adler, wagner - x-ray diffraction study of the effects of solutes on the occurrence of stacking...

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JOURNAL OF APPL IED PHYS ICS VOLUME 33 . NUMBER 12 DECEMBER 1962

X-Ray Diffraction Study of the Effects of Solutes on the Occurrence ofStacking Faults in Silver-Base AlloysR. P. r. ADLER AND C. N. J. WAGNERHammond Metallurgical Laboratory, Yale Univtrsity, New Haven, Connecticut

(Received June 21, 1962)The addition of the solutes cadmium, indium, and tin to silver increases the probability of deformationfaults a in filings from 3 X 10-3 in pure silver to a maximum value of 45 X 10-3 at the highest concentrationsof solute. In addition, the twin fault probability f3 measured from center of gravity displacements variesfrom lOX 10-3 for pure silver to 30X 10-3 for the alloys highest in tin or indium concentration. Latticeparameters ahkl were determined from all available reflections of the cold-worked and annealed specimensand plotted as a function of cos28/sinO. By relating the large scatter of the individual ahkl to the occurrenceof deformation faults in the deformed material, the true lattice parameter, ao(CW), and the deformationfault probabilities a of cold -worked materials could be determined. There was an apparent decrease in latticeparameter of the deformed Ag-Sn alloys which was largest (,,-,0.1%) for the greates t tin concentrat ion(Ag-9%Sn). Using Fourier analysis of line profiles, the effective particle sizes (D,lhkl and root mean squarestrains [(SL2)avJhklt were determined. The measured effective particle sizes were anisotropic [(D')l\l/

(D,bo=1.7J and are primarily a consequence of deformation and twin faulting. The values for the compound fault probability (1.5a+f3) from peak shift and asymmetry and from anisotropic particle sizes, i.e.,from peak broadening, agreed rather well.

1. INTRODUCTION

STACKING faults can exert an important influenceon the structural and mechanical properties of facecentered cubic (fcc) metals and alloys. Recrystallization/2 texture formation,a,4 and microstructure5- S canbe related to the stacking fault energy of the material.Work hardening5,6,S-u as well as low temperature creep12theories also predict a relationship between cross-slipand stacking fault energy,Direct observations of stacking faults can be accomplished by using electron transmission microscopyor x-ray diffraction techniques. Using the electronmicroscope, the stacking fault energy 'Y can be measured5 ,13,14 for metals and alloys of low stacking faultenergy provided that the amount of plastic deformationis small (i.e., low concentration of stacking faults).Using x-ray diffraction15 ,16 the stacking fault probabilitycan be obtained in heavily deformed metals. For a givenamount of deformation, the relative change in stacking

1 H. Hu, R. S. Cline, and S. R. Goodman, Trans. AIME 224, 96(1962).2 N. Brown, Trans. AIME 221, 236 (1961).3 H. Hu and R. S. Cline, J. Appl. Phys. 32, 760 (1961).4 H. Hu, R. S. Cline, and S. R. Goodman, J. Appl. Phys. 32,1392 (1961).6 M. J. Whelan, P. B. Hirsch, R. W. Horne, and W. Bollmann,Proc. Roy. Soc. (London) A240, 524 (1957).6 N. F. Mott, Trans. AIME 218,962 (1960).7 C. S. Barrett, Imperfections in Nearly Perfect Crystals UohnWiley & Sons, Inc., New York, 1952), p. 97.8 P. B. Hirsch, Internal Stresses and Fatigue in Metals (ElsevierPublishing Company, Inc., New York, 1959), p. 139.9 A. Seeger, Defects in Crystalline Solids, Bristol Conference (ThePhysical Society, London, 1955), p. 328.10 A. Seeger, Dislocations and Mechanical Properties of CrystalsUohn Wiley & Sons, Inc., New York, 1956), p. 243.11 A. Seeger, R. Berner, and H. Wolf, Z. Physik 155, 247 (1959).12 P. R. Thornton and P. B. Hirsch, Phil. Mag. 3, 738 (1958).13 M. J. Whelan, Proc. Roy. Soc. (London)IA249, 114 (1959).14 A. Howie and P. R. Swann, Phi!. Mag. 6, 1215 (1961).15 M, S. Paterson, J. App!. Phys. 23, 805 (1952).16 B. E. Warren, Progress in Metal Physics (Pergamon Press,New York, 1959), Vo!' 8, p. 147.

