Revised Grain-Scattering Formulas and Tables

THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 37, NUMBER 4 APRIL 1965

Revised Grain-Scattering Formulas and Tables

EMMANUEL P. PA?ADAKIS

Bell Telephone Laboratories, Inc., Allentown, Pennsylvania (Received 30 November 1965)

The current theory of Rayleigh and stochastic scattering in polycrystalline materials is reviewed and compared with (1) former theory and (2) experiment. Rayleigh scattering giving ultrasonic attenuation equal to STf 4 (S is the Rayleigh scattering factor, T the average scattering volume, f the frequency) occurs when X > 2,r• (X is the wavelength, D the average grain diameter); stochastic scattering yielding ultrasonic attenuation equal to •Df 2 (• is the stochastic scattering factor) occurs when X <2•r/5. The average scattering volume and average grain diameter must be evaluated by taking their averages over the grain-size distribution in the metal. When this is done, the current theory accounts rather well for the scattering component of the ultrasonic attenuation in polycrystalline metals. Former theory underestimated the scattering. A tabulation is made of the scattering factors S and E in various materials. The computed scattering factors show that polycrystalline samples of the following materials should have low attenuation' aluminum, chromite, chromium, magnesium, magnetite, silicon, strontium nitrate, tungsten, vanadium, and YIG.

INTRODUCTION

INCE the advent of RADAR and the application of its ranging methods to mechanical radiation in ultrasonic pulse-echo measurements in the megacycle- per-second region, the problem of scattering of ultrasonic waves by grains in metals has been of interest. Early workers •-ø showed that scattering by grains con- tributed a large part of the ultrasonic attenuation of polycrystalline metals and that the functional dependence of the scattering on frequency f and grain diameter D varied with the ratio of wavelength X to grain diameter. The functions are tabulated in Table I.

TA•L•. I. Functional

dependence of scattering on D and f as the ratio X//5 is varied.

Type of scattering Formula

>> 1.0 Rayleigh Daft •1.0 Stochastic Dff <1.0 Diffusive

• W. P. Mason and H. J. McSkimin, J. Acoust. Soc. Am. 19, 464-473 (1947).

2 C. L. Pekeris, Phys. Rev. 71, 268-269 (1947). a W. P. Mason and H. J. McSkimin, J. Appl. Phys. 19, 940-946

(1948). 4 W. Roth, J. App!. Phys. 19, 901-910 (1948). 5 H. B. Huntington, J. Acoust. Soc. Am. 22, 362-364 (1950). 6 R. L. Roderick and R. Truell, J. Appl. Phys. 23, 267-279

(1952).

An attenuation mechanism linear in frequency was also found •.4 and was attributed • to internal friction

found previously • in kilocycle-per-second resonator measurements. In subsequent work, many authors s-n have expressed the attenuation in the various regions of scattering (particularly the Rayleigh region) as the sum of a term linear in frequency and the proper term from Table I. More recently, a term proportional to the square of the frequency has been introduced to account for dislocation damping. • This term is particularly strong during cyclic stress approaching fatigue TM but also can be present in dormant metals. •4

In the early work, the expressions derived for the attenuation from Rayleigh scattering by Mason • and Mason and McSkimin a were qualitatively correct but omitted mode conversion. These formulas give results differing up to two orders of magnitude from the results

• R. L. Wegel and H. Walther, Physics 6, 141-157 (1935). 8 K. Kamigaki, T6hoku Univ. Res. Inst. Sci. Rept. RITU A9,

48-77 (1957). 0 T. Hirone and K. Kamigaki, T6hoku Univ. Res. Inst. Sci.

Rept. RITU A10, 276-282 (1958). •0 L. G. Merkulov, Soviet Phys.--Tech. Phys. 1, 59-69 (1956)

[-Transl. of Zh. Techn. Fiz. 26, 64-75 (1956)•. n E. P. Papadakis and E. L. Reed, J. Appl. Phys. 32, 682-687

(1961). •= A. Granato and K. Lticke, J. Appl. Phys. 27, 583-593, 789-

805 (1956). •a A. Granato, A. Hikata, and K. Liicke, Acta Met. 6, 470-480

(1958). • E. P. Papadakis, J. Appl. Phys. 35, 1474-1482 (1964).

703

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704 E. P. PAPADAKIS

of more-recent equations. The new expressions are reviewed and their predictions compared with experiment in the next two sections.

I. CURRENT THEORY ON RAYLEIGH SCATTERING

Merkulov 1ø has expanded upon the work of Lifshits and ParkhomovskiP 5 to formulate the attenuation a due

to Rayleigh scattering in hexagonal as well as cubic 15 metals. The formulas for longitudinal (0 and transverse waves (t) are presented here in neper/unit length. The requisite ratio of wavelength to grain diameter is X> 2•rD.

A. Cubic Crystals

8•rat•2Tf4 2+2 2•ra•2T f4 { 2 + 3 ] ae=----! 1' at ..... , (1) 37502vd [vt • ,?J ' 125p2vta vg s vt •

with

t1=611--612--2644, (2)

where the cq are the elastic moduli of the cubic crystallites, T is the average crystallite volume, f is frequency, p is density, and the v's are ultrasonic velocities.

