The Critical Voltage Effectin High-Voltage

Electron' MicroscopyR.M. Fisher

U.S. Steel Corporation, Research LaboratoryMonroeville, Pennsylvania

C.G. ShirleyMotorola, Inc., Semiconductor Research

and Development LaboratoriesPhoenix, Arizona

Reprinted from JOURNAL OF METALS, Vol. 33, No.3, March, 1981, p. 26-30.

The Critical Voltage Effectin High-Voltage

Electron MicroscopyR.M. Fisher

U.S. Steel Corporation, Research LaboratoryMonroeville, Pennsylvania

SUMMARY

In addition to revealing the internal microstructures ofmaterials, the high-voltage electron microscope can also beused to characterize "invisible" features such as electronscattering, site occupancy, and atomic displacement ampli­tudes on a quantitative basis. The physical basis of theunique "critical voltage effect" is the occurrence of de­structive interference between current waves within thespecimen at a particular accelerating voltage. Potentialapplications to studies of random and ordered alloy solidsolutions largely depend on the magnitude of the size dis­parity between solute and solvent atoms.

INTRODUCTION

Research-oriented metallurgists are generally aware thatincreasing the accelerating voltage of the electron micro­scope can result in a significant increase in specimen pene­tration.1.2 This ability to examine thick specimens intransmission has been utilized extensively to study the detail­ed features of microstructures developed during transforma­tion, precipitation, and deformation, using either post­mortem sectioning or dynamic in-situ observations on pre­thinned specimens. At accelerating voltages above the dis­placement threshold for crystals, electrons in the beam aresufficiently energetic to knock atoms from their lattice sitesto create vacancies and interstitials. Study of the agglomera­tion of vacancies to form planar faults or voids has been amajor application of the HVEM for a number of years. Fi­nally, advances in electron microscope lens design andpainstaking attention to mechanical and electrical stabilityhave led to the development of very high resolution instru­ments which should produce atomic-level resolution in thenear future.

All these features of high voltage electron microscopyhave been discussed in detail in various AIME symposia onmetallography, as well as in previous special editions ofJournal of Metals which were devoted to microscopy.3.4However, there is one additional and unique aspect of high­voltage electron microscopy that is still not generally familiarto materials scientists, namely, the "critical voltage" ef­fect. 5•B This phenomenon takes its name from the fact thatsubtle but characteristic changes occur in the image and cor­responding electron diffraction patterns from crystals at cer­tain critical voltages which are determined by the composi­tion, structure, temperature, and other specimen factors.Careful determination of the critical voltage (Ve) can pro­vide quantitative information about departures from ran­domness in lattice site occupancy, i.e., clustering or short­range order, Debye temperatures, bonding, and charge trans­fer in alloys. These basic parameters of crystal physics areusually derived from x-ray or neutron scattering studies orelastic wave propagation experiments and are not consideredamenable to analysis by electron microscopy.

An important advantage of the critical voltage tech­nique over the other experimental methods is that it is notnecessary to work with large single crystals, as the mag-

26

C.G. ShirleyMotorola, Inc., Semiconductor Research

and Development LaboratoriesPhoenix, Arizona

nifying power of the electron microscope can be used to lo­calize the analysis on small grains (~1 /Lm) or alloy phases,or even selected areas within a larger grain if inhomo­geneity in composition is suspected. However, efforts tocapitalize on the Ve effect for alloy studies have been handi­capped until recently by the lack of a suitable theoreticalframework for interpreting such measurements in terms offundamental parameters.

The purpose of this brief article is to draw attention tothe salient features of the Ve effect-particularly in terms ofits limitations as well as the type of information which can(or cannot) be derived.