fault probability with solute concentration in an alloycan be related to the change in stacking fault energy 'Y.Values of 'Y can also be measured indirectly frommechanical properties (stress-strain relationshipslO,l1and low temperature creepl2) by assuming that thetheories of work hardening involving the coalescing ofextended dislocations are correct. Using twin boundaryenergies,17 'Y may also be found if the assumption is madethat the stacking fault energy is twice as large as twinboundary energy.

I t has been shown that solute additions to puremetals alter the stacking fault energy. Using the electronmicroscope, Howie and Swannl4 found that the stackingfault energy decreases with increasing solute concentration. In terminal copper alloy systems1S- 24 x-ray investigations have shown that the stacking fault probabilityincreases with increasing solute concentration. Similarresults23-25 have been found for terminal silver alloys.Plastic deformation of fcc metals may produce twotypes of stacking faults which can be detected by x-raydiffraction. I f the normal stacking sequence26 ,27 of (111)planes is ABCABC, then a deformation fault26 ,27 is abreak in this sequence A B C ~ B C A B C where the arrowindicates the fault plane. A reversal in the sequenceABCACBA represents a twin fault. 26 ,27 Deformationfaults produce a shift of the position of the powderpattern peak, whereas twin faults cause the line profile

17 R. Fullman, J. App!. Phys. 22, 448 (1951).18 B. E. Warren and E. P. Warekois, Acta Met. 3, 473 (1955).19 H. Otte, J. App!. Phys. 33, 1436 (1962).20 J. C. Helion, M. Eng., thesis, Yale University, 1961.21 C. N. J. Wagner, Acta Met. 5, 427 (1957).22 R. E. Smallman and K. H. Westmacott, Phil. Mag. 2, 669(1957).23 D. E. Mikkola and J. B. Cohen, J. App!. Phys. 33,892 (1962).2. R. G. Davies and R. W. Cahn, Acta Met. 10, 621 (1962).2. L. F. VassamiIIet, J. App!. Phys. 32, 778 (1961).25 C. N. J. Wagner, Acta Met . 5, 477 (1957).27 A. H. Cottrell, Dislocations and Plastic Flow of Crystals(Clarendon Press, Oxford, 1953), p. 73.

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Alloy(atomic percent)--- AgAg-2.5 InAg-5.0 InAg-7.5InAg-l0.0 InAg-12.5 InAg-15.0 In----

STACKING FAULTS IN S ILVER -BASE A L L O Y S 3453TABLE II . Experimental values for silver-indium alloys.

Peak shift Lattice parameter(aX 103) (aX 103 ) Annealed Cold-worked(Lattice parameter (Peak (1.50+{:I)M+Ashift) shift) ({:IX 10') X 103 ao(Ann.) au(C.w.)(A) (A)

5 3 11 16 4.0866 4.08636 6 17 26 4.0941 4.093814 12 16 35 4.1030 4.103019 17 27 53 4.1104 4.109725 27 10 51 4.1188 4.11!lO28 29 30 73 4.1268 4.126438 40 31 91 4.1356 4.1350

Peak broadeningAlloy(atomic percent) (D,)111 (A) (D,hoo (A) Tmin (A) [(l:h_50).v J1llt [(Eh_50).vJtooi(1.50+{:I)BX 10' X 103 X 10'AgAg-2.5 InAg-S.O InAg-7.5 InAg-10.0 InAg-12.5 InAg-15.0 In

Alloy(atomic percent)AgAg-1.5 SnAg-3.0 SnAg-4.5 SnAg-6.0 SnAg-7.5 SnAg-9.0 Sn

240 135 370 23215 110 500 32150 90 200 33140 80 205 39120 70 165 4390 60 100 4570 45 76 58