B. Hexagonal Crystals

450p2vt •[vt • vt •1 450p2vt •[vt • vt •] with

88 80 128 320

15 3 3 3

82 272 112 b 1 = •2 + 30x2 +_•v2_Jr_ 30x3' +---•v q- 80xv,

15 3 3

41 136 56 a2-- •2-1-15X2-1---•r/2-1-10x-•+•vv+ 40xv,

15 3 3

(3)

(4)

8

b •. = _,y2_jr_ 28v 2-Jr- 8'yv 5

where '= ell"{- C33-- 2 (613"3- 2644),

X- c1•-- c12, and v- c44+ (c12- c11)/2.

The various elastic moduli are usually obtained from pulse-echo velocity measurements on single crystals, while v t and vt are found similarly from isotropic, homogeneous polycrystalline specimens with equiaxed grains. (For deformed materials with elongated, oriented grains, the equations must be modified in as-yet-unde- termined manner.) The density is easily found and the frequency is variable in the experiment, being limited to the odd harmonics of the transducers available.

15 I. M. Lifshits and G. D. ?arkhomovskii, Rec. Kharkov State Univ. 27, 25 (1948); Zh. Eksperim. i Teor. Fiz. 20, 175-182 (1950).

Bhatia and Moore 'ø have also presented a theoretical discussion of the Rayleigh scattering of sound in polycrystals. Their results can be rewritten for the attenuation in sound pressure as

4•r•Tf 4 -- -- ,andat .... . (6) 2vt • vt 5 5p%t • [2re • vt •

The coefficients A 1, B1, 2t0, and Bo were evaluated for orthorhombic symmetry, and are

2 1

A 1 = Bo=---P2+--(24a--[ - 7b-l- 13c+d), 225 135

and

8 8

B 1= ---/7 2-Jr- -- (4a-Jr - 2b-I- 3c-+- d), (7) 675 135

1 1

A 6 =---P2+-- (12a+ b+4c- 2d), 150 90

where

J'= 2

a= (644+655+600) 2-- 3 (c44c•+c•scoo+

b= (611+622+C33)2--3(611622+622633+611C33), (8)

c= (Cl2+Cla+C2a) 2-- 3 (c12Ola+ c•ac2a+ Cl2C2a), and

+c•(Cl•+C•a- 2c12).

When the conditions of cubic or hexagonal symmetry are applied, the formulas of Lifshits and Parkhomov- skii • and of Merkolov, TM Eqs. (1) and (3), reduce to the same expressions as do the formulas of Bhatia and Moore, Eq. (6).

Co•on to all the formulations is the problem of finding the average grain volume T. The author •7 has given an expression for it as

4• r=-- (9)

3 (Ra)•,. '

where the averages of the powers of the grain radius are taken over the grain-size distribution Nv(R). This distribution 1• is finite or zero at R=0 and has a de-

creasing tail for large R. A further analysis •8 of Nv(R) on the basis of the integral grain-image area distributions of N• (r) on photomicrographs of metals showed that Nv(R) could be expressed approximately as

Nv(R)=A (R--Ro) exp(--aR) (10)

in most cases. Although the averages (Rø)• and can be expressed analytically in this case, it is more convenient to find the averages by numerical integration

• A. B. Bhatia and R. A. Moore, J. Acoust. Soc. Am. 31, 1140- 1142 (1959), based on A. B. Bhatia, ibid. 31• 16-23 (1959).

•* E. P. Papadakis, J. Acoust. Soc. Am. 33, 1616-1621 (1961). •8 E. P. Papadakis, J. Appl. Phys. 35, 1586-1594 (1964).


GRAIN-SCATTERIN G FORMULAS AND TABLES 705

after the distribution Nv(R) has been found is from the differential grain-image area distribution n.4(r) by a matrix inversion. N,i (r) is the fraction of grain images smaller than or equal to the image radius r. The differential grain-image area distribution n,i (r) is defined as N,i(r)--N,i(r--Ar). The two distributions are related by the matrix equation n,i (ri) = A •iNv(Ri). A computer program has been written is that computes the matrix elements, inverts the matrix, multiplies it by the vector nA(r) giving _/Vv(R), and finds the averages (R6)av and (R3)•v, and computes T by Eq. (9).

Further analysis •8 showed that a good approximation to the scattering volume T could be found directly from the photomicrographs. The volume T is equal to twice the volume of a sphere of diameter equal to the diameter of an image in the 95th percentile. That is, the diameter used is the diameter of the largest remaining image after the largest 5% of the images are discounted. These 5% cover about 25% of the area of the photo- micrograph, so it is sufficient to block off the largest images until 25% of the area is covered and to take the diameter of the next largest image as dos, representative of the 95th percentile. Then the scattering volume is given by

T•'•r(dos)3/3. (11)

It is preferable to use Eq. (9) if enough images or (200 over) are shown on the micrographs so that a reasonable distribution can be produced. Otherwise Eq. (11) must be used. However, when only a few grains have been photographed, it is uncertain whether a representative group is shown. Of course, Eq. (11) can be used with large samples of grains if it is not convenient to compute •¾v(R) and the averages.