ORIGIN AND OBSERVATIONS OF THE Ve EFFECT

Detailed discussion of the basic electron scattering pro­cesses that give rise to the critical voltage effect and themathematical formalism is beyond the scope of this ar­ticle. 7•8 However, the general features may be of interest tosome readers, especially those who may already be aware thatupon entering a crystalline specimen the energy of the elec­tron beam, i.e., the current flow, is distributed among aseries of standing waves. These recombine at the exit sur­face to form the directly transmitted and diffracted beamswhich form the various "spots" in the diffraction pattern.The amplitude and symmetry of the various Bloch wavesdepend on the electron beam energy, i.e., the acceleratingvoltage of the microscope, as well as the crystallographicorientation, average atomic weight (composition), andtemperature of the specimen. At the "critical" voltage,pairs of highly excited Bloch waves may be equal in am­plitude but exactly out of phase, leading to complete ex­tinction of the corresponding diffracted beam, and producingrelated changes in the image.

In terms of diffraction physics, the criterion for the oc­currence of Ve is the point of contact between two branch­es of the dispersion surface at a Brillouin zone boundary. Thiscondition results when the values of the crystal structurefactors Fg , F2g , etc., bear a fixed ratio to one another. Itis only through the structure factors that Ve depends oncrystal parameters such as temperature, composition,short-range order (s.r.o), long-range order (l.r.o.), etc.

The consequences of destructive interference are only ap­parent within ±5% of Ve , which can range from 30 kV to3,000 kV (Table I). Under favorable conditions, Ve can bemeasured to ±0.5%. Examples of changes in the image anddiffraction pattern very near Ve are illustrated in Figures1 and 2.*

The diffraction process and the physics of the dependenceof structure factors on crystal parameters are completelydecoupled, and the former is quite accurately understood.For applications of the Ve effect, it is necessary to have amodel for the dependence of structure factors on crystal para-

~The critical voltage can also be measured using convergent beam diffraction techniquesIn HVEMs modified to operate in this mode. See references 19 and 20 for theoretical andexperimental papers on this method.

JOURNAL OF METALS· March, 1981

Figure 1. Illustration of the disappearance of the 400 bend contour in Fe·30%Cr at the critical voltage (285 kV). (a) 299kV;(b) 285 kV; (c) 275 kV.

Table I: Representative Values of Vc at 300K

.. Debye Temperature 9-11) in K.. * Indices of diffracting planes

Elements

The structure of the theory for elements is illustrated inFigure 3. Note that the item in any box is directly influencedby items connected by lines leading into the box. For ex­ample, the structure factors are determined by the latticeparameters, the mean-square displacements (commonlyrepresented by <112», and the atomic scattering factors,f. The atomic scattering factors depend on the detailedelectron distribution of each atom.

What is not shown in Figure 3 is the relative sensitivitiesof the influences. It turns out that Vc is particularly sensi­tive to variations in Fg at low scattering angles (correspond­ing to rearrangement of outer atomic electrons), so the tem­perature variation of Vc can give highly accurate values oflow-angle atomic scattering factors at reciprocal latticepoints as well as the crystal Debye temperature, en; or ifOn is regarded as known (from x-ray diffraction data), onlyroom temperature Vc data are needed. Deviations of meas­ured values of f at low angles from free-atom computedvalues give insight into the rearrangement of outer atomicelectrons in the crystalline environment. Many such de­terminations exist in the literature; two examples are givenin Table II.

kV

222 004**436 911279 555297 589316 60254 2210 110

220 004

264 ~1298

304 ~1294

35 7890 660

00.4 22.2

678 310236286

333 440

925 1028

fccAl (395 K)*Co (380 K)Ni (390K)Cu (320 K)Ag (220 K)Au (185 K)

beeCr (495 K)Fe (425 K)Mo (380K)W (315 K)

hcpMg (330 K)Ti (355 K)Co (380 K)

.diamond cubic

Ge (290 K)Figure 2. Illustration of the change in the symmetry of theKikuchi lines for Cu at the critical voltage.

APPLICATIONS OF THE Vc EFFECTS

Although Vc can be measured quite precisely, the experi­menter usually has available the temperature variation ofonly one or two critical voltages at (for alloys) one or twocompositions. Hence, the need for a few-parameter model ofthe structure factor. Such a model was immediately avail­able to interpret elemental Vc data, but a model was not atfirst available to interpret alloy data.

meters. In the following section, we give a heuristic descrip­tion of models of the structure factor tailored to interpreta­tion of Vc data.

