Ultramicroscopy 36 (1991) 297-305 North-Holland
Compositional variations in apatites with respect to preferential ionic extraction F.J.G. Cuisinier a, R . W . G l a i s h e r a n d R.M. F r a n k a
J.-C. V o e g e l
J.L. H u t c h i s o n c, E.F. Br6s a
Facultd de Chirurgie Dentaire, I N S E R M U157, 1 Place de l'HOpital, F-67000 Strasbourg, France b Electron Microscope Unit, The Sydney University, N.S.W. 2006, DX 1154, Sydney, Australia ¢ Department of Metallurgy and Science of Materials, University of Oxford, Oxford OXI 3PH, England, UK
Received 4 February 1991; in final form 19 March 1991
Detection of ionic losses from the apatitic structure (Calo(PO4)6(OH)2) by high-resolution electron microscopy was investigated theoretically. Linear image analysis showed the need for an objective aperture of at least 3.7 nm-1 to visualize four different coordinates (CalI, OH-, P and mid-point between CalI-P bond). High-resolution image analysis and plotting of OH- column intensity against specimen thickness showed an inverse proportionality between composition and OH- image intensities for very thin specimens (less than 2 nm). Image intensity variation would be detectable experimentally, but the preparation of such thin specimens by ultramicrotomyis impossible.
1. Introduction The destruction of biological apatite crystals during pathological processes such as dental caries as well as the dissolution of synthetic hydroxyapatites in the presence of acidic media was generally interpreted as the consequence of local ionic variations [1-3] or the presence of structural defects within the apatite crystallites . Moreover, the progress of the lesion at the monocrystallite level was itself related to dislocations . It is also recognized that non-metallic solids are particularly sensitive to electron beam damage leading to local atomic disorder. Also Takuma  demonstrated electron-transparent areas along the central dark line in certain biological apatite crystals which, in fact, visualized preferential ionic extraction induced by the electron beam. This higher reactivity of certain local areas within apatite crystals may be connected directly with the dissolution initiation as visualized for biological apatites . Moreover, comparing the kinetics of void formation for two different apatite samples (iodo-
and hydroxy-apatite) , a striking correlation between composition and the process itself was found. The presence within the crystallite of structural defects like grain boundaries or dislocations may itself favour local ionic extraction. Biological apatites are presented as calcium-deficient compounds. The observed deficiency leads to the formation of vacancies or unoccupied ionic sites in order to compensate charge losses. The aim of this theoretical work was to consider several reasonable and simple ionic extraction processes for the apatite structure and to investigate whether a correlation between contrast variation in conventional high-resolution electron microscope images and composition variation could be found.
2. Crystal structure 2.1. H y d r o x y a p a t i t e
Hydroxyapatite H A (Cal0(PO4)6(OH)2) possesses a hexagonal structure with space group
0304-3991/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
F.J.G. Cuisinier et al. / Compositional oariations in apatites ©-
P63/m. Lattice parameters are a = b = 0.9432 nm and c = 0.6881 nm . The projection along the (001) zone axis of hydroxyapatite (fig. la) shows hydroxyl groups ( O H - ) situated on the six-fold c-axis. These ions are slightly below and above the mirror planes at z = 1 / 4 and 3/4. Calcium cations are present in the lattice in two different site types: four calcium I ( C a I ) on the four-fold axis and six calcium II (Ca II) forming the apexes of equilateral triangles around O H - located on the six-fold axis.
Eo(E o + 1022)/460A,
........... - .... ..........