II. EXPERIMENTS ON RAYLEIGH SCATTERING

Complete longitudinal-wave data on attenuation versus frequency and on grain-size distribution have been taken on one specimen of iron-30 nickel alloy. n.18 This specimen (No. AU1 equiaxed) had single-crystal grains, most of which were of the proper size to produce Rayleigh scattering in the frequency range used (some grains were too large, however). In addition, the elastic constants of the crystallites were known? The attenuation data after correction for diffraction •ø were analyzed n in terms of the equation

e=af+af4. (12)

The coefficient a4 for Rayleigh scattering was also computed theoretically is from Eqs. (1) and (9). The experimental and theoretical values of a4 compare as follows'

•0 G. A. Alers, J. R. Neighbors, and H. Sato, Phys. Chem. Solids 13, 40-55 (1960).

•0 (a) H. Seki, A. Granato, and R. Truell, J. Acoust. Soc. Am. 28, 230-238 (1956). (b) E. P. Papadakis, J. Acoust. Soc. Am. 31, 150-152 (1959).

Equiaxed Iron-30 Nickel No. AU1

Experimental a4 = 3.6X10 -•

Theoretical a4 = 10.2X 10 -• (13)

[-Units are N/cm. (Mc/sec) 4-]

The theory is high by a factor of 3. Part of this may lie in the diffraction corrections or in the elastic hysteresis term, part in the scattering theory, and part in the grain-counting method. There is also the possibility that the largest grains in the sample are outside the Rayleigh scattering region.

This specimen of iron-30 nickel and another of a larger grain size were quenched in liquid nitrogen to produce 90% martensite. l• In the process, the grains were broken up into many platelets. The grain-size distributions of both specimens were found is and the theoretical a•'s computed. In both specimens, the theoretical a4's were larger than the experimental a•'s by a factor of 30 (+10%). This indicates that the martensitic transformation lowers the scattering by a factor of 10.

Hirone and Kamigaki s measured stainless steel. Their data include micrographs showing only a few grain images. The elastic anisotropy of the grains is not known. The anisotropy was assumed to be the same as that of iron, and an approximate dos was used in Eq. (11)in the case of their specimen No. 200, which showed the most grain images. The theoretical a• compared rather favorably with the experimental, as shown below.

Stainless-Steel Specimen No. 200

Experimental a4---- 3.6X10 -•

Theoretical Ct4-- 18.9X 10 -'• (14)

•Units are N/cm. (Mc//sec)4•

In this case, the difference is a factor of 5. Hirone and Kamigaki's data are not corrected for diffraction, which can change the slope of an attenuation-versus-frequency curve. Some error may enter from this omission. Also, the estimate of d0• may be in error by 50%.

Seemann and Bentz 21 have attenuation data at only one (at most, two) frequencies on several specimens on which they made careful micrographic studies. From their grain counting, the author is computed the grain- size distributions of their specimens and obtained the scattering volumes by Eq. (9). After correction for diffraction, but without the elimination of the unknown hysteresis part of the attenuation, the experimentally measured attenuations were larger than the predictions of Eq. (1) in all cases but one. Unfortunately, attenuation data were meager.

The best agreement so far between theory and experiment has been found in the iron-30 nickel specimen mentioned. The data on stainless steel are also encourag-

• H. J. Seemann and W. Bentz, Z. Metallk. 45, 663-669 (1954).


706 E. P. PAPADAKIS

TABLE II. Rayleigh scattering factors in cubic materials.

Material

dB/•sec. (Mc/sec) 4'cma dB/cm. (Mc/sec) 4'cma S t• St S t• St

Aluminuma 25.8 86.5 - 40.2 284 Aluminum antimonide b 475 1 170 996 4 450 Ammonium bromide ø 1 790 8 310 5 480 19 600 Ammonium chloride c 637 1 320 1 340 4 690 Barium nitrate ø 430 1 050 1 040 4 610 Chro mite o 15.2 38.5 17.7 82.3 Chromiumd 33.4 67.7 49.3 164 Copper a 1 530 5 600 3 060 24 600 CuaAu e 2 400 8 900 5 850 48 100 Fluorsparo 86.6 206 126 536 Galena o 787 1 720 2 040 7 620 Germaniuma 229 481 443 1 550 Gold a 5 300 29 000 16 300 241 000 Indium antimonide f 1 460 3 880 4 100 20 300 Iron• 417 1 050 700 3 260 Irong 569 1 520 977 4 920 Iron-30% nickel h 1 910 3 870 4 210 14 000 Lead• 10 600 64 700 54 500 938 000 Lead i 32 500 166 000 147 000 1 960 000 Lead nitrate ø 1 860 6 940 5 670 47 200 Lithium fluoride ø 200 438 305 1 140 Magnesium oxideo 24.3 47.8 26.1 83.1 Magnetite o 13.4 32.0 18.3 77.7 Nickela 541 1 640 896 5 480 Niobiumi 416 1 600 801 6 990 Palladium k ! 940 8 330 4 240 43 600 Potassium • 27 600 85 000 111 000 696 000 Potassium bromide ø 2 700 5 630 8 250 28 700 Potassium chloride ø 1 180 2 410 2 860 9 580 Pyrites o 97.9 138 125 243 Silicona 37.5 81.7 42.5 158 Silver• 3 580 13 800 9 810 85 700 Silver bromideo ! 380 5 690 4 800 46 200 Silver chloride ø 1 860 8 300 5 850 63 400 Sodium • 15 900 37 900 52 600 223 000 Sodium bromide c 469 1 010 1 390 5 120 Sodium chloride ø 139 300 302 1 110 Strontium nitrate • 39.1 104 97.5 494 Tantalumi 447 1 470 1 080 7 530 Thallium bromide ø 2 120 5 770 9 790 50 600 Thallium chloride ø 2 330 6 290 10 100 51 500 Thorium • 9 550 24 800 33 700 163 000 Tungstena 0.013 0.040 0.023 0.151 Vanadiumi 38.2 127 63.1 445 YIG TM 8.0 20.2 11.0 52.0 Zincblende o 512 1 750 991 7 250