JOURNAL OF METALS· March, 1981 27

Table II: X-Ray and Electron Sca«erlng Amplitude and 100 kVExtinction Distance Calculated for Free Atoms and Determinedfor Crystals From Vc Measurements·

ATOMIC SCATTERINGFACTOR

Reflection f x f e ~.AFe (bee)

110 (18.46) 18.42 (2.97) 3.01 (270) 286.7200 (15.26) 15.22 (2.11) 2.12 (395) 411.8211 (13.15) 13.28 (1.68) 1.67 (503) 534.6220 (11.64) 11.91 (1.41) 1.39 (606) 657.6

eu (fcc)111 (22.06) 21.75 (2.89) 3.02 (242) 285.8Figure 3. Schematic Illustration of influence of intrinsic crystal200 (20.67) 20.44 (2.60) 2.68 (281) 326.3

I:'parameters (Debye temperature - 8 D and atomic scattering220 (16.79) 16.69 (1.91) 1.93 (416) 472.5factor - fe) and experimental parameters (specimen tempera-311 (14.79) 14.76 (1.62) 1.62 (505) 579.0ture and orientation) on the critical voltage for elements.

Measurements of Vc at several temperatures have now provided*f~ for x-rays; fe for electrons (free atoms) - crystalaccurate values of 8 D and fe for most elements.

FIXED lLLOY PARAMETERS . ."DE~YE TEMP INTERPOLATION PARAMETER (TAU)=STATIC DISPL A~PLfTUDE FACTQR(GAM~A) =VEGAROS LAw DEVIA ION PARA~ETtR (6) =110 SYSTEMATICS uF BCC

Table III: Representative ComputerPrintout Illustrating Typical Values ofParameters Used in the Calculations

FOR FF _ ._DEBYE TEMPERATJRt=AhlMIC wE11>HT _ =LATTI~E PARAMETER=

FOR S1 _ ,DEBYE TfMPERATJRE=ATOMIC WEIGHT =EFFEt.TIVE LAT PAR=

424.000 ([)EG K):'5.84'72.b664 (ANGSTQOMS)

~OO.OOO WEG K)2b.ObF,c.4b7c (ANGSTROMS)

FE. I

(OEG K)(At\lGS)

(MOLE FR OFCANGS**21(ANGS**2)CANGS**21(ANGS**2)(ANGS**21<DEG K)

(KVI(KVICr< V)

KVKV/OEG CKV

2 .. 5.002.8557

.0,+61

.90001.5631E-021.5831E-021.4627E-021.2044E-031.2044E-03418.36

.0'00

.000

.000

=

=

=

=

=

= 341018j09.59306.73

1.934bE 01= -2.1270E-Ol= -2.9053E 00

=

TEMPERlTURELATTICE PARAMETeRE.TACOMPOSITIor~MSD OF Fi=.: ATOMSMSO OF SI ATOMSTHERMlL MSD OF FE ANU SI ATOM~=STATIC MSD OF FE ATOMSSTATIC MSD OF 51 ATUMSALLOY ~E8YE TEMPERATURESRORT-RANGE ORUER PARAMETERS

ALPrfAIALPrfA2ALPrfA3

CRITICAL VOL TAGE(1) 3-8EA~ ESTIMAtE(2) AETHE ESTI~ATE(]) 17-8EAM ESTIMATE =

CRITICAL VOLTAGE SENSITIvITIESDVC/OCOMPOSITI~NDVC/DTEMPERAT0REDVClDALPHAI

Figure 4. Schematic representation of in­fluences of various crystal parameterson the critical voltage for alloys.Measurements of Vc for a range ofcompositions, specimen temperatureand orientation can yield accuratevalues of the Intrinsic parameters forthe alloy systems (r,y and f~,

f~) and a quantitative value of thestate of order (ex or S) for particularspecimens. ,

,I1

1

1

I,,L ...J

1,I

____ J

28 JOURNAL OF METALS • Marc~,c 1981

Table IV: Examples of the Effect of Alloying on Vc*

*Illustrates magnitude and direction of Vc changes with alloying for room temperaturemeasurements.