The induction of voids in a crystallite under electron b e a m irradiation is mainly a consequence of ionic extraction from the crystalline structure. This process is, among others, a function of the nature of the atoms, the structure of the solid, inter-atomic bonding, the localization of the atom and the specimen orientation itself. The m a x i m u m energy Emax (in eV) which can be transferred to an atom by the incident electrons according to ref.  is Ema x =
2.2. Deficient hydroxyapatite
Hydroxyl groups @
Fig. 1. Structure of hydroxyapatite seen (a) along (001) zone axis, full line = unit cell, dashed line = C a I I triangles; (b) in perspective showing the arrangement of ions around O H columns. Arrows indicate C a l I with an occupancy factor of 5/6.
with Emax in eV, E 0 energy of the incident electrons (in keV) and A atomic mass of the target atom (in a.mu.). In order to induce atomic displacement Emax must be larger than the minimum kinetic energy needed for its extraction. In strongly bonded ionic structures like H A this critical energy value is around 60 eV . E m a x values for calcium and phosphorus of H A (table 1) estimated according to eq. (1) are below the minimum value, whereas Emax values for H and O are highest and both are above the minim u m kinetic energy needed for ion extraction . This rough estimation shows clearly that O H appears to be the most easily extracted ion from the H A structure. Moreover, Schottky defects (stoichiometric defects in ionic crystals) are thermodynamically favoured in calcium apatites . Thus in H A the formation of two O H - vacancies is compensated
by a calcium vacancy. Electron neutrality must be re-established in the closest environment of the formed O H - vacancy . A simple geometric consideration (fig. l b ) suggests that compensation will be due to Ca II ion loss rather than loss of Ca I, the loss of one O H - ion being accompanied by the loss of 0.5 Ca II. This point is taken into consideration in image simulation by applying an occupancy factor of 5 / 6 for each Ca II surrounding an extracted O H - .
Table 1 M a x i m u m energy transferred (Emax) to the different atoms (A: atomic mass) of H A for incident electron with an energy of 400 keV Atom
Hydrogen Oxygen Phosphorus Calcium
1 16 31 40
1221 76.3 39.4 30.5
F..J. G. Cuisinier et aL / Compositional variations in apatites
Thus, hydroxyl-deficient hydroxyapatites may be described by the formula: ( C a I ) 4 ( C a I I ) 6 - ( x / 2 ) ( P O 4 ) 6 ( O H ) x,
with 0 < x < 2. Presently only three limiting cases are considered with x = 0 (HA2), x = 1 (HA1) and x = 2 HA0.
3. Analysis procedure High-resolution electron microscopical investigations in which composition changes have to be analysed require preliminary resolution and specimen thickness estimations leading to image contrast which is sensitive to compositional change. In a first approach, linear images (corresponding to weak-phase-object !mages) are often calculated for different objective apertures in order to define the needed resolution for imaging atomic columns in which composition change occurs. The estimation of the required specimen thickness is given by the diffracted beam intensities. Then, simulated H R E M images (non-linear images corresponding to phase objects) are used to study the effects on image intensities of composition changes within the crystal unit celL This study is mainly devoted to the detection in HA of composition changes observed along the c-axis. Hence, linear and non-linear images are both simulated for this zone axis.
Fig. 2. Simulated diffraction pattern of [0011 HA. Schematic objective apertures are drawn on the pattern; aperture: (1) r = 1.7 n m - 1 , (2) r = 2.2 n m - 1 , (3) r = 3.7 n m - 1 , (4) r = 6.6 r i m -
demands that the spherical aberration coefficient Cs is effectively zero, an objective lens defocus value of - 4 nm and a specimen thickness of 0.68 nm for hydroxyapatite observed along . The spatial frequency cut-offs are determined by the objective lens aperture radius (fig. 2). Based solely upon visual examination of the linear images, the identification of the objective 0 7
3.1. Linear images
p~ool ', 'Z'_J ....
Linear images are simulated with the program EM256  for increasing objective aperture diameters in order to choose the best condition for composition change monitoring . The linearity of the imaging process ensures that the image contrast is directly related to the atomic potentials of the crystals. Specific structural features such as atomic columns can be associated with particular reflections. Conditions for simulation are defined by sin X = - 1 (X: wave aberration function) for all beams included in the objective aperture . For a H R E M operating at 400 kV, this value of X
5o01 r .-.I ._._J 02'
I f,OOJ ~'.;
13001 t j 01
; ........... 7 " " - } . . . . 4 - i .-A . . . . . . . . . . . .......... -~,._.__T ~- . . . . . . . . . . . . . . .
Fig. 3. Relative intensity (Irl) versus resolution (nrn - ] ) at different centers of the unit cell. Chosen centers are O H - , CaII, P and the middle point between C a I I and P columns.