TABLE III. Experimental values for silver-tin alloys.Peak shift

(aX 103) (aX 10')(Lattice parameter (Peak (l.5a+.B)M+Ashift) shift) ({:IX 10') X 10'S 3 11 167 6 6 1511 9 14 2820 18 19 4624 23 18 52

38 36 21 7542 45 37 104Peak broadening

2.0 2.83.2 4.53.1 4.83.1 5.74.2 5.04.2 5.04.5 5.6

Lattice parameterAnnealed Cold-workedao(Ann.) ao(C.W.)(A) (A)

4.0866 4.08634.0932 4.09214.0998 4.09824.1057 4.10424.1122 4.11144.1191 4.11724.1265 4.1222[( Eh_50).vJI11illoy(atomic percent) (D,)1l1 (A) (D,hoo (A) Tmi n (A) [(e2L_50).vJl001(1.5a+i3hX lO' X 103 X1O'

AgAg-1.5 SnAg-3.0 SnAg-4.5 SnAg-6.0SnAg-7.5 SnAg-9.0 Sn

24022018514013510571

13512011080756537

peaks.l8,21 ,31 The deformation fault probabilitiy ex is thusrelated to these shiftsI8 ,26:(6.(2(Jhkl)- (6.(2(Jh'k'l' )=Ha, (2)

where H is a parameter defined by Wagner.31 The deformation fault probability a was calculated from thefollowing neighboring pairs (111)-(200), (200)-(220),and (220)-(311) for the three solutes series studied; theaverage value is recorded in Tables II , III, and IV. Thecontinuous variation of the average 0: with solute concentration is shown in Fig. 1.To determine the true lattice parameter of anannealed sample, ao(Ann.), the lattice parameters

31 C. N. J. Wagner, Z. Metallk. 51, 259 (1960).

370 23 2.0 2.8390 28 3.5 4.4260 29 3.7 5.0220 42 3.4 4.9220 43 3.8 5.5125 43 4.1 6.2150 94 4.6 5.3

calculated from all available (hkl) reflections ahkl areplotted versus the extrapolation function, cos2(J/sin(J(this function was chosen because it eliminates thegreatest estimated error due to the displacement of the

TABLE IV. E x p e r i ~ e n t a l values for silver-cadmium alloys.Alloy (aX 10')(atomic percent) (Peak shift) (i3X 10')

Ag 3 11Ag-lO Cd 9 15Ag-20 Cd 19 26Ag-30 Cd 23 28Ag-40 Cd 30 21

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STACKI . \ JG FAULTS IN S ILVER -BASE ALLOYS 3455o

oinA

4.13 cold worked

' - : ~ I - ~ ; ~ ~ ; d - : - - ,4.13

4.12 - - - r - - - - A q : 7 . 5 S ~

t - ~ ~ r 09E I ! I ! _. 0 I 2 3coste/sineFIG. 3. Determination of extrapolated lattice paramet.ers forseveral annealed and cold-worked Ag-Sn alloys. DeviatIOns ofindividual cold-worked lattice parameters, ahkl , from the extrapolated line are the result of deformation faulting.

ones calculated from relative peak shift of neighboringreflections of cold-worked and annealed samples (TablesII and III).

c. Peak Asymmetry MeasurementsTwin faults asymmetrically broaden a diffractionpeak. Since twin faults are the only cause of asymmetryarising out of cold work for these alloys, any differencein position of the center of gravity and the peak maximum of a cold-worked peak would give an indicationof twin faulting. Cohen and Wagner34 have derived arelationship between center of gravity shift and twinfault probability {3. By combining AC.G. (20), which isthe displacement of the center of gravi ty from the peakmaximum, for (111) and (200) reflections partialcompensation for systematic and instrumental effectscan be achieved. The following equation allows thedetermination of twin fault probability {334;

AC.G. (20) 111 - AC.G. (28) zoo{3 11 tan0l11+14.6 tan8zoo (4)

