Simulation and interpretation of STEM side-band holograms

Ultramicroscopy45 (1992) 281-290 North-Holland

nlilh~vmil~rkoa~un~m

Simulation and interpretation of STEM side-band holograms John Konnert and Peter D'Antonio Laboratory for the Structure of Matter, Naval Research Laboratory, Washington, DC 20375, USA

Received 17 June 1992

Simulations of side-band holograms produced with a STEM equipped with a coherent, point electron source and a beam splitter are presented and interpreted. Holograms are calculated for various electron voltages, beam separations and thicknesses of GaAs [110]. The results illustrate features in a hologram associated with a stable, well-adjusted microscope. The simulations also provide data on the dynamic range and resolution necessary for recording the patterns and indicate techniques for removing from the reconstructed images interferences from unscattered beams.

I. Introduction

Image reconstruction using in-line holograms was first suggested by G a b o r [1,2] as a means for improving the resolution of electron microscope images. Such a hologram is produced by the interference of the direct beam transmitted through the sample with the wave scattered from the sample. Lin and Cowley [3] demonstrated with STEM data that some resolution enhancement is possible; however, the presence of a defocused, aberrated, conjugate image along with the desired image limits the usefulness of the technique. Lichte [4] has made progress with the use of an electrostatic bi-prism to split the beam in a T E M into two components, one of which passes through the sample, and has obtained ultra-high-resolution side-band holograms [5]. In a similar spirit, Cowley [6] has proposed a scheme for the use of a bi-prism in a STEM to produce side-band holograms possessing 1 .~ resolution or better. Again, two focused beams are to be produced at the sample level, one passing through the sample, and the other passing through the vacuum adjacent to the sample. Cowley also indicated how a second bi-prism could be used after the sample to decrease the apparent separaElsevier Science Publishers B.V.

tion of the beams and correspondingly increase the spacing of the interference fringes. This paper presents computer simulations of holograms produced by such a STEM. Some of the effects of accelerating voltages, b e a m separations and specimen thickness have been investigated for a GaAs [110] sample. It is hoped that the results will be useful to the experimentalist in several respects. Features may be identified in the holograms that indicate the size, coherence and astigmatism of the beams. These features should prove useful in the adjustment of the microscope. Analyses of the simulated holograms suggest small additions to the procedure suggested by Cowley [6] and Gribelyuk and Cowley [7]. The isolation of the component of interest in the reconstructed image requires that the component due to the unscattered wave passing through the sample be removed. Also, for small apparent beam separations and extremely thin samples, the interfering components arising from the "tails" of the beams, weak intensity distance from the beam centers, must be removed. The p a p e r first lists the equations used to compute holograms for a sample of arbitrary thickness. It then presents computed holograms, along with illustrations and discussions of the visible components in these

282

J. Konnert, P. D'Antonio /STEMside-band holograms

holograms. Finally, components of the reconstructed images are presented and discussed.

Ap(u) is the aperture limiting the angle of beam convergence. The UBs(U) component at the exit surface of the sample is:

2. Equations

N 1/)'BS(L/) = B ( u , aexit .... face,ro=rs) I-I r e a l [ Q , ( 0 ) ] . n=l

It is useful to factor the exit surface wavefunction, U(u), into components corresponding to the wave scattered from the sample, qts(U), the beam passing unscattered through the sample, UBs(U), and the beam passing through the vacuum and displaced from the sample, UBD(U).

U ( u ) = Us(U ) + UBs(U ) + qtBD(U ).

(1)

For a model represented with N slices, the sample exit surface wavefunction, UN(U) = Us(U) + UBs(U), may be computed with a conventional multislice calculation [8]. The wave leaving the ( n - 1)th slice is converted to the wave leaving the nth slice.

U,,(u) = [U,,_,(u)P,,_,(u)]

• Q,,(u).

(2)

The operation is carried out ( N - 1 ) times. The convolution operation is indicated with * between terms, the phase grating for the nth slice is Q,(u), and the propagation function is P(u). The incident beam wavefunction at the entrance surface of the sample convoluted with Ql(u) is qtl(U). For the computations reported here, ~Bs(U) will be defined later as the component of U(u) that is a function of only the Q,(0) phase grating terms. It will, thus, contain a component that is scattered back from the diffracted beams. For a point electron source, the beam wavefunction at the sample, B(u, 6, ro), for a beam centered at r 0 with defocus 6, is taken to be the objective lens contrast transfer function.