F.J.G. Cuisinier et al. //Compositional oariations in apatites
aperture limit which first resolves the atomic columns of interest is difficult• To overcome this problem the relative intensity I,L at a given point of the linear image is plotted for particular coordinates of the image against increasing resolution (fig. 3). /+] is given by: /rl = Im, -- Ipl,
where I~] is the mean intensity over the whole simulated linear image and Ipt the intensity at a given point of the linear image. A linear image dot with an Iri value different from zero must be clearly observable. Four different particular coordinates of the linear image have been considered: the centers of O H - , Ca II and phosphorus columns and the mid-
Hilt ImJlr ilIHI
+IHml l + m ; £ l l
lPll +iii' ,lill+
I J A +lU=
t i l l mini
] l ~ *lnml+
,ll+ I l L +HHt
~IIIII iHIHi LIHHf
Hill + J +lllll
+lll IHlll +Hill
i l l Ohm
! 1 limit
a l l ,Imllly
~ 1 t ,mime+
;lllli+ lflfll IIit
0 :1111 imllll£ :iii|~
llill+ I l l :Olli
illlll IlIINIII IIIIIIIL~ I~ rllnlE
++++ +.++ . . . . :++~++,+,~,,+,+,++.+~+ ...... ~;+~++,,,,+,+++++++.++++ tlltl++,7+ -,+~+IIIIHIItlIIW~+++ -+.'+)+~+tlllHllllllttlit.++J+ ttlllt+l+;+ +;tG+ilillHlllltllllltltlill~l+',,+t++~lltlllilllillllllllillll+t
illl troll 11111~II ~III
lllllltll~ l i l t +11~1
i~il 11411+I+ 111111
i+~ ilflll, + ~ +
+ +*4~++++l+++ll.~m++~<+~ ) +++++m+l i H l i l l l l i + + ~ , ~ +
+++++. . . . . . . . . .
+ ~ + ~..... + + + +~+++++ll+++,,++++~mv +~ • . . + t +~m++,++ ~t l l l H l l d l + t i l m ~ +
+ -it + +i+
, +I l H I I ~ I I I I ~ < + IHII~IIII li+
.i i i i.i . . i. .
. +i I I I I I I ~ I I I I I I +I l l l l l l l l l l l
. + H I H + .
, ++m;: i++li+
= +i. ,,,,..m.~w..,
., ,. . . . . .~.l l.l l.l i.i l.+ :
-+ + + +-+ i + ~ . . . . . .+.l l.u.l l.n.l a. . . .
++ l l l l l l l ~ +
l.l .l +. . . m. ; + 1 1- -H-I I,H, I I I + I +...+. . . . . . . t.J. .I ~. .H.I .I I.i .I I.+.G + ~ I + I I m )+++ Illll lll+~ Y~ I I I I I 1 4 1 1 1 1 1 1 1 1 + Ill+ +I I I II, +lj I
+ l l l l l l . ~ . / l l
. +llllll I illltll.
- - ~
)::. . . .
+I l l l l l l + + l l ,
.~llllll i Itlllll
i;+ |ll iiii.
: +I I I I H I I I I . ,
. . . . . . lfl
~l lla Ii+
.=ff ~ ' ~ , .
,, + "
::: : +
~IIIIIU|INIIII~. <-*+: ~IIII ,i l i l l l l l i ~ i Xitlli < +illlll l l . . . . . . . . . . . . . vlIIHHII + lllllllllH+ : ++. + +lllllllllll. +i l l l l l l H I
+ ~ . . . .
' = ~
. . . .
. . . . . . . . . . . . . . . . . . . . . . .lllmllllmll~l+ . .~c; l l l l l l l m l l +ltll+". . . . . . . . . . . . . . i lilllll I IIIUIHII+ ~++ + +II IIIIII~II
.N+ = ' + ,
I t ~> : +III~4111111 ~i l I M I I
++t II t
. .+.l .i l.l +. + + ~ I I I ¢ I. . .".+.+. .+.I. l.l .l l.l m
ill+ +**Im+++~ llll .+
l ~ +f lI + 'fill
rim+ .+ . . . . . . .
l+a+, no*m ill
AI # ++
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+m+ll~ll + ++i+
,+. . . . . "
M1 + +lilt +fill
. i ltill IIII
lflll 114111+ +lUl
i++ , I011~ +lmlll+
.: . . . . .