For all three solute series the twin fault probability {3increases with alloy content (Tables II-IV). To verifythat the combination of the (111) and (200) center ofgravity displacements compensated for the systematicand instrumental effects, the value CAe.G. (028 111 )-AC.G.(028200)] (or abbreviating, AAC.G.) was also

M J. B. Cohen and C. N. J. Wagner, J. Appl. Phys. 33, 2073(1962).

measured for three annealed samples (pure Ag, AAC.G.=0.005; Ag-9 Sn, AAC.G.=0.009; Ag-15 In, AAC.('.= 0.001 0); these values were negligible compared totypical values of AAC.G. between 0.100 and 0.40 forthe cold-worked materials. Scatter in values of AAC.G.was due to errors in evaluating peak tails and indetermining the peak maximum.

D. Peak Broadening MeasurementsStacking faults produce peak profile broadening thatis independent of the order of reflection (i.e., anapparent particle size effect); therefore, any measuredparticle size will contain the true domain size and ameasure of the stacking fault probabilitiesI6 a and {3. Todetermine the effective particle sizes and strain effectsa Fourier analysis was performed on the K"l peak profiles using either Beevers-Lipson strips or a Mader-Ottharmonic analyzer. Initially correcting for instrumental

broadening with an annealed tungsten powder standardusing the Stokes method,25 the separation of the strainterm from the particle size term was then accomplishedby using the Warren-Averbach technique16 0n the (111),(222), (200), and (400) reflections. The intercept of theinitial slope of the order independent Fourier coefficientALsF with the abscissa L gives the effective particlesize36 (De)hkl for a given set of planes (hkl). For the(111) reflection the following relation holds

1 1 1 v'3- -= -+ -+ - ( 1 . 5 a+ {3 ) (5)(D e)1l1 D VlT 4aand for (200)

1 1 1 1- -= -+ - -+ - ( 1 . 5 a+ {3 ) , (6)(De) 200 D ( 1 . 5 ) ~ T awhere D is the coherent domain size normal to thereflecting plane, a and {3 are fault densities, a is thelattice parameter, and T is the domain size in thefaulting plane. The last term in Eq. (5) and (6) containing the compound faulting probability (1.5a+{3) canbe considered as the reciprocal of the stacking faultparticle size in the defined direction (hkl). There is adefinite decrease in effective particle size as the amountof either solute (In, Sn) increases (Tables I I and III,Figs. 4 and 5). As defined by Warren36 a minimum valueof T, Tmi n can be found from Eqs. (5) and (6);

Tmin=0.82 / [ ( ~ ~ : 1 1 - ( D ~ ) 2 0 J (7)These values are given in Tables II and III and indicatea definite decrease in Tmi n with increasing amounts ofsolute (Figs. 4 and 5).The root mean square strain component (rmss) as a

,6 A. R. Stokes, Proc. Phys. Soc. B61, 382 l1948)..& B. E. Warren, ]. Appl. Phys. 32, 2428 (1961).



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STACKING FAULTS IN S ILVER-BASE ALLOYS 3457the anomalously large values of ex to become smallerand approach the values of ex calculated from (111),(200), and (220) peaks, but does not appreciably affectthe cold-worked lattice parameter extrapolation. Although spacing faults successfully explain the largeshifts of the (420) and (331), the relative shifts of the(400), (222), and (311) peaks which do not agree inmagnitude with the predicted values from the Paterson l5theory must be the result of some other effect.Concurrently with the determination of deformationfault probabilities from relative lattice parameter shifts,the apparent lattice parameter of the heavily deformedspecimens were measured (Tables II and III). Thepresence of tin in deformed silver alloys causes adecrease in lattice parameter; the maximum decreasefound was ~ O . 1 percent of Ag-9 Sn. Alloying indiumwith silver has no significant effect on cold-workedlattice parameters. Similar effects have been observed incopper-base alloys; in fact for copper-zinc alloys anexpansion has been observed. 19 ,20 The fact that thelattice parameters of deformed ti n and indium alloysshow different trends, although they have similar linebroadening characteristics, indicates that this is a realeffect whose magnitude depends on the type andquantity of solute present in silver. A greater numberof alloy systems and pure metals must be studied beforeany firm conclusions can be made; however, the effectis probably related to the variation of compositionaround a stacking fault (e.g., Suzuki effect37 ) .