B(u, 6, r0) = A p ( u ) e [-ixu')] e (-2~ri''r°)

(3)

(4) If all slices are identical, as is the case for the computations described in the paper, qtBs(U) is the beam wavefunction computed for the defocus of the exit surface of the sample times the real part of Q(0) raised to the Nth power. The wavefunction for the displaced beam at the plane of the exist surface of the sample is: 1/.tos(//) = n ( u , 6exit.... face,r0--rD)" The

diffraction pattern,

the

(5) hologram,

u(,)U*(u) = [ U s ( r ) + UBs(U ) + UBD(U) ]

x [,/,~(r) + UBs(-) + % ° ( . ) ]

*

= I UBs(U) I 2 + I UBD(U) I 2 + I qts(U) I 2

+ %s(U)~e~D(u) + + %s(U)%*(u)

U~s(U)%D(u)

+ U~s(u)Us(u)

+ q%D(U)U~(u) + U~D(U)Us(U )

x(u) = ~r6,~ I u 12 +

(6)

where the superscript * indicates a complex conjugate. The hologram may also be grouped into components, the details of which are apparent in the pattern and should serve to adjust the microscope and evaluate the quality of the hologram. Constant component

= I UBs(U) 12 + I U ~ o ( u ) 12 where

is

%,u*(,).

(7)

B e a m - b e a m component ½'nCsA31 u 14

l ul = 2 sin(O)/A, O is the angle between the center of the main beam and the scattered electrons, A is the electron wavelength, C~ is the objective lens spherical aberration coefficient and

= %s(U)q,~o(u) + U & ( u ) % o ( u ).

(8)

In-line hologram component

= lUs(u ) 12 + UBs(U)Us*(U) + U~s(U) Us(U ) .

(9)

J. Konnert, P. D'Antonio / STEM side-band holograms

Sample-displaced beam component =

+

(10)

3. Visible features in side-band holograms

Modified versions of the ASU multislice programs [9], in which the convolutions in reciprocal space indicated in eq. (2) are carried out as multiplications in real space, were used to evaluate the electron wave as it passed through the sample. Since it was desired both that the wave be sampled at a fine increment in real space ( ~ 0.2 A) and that the two electron beams could be separated from eacho other and their periodic images by at least 50 A, the computations were carried out with 512 x 512 arrays. The slice thicko ness was taken to be 3.96 A, the periodicity of GaAs parallel to the beam direction [110]. Atoms corresponding to a small particle were placed in a cell 112.5 ,g, x 112.5 ,~ x 3.96 A and the scattering potential was calculated at (112.5 A / 5 1 2 = 0.22 ~,) intervals. The amplitude of the scattering potential for ¼th of the unit cell is shown in fig. la. Fig. lb is an enlargement of the center of fig. la. Fig. 2a displays the hologram computed for a 48 A, 12-layer-thick sample of GaAs [110]. Figs. 2b-2d illustrate the components of the hologram. For the computation of this hologram one beam was centered at the middle of fig. la (x = y = 0.0), and the second, displaced beam was centered at r D (x = - y = 0.1), a separation of 11 A. While actual experiments would probably be carried out with a larger beam separation, Cowley [6] has pointed out that a second bi-prism could reduce the apparent separation to this value. The diffraction phase grating for this slice of atoms, which is the Fourier transform of the scattering potential, consists of a square array of Bragg reflections with a spacing of (1/(112.5 A,)= 0.00885 A - l ) . The accelerating voltage was 300 keV (A = 0.0917 ,~), the defocus of the beam was - 5 2 5 ~, at the entrance surface of the sample, C s equals 0.85 mm and Ap has a radius of 1.0 A-~. The beam was propagated through the sample using scattering vectors of magnitude less than