Reference 23. e Reference 29. h Reference 32. k Reference 35. Reference 27. f Reference 30. i Reference 33. • Reference 36. Reference 25. g Reference 31. i Reference 34. m Reference 37. Reference 28.

ing. More experimental work must be done on this problem in polycrystals with single-crystal grains.

III. TABLES OF RAYLEIGH SCATTERIl•IG FACTORS

Because of the moderate agreement between theory and experiment, it is worthwhile to tabulate the scattering factors of Eqs. (1) and (3). The equations can be rewritten

where S is the Rayleigh scattering factor to be tabulated. A short computer program was written to calcu- late S. The results are given in Tables II and III for cubic and hexagonal materials, respectively. Because measurements on polycrystalline samples were lacking,

it was necessary to compute vz and vt from the elastic moduli in some cases. For this purpose, the method of Voigt 22 was employed. The Voigt velocities of the un- tested materials appear in Table IV.

Data for these analyses come from various sources. The materials listed by Mason 2a'24 were all used. Other materials listed by Hearmon •5 were included. When Hearmon gave more than one reference to a particular crystal, the data of the most-recent investigations were used. Where the crystal densities were listed in the compendia, they were used. Otherwise, handbook values •ø were used. When the choice had to be made

among various forms of a metal or alloy, the density of the wrought form was used. Besides the materials listed by Mason and Hearmon, several newly measured materials were found in the literature. 27-47 These were

included in the present computation; their moduli and densities are summarized in Table V.

2= R. F. S. Hearmon, Introduction to Applied Anisotropic Elas- ticity (Oxford University Press, London, 1962), Sec. 3.4, pp. 41-44.

2a D. E. Gray, Ed., American Institute of Physics Handbook (McGraw-Hill Book Co., Inc., New York, 1957), Sec. 3f, pp. 3/74- 3/88.

2• W. P. Mason, Physical Acoustics and the Properties of Solids (D. Van Nostrand Co., Inc., Princeton, N.J., 1958), pp. 17-20, 358-359.

2• R. F. S. Hearmon, (a) Rev. Mod. Phys. 18, 409-440 (1946); (b) Advan. Phys. 5, 323-383 (1956).

26 C. D. Hodgman, Ed., Handbook of Chemistry and Ph)sics (Chemical Rubber Publishing Co., Cleveland, Ohio, 1958), 40th ed.

2, D. I. Bolef and M. Menes, J. Appl. Phys. 31, 1426-1427 (1960).

28 D. I. Bolef and J. deKlerk, Phys. Rev. 129, 1063-1067 (1963). 20 p. A. Flinn, G. M. McManus, and J. A. Rayne, Phys. Chem.

Solids 15, 189-195 (1960). a0 L. J. Slutsky and C. W. Garland, Phys. Rev. 113, 167-169

(1959). a• j. A. Rayne and B. S. Chandrasekhar, Phys. Rev. 122, 1714--

1716 (1961). a• G. A. Alers, J. R. Neighbors, and H. Sato, Phys. Chem.

Solids 13, 40-55 (1960). aa D. L. Waldorf and G. A. Alers, J. Appl. Phys. 33, 3266-3269

(1962). a4 D. I. Bolef, J. Appl. Phys. 32, 100-105 (1961). a• j. A. Rayne, Phys. Rev. 118, 1545-1549 (1960). a• p. E. Armstrong, O. N. Carlson, and J. F. Smith, J. Appl.

Phys. 30, 36-41 (1959). a7 A. E. Clark and R. E. Strakna, J. Appl. Phys. 32, 1172-1173

(1961). a8 j. F. Smith and C. L. Arbogast, J. Appl. Phys. 31, 99-102

(1960). • J. A. Rayne and B. S. Chandrasekhar, Phys. Rev. 120, 1658-

1663 (1960 ). •o E. P. Papadakis and H. Bernstein, J. Acoust. Soc. Am. 35,

521-524 (1963). • B. S. Chandrasekhar and J. A. Rayne, Phys. Rev. 124, 1011-

1014 (1961). • G. L. Vick and L. E. Hollander, J. Acoust. Soc. Am. 32, 947-

949 (1960). • J. F. Smith and J. A. Gjevre, J. Appl. Phys. 31, 645-647

(1960). • T. B. Bateman, J. Appl. Phys. 33, 3309-3312 (1962). • L. J. Slutsky and C. W. Garland, Phys. Rev. 107, 972-976

(1957). • D. Berlincourt, H. Jaffe, and L. R. Shiozawa, Phys. Rev.