Table V: Examples of the Effect of Short Range Ordering orClustering on Ve (400)*

Alloy Ye(kY) (a ' =O) AYe (a,=0.2)

Aus• Cus. 237 -69Ag,s Mn,s 265 -0.16COBS Fe,s 558 -1.1Cu•• Sn lO 464 -120Crs• Mos 867 20.8FeB. Si,. 1342 8.8VB. Pd,. 1068 6.4

Solute (10 mol.%)

Si 346.0 kVAl 345.5Ni 316.7Au 182.0Sn 156.0

Al 620.9 kVCu 591.5Fe 586.2Mo 533.5Au 434.5

Al 320.0 kVSi 314.0Co 307.0Cr 296.5Mo 235.0

Solvent

Cu Ve (222) 316 kV

Fe Vc (220) 304 kV

Ni Ve (222) 589 kV

Disordered Solid SolutionsFor solid solutions, the structure factors depend not only

on the time-varying vibrational (thermal) mean-squareatomic displacements (m.s.d.) but also on static m.s.d. dueto atomic radius differences. The structure of this more com­plicated model is shown in Figure 4; details are given inreference 11. In general, the two-model parameters T andI' may be determined by analysis of the temperature varia­tion of Vc at one composition in a fully disordered alloy.From a theoretical standpoint, T and I' are expected in thesimplest model to have values of unity. This expectation hasbeen used in two ways: I' and T fitted to a system's Vc datamay have deviations from unity which can be interpretedphysically, whereas for unknown systems, I' = 1, T = 1 givesa best estimate for the sensitivity of Vc to crystal parameters.

A computer program embodying this theory has been writ­ten, a sample printout of which is shown in Table III. Theprogram has been used to compute the sensitivity of Vc tocomposition (via thermal and static m.s.d., lattice parame­ters and explicitly structure factors; see Figure 4) ; someexamples are given in Table IV. The short-range orderdependence of Vc comes from the thermal and static m.s.d.and, in principle, from the atomic scattering factors, al­though the theory does not encompass this last effect. Foralloys with large atomic radius difference, the eff~ct throughthe static m.s.d. is dominant, especially in f.c.c. alloys (e.g.,CuAu, NiAu, CuSn); see Figure 5. Sensitivity of Vc tos.r.o. is generally less in bee alloys. Some examples of Vc

sensitivity to s.r.o. are shown in Table V.When the atomic radius difference is negligible, the static

displacements and s.r.o. (since its effect on thermal m.s.d.is weak) can be dropped from Figure 4, giving a simplifiedone-parameter (T) model. An example of such a single­parameter fit to Vc data is shown in Figure 6. The case ofFeCr is one in which electron transfer effects (dotted lines inFigure 4) are not obscured by static m.s.d. -this could be aninteresting system to examine more carefully. When staticm.s.d. are appreciable, one must use a more complicatedtwo-parameter fit to I' and T. The essential idea is the same,but too complicated to discuss here. See reference 10 for ex­amples.

Ordered AlloysFor ordered alloys, the structure of the theory is similar to

that for disordered alloys with short-range order, but withthe additional feature that the long-range order parameterexplicitly affects the structure factors. 14 Examples of theeffect of l.r.o. on Vc are shown in Table VI. Generally, theeffect of Vc on l.r.o. is smaller than theory predicts. IS - 17

In long-range-ordered structures the effect of S and/ or com­position on the atomic scattering factors and bondingstrengths (not explicitly taken into account by the model)may have a major effect on Vc .18 That is, the dotted linesin Figure 4 are actually major influences. This will be a fruit­ful area for future study.