283

1.0 ~ - i for the incident beam and the 192500 beams of magnitude less than 2.2 A - I for the scattering potential of the layers. The visible features in this hologram may be associated with the components identified in eqs. (8,9) and (10). The concentric rings in the hologram belong to the in-line hologram component, fig. 2b. The circularity of these rings reflects the focus of the beam passing through the sample and the adequacy of the correction for astigmatism. This may be seen by looking at fig. 3a which illustrates the imaginary part of the unscattered beam at the o exit surface, 3 = - 4 7 7 A, of the 12-layer-thick sample. The real space amplitude of the same beam is shown in fig. lc. Since the intensity is taken as constant for all angles within Ap(u), the distance between rings in fig. 3a indicates the range in u over which the phase of the wave changes by 27r. The phase changes quite graduo 1 ally for u values b e t w e e n 0 A 1 and ~ 0 . 5 A . The maxima become progressively closer reaching a spacing of ~0.02 ,~-~ at 1.0 ,~-~. Were astigmatism present, (5 would vary with the angle of the scattering vector, and the rings would not be circular. Since it is necessary to know the phase of the incident electron waves out to the limit of the desired resolution, it is desirable to be able to detect the maxima and minima of the rings out to that resolution. Thus a 200 x 200 detector array would be adequate to obtain 1 resolution data with a 300 keV beam. The parallel fringes visible in the hologram are the beam, beam component depicted in fig. 2c. They indicate both the interbeam spacing and the equivalence of the two beams. If the beams had different defocus values, the interference fringes would be curved, rather than straight. As the interbeam spacing increases, the fringes move closer together. Finally, some of the features of the sample, displaced beam component, fig. 2d, are visible in the hologram. It is this component from which the side-beam image is reconstructed. Not all of these components would be easily observed for an extremely thin sample. Fig. 4a displays the hologram calculated with the same incident beam as for fig. 2a for a single layer of atoms. Individual components of this hologram

284

J. Konnert, P. D'Antonio / STEM side-band holograms

are very similar in features to those in fig. 2 and are not pictured again. However, their relative magnitudes differ. Whereas all components are visible in the hologram, fig. 2a, for the 48 .~ thick sample, the only features readily observed in the SCATTERING

h o l o g r a m for t h e 4 ,~ thick s a m p l e , fig. 4a, are the fringes associated with the beam-beam comp o n e n t . T h i s is b e c a u s e a very small f r a c t i o n of t h e b e a m has b e e n s c a t t e r e d by t h e t h i n s a m p l e . T h e largest f e a t u r e s in t h e d i s p l a c e d b e a m c o m -

POTENTIAL

SCATTERING

(a)

' ' ' ~ /

POTENTIAL

...............

~ ' ' '

ib')J_ 3

L

i4 2v(k)

- 20 i.@O

- 10

-0

L

1.0

i

!o

L

.

@

--10

L-1

--20

I

--2

,, i--3

,i,,,Jlltl ,,, i

i

-20

-10

i

i

b

0

10

20

-3

-2

300 keV B E A M

-1 100

0

1

2

3

keV B E A M

3 ~(,~) 2

2

-

0

7

,

,

i

i

~

1

i

-3

-2

-1

0

1

2

3

x(.~)

-3

-2

-1

0

1

2

3

x(~.)

Fig. 1. (a) Amplitude of the scattering potential for a 3.96 A. thick slice of GaAs [110]. One fourth of the cell employed for the computations is shown. (b) Enlargement of the center of (a). (c) Real space amplitude of unscattered, 300 keV electron beam wavefunction at the exit surface of the 12-layer-thick sample. (d) Real space amplitude of unscattered, 100 keV electron beam wavefunction at the exit surface of the 12-layer-thick sample.

J. Konnert, P. D'Antonio / STEM side-band holograms

285

a r b i t r a r i l y a s s u m e t h a t r a n g e to be 25-fold, we c o n c l u d e t h a t r e s o l u t i o n o f single a t o m s would r e q u i r e t h a t t h e intensities b e r e c o r d e d over a r a n g e o f at least 16 × 25 o r 400. E x t r a n e o u s backg r o u n d intensity p r e s e n t in real d a t a a n d / o r lighter e l e m e n t s indicates t h a t a d e t e c t o r r a n g e o f at least 1000-2000 w o u l d be d e s i r a b l e . T h e req u i r e m e n t s w o u l d b e less d e m a n d i n g for the 12layer s a m p l e in which c o l u m n s o f 12 a t o m s are superimposed.