+ i k l l ~
~ ' H I I I I I
i+ lllfl+ +lml+
f l ~ l i l l t +
: +i11<.+ +,,...:
l+l ll+l l l l l / ++++
~ .~. . . . . . . . . . .
tu till +11
.1Ii thin +lllfl ++lJ il.l ~alllllg
~tl l~ll. ++lfllg+
t i l7l1l 1 +ml
+. . . . . , ,
t~ + ~
+ l l l l l l l l -
; + I I I +
: +~ ~++i + + +.~+ml++ll
......... 0.+ ....
+ ] ..... ) +++IHII~ II++ > + + ) ~+ II II~IIII m+. • t s, +IIII4BIIIII I~+ + + +t +I l l l l l H I L +++ +~ + m+ l l E I l I I I I
L+'Y??;Y+:+ i ; . + % ; . ; i ~
•. . . . . . . . . . . . . . .
+ ; ..... +i ++
~ + A + f l l l H ~ f l l +t+ ¢ i + + + + + ~ + l i l l l ~ l + l l l l , + ;I lmlmlllglllllBllll~ C ) + * t +++lllllmllllllllllllllH +i llmlilllllllldnllllli + + + T + + ~+I i I I ~ I H I I I I I I I I I I I H H i +ill + --IIII+~+ + + ~+& I I I ~ H I I + t i l l l l l l l l t l ( + + ++ + ; + l l l O l l l l l l l l l l l l l l l l l l t
~lllll~ l i l t ~llfl.
mllll mlnl IH
i i i i mlllllalll +H i l l +I$
~l i1~11, ~llllal~
Iml~ I l i l imll
Hll+mt~ )+ ~ + t ++++llOfllm++l+wA+ + +. + " + t ++,+ llHltfl~IHl ll+J + + + ++i++l+ + ..... + ++ A + . + A [ I I I H i l l + l+++A + ~ . . . . . . ++ + + W A t + + R I I H J IIKI++A " ; ~ + + + + ~ + > + ~ ~ • + + + + + + + + m~ + + m m m l + A , m ~ + m ~ + + + l t + • • * • + r ; + . ~ ^ [ + ~ W , I+~++++ ~++.
.rdl+ + H i Itll
Fig. 4. Linear images of  hydroxyapatite as a function of resolution: (a) Structure image of H A projected along  axis, M middle point between C a l I - P , atom symbols as in fig. 1; (b) 6-beam linear image, objective aperture (fig. 2(1)) including all 100-type beams; (c) 12-beam linear image, objective aperture (fig. 2(2)) including all beams 200; (d) 36-beam linear image, objective aperture (fig. 2(3)) including all beams to 300; (e) 90-beam linear image, objective aperture (fig. 2(4)) including all beams to 500; (f) linear image with a structural resolution of 9.3 n m - ] .
F.J.G. Cuisinier et aL / Compositional variations in apatites 50
ber since the projected specimen potential is related to the atomic scattering amplitudes. With increasing specimen thickness, dynamical diffraction effects cause the intensities of the diffracted
]l. 45 -
(300) (100) ~ 10001 -
~,~ 0 30 0 25
i,/"'x. OI'O ' 0 05
k / e",
', __ X
/ ~ .
THICKNESS (nm) Fig. 5. Beam intensities versus thickness for (000), (100), (300) beam families of HA. Accelerating voltage: E = 400 kV.
dle point of the boundary between Ca lI and phosphorus columns. This last center was chosen because the comparison of its 1 d value with those of C a l l and O H - is a good estimation for the resolution possibility of Ca II and P. 3.2. Diffracted beam intensities In a linear image, dark contrast corresponding to atomic columns increases with the atomic num-
THICKNESS (nm) Fig. 6. Mean image intensity (Im) versus thickness for different hydroxyl-deficient apatite (HA2; x = 2, HA~: x = 1 and HA0:
x = 0).
