A ZONEPLATE APERTURE FOR ENHANCING RESOLUTION IN PHASECONTRAST ELECTRON MICROSCOPYClaire B. Eisenhandler and Benjamin M. Siegel Citation: Applied Physics Letters 8, 258 (1966); doi: 10.1063/1.1754427 View online: http://dx.doi.org/10.1063/1.1754427 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/8/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Controlled electroplating for high-aspect-ratio zone-plate fabrication J. Vac. Sci. Technol. B 24, 2592 (2006); 10.1116/1.2362761 Maskless, parallel patterning with zone-plate array lithography J. Vac. Sci. Technol. B 17, 3449 (1999); 10.1116/1.591028 Erratum: Zoneplate coded imaging on a microscopic scale J. Appl. Phys. 48, 4416 (1977); 10.1063/1.324300 Zoneplate coded imaging on a microscopic scale J. Appl. Phys. 48, 1563 (1977); 10.1063/1.323833 Erratum: A ZonePlate Aperture for Enhancing Resolution in PhaseContrast Electron Microscopy Appl. Phys. Lett. 9, 217 (1966); 10.1063/1.1754716

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Volume 8, Number 10 APPLIED PHYSICS LETTERS 15 May 1966

A ZONE-PLATE APERTURE FOR ENHANCING RESOLUTION IN PHASE­CONTRAST ELECTRON MICROSCOPY*

(computer evaluation; single-atom imaging; electron microscope design; T)

Our recent studies on phase contrast in electron-. microscope imaging of single atoms' gave a char­acteristic behavior of the phase factor indicating the criteria which must be used in designing the annular type zone plates which have been suggested by Hoppe2,3 and by Lenz.4 As Hanzen et a1.5 ,6

have pointed out, the Hoppe-Lenz zone plates must be modified to take account of the phase shift of about rr/2 which occurs on scattering if phase con­trast in the reconstructed image is to be obtained. Hoppe and Langer7 have recently reported on calculations in which they now consider this essen­tial factor, but they use the same defocusing of the lens as for their previous zone plates. Actually their defocus criterion will not give optimum contrast or resolution. The phase contrast exhibited by single atoms indicates that optimum contrast and resolu­tion of the atomic image depend on the shape over the back focal plane of the phase factor in the Kirch­hoff diffraction integral, which can be adjusted for a given lens by suitable choice of defocus and aper­ture size. Our calculations' indicate that proper optimization of defocus using a zone plate with only two annular rings would improve resolution from 3.1 A to 2.1 A for 1 OO-k V electrons with a presently available lens having a spherical aber­ration coefficient Cs = 0.092 cm.

The optimum shape of this phase factor will now be considered in determining the actual con­figuration of the annular zone plate for a particular lens. Attention is directed toward the possibility of distinguishing individual atoms or characteristic atomic groupings within biological molecules. The calculations in this Letter and the previous Letter' apply to specimens for which phase contrast is the dominant contrast mechanism. Effects of inelastic scattering are neglected. It is assumed that the specimen is at liquid helium temperature, mini­mizing the effect of thermal vibration of its atoms.

Derivations of equations used in calculating the

*Work supported by grants from the National Institutes of Health, The National Science Foundation, and the Advanced Research Projects Agency.

258

Claire B. Eisenhandler and Benjamin M. Siegel Dept. of Engineering Physics and

Laboratory of Atomic and Solid State Physics Cornell University, Ithaca, New York

(Received 3 January 1966; in final form 8 April 1966)

phase contrast exhibited by a single atom can be found in previous papers.1,8 If the incident elec­tron intensity is taken as unity, the total intensity, l'I'tot (Xi ,yi) 12 at the image plane, (Xi ,yi), can be given in the kinematic approximation where the total cross section is small as

l'I'tot (Xi,yi)12=M-2 +2M-'Re['I'(xi,yi)] , (1)

where M is the magnification, and Re['I'(xi,yi)] is the real part of the amplitude distribution of the scattered electrons. For a single atom at the origin of the object plane (axial illumination assumed),

Re['I'(";)] = 2rr(1-(32)-'/2 (MA)-' I rEsin (a/2)/A] aperture

cos x(a) lo[(2rra";/A)ada] , (2)

where A is the wavelength of the incident electrons, (1 - (32)-1/2 is the factor correcting the relativistic mass effect, r [sin (a/2)/A] is the atomic scattering amplitude9 for electrons, a is the scattering angle, ri is the radial distance in the image plane, and x(a) is the phase of the scattered electrons.

using the first Born approximation, which neglects a small phase shift depending on the interaction of the electron and the atomic scatterer. Cs is the spherical aberration constant. M2!:!if is the distance between the Gaussian image plane and the plane at which the image is observed and is equivalent to a defocus, !:!if, referred to object space. The inte­gration in Eq. (2) is carried out over the aperture in the back focal plane or over the transparent rings of a zone plate placed in the aperture plane.