p o n e n t at s c a t t e r i n g angles n e a r 1 A - J a r e app r o x i m a t e l y ~6 o f t h e intensity r a n g e o f the visible beam, beam component of the hologram. T h e relative sizes o f the f e a t u r e s in t h e onelayer h o l o g r a m a r e significant for the e x p e r i m e n t t h a t seeks to resolve single a t o m s as might b e the case for d i s o r d e r e d regions o f s a m p l e . T h e samp l e - d i s p l a c e d b e a m c o m p o n e n t o f the h o l o g r a m s h o u l d be r e c o r d e d a c c u r a t e l y over a r a n g e of intensity at the limit o f resolution, l ,~-1. If we

(C

1A

t

Fig. 2. (a) STEM side-band hologram computed for a 48 ,~ (12 layers) thick sample of GaAs with one 300 keV beam centered at (x = 0, y = 0) of fig. la and another beam centered at (x = 0.1 (11 ,~), y = -0.1 ( - 11 ,~)). (b) In-line hologram component of (a). (c) Beam-beam component of (a). (d) Sample, displaced beam component of (a).

286

J. Konnert, P. D'Antonio /STEMside-band holograms

A hologram is computed for 12 sample layers, with all conditions identical to those of fig. 2, except that the displaced beam is at (x = - y = 0.333) for a beam separation of 53 .~. The separation of the interference fringes is 0.019 , ~ - 1. This is approaching the maximum beam separation usable when the hologram is recorded as a square array with spacings of 0.0088 A-1. Thus, a 53 ,~ beam separation requires about the same detector resolution as the resolution of oscillations in the incident b e a m wavefunction for a 300 keV beam at a resolution limit of 1 ,~. Detector requirements would be much more demanding for a 100 keV beam. Figs. ld and 3b illustrate a 100 keV beam at Scherzer focus (3 = - 6 7 5 A) for C S equal to 0.85 mm. The oscillations begin at ~ 0.35 .~-~ and have a spacing of 0.002 A - ~ at 1 ,~-~. The details without rotational symmetry are due to the inadequacy of sampling the pattern at 0.0088 A-~ intervals. (It may be noted that Lichte [10] has shown that Scherzer focus is not the optimum focus for holograms obtained with a T E M where small incident beam size is not a consideration. In that case a focus is chosen for which the oscillations in the

transfer function begin at a low scattering angle, but do not become extremely rapid until a much higher value is reached.) Fig. 4c displays a hologram computed for 48 A thick sample with this beam. The oscillations are much more rapid at large angle than can be portrayed with the resolution at which the pattern was computed. A 2000 x 2000 detector array would be necessary to resolve the intensity oscillations at 1 ,~- ~ resolution with a 100 keV beam.

4. Image reconstruction Image reconstruction is carried out as described by Cowley [6] with some variations designed to isolate the component of the image associated with the sample. The procedure involves three steps. (1) The average value for the intensity is subtracted from the hologram. This removes the constant component, eq. (7), from the hologram. Were this component not removed, the "tail" from the Fourier transform of these features could mask individual atom details in the reconstructed image. (2) The remaining hologram

Fig. 3. (a) Imaginary part of 300 keV electron beam wavefunction at the exit surface of the 12-layer-thicksample. (b) Imaginary part of 100 keV electron beam wavefunction at the exit surface of the 12-layer-thicksample.

287

J. Konnert, P. D'Antonio /STEMside-band holograms

2O Y(A) 10

o,

~

.,

~

/

> '~

-20

ill

2,

e

-10

0

10

"

-10

~-~

--20

20

X(A) Fig. 5. Reconstructed image with side-bands computed for a 4 A (one layer) thick sample of GaAs with one 300 keV beam centered at (x = 0, y = 0) of fig. la and another beam centered at (x=0.1 (11 A), y = - 0 . 1 ( - 1 1 ,~)). One fourth of cell used for computations is shown.

is m u l t i p l i e d by t h e w a v e f u n c t i o n o f t h e d i s p l a c e d b e a m c o m p u t e d in t h e p l a n e of t h e exit s u r f a c e o f t h e s a m p l e . F o u r i e r t r a n s f o r m a t i o n of t h e r e s u l t ing f u n c t i o n n e a r l y isolates t h e w a v e f u n c t i o n at t h e exit s u r f a c e o f t h e s a m p l e . (3) T h e i m a g e c o m p o n e n t a r i s i n g f r o m t h e e l e c t r o n s t h a t pass u n s c a t t e r e d t h r o u g h t h e s a m p l e is s u b t r a c t e d . Fig. 5 displays t h e a m p l i t u d e of t h e F o u r i e r t r a n s f o r m o f t h e h o l o g r a m s h o w n in fig. 2, a f t e r t h e a v e r a g e v a l u e has b e e n s u b t r a c t e d a n d t h e r e s u l t m u l t i p l i e d by t h e w a v e f u n c t i o n for t h e disp l a c e d b e a m . T h e f e a t u r e s at t h e c e n t e r a r e d u e p r i m a r i l y to q J s ( r ) + q % s ( r ) , w h e r e ~bs(r) a n d ~bBs(r) a r e t h e F o u r i e r t r a n s f o r m s of qts(U) a n d