Fig. 7. Compositional series of high-resolution images for  H A for crystal thickness 45 nm (a) and 88 nm (b). O H present in the crystal from top to bottom: 0%, 25%, 50%, 75% and 100%.
F.J.G. Cuisinier et aL / Compositional variations in apatites
beams to oscillate . Therefore, diffracted beam intensities are plotted versus specimen thickness, and when specimen thickness reaches the extinction distance of the transmitted beam, the image contrast becomes minimal. Since phase contrast is minimal for this thickness, perturbation of the diffracted intensities due to defects should be stronger . 3.3. H R E M image simulation Simulated high-resolution electron microscope (HREM) images were calculated using the EM256 programs written by Waddington . hnage calculation and intensity plots are calculated for a H R E M operating at 400 kV with a Cs of 1 mm, an energy spread of 6 nm and an illumination halfangle of 0.5 mrad. Image simulation was realized at Scherzer defocus for the objective lens. For this value the information contained in an image can easily be compared with the structure itself . A phase grating slice thickness of 0.34 nm was chosen in order to represent a half interplanar spacing along the  axis parallel to the observed direction. Image intensities were studied for the whole unit cell and for the O H - columns. Intensity average of the 65536 pixels constituting a whole image was considered as the image mean intensity (Im)" Im which is certainly not particularly sensitive to composition changes was chosen because it can be related to experimental data. In numerous experimental images white areas are observed near the center of the crystal. In these areas the image intensity of the whole unit cell decreases and,
apparently, not intensities of particular points of the unit cell.
4.1. Resolution When the objective aperture included the six 100 beams (fig. 2, aperture 1) only hydroxyl columns had a positive I a value, whereas the corresponding linear image (fig. 4b) was simply formed by O H - columns at the four comers of the unit cell. Addition of high-order reflections modified this basic six-fold image, and their respective contributions were modulated by their structure factor magnitudes and phases . Addition of 200 beams in the objective aperture (fig. 2, aperture 2) allowed the visualization of Ca I columns (fig. 4c). The inclusion of the 300 reflections (fig. 2, aperture 3) gave a positive Irl for the four considered coordinates (fig. 3), but the corresponding 36-beam linear image showed a complete overlapping of Ca II and P dots (fig. 4d). With a structural resolution of 6 nm -1 (90-beam image) it was again impossible to distinguish between Ca II and phosphorus colunms (fig. 4e). The separation of P and Ca II columns even with a structural resolution of 9.3 nm -1 (fig. 4f) was impossible because oxygen atomic columns (from phosphate groups) overlapped C a l I columns (fig. la) when projected along the  zone axis. The linear image analysis showed that an objective aperture of at least 3.7 nm -1 is needed to distinguish O H - and C a l I - P . The experimental
Fig. 8. Mean image intensities for (a) 45 n m and (b) 88 n m thickness against different O H - percentages within H A samples.
F.J.G. Cuisinier et al. / Compositional variations in apatites
resolution limit of 6 nm-~, and even the theoretical resolution of 9.3 nm -~, would not allow the separation of P and Ca lI dots. The use of a resolution higher than 3.7 nm-1 would furnish no useful structural information since composition variations in hydroxyl-deficient apatites concerned only O H - and C a l l atomic columns. Consequently in the present study, composition variations were only studied with a Scherzer resolution of 3.7 n m - t .