SUMMARY AND DISCUSSIONDiligent readers who have persisted to this point should

now be aware of the existence of a unique critical voltagephenomenon and appreciate the manner in which it can beused to obtain quantitative information about alloys. A fewmay wish to look up some of the key references (es­pecially No. 10) to examine the theoretical aspects in detailand perhaps be inspired to gain access to one of the nation'sHVEMs and make some measurements on alloys of specialinterest. The effort could be very rewarding, especially ifpains are taken to do such a study thoroughly, i.e., makeVe measurements at several different temperatures andspecimen orientations and for a range of alloy composi­tions and states of short-range order or clustering.

*al is Cowley-Warren first nearest neighbor occupancy parameter 0 for random solidsolution, ±O.25 for maximum short-range ordering or clustering. _Crystal oriented for400 reflections.

Table VI: Effect of Long Range Ordering on the Critical Voltage

Alloy S=O* S=l**

CU3Au 222 -166 -175400 -381 -410

Ni3Fe16 222 -272 -280FeCo'6 220 -314 -314Ni,Mo" 400 -460 -490

·S=G-random solid solution.* S=l-perfect long range order

The purpose, of course, will be to gain new insight into thefundamental nature of alloys, with a view to discovering newcombinations with unusual magnetic, electronic, elastic, ortransport properties. One such alloy could improve on or sub­stitute for alloys which utilize components already in shortsupply.

JOURNAL OF METALS· March, 1981 29

,-----------------------,340

260

330

8040 60mole % Cr.

20'-----'---'--_.L-_-'----_-'----_--'---_--'---_-'--_-'--_-'---------J 250

CrFe

1.6

1.4

~ 1.2b~

'0 1.02J'c

0.82-

'"-3 0.6

0.4

0.2

Ni(O.8) AU(O.2)

100

300

Figure 5. Illustration of the reduction of the V, for Ni at K (595kV) due to 1) thermal displacements at room temperature;2) alloying with 20% Au; and 3) aging to produce maximumclustering.

Figure 6. Illustration of the change in V,. with compositionfor Fe-Cr alloys. The parameter T is related to the spring con­stants between adjacent atoms.

ABOUT THE AUTHORS

ACKNOWLEDGMENTS

A number of present or former staff members, as well asvisiting scientists, at the U.S. Steel Research Laboratoryhave participated in critical voltage research studies. Theseinclude J.S. Lally, A. Szirmae, L.E. Thomas (now at Han­ford Engineering Development Lab.), A.J.F. Metherell(Cambridge University), and C.J. Humphries (OxfordUniversity).