In the previous work,' it was found that optimum contrast with minimal distortion could be obtained for single atoms by adjusting the defocus so that the maximum value of the phase angle, Xmax,

would satisfy:

Xmax = rr[(1/2 + !:!if2)/(2CsA)] = 1.21rr (4)

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Volume 8, Number 10 APPLIED PHYSICS LETTERS 15 May 1966

or, solving Eq. (4) explicitly for llf,

llf = (1.42Cs.\)1/2 . (5)

The aperture angle, a max, was set at the nonzero value of a where X = 1/217" or

(6)

Figure la shows the phase factor at 100 kV for a lens having Cs = 0.092 cm where Eqs. (5) and (6) are satisfied. As a result of choosing the defocus according to Eq. (5), the phase factor is ~-1 over a considerable portion of the aperture, thus allowing portions of the diffraction pattern of the object appearing in that portion of the aperture to con­tribute to the image with minimal distortion caused by lens aberrations. Choosing the aperture angle to satisfy Eq. (6) ensures that the phase factor does not change sign within the aperture.

If a zone plate is used to block out regions of the aperture where the phase factor is positive, the conditions set by Eqs. (4), (5), and (6) can be ex­tended so that

Xmax = 17"[(1/2 + llP)/(2Cs.\)] = (K + 0.21)17" K = 1, 3, 5 ...

ex.

(7)

I O~-----=,O'F5_-;---:-:;=---:::;

o

Fig. 1. Image figure of a single sulfur atom imaged by 100-kV electrons and an objective lens having Cs = 0.092 cm shown with phase factor and zone widths of zone plates (j =

1.5 mm) corresponding to different defocus and K values: (a) K = I, Af= 695 A; (b) K = 3, Af= 1360 A; (c) K = 5, Ilf= 1790 A.

or, in solving Eq. (7) explicitly for llf,

llf= [(2K - 0.59)Cs.\] 112 . (8)

Figures Ib and Ie indicate the shape of the phase factor satisfying Eqs. (7) and (8) for K = 3 and K = 5, respectively. The broad region where Cos X ~ -1 appears at larger scattering angles than in Fig. la, thus allowing a larger effective aperture size and thus better resolution. For even values of K in Eqs. (7) and (8), Cos X ~ + lover a broad region. Since Cos X < 0 at the origin and since for phase contrast the unscattered beam must reach the image plane, a zone plate suitable for phase contrast must be opaque over annular regions where Cos X > 0, if the considerable distortion caused by having the phase factor change sign within the aperture is to be avoided. Thus defocusing to satisfy Eq. (8) for even values of K is not useful for phase contrast. In Figs. Ib and Ie the dimensions and configuration of the zone plate designed to accompany the defocus set by Eq. (8) (for K = 3 and K = 5) are indicated. The scattering angles, an, at which zones begin are given by

an = {llf/Cs - [(llP/C;- (n - 1).\)/(Cs)]112}1/2

for n = 1, 5, 9 ... 2K - 1 (transparent);

for n = 3, 7, 11 ... 2K - 3 (opaque). (9)

The maximum aperture angle is set at

a max = {llfiCs + [(llP/Cs2 - (2K - 1).\) / (Cs )] 1I2p/2 •

( 10)

It would be possible, in principle, to extend the zone plate past this point so as to include more regions where the phase factor is negative but, in general, these regions are so narrow as to cause serious difficulty in manufacture as well as increased sensitivity to fluctuations in defocusing, etc.

Calculations were performed for a lens having Cs = 0.092 cm and an incident electron energy of 100 kV. The image figure for a single sulfur atom with no zone plate (K = 1) satisfying Eqs. (4) and (5) was calculated previously/ but is shown in Fig. la with a plot of Cos x(a) vs a for purposes of comparison. Figures Ib and Ie show image figures of single sulfur atoms imaged using zone plates characterized respectively by K = 3 and K = 5 in Eqs. (6) - (8), where the defocus is accordingly in­creased to satisfy Eq. (6) for these values of K. While the improvement in contrast with the use of a zone plate is only :S 1 % for both zone plates considered here, a significant improvement in resolution, as indicated by the decrease in halfwidth, is obtained

259

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Volume 8, Number 10 APPLIED PHYSICS LETTERS 15 May 1966

with the zone plates. The halfwidth of 2.2 A ob­served for no zone plate (K = 1) is decreased to 1.7 A for a zone plate characterized by K = 3 and to 1.5 A for one having K = 5. Estimates based on work on molecular images now in progress indicate that these halfwidths correspond to smallest re­solvable interatomic spacings, ares, of 3.1 A, 2.3 A, and 2.1 A, respectively (for 5% peak to valley contrastl).