+,~+..,~ +~++ + + + + , , . m . m u ++~ +,~++++++~+~+ ++,~+ +.~ • +++++ram++ .~m +., +
4.2. Optimum specimen thickness
The dependence of the beam intensity on the specimen thickness of HA (fig. 5) showed that (000) beam is minimal for 25 and 85 nm. For these two thicknesses the image contrast is low and thus the defect visibility is maximum in the atomic columns where composition changes occur . The intensities of 300 beams were minimal in the 4 0 - 5 0 nm range whereas the intensities of 100 and (000) beams became dominant. Thus, this thickness range appeared to be useful for studying
T +++.ii+~;~ +;iS +.++ + +:+ ++++i+++:~+;+i+:r:+~++++++
¢+i+* +I~;*+ ,++mH+ ~+mm+ t.~,++++ ++++=+imll+ +++.m++~++mt+ ++ira++++, ++++i~+ +m++ ++iii,++++l++e+, ++lltlll+ +=mini *ram+*++ml.*t~+,
+;+~++"+i:+~?++ ++,.+++++?i+i++++'++.++=i+++.+,.!i~++ +,.~; +++++m+~Im|m +itnl++ +111,:+ ~++++++-+m+,. . . ..+11.*+, . ,+ ++*~+++ ++++.+.++.,,+ +++,+~ ++11 .+ +. . . . . . . . . . . . . =.+. . . . . . . . . . +,++,•
!++:i~:++'!++i+~++++!++++++!: ++!+:.+!+tt+++t++++m+++ +;;++ ~++~;+ + ;;+++
,++ ++ ++L.::+ +'+++++++'+',+++ ++
~++7~,+5i!+++!:+++;~+'+,+5+:+y A+++++~++~+?+ +t+~+
-+'.,,-.~+". . . . .
~ m ,
++i.i+ + . m ~ ++emmmmltllml~-ram +~.+++++.=+~+ ++m+++ ,;+,.~+++.+~. . . . ,+~+, +~ , ++~+,++++p . . . . . . ,+~+++.++.+++t+~++~. . . . . . . .=,+ +~.~
3.5 4"01 3.0 2.5 0 2.0 1.5 / 1.0 0.5
....... ' " /
Compositional for crystal in the crystal
series thickness from
16 nm 25%,
THICKNESS (nm) Fig.
Variation versus apatite
x = 2, HA~:
for different x =1
the influence of the first-order reflections on image formation as well as their ability to take into account composition changes .
F.J. G. Cuisinier et aL / Compositional variations in apatites
4.3. Image intensity variation with compositional change
4.3.2. linage intensity at the O H - sites The image intensity at the O H - site is the image intensity of the pixel situated exactly at the corner of the unit cell on the six-fold axis. When the intensity along O H - columns was considered, completely different shapes were observed for the three investigated samples (fig. 9), the only common point being an intensity minimum reached for 80 nm. However, two thickness domains (1-3 and 10-25 nm) showed an inverse proportionality between composition and image intensities for O H - . When we considered two particular thicknesses in these domains (for example 1.6 and 16 nm) (figs. 10 and 11) it appeared that for very thin crystallites a linear relationship between intensity and O H - composition existed. A variation of intensity of at least 3% is necessary to be detectable , but 10% is a more reasonable value . In the present system and for a crystal of 1.6 nm in thickness the image intensity varied by 10% when O H - changed by about 20% (fig. 10a). In this case, for H A a composition change of 20% seems to be observable. This is not surprising; such a thin crystal with low-mass atoms is a weak-phase object. The image formed by a weakphase object is strongly dependent on kinematical scattering , and a correct estimation of the intensity of O H - columns against composition changes had theoretically to take them into consideration. For a thickness of 16 nm (fig. 10b) the relationship between composition and O H - col-
4.3.1. Image mean intensity The image mean intensities for HA2, HA 1 and H A 0 indicated comparable variations depending on sample thickness (fig. 6). The three curves showed minima around 25, 55 and 85 nm. With respect to fig. 5 thicknesses of 25 and 85 nm corresponded to a weak contribution of the (000) beam. In two thickness domains (between 40-55 and 80-100 nm) a certain proportionality between mean intensity and atomic concentration seemed to exist. When the two particular thicknesses of 45 and 88 nm are considered, the compositional series of H R E M images (figs. 7a and 7b) and the plot of the mean image intensities against composition changes (figs. 8a and 8b) clearly show that no linear straightforwad correlation exists between atom concentration and mean image intensity. In the case of a specimen thickness of 45 nm the 100 beams mainly responsible for the formation of the O H - dots (fig. 4b) must be affected by composition changes, but their contribution to the mean image intensity is too low to induce a simple relationship with O H - . For 88 nm, apparently a more favourable thickness where composition variations were generally apparent as supposed by high-resolution image simulation [14,16], the absence of a direct relationship is more surprising.