Fig. 4. (a) STEM side-band hologram computed for a 4 ,~ (one layer) thick sample of GaAs with one 300 keV beam centered at (x =0, y = 0) and another beam centered at (x = 0.1 (11 ,~), y = -0.1 ( - 11 ,~)). (b) Hologram computed for a 48 ,~ (12 layers) thick sample of GaAs with one 300 keV beam centered at (x =0, y = 0) of (a) and another beam centered at ( x = ~ (38 A), y = ~ ( - 3 8 ,~)). (c) Hologram computed for a 48 ,~ (12 layers) thick sample of GaAs with one 100 keV beam centered at ( x = 0 , y = 0 ) of (a) and another beam centered at (x = 0.1, y = 0.1).

J. Konnert, P. D'Antonio / STEM side-band holograms

288

)/' / /f~

~I,

~ /

,' >':. / ]

':,j ( (b):

, ,

'( H"I 'I ,

"~

CO, J' <.~. . ./

._/~x__.~

t

.. / s

.00

(\

{

.O0~:

f

~J

/

/

'

,"

]

,

F

L I

}

w--, L

. Cc2

•-'oe

•-'\ •

~ 2 ~

/

_.

/

1

/

") ' ~

I

L_

/

(;

.03

/1 L • 006

-

,

~

,

, /¢-~1

i

i

i/

i

: . OCl~

i

i

i

,

i-,

~

~/ i-- T- i -T

q

/ /

I

,'

"

•

" ~--

'

tr\

o

, i ,'(,(~,1, l ' ~

(A)

,

-~ 2

'~- "x

"J

\

'>(

,,/;

1

/",

I,/ /

)

¢

/~

!;

7

F',

I

o

f ,~,/

} 'i" " "

', /

-1

/~

f

i ,

/

\

f-

;

/

f

-;

,(£5

o

c o

i

I\x I \

"'1

\

"

0~0

I

J

/

~

, /

,,

-2

-1

0

•

i

-3

1

q

/

c-~>

2

i

~

r

i

i

t

t

i

t

I

\ I

'."

/ I

I

',

! J

0

I

L •',

/ I

~

i I

h

3

x(X) Fig. 6. (a) Real space amplitude of the scattered component at the exit surface wave of the 12-layer-thick sample. 300 keV beam centered at (x = 0, y = 0) of fig. la. This information is the goal of the reconstruction based on the hologram in fig. 2a. (b) Real space amplitude of the exit surface reconstructed from fig. 2a including the unscattered beam passing through the sample and "tails" from the details centered at (x = 0.1 (11 ,~), y = - 0.1( - 11 J,)) and (x = 0.2 (23 fi0, y = - 0.2 ( - 23 A)) in fig. 5. (c) Same as (b) with the unscattered beam passing through the sample removed. (d) Same as (c) with average hologram intensity subtracted before image calculation•

J. Konnert, P. D'Antonio / STEM side-band holograms

~ - - ~

.

.

.

.

.

.

.o~o

.

.

.

.

.

.

.

.

.

.

"

~\~

.000

-3

•

....

-2

I~_,,

. . . . . . . . . . . .

-1

0

X(J.)

:

/

/

' ////

//

"~\~

i

/

i /~',I

i

p

i

i(

,'\A'\'-~>

i

i

-K~/ '

L

L 000

,,,('

i ml-

ta5

.

'J

~ •

289

1

t,,

\\~,,~t

"l/

"

(W~D]'$)]},))il///)/' / t~/

L 000

,_q . . . .