The particle size due to faulting [Eqs. (5) and (6) ] isinversely proportional to compound faulting probability(1.5ex+,S). Since the effective particle size decreases asreciprocal stacking fault probability decreases (i.e.,with increasing solute concentration) (Figs. 4 and 5),stacking faults do contribute significantly to the particlesize broadening. Anisotropic effective particle sizes alsoyield information about the particle size in the planeof the stacking fault T. I f stacking faults were completely extended across a coherent domain dimension D,then the particle size in the faulting plane T and Dwould be approximately equal and relatively largecompared to the domain size due to faulting [i.e., the"1.5ex+f:l" term in Eqs. (5) and (6)]. In this limiting casethe effect of D and T on effective part icle size can beneglected to a first approximation so that the effectiveparticle size depends essentially only on the fictitiousdomain size due to faulting; the ratio for this conditionof completely extended stacking faults can then beobtained [(D e)I1I/(De)200]= 2.3. The average experimental ratio is 1.7. This indicates that D and morelikely T [because Tmin is slightly larger than (D e)111] doinfluence the magnitude of the effective particle size. Asseen from Tables II and III and Figs. 4 and 5, thisminimum particle size in the plane of the stacking faultTmi n decreases with increasing alloy content and might

37 A. H. Cottrell, Rela,tion of Properties to Microstructure(American Society for Metals, Cleveland, 1954), p. 131.

be the result of a greater number of stacking faults on agiven (111) faulting plane.The rms strain distributions for columns of unitcells normal to both (111) and (200) planes weremeasured using Fourier analysis. For all deformedspecimens in both measured directions the strains

decrease asymptotically with increasing distance Lnormal to the reflecting plane. For both the tin and theindium series using an arbitrary distance L=50 A, therewas a gradual increase in rmss in the (111) and (100)directions as solute content was raised (Tables II andIII). In general for a given amount of solute present andin either direction, the measured value of rmss[( 82L=50 ;..)nv]hkZ! in the tin series was higher than in theindium alloy of corresponding concentration. This,coupled with the fact that for the same alloy composition range the tin alloy had the higher value of deformation fault probability, suggests that the higher densityof dislocations contributed this additional amount ofstrain. The existence of the measured anisotropic rmsswithin the material cannot be rationalized in terms ofan isotropic stress field where the differences in strainsarise from the variation of the elastic constants withcrystallographic direction. For this ideal isotropic stresscase using the elastic constants for silverS the ratiowould be (8100/8 111)= (E 111/E 200 ) =2.7. The averageexperimentally observed value (8 100/8 111) = 1.4 falls between the ideal cases of isotropic stresses and isotropicstrains. Thus the presence of dislocations and stackingfaults which do have stress directionality27 probablyinfluence this experimentally measured strain ratio.Since the addition of polyvalent solutes to a monovalent solvent metal increases the deformation faultprobability, the influence of the average number ofchemical valence electrons per atom on the stackingfault energy was considered. 14 ,23,24 Plotting ex as a function of electron concentration e/ a for the solutes tin(Sn+4) , indium (In+3) , and cadmium (Cd+2) in silverbase alloys, the values for ex of the indium and tin alloysfell on the same curve whereas those for the cadmiumalloys at higher concentrations deviated below thecommon curve for indium and tin alloys (Fig. 6).Interestingly, if the recent results for copper base-tinalloys20 are also plotted in Fig. 6, the values of afall on the same common curve as that for the silver-tinand -indium alloys. This phenomenological behaviorindicates the importance of both the solute specie andthe electron concentration on the occurrence of stackingfaults. However, the results for the silver-cadmiumalloys show that other variables (e.g., atomic misfit)should also be included to adequately describe thevariation of the deformation fault probability in coldworked fcc alloys with solute concentration.

I t is of interest to compare the values for compoundfaulting probabilities (1.5o:+f3)M+A obtained from peak38 C. S. Barrett, Structure of Metals (McGraw-Hill BookCompany, Inc., New York, 1952), p. 533.