129, 1009-1017 (1963). 47 H. J. McSkimin, T. B. Bateman, and A. R. Hutson, J. Acoust.

Soc. Am. 33, 856(A) (1961).


GRAIN-SCATTERING FORMULAS AND TABLES 707

IV. THEORY ON STOCHASTIC SCATTERING

The scattering theory of Lifshits and ParkhomovskiP 5 yields results for stochastic phase scattering when the condition X < 2•rD is applied. These authors worked out the expressions for attenuation in cubic metals, and again Merkulov 1ø worked out the hexagonal case. Hunt- ington 5 did earlier work on the subject. The expressions cited by Merkulov involve the anisotropy in a slightly different way but are simpler than those for Rayleigh scattering since mode conversion is absent. All the experimental quantities and combinations of elastic moduli were defined previously in Eqs. (1) through (5). The units are neper/unit length.

Cubic Crystals

at =---- , at= . (16) 525v•6p s 21Ovt6p s

Hexagonal Crystals

16•r•Df •' ae= • (7•'-3 - 35x•-3 - 14Or?+ 140xr•q- 30x•-+- 60r•).

1575vdf (17) No expression for the hexagonal stochastic scattering at has been given.

Here, D is the average grain diameter in the metal. It is also to be found by averaging over the grain-size distribution •s Nv(R). D is equal to 1.45d•0, where ds0 is the median image diameter. The attenuation data of Merkulov •ø is thorough but he does not show photomicrographs, so the grain-size distribution (and hence D) is unknown. Other data s are in the same condition. More work is essential. However, D is known in two Fe-30 Ni specimens. These are analyzed later in this paper.

V. TABLES OF STOCHASTIC SCATTERING FACTORS

For completeness, to give impetus to further work, and to indicate the magnitude of the attenuation of

TABn• III. Rayleigh scattering factors in hexagonal materials.

IV. Voigt velocities in some polycrystals. Units' 105 cm/sec.

Material vl vt

A. Cubic

Aluminum antimonide • 4.77 2.63 Ammonium bromide • 3.27 1.94 Ammonium chloride• 4.72 2.83 Barium nitrate b 4.12 2.28 Chromite • 8.60 4.68 Chromium ø 6.77 4.12 CuaAu d 4.11 1.85 Fluorspar b 6.84 3.84 Galena b 3.86 2.26 Germanium a 5.18 3.10 Indium antimonide • 3.59 1.91 Iron • 5.83 3.09 Iron-30% nickelg 4.55 2.77 Lead h 2.21 0.85 Lead nitrate b 3.28 1.47 Lithium fluoride b 6.56 3.84

Magnesium oxide b 9.32 5.76 Magnetite b 7.34 4.12 Niobium i 5.19 2.29 Palladiumi 4.57 1.91 Potassium a 2.47 1.22 Potassium bromide b 3.27 1.96 Potassium chloride• 4.15 2.52

Pyrites b 7.82 5.69 Silicon a 8.81 5.17 Silver bromide b 2.88 1.23 Silver chloride b 3.19 1.31 S odium a 3.03 1.70 Sodium bromide b 3.36 1.98 Sodium chloride b 4.60 2.71 Strontium nitrate b 4.01 2.12 Tantalum i 4.11 1.96 Thallium bromide •' 2.17 1.14 Thallium chloride }' 2.31 1.22 Thorium k 2.83 1.52 Vanadium i 6.06 2.87 YIG t 7.29 4.4l Zincblende b 5.17 2.42

_

B. Hexagonal

Apatite b 7.21 4.41 Beryl b 9.64 5.56 Beryllium m 13.00 8.97 /•-tin • 3.26 1.52 Cadmium a 3.33 1.69 Cobalt a 5.88 3.10 Graphite o 4.21 2.03 lndiump 2.56 0.74

Magnesiumq 5.77 3.15 Rutile • 8.72 4.44 Yttrium * 4.10 2.38 Zinc oxide t 6.00 2.84

Material dB/t•sec-(Mc/sec)4.cm a dB/cm-(Mc/sec)4.cm a

St St St St

Apatite • 92.2 686 127 1 550 Beryb 18.5 40.4 19.2 72.6 Beryllium b 1.94 --0.133 1.49 --0.148 •-tin • 4 990 38 100 15 300 251 000 Cadmium a 6 040 --237 18 100 -- 1 400 Cobalt a 196 1 720 333 5 550

Graphite • 6 770 -- 7 750 16 000 --38 200 Indium f 16 300 11 100 64 000 150 000

Magnesium a 54.3 285 94.2 937 Rutileg 371 258 425 581 Yttrium h 69.1 536 168 2 250 Zinc a 2 420 2 880 5 760 11 800 Zinc oxide i 41.0 460 68.4 1 620

Reference 25. d Reference 23. f Reference 41. h Reference 43. Reference 38. • Reference 40. g Reference 42. i Reference 44. Reference 39.

