing a on a material with linear
radiations in order to obtain a complete pole figure.
absorption coefficient μ, indicate that the product μa should be
4.4 Specimen Holder—Transmission Method—If the trans-
less than 0.5 if significant intensity losses are to be avoided.
mission method is used, the specimen holder shall employ the
For an iron sample with cobalt K-alpha radiation, μ = 416
geometry of Decker, et al (26), shown in Fig. 2 and described
−1
cm , corresponding to a ≤ 12 μm.
in 2.3. It is desirable that the specimen holder be equipped with
a means for oscillating the sample in the plane of its surface 5.2 For the transmission method, maximum intensity is
without changing the orientation of the sample. The specimen obtained for a linear absorption thickness product equal to cos
holder shall preferably be providedwith automatic means for θ. For an iron sample with molybdenum
...