There is not much advantage in adjusting de­focus and designing zone plates for K > 5, since the improvement in resolution becomes progres­sively smaller as K increases. Also, the width of the zones becomes smaller with larger K and thus more difficult to make and more sensitive to focus­ing errors, etc. The widths of the zones required for the K = 3 and K = 5 plates are indicated in Fig. 1 under the corresponding phase factors and are calculated by assuming / = 1.5 mm in the relation p = a/ where p is the radius in the back focal

plane. These measurements indicate that a plate of this design could be made today. Further, calcu­lations made for the K = 5 zone plate indicate that a fluctuation in defocusing of ±30 A would not significantly affect the resolution of the plate and would only lower contrast of the sulfur image -1 %.

'c. B. Eisenhandler and B. M. Siegel,]. Appl. Phys. 37, to be published.

2W. Hoppe, Naturwiss 48,736 (1961). 'w. Hoppe, Optik 20, 599 (1963). 4F. Lenz, Z. Physik 172, 498 (1963). 5K. J. Hanszen, B. Morgenstern and K. J. Rosenbruch, Z.

angew. Physik 16, 477 (1964). 6K. J. Hanszen and B. Morgenstern, Z. angew. Physik 19,

215 (\965). 7W. Hoppe and R. Langer, Internal. Conf. on Electron Dif­

fraction and Cryst. Defects, Paper I, ()-5 (Melbourne 1965). "R. D. Heidenreich and R. W. Hamming, Bell System Tech.].

44, 207 (\ 965). 9The values of f"[sin (a/2)/A] were obtained from Ibers and

Vainshtein, International Crystallographic Tables (Kynoch Press, Birmingham, England, 1962), Vol. III, Table 3.3.3 A(I).

SENSITIZATION OF Tb3+ LUMINESCENCE BY Sn2+ AND Cu+ IN ALKALINE EARTH PHOSPHATES

(photoluminescence; phosphors; EfT)

In the literature several examples are given of the sensitization of Tb3+ luminescence by other rare-earth ions. For example in CaF2 : (Ce, Tb)l energy transfer from cerium to terbium was found, whereas in alkaline-earth alkali borates,2 doped with Gd3+ and Tb3+, energy can be transferred from gadolinium to terbium.

Recently some examples of terbium sensitization have been reported, in which energy is transferred from ions other than rare-earths. Mizuno and Mas­uda3 observed Tb sensitization by Cu+ and TI+ re­spectively in Sr2.5Mgo.5(P04h. Shionoya and N aka­zawa4 observed sensitization of Tb luminescence by Ce3+ and Cu+ in calcium metaphosphate glasses. These authors assume a nonradiative energy trans­fer (presumably by a resonance process). Rimbach5

described Sr3(P04h : (Cu, Tb) in which phosphor energy transfer from Cu+ to Tb3+ probably also takes place.

The question arises whether sensitization of the Tb3+ luminescence by Cu+ ions can also occur in

260

W. L. Wanmaker, A. Brit and]. W. ler Vrugt N. V. Philips' Gloeilampenfabrieken

Eindhoven, Netherlands (Received 15 March 1966; in final form 18 April 1966)

other host lattices. Moreover it seems worthwhile to look for sensitizer ions other than Cu+ and Tl+.

In order to achieve sensitization of terbium lumi­nescence for 254 mf.-t excitation, it is desirable that the sensitizer ions have a strong absorption in this region. Furthermore it is necessary that the energy level scheme show an overlap of some excited levels of the sensitizer and of the terbium ions, between which the transfer can occur.

Since (Sr,Mg)a(P04h : Sn,6 (Sr,Mg)a(P04h : Cu (ref. 7) and SrLiP04 : Cu (ref. 8) are efficient photo­luminescent phosphors (proving that with 254 mf.-t excitation the exciting radiation is readily absorbed and radiation less transitions are negligible), we examined the sensitization of the Tb luminescence by Cu and Sn respectively in these host lattices. In addition CaLiP04 phosphors were investigated.

The phosphors were prepared by reactions in the solid state at temperatures ranging from 1000° to 1200°C. The starting materials were luminescent­grade SrHP04 , Li2C03, CaHP04 , MgO and TbP04 •

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