1.25 ~ 1.oo.
% OHFig. 11. Image intensities of O H - columns ( l o l l )
OmO " 15
versus different O H samples.
percentages for (a) 1.6 n m and (b) 16 n m thick H A
F..J. G. Cuisinier et al. / Compositional variations in apatites
u m n s i n t e n s i t y is n o t at all s t r a i g h t f o r w a r d , a n d the access to c o m p o s i t i o n of O H - c o l u m n s is i m p o s s i b l e for thick crystals.
5. Conclusion T h e theoretical a p p r o a c h d e v e l o p e d in this p a p e r for the s t u d y of O H - e x t r a c t i o n f r o m a p a t i t e structure, l e a d i n g to c o m p o s i t i o n changes w i t h i n O H - a n d C a l I columns, c o n f i r m e d t h a t this p r o b l e m is n o t easy to solve practically. A straightforward relationship between composition a n d i m a g e intensities seems to exist o n l y for very thin specimens (less t h a n 2 n m ) w h i c h c a n n o t , unfortunately, be prepared by ultramicrotomy. W e recognized that the multislice a p p r o a c h is b a s e d on an a p p r o x i m a t i o n r a t h e r t h a n o n a regular c a l c u l a t i o n of d y n a m i c a l l y d i f f r a c t e d b e a m intensities. H o w e v e r , the multislice s i m u l a t i o n p r o g r a m s n o w w i d e l y used in m a n y l a b o r a t o r i e s app e a r to give c o n s i s t e n t l y g o o d a g r e e m e n t with e x p e r i m e n t a l images.
References  J. Arends and W.L. Jongebloed, Caries Res. 11 (1977) 186.  J.C. Voegel and R.M. Frank, Calcif. Tissue Res. 24 (1977) 19.
 E.F. Bres, W.G. Waddington, J.C. Voegel, J.C. Barry and R.M. Frank, Biophysical J. 50 (1986) 1185.  E.F. Bres, J.C. Barry and J.L Hutchison, Ultramicroscopy 12 (1984) 367.  S. Takuma, H. Tohda, N. Tanaka and T. Kobayashi, J. Electron Microsc. 36 (1987) 387.  G. Hirai and R.W. Fearnhead, in: Proc. 6th Int. Conf. on X-ray Optics and Microanalysis, Eds. G. Shinoda, K. Kohra and T. Inchi Okawa, Tokyo University, 1972, p. 863.  M.I. Kay, R.A. Young and A.S. Posner, Nature 204 (1964) 1050.  R.F. Egerton, P.A. Crozier and P. Rice, Ultramicroscopy 23 (1987) 305.  L.W. Hobbs, Ultramicroscopy 23 (1987) 339.  J. Arends, B.S.H. Boyce, D.O. Welch and R. Smoluchowski, in: Proc. Int. Symp. on the Structural Properties of Hydroxyapatite, Ed. R.A. Young (Benjamin, New York, 1971) p. 320.  J.C. Labarthe, M. Therasse, G. Bonel and G. Montel, C.R. Acad. Sci. 276 (1973) 1175.  E.F. Bres, J.C. Voegel, J.C. Barry, W.G. Waddington and R.M. Frank, J. Appl. Cryst. 19 (1986) 168.  D.J. Smith and M.A. O'Keefe, Acta Cryst. A 39 (1983) 139.  R.W. Glaisher and A.E.C. Spargo, Ultramicroscopy 27 (1989) 19.  J.M. Howe, D.P. Basile and N. Prabhu, Acta Cryst. A 44 (1988) 449.  P.G. Self, R.W. Glaisher and A.E.C. Spargo, Ultramicroscopy 18 (1985) 49.  N. Tanaka and J.M. Cooley, Acta Cryst. A 43 (1987) 337.  S. Iijima, Optik 48 (1977) 193.  J.C.H. Spence, Experimental High-Resolution Electron Microscopy (Oxford University Press, Oxford, 1988) p. 72.