References1. R.M. Fisher, "Electron Microscoy-l Million Volts and Beyond," Research and De­velopment, 23 (1972) p. 18-22.2. G. Thomas and M.J. Goringe, Transmission Electron Microscopy of Materials, Wiley­lnterscience, New York, 1979.3. "High Voltage Electron Microscopy," TMS Symposium, AIME 109th Annual-Meeting,and J. Metals 31 (12) (1979) p. 37 and 57.4. G. Thomas, "High Voltage and High Resolution Electron Microscopy in MaterialsScience," J. Metals 29 (2) (1977) p. 31. (Special issue on advances in electron microscopy)5. F. Nagata and A. Fukuhara, "222 Electron Reflection from Aluminum and System­atic Interaction," Japan J. Appl: Phys. 6 (1967) p. 1233-1235.6. R. Uyeda, "Dynamical Effects in High Voltage Electron Diffraction," Acta CrysLA24 (1968) p. 175·181.7. J.S. Lally, C.J. Humphreys, A.J.F. Metherell, and R.M. Fisher, "The Critical VoltageEffect in High Voltage Electron Microscopy," Phil. Mag. 25 (1972) p. 321-343.8. L.E. Thomas, C.G. Shirley, J.S. Lally, and R.M. Fisher, High- Voltage Electron Mi­croscopy, Academic Press, Lorrdon, 1974, p. 38.9. A.J.F. Metherell and R.M. Fisher, "Consequences of Bloch's Theorem on the DynamicalTheory of Electron Diffraction Contrast," Phys. Stat. Sol. 32 (2) (1969) p_ 551-562.10. e.G. Shirley and R.M. Fisher, "Atomic Mean-Square Displacements and the Crit­ical Voltage Effect in Cubic Solid Solutions," Phil. Mag. 39 (1979) p. 91-117.II. K. Kuroda, Y. Tomokiyo, and T. Eguchi, Proceedings of the 5th [ntl. Conf on HVEM,Kyoto, Japan, 1977.12. KP. Butler, "Critical Voltage Analysis of Precipitation in Certain Nickel-Base AI·lays," Phys. Stat. Sol. a (18) (1973) p. 71·83.13. K. Kuroda, Y. Tomokiyo, and T. Eguchi, "Temperature Dependence of Critical Volt­ages in Cu-Base Alloys," Proceedings of the 6th [ntl. Conf. on HVEM, Antwerp, 1980,p. 112-115.14. C.G. Shirley and R.M. Fisher, "Application of the Critical Voltage Effect to AlloyStudies," Proceedings of tileBth Intl. Conf. on HVEM, Antwerp, 1980, p. 88.15. Y. Tomokiyo, K. Kuroda, H. Matsubata, and T. Eguchi, "Critical Voltage Effectin Ordering Alloys," Proceedings of the 6th [ntl. Conf. on HVEM, Antwerp, 1980, p.108-111.16. R. Sinclair, M.J. Goringe, and G. Thomas, "Application of the Critical Voltage Effectto the Study of Ordering in Alloys," Phil. Mag. 32 (1975) p. 501·512.17_ A. Fox, "The Critical Voltage Effect in B 2 Alloys," Proceedings of the 6th [ntl. Conf.on HVEM, Antwerp, 1980, p. 104-107.18. C.G. Shirley, J.S. Lally, L.E. Thomas, and R.M. Fisher, "High Voltage ElectronDiffraction Measurements of Debye Temperatures of Cr-Fe and Their Disordered Alloys,"Acta. Cryst. A31 (1975) p. 174-177.19. J.W. Steeds, "Information About the Crystal Potential From Zone Axis Patterns,"Proceedings of the 6th [ntl. Conf. on HVEM, Antwerp, 1980, p. 96-103.20 ..J.R. Sellar, D. {meson, and C.J. Humphreys, "Experimental and TheoreticalStudy of the Convergent-Beam Critical Voltage Effect in High Voltage Electron Dif­fraction," Proceedings of the 6th [ntl. Conf. on HVEM, Antwerp, 1980, p. 120~123.

R.M. Fisher is senior research consultantin Corporate Research, with U.S_ Steel Re­search, Monroeville, Pennsylvania. He re­ceived his BSc from the University of Michi­gan in 1949, and then joined U.S. SteelResearch in Kearny, New Jersey. He receivedhis PhD from the University of Cambridgein 1962 under the supervision of Professor IA. H. Cottrell (F.R.S.). He is an Overseas Fellow ifof Churchill College, Cambridge, and has served as president ofthe Electron Microscopy Society of America. His research activ­ities have largely centered on the characterization of materialsby electron microscopy and other analysis techniques, leading tonumerous technical and scientific publications.

C. Glenn Shirley is senior scientist at theMotorola Semiconductor Research and De­velopment Laboratory, Phoenix, Arizona. Hereceived his BSc (1969) and MSc (1970) fromthe University of Melbourne (Australia), andhis PhD in physics from Arizona State Uni­versity in 1974. He was a Post-Doctoral Fel­low at Carnegie-Mellon University from1973-76, and a visiting scientist at U.S.Steel Basic Research until he joined Motorola in 1977. His in­terests include theoretical and experimental analyses of latticeproperties of materials, currently with emphasis on electronicproperties of thin films. He has published 12 papers in thesefields.

30 JOURNAL OF METALS' March, 1981