Reference 27. f Reference 31. k Reference 36. p Reference 41. Reference 25. g Reference 32. 1 Reference 37. q Reference 45. Reference 28. h Reference 33. m Reference 38. r Reference 42. Reference 23. i Reference 34. n Reference 39. • Reference 43. Reference 30. i Reference 35. o Reference 40. • Reference 44.

ultrasound to be expected, the stochastic scattering factor 2• is defined in Eq. (18) and tabulated for cubic materials in Table VI. Equations (16) and (17) are rewritten as

a=D fS2• (18) for this purpose.

VI. ANALYSIS OF THE TABLES

Several things should be noted concerning the tabulated values. First, the metals having low attenuation


708 E. P. PAPADAKIS

TABLE V. Moduli and densities of some crystals. Moduli units are l0 n dyn/cmh

p

Crystal gm/cm • 611 612 618 688 644

A. Cubic

A1Sb a 4.36 8.94 4.43 4.16 Chromium b 7.20 35.0 7.60 10.1 Cu3Au c 12.3 18.5 13.5 6.87 In Sb a 5.78 6.67 3.64 3.02 Iron ø 7.92 23.3 13.6 11.8 Fe-30%Ni f 8.22 17.7 12.2 10.5 Leadg 11.3 4.95 4.23 1.49 Niobium h 8.58 24.6 13.4 2.87 Palladium i 12.0 22.7 17.6 7.17 Tantalum h 16.7 26.7 16.1 8.25 ThoriumJ 11.7 7.53 4.89 4.78 Vanadium h 6.02 22.8 11.9 4.26 YIG k 5.17 26.9 7.64 10.8

B. Hexagonal

Beryllium • 1.85 29.2 2.67 1.40 33.6 16.2 fi-tin m 7.28 7.23 5.94 3.58 8.84 2.20 Cd Se n 5.81 7.41 4.52 3.93 8.36 1.32 Cd S n 4.82 9.07 5.81 5.10 9.38 1.50 Cd S ø 4.82 8.58 5.33 4.62 9.37 1.49 Graphitep 2.17 5.21 1.88 1.80 2.58 0.15 Indiumq 7.28 4.54 4.01 4.15 4.52 0.65 Magnesium r 1.75 5.94 2.56 2.14 6.16 1.64 Rutile s 4.26 24.8 20.0 14.0 45.2 12.0 Yttrium • 4.47 7.79 2.85 2.10 7.69 2.43 ZnO u 5.68 21.0 12.1 10.5 21.1 4.25

Reference 27. g Reference 33. 1 Reference 38. q Reference 41. Reference 28. h Reference 34. m Reference 39. r Reference 45. Reference 29. i Reference 35. = Reference 46. • Reference 42. Reference 30. i Reference 36. o Reference 47. • Reference 43. Reference 31. k Reference 37. v Reference 40. u Reference 44. Reference 32.

in previous work, a,•ø.•a aluminum, magnesium, and tungsten, have low scattering factors. Second, some other materials, including vanadium, chromium, and polycrystalline "YIG," chromite, magnetite, strontium nitrate, and silicon, also have low scattering factors. Third, the transverse-wave scattering factor for a cubic material is larger than the factor for longitudinal waves. Fourth, the hexagonal scattering formulas, Eqs. (3) and (17), can break down since the crossterms (involving •x, xr/, and r/•) can remain negative if one or two of the factors •, x, and r• are negative. In fact, the whole expression for the scattering factor can be negative, as in the case of the shear-wave scattering factors of beryllium, cadmium, and graphite. Negative attenuations due to scattering are meaningless; theoretical work should be done to find the proper formulation to eliminate the negativity. The negative terms seem to be influential in both Rayleigh scattering factors in beryllium, giving factors smaller in magnitude than those for magnesium. One would expect an extremely anisotropic material such as beryllium to produce large scattering factors. Such should be the case if the abso- lute values of the crossterms (in •'x, etc.) were added to the squared terms (•,2, etc.) to yield the coefficients. Fifth, in the cubic case in which there is no negativity problem, the transition from Rayleigh to stochastic scattering is smooth. That is, on a logarithmic plot of

attenuation versus frequency in a particular material, one can draw a curve of slope 4 through a point in the Rayleigh region and connect it with a curve of slope 2 through a point in the stochastic region; the connecting curve will have monotonic decreasing slope. As an example, the data on the equiaxed iron-30% nickel specimen No. AU1 are plotted in Fig. 1. The parameters TM are /)=0.00561 cm, T=5.76X10 -• cm •, v•=4.65X10 • cm/sec for longitudinal waves, and vt=2.69X10 • cm/sec for transverse waves. The boundary between the two regions for longitudinal propagation is at X= 2•r/)= vdfB; the boundary frequency fB is then 12.9 Mc/sec. The Rayleigh point was computed at 1.29 Mc/sec and the stochastic point at 129 Mc/sec. Attenuation data u also appear on the graph. The agree-