2

3

Fig. 7. (a) Real space amplitude of the scattered component at the exit surface wave of the one-layer-thick sample. 300 keV beam centered at (x = 0, y = 0) of fig. la. This information is the goal of the reconstruction based on the hologram in fig. 2a. (b) Real space amplitude of the exit surface reconstructed from fig. 3a including the unscattered beam passing through the sample and "tails" from the details centered at (x = 0 . 1 (11 J,), y = - 0 . 1 ( - 1 1 ,~)) and (x = 0 . 2 (23 A), y = - 0 . 2 ( - 2 3 A)). (c) Same as (b) with the unscattered beam passing through the sample removed. (d) Same as (c) with average hologram intensity subtracted before image calculation.

290

J. Konnert, P. D'Antonio /STEMside-band holograms

I/-rBs(U). The other features are grouped about r = r D and r = 2r D (see Cowley [6]). The goal of the image reconstruction will be to obtain the best possible representation for ~bs(r). For the sake of brevity, we will examine only the amplitude of the relevant functions, rather than the full complex functions. Fig. 6a displays the amplitude of ~bs(r) computed for the 12-layerthick sample. We seek to obtain this function as accurately as possible from the hologram in fig. 2a. Fig. 6b displays the result obtained by Fourier-transforming the hologram in fig. 2a after having multiplied it by the wavefunction of the displaced beam. As mentioned, the main features are associated with ~/,s(r) and 0BS(r). Also present are smaller details associated with the major details centered at r = r D and r = 2r D. Fig. 6c is obtained by subtracting from fig. 6b the contribution from the unscattered b e a m passing through the sample, ~ a s ( r ) - Finally, fig. 6d is obtained by subtracting the average intensity from the hologram and then proceeding as for fig. 6c. Figs. 7 a - 7 d are the images corresponding to figs. 6 a - 6 d computed for the sample one layer thick for which the hologram is shown in fig. 4a. The dominant features in fig. 7b are due to the unscattered b e a m passing through the sample, ~bBs(r). Subtraction of ~bBs(r) reveals a somewhat distorted image, fig. 7c, of the central atoms. Subtraction of the average intensity component before reconstruction results in a slightly improved representation of the central atoms. Features associated with the major details centered at r = r D and r = 2r o still obscure detail away from the central portion of the image. 5. Summary Simulations of side-band holograms produced with a STEM have been presented and interpreted. Conditions necessary for the reconstruction of an image possessing 1 .& resolution have been determined. These simulations correspond to the experimentally unattainable limit of a com-

pletely stable thin sample irradiated with a perfectly stable STEM equipped with a completely coherent point electron source. Thus, the conditions determined must, at best, be considered as minimum ones. For a 300 keV electron beam, the detector resolution should be 0.01 A-~ or better in order that the oscillations in the incident beam wavefunction be detected to the limit of resolution. The ability to detect these oscillations will be useful in determining the coherency, stability and focus of the beam. In order to obtain an image with 1 A resolution using a 300 keV beam, it is desirable to record the holograms with a detector array of at least size 200 × 200. A 100 keV electron beam would require an order of magnitude greater detector resolution in order to detect the oscillations in the incident beam waveo 1 function a 1 A . The resolution of single atoms similar to Ga and As (31 and 33 electrons) requires that the hologram intensities be recorded over a range of least 400 in the ideal limit. Extraneous background a n d / o r lighter elements would make a range of 1000-2000 more desirable. Finally image reconstruction based on the simulated holograms has demonstrated that interferences from unscattered beams must be removed in order to resolve single atoms.

References [1] D. Gabor, Nature 161 (1948) 777. [2] D. Gabor, Proc. Roy. Soc. (London) Ser. A 197 (1949) 454. [3] J.TA. Lin and J.M. Cowley, Ultramicroscopy 19 (1986) 179. [4] H. Lichte, Ultramicroscopy 20 (1986) 293. [5] E. V61kl and H. Lichte, Ultramicroscopy 32 (1990) 177. [6] J.M. Cowley, Ultramicroscopy 34 (1990) 293. [7] M.A. Gribelyuk and J.M. Cowley, in: Proc. 49th Annual EMSA Meeting, San Jose, 1991 (San Francisco Press, San Francisco, 1991). [8] J.M. Cowley, Diffraction Physics (North Holland, Amsterdam, 1975). [9l Multislice Computer Program, High Resolution Electron Microscope Facility, Arizona State University, Tempe, Arizona 85281. [10] H. Lichte, Ultramicroscopy 38 (1991) 13.