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3458 R. P. I . A D L E R AND C. N. J. W A G N E R

'0" .40;.;:t:

30.. 0oL-a..= 20u:8'B 10E -..

' .-++

+

Alloy; Aq-CdAq-In+ Aq-Sn-Cu-Sno 1.0 1.1 1.2 1.3 114Electron Concentration, eja

FIG. 6. Variation of deformation fault probability'" as a function of electronconcentration e/a. Valuesfor copper base-tin alloystaken from reference 20 .

shift (a) and asymmetry ({:/) with the correspondingterm (1.5a+{:/)B calculated from anisotropic particlesizes (Tables II and III). Since the faulting probabilitieswere derived by two different methods (recognizing thelimits of accuracy involved), the essential agreement isan experimental verification of the theory and substantiates the experimental measurements.

IV. CONCLUSIONS(1) Alloying cadmium, indium, and tin with silverincreases the probability of stacking faults in these coldworked alloy filings with respect to pure silver.(2) Corresponding to the increase in stacking faultfrequency as a function of alloying, there is a significant

decrease in the effective particle sizes and the domainsize in the plane of the stacking fault.(3) Using the relative changes in lattice parametercalculated from the individual reflections, both deformation fault probability and lattice parameter of adeformed material can be determined.(4) After cold-working there was a detectable decrease in lattice parameter of the tin but not the indiumailoys.(5) Values of the compound faulting probability(1.5a+{:/) obtained from asymmetry and peak maximameasurements were consistent with the ones obtainedfrom anisotropic particle size measurements.

ACKNOWLEDGMENTThis work was conducted under contract with theOffice of Naval Research.

J O U R N A L O F A P P L I E D P H Y S I C S V O L U M E 3 3 , N U M B E R 12 D E C E M B E R 1962

Formation Conditions and Structure of Thin Epitaxial GermaniumFilms on Single-Crystal Substrates*BILLY W. SLOOPE AND CALVIN O. TILLERVirginia Institute for Scientific Research, Richmond, Virginia

(Received April 4, 1962)An experimental investigation of the effects of formation conditions on the structural characteristics ofthin Ge films vacuum deposited onto synthetic single crystals of CaF 2, NaCI, NaF, and MgO is reported.Formation conditions include substrate temperature during deposition, rate of deposition, and heat treatment. The amorphous to crystalline transformation of Ge was found to occur in the 300-350C substratetemperature range. It is shown that single--crystal films, 1500 A thick, can be formed on CaF2 substratesat temperatures between 4500 and 700C by proper choice of rate of deposition. Crystalline structure~ r o s i t y , complexity of imperfections, and film adhesion are dependent on the rate of deposition and d e p o s i ~ tlOn temperature,

INTRODUCTIONEXTENSIVE research has been carried out onvarious aspects of growth, structure, and physicalproperties of thin evaporated films. Many of thesestudies have made use of single-crystal films formed bydeposition onto single-crystal substrates, a processcalled epitaxy. In a previous study on epitaxial Agfilmsl it was found that formation conditions such as therate of deposition, deposition temperature, and annealing treatments controlled the microstructure as well asthe crystalline structure of the deposited film. Therefore, in order to produce films of a particular material* Supported by the United States Department of Defense.1 B. W. Sloope and C. O. Tiller, J. Appl. Phys. 32, 1331 (1961).

with specific structural characteristics, it is necessary tomake use of particular formation conditions. Thesestructural characteristics also depend upon the substrate on which the film is deposited and, at present, itis not possible to choose a substrate with assurance thata single-crystal film can be formed epitaxially. This isdue to the fact that a satisfactory theory of the mechanism of epitaxial growth has not been developed. Thisreport contains additional experimental information onthe relation between orientation and formation conditions which may be of value in understanding the epitaxial process.Recently, researches have been directed toward theuse of epitaxial semiconductor films because of their

adler, wagner - x-ray diffraction study of the effects of solutes on the occurrence of stacking...

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