TaBL• VI. Stochastic scattering factors in cubic materials.

dB/t•sec. (Mc/sec)•.cm dB/cm. (Mc/sec)•.cm Material •:• •:• •:t •

Aluminuma 0.038 0.994 0.059 3.27

Aluminum antimonide b 0.807 9.90 1.69 37.6

Ammonium bromideo 2.03 17.3 6.22 89.2

Ammonium chlorideo 1.58 12.7 3.35 45.1

Barium nitrateo 0.555 6.68 1.34 29.3

Chromitee 0.079 1.03 0.092 2.20

Chromium d 0.183 1.37 0.271 3.33

Copper• 1.10 36.0 2.19 158 CuaAuo 1.12 38.0 2.73 205

Fluorsparc 0.330 3.70 0.482 9.63 Galena o 1.17 10.6 3.03 47.0

Germaniu m• 0.682 5.56 1.31 17.9

Gold• 0.586 52.5 1.80 438

Indium antimonidef 1.18 17.4 3.30 91.2

Iron• 1.03 13.6 1.73 42.0

Irong 1.19 17.8 2.04 57.7

Iron-30% nickel h 4.76 35.6 10.4 128 Leada 0.335 38.7 1.71 562

Lead i 1.98 149 8.96 1760

Lead nitrate o 0.542 18.7 1.65 127

Lithium fluoride o 0.857 7.80 1.30 20.3

Magnesium oxide • 0.273 1.89 0.292 3.28 Magnetiteo 0.059 0.662 0.080 1.60 Nickel• 0.887 18.3 1.46 61.1

Niobium • 0.280 10.4 0.541 45.8

Palladium k 0.777 38.0 1.70 199

Potassium • 7.37 156 29.8 1280

Potassium bromide o 3.21 26.0 9.84 132

Potassium chlorideo 2.42 18.3 5.84 72.8

Pyrites o 1.63 4.99 2.08 8.79 Silicon• 0.293 2.63 0.333 5.09

Silver • 1.19 44.7 3.27 277

Silver bromide o 0.245 10.7 0.850 87.6

Silver chloride o 0.333 17.8 1.04 136

Sodiuma 11.8 133 39.2 786

Sodium bromideo 0.545 4.79 1.62 24.2

Sodium chlorideo 0.302 2.66 0.658 9.83

Strontium nitrateo 0.038 0.579 0.096 2.73

Tantalumi 0.278 7.04 0.676 35.9

Thallium bromideo 0.591 9.23 2.72 81.0

Thallium chloride o 0.756 11.5 3.27 94.3

Thoriu ml 5.03 70.4 17.7 46.3

Tungsten• 1.5 X10-• 3.4 X10 -•4 2.8 X10-•5 1.3 X10 -la Vanadium i 0.050 1.31 0.082 4.56

YIG TM 0.028 0.375 0.039 0.965

Zincblende o 0.459 12.7 0.889 52.8

Reference 2 •. • Reference 29. h Reference 32. k Reference 35. Reference 27. f Reference 30. i Reference 33. • Reference 36. Reference 25. g Reference 31. i Reference 34. m Reference 37. Reference 28.


GRAIN-SCATTERING FORMULAS AND TABLES 709

ment is much better than Eq. (13) had indicated. The reason is that data at 6, 7, and 9 Mc/sec are near the boundary between regions, 12.9 Mc/sec. There, the slope of the curve has decreased to about 3, and the magnitude is not as great as it would have been had the curve continued up with a slope of 4. The physical meaning of the gradual transition is that the grains leave the Rayleigh region and enter the stochastic region in order according to size, largest first, as the frequency is increased. Thus, in most cases the current Rayleigh scattering theory should predict somewhat excessive attenuations. In addition, the slope of the logarithmic plot of scattering attenuations versus frequency should be less than 4, except very deep in the Rayleigh region.

Thus, the proper way to analyze attenuation data on polycrystalline metals involves several steps, as follows'

(1) Find the grain-size distribution •¾v(R) of a metal and its moments T and D from analysis •s of its photomicrographs.

(2) Plot the Rayleigh point using T and the stochastic point using O. Connect these with a curve (on logarithmic paper) having a slope of 4 at the Rayleigh point and 2 at the stochastic point.

(3) Correct the attenuation data for diffraction 2 and also for other effects TM if possible.

(4) Compare the corrected data with the curve of step (2).

In Ref. 18, the scattering volume T of fine-grained metals was found to be 1X10 -7 cm 3 or larger. If 10 Mc/sec is used as the ultrasonic frequency, then the Rayleigh scattering will be 10-3S in such a metal. If S is 1000 in dB/usec. (Mc/sec) 4. cm 3, then the attenuation of this metal is 1 dB/usec at 10 Mc/sec. It is difficult to measure attenuation higher than 2 dB/usec with the pulse-echo method; therefore, S-- 2000 is a rough upper limit on the fine-grained materials that can be investi- gated at 10 Mc/sec. At 20 Mc/sec, the limit is near S= 100 dB/•sec. (Mc/sec)4.cm •. In the sample 18 with T=lX10 -7 cm a, the average grain diameter was /5=0.002 cm. This makes the boundary frequency about 40 Mc/sec between the Rayleigh and stochastic regions. Thus, even in very fine-grained material one begins to leave the Rayleigh region around 20 Mc/sec. In fine-grained material with S < 100, one can investi- gate the transition from Rayleigh to stochastic scattering in a frequency range above 10 Mc/'sec in which the ultrasonic measurements are accurate and diffraction corrections are small.

VII. SUMMARY

Current theory on scattering of ultrasound in metals indicates that in the Rayleigh scattering region (2,> 2•rD) the attenuation is proportional to the fourth power of the frequency, to terms expressing the square

FIG. 1. Rayleigh and stochastic scat-

tering in equiaxed iron-30% nickel alloy specimen No. AU1. n The Rayleigh and stochastic points were computed at frequencies differing by factors of 10 from the boundary frequency fB defined by X= 2•rD. The average scattering volume T and the mean

grain diameter /9 were found by ana- lyzing a photomicro- graph of specimen No. AU1 by the method developed by the author. •s These parameters were used in Eqs. (15) and.(18) for the attenuation

where the scattering parameters S and • for the Rayleigh and stochastic regions, re-

l0 6

l0 5

l0 4

I I I I

- CURVE rø•/•• ..r•r, • - c STOCHASTIC ONNECTING REGIONS / ""•iNT - -

•' _

]/s,..- ATTENUATION

I -

/•/.---R•Y LEIGH POINT t• II I I

I.O IO IOO IOOO IOOOO

FREQUENCY, Mc

spectively, were obtained from the Lifshits-Parkhomovskii- Merkulov theory. TM The measured longitudinal-wave attenuation falls on the curve connecting the regions.

of the elastic anisotropy of the grains, and to the average grain volume. This volume is to be computed through averages of powers of the grain radius over the grain-size distribution. Mode conversion occurs at the grains, so the scattered energy is carried away by both longitudinal and transverse waves. The transverse- wave term in the attenuation is dominant for either

type of incident wave. The current theory using the procedure outlined for finding the average scattering volume T predicted attenuation a factor of 3 larger than the experimental result in one case. Graphical analysis of the same data showed better agreement. Other experiments are inconclusive because of lack of grain-size data.

Theory shows that in the stochastic phase scattering region the attenuation is proportional to the square of the frequency, to terms expressing the square of the elastic anisotropy of the grains, and to the average grain diameter. This quantity is an average of the diameter over the grain-size distribution. There is no mode conversion in this type of attenuation.

Since there are some relatively large grains in every sample, it often happens •ø that, at the upper end of the frequency range used, the largest grains are scattering by the stochastic process, while the smaller ones are still in the Rayleigh region. Thus, the frequency dependence of the attenuation will often appear to be less than fourth power even though Rayleigh scattering is occurring. The largest exponent that the author has ever observed TM in steel, iron-nickel alloys, various other metlas, and ceramics is 3.8. More often, it is around 3.0. Of course, other effects such as elastic hysteresis and dislocation damping are contributing


710 E. P. PAPADAKIS

terms lowering the exponent of the frequency in the gross attenuation.

The Rayleigh scattering formulas have been broken down into the form a= Tf4S and the stochastic scattering formulas into a= 1}f•I;. The variables T and/3 are controlled metallurgically, while f is controlled elec- tronically in any experiment on ultrasonic attenuation. The Rayleigh and stochastic scattering factors S and E, respectively, are characteristics of the metal, inde- pendent of the grain size. S and I; are tabulated for several metals according to the formulas of Lifshits and Parkhomovskii •5 and of Merkulov?

VIII. SUGGESTIONS

For confirmation of the scattering theory now extant, more experimental work is needed. It is necessary to start with homogeneous polycrystalline metals equiaxed and isotropic in a macroscopic sense. The grain-size distribution should be found from photomicrographs. To make the grain-image counting compatible with the transformation of Ref. 18, the grain images are counted in 10 categories of size, starting at 0.1Dmax and going to 1.0Dmax in steps of 0.1Dm•.x, where Dm,.x is the diameter of the largest image ob-

tained. The grain-size distribution Nv(R) is computed from this histogram of images, and then T or /3 are found from Nv(R). Otherwise, d95 should be found where dg• is the diameter of a grain image, which is as large as or larger than 95% of the grain images. T is found from d• while /3 can be found from d•0, the median grain-image diameter. The ultrasonic-attenuation data should be taken over as large a frequency range as possible for both longitudinal and transverse incident waves. Corrections for diffraction sø must be made, so data on specimen thickness, transducer diameter, echoes used, and wave velocities should be re- corded in addition to the frequency and attenuation figures. After this correction, the elastic hysteresis and/or dislocation damping must be separated out either through numerical analysis or by other manipula- tions such as graphical comparisons among samples or experimental techniques for altering the unwanted damping. Once the scattering component is isolated, it can be compared with theory.

ACKNOWLEDGMENTS

The author is indebted to Althea Yates for help with the computer codes and data processing.


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Revised Grain-Scattering Formulas and Tables