Earth’s nanomaterials

Digital / Information Revolution

Molecular Biology Revolution

Nanotechnology Revolution

Cognition Revolution, “Decades” of the Brain

Major Technological Revolutions in Human History (cont.)

20101970 19901950 1960 1980 200019401930

The President’s 2019 Budget supports nanoscale science, engineering, and technology R&D at 12 agencies (approx. $1.5B). The five Federal organizations with the largest investments (representing 95% of the total) are:

HHS/NIH (nanotechnology-based biomedical research at the intersection of life and physical sciences). NSF (fundamental research and education across all disciplines of science and engineering). DOE (fundamental and applied research providing a basis for new and improved energy technologies). DOD (science and engineering research advancing defense and dual-use capabilities).

Melting temperature = 1,064°C

Melting temperature= 427°C

Gold

Color = gold

Color = pink, orange,red, purple, violet

Gold

Gold

Inert gold in 1,000 year old shipwreck.

Catalytic gold in gas reaction research

KQED San Francisco

1 nm = 10-9 m

Piece of paper

= 100,000 nm thick

Human hair = 80,000 nm thick

DNA = 2.5 nm diameter

Gold atom = 0.3 nm diameter

How small is “nano”?

108 meters 108 meters

Hochella et al. (2008) Science, v. 319, p. 1631-1635.

Phytoplankton: half of all photosynthesis on Earth!

So what’s the big deal for the Earth sciences!? OK, an example . . .

Hochella et al. (2008) Science, v. 319, p. 1631-1635.

Nano-hematite, HAADF-STEM tomography, aggregate, 30 nm

The geometry of the transform is illustrated inFig. 1. A discrete sampling of the Radon trans-form is geometrically equivalent to the sampling ofan experimental object by some form of trans-mitted signal or projection. The consequence ofsuch equivalency is that the reconstruction of thestructure of an object f ðx; yÞ from projections Rfcan be achieved by implementation of the inverseRadon transform. All conventional reconstructionalgorithms are approximations, with varyingaccuracy, of this inverse transform.

The Radon transform operation converts thedata into ‘Radon space’ (l;y), where l is the lineperpendicular to the projection direction and y isthe angle of the projection. A point in real space(x ¼ r cos f; y ¼ r sin f) is a line in Radon space(l;y) linked by the equation l ¼ r cosðy$ fÞ: Asingle projection of the object, a discrete samplingof the Radon transform, is a line at constant y inRadon space. A series of projections at differentangles provides an increased sampling of Radonspace. Given a sufficient number of projections, aninverse Radon transform of this space shouldreconstruct the object. It becomes evident that anyexperimental sampling of (l;y) will be discrete andas such any inversion will also be imperfect. Theproblem of reconstruction then becomes achievingthe ‘best’ reconstruction of the object given thelimited experimental data.

4. The central slice theorem and Fourier spacereconstruction

In practice reconstruction from projections isaided by an understanding of the relationshipbetween a projection in real space and Fourierspace. The ‘central slice theorem’ or the ‘projec-tion-slice theorem’ states that a projection at agiven angle is a central section through the Fouriertransform of that object. This is of course exactlythe theorem used for the ‘projection approxima-tion’ relating the intensity of zero order Laue zone(ZOLZ) reflections to the crystal potential pro-jected in a direction parallel to the zone axis. A fullexploration of the relationship between the Radonand the Fourier transforms is covered in Ref. [8].

Thus, if a series of projections are acquired atdifferent tilt angles, each projection will equate topart of an object’s Fourier transform, sampling theobject over the full range of frequencies in acentral section. The shape of most objects will bedescribed only partially by the frequencies in onesection but by taking multiple projections atdifferent angles many sections will be sampled inFourier space. This will describe the Fouriertransform of an object in many directions, allow-ing a fuller description of an object in real space.In principle a sufficiently large number of projec-tions taken over all angles will enable a completedescription of the object.

Therefore tomographic reconstruction is possi-ble from an inverse Fourier transform of thesuperposition of a set of Fourier transformedprojections: an approach known as direct Fourierreconstruction [5] which was used for the firsttomographic reconstruction from electron micro-graphs [9]. This theory also allows a logicaldescription of the effects of sampling deficienciesin the original dataset. If projections are missingfrom an angular range, brought about by a limiton the maximum tilt angle, then Fourier space isunder-sampled in those directions and as aconsequence the back transform of the object willbe degraded in the direction of this missinginformation.

Unfortunately a practical implementation of theFourier space reconstruction approach is not assimple as an inverse transform. The projection

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l

f (x,y)y

xD

L

θ

Fig. 1. The Radon transform R can be visualised as theintegration through a body D in real space f ðx; yÞ along allpossible line integrals L; with its normal at an angle y to thehorizontal.

P.A. Midgley, M. Weyland / Ultramicroscopy 96 (2003) 413–431416

important to the study of specimens found inmaterials research.

The need to obtain higher dimensionality‘structures’ using lower dimensionality data ispresent in many different fields of physical andlife sciences. The first real application of thishowever, was in the field of astronomy whenBracewell [6] proposed a method of reconstructinga 2D map of solar microwave emission from aseries of 1D ‘fan beam’ profiles measured by aradio telescope. This work covered the mathema-tical formulation of projection and reconstruction,introducing a number of important relationshipsthat will be discussed later. Surprisingly, despite itsobvious potential, this work had little impactbeyond its immediate field. However, in 1963,interest in tomographic reconstruction was stimu-lated again by its possible application in medicine[7]. This paper lead directly to the development ofthe X-ray computerised tomography (CT) scanner[8], routinely known as the CAT-scan (computerassisted tomography or computerised axial tomo-graphy). This phenomenally successful technique,which led to a joint Nobel Prize for Cormack andHounsfield in 1979, is undoubtedly the most well-known application of 3D tomographic reconstruc-tion. The success of the CAT-scan was mirrored inthe development of other medical analysis techni-ques such as positron emission computedtomography (ECT), ultrasound CT and zeugma-tography (reconstruction from NMR imaging).This success led to tomography being applied inmany other disciplines to allow, for example, 3Dstress analysis, geophysical mapping and non-destructive testing. An exhaustive review of thedifferent applications for tomographic techniquescan be found in a review by Deans [9].

Interest in tomographic reconstruction fromelectron micrographs started with the publicationof three seminal papers in 1968. The first was by deRosier and Klug [10] in which the structure of abiological macromolecule was determined whosehelical symmetry allowed full reconstruction froma single projection (micrograph). The Fourierreconstruction methods used in this paper were anatural progression from those developed for theanalysis of atomic structure by X-ray crystal-lography [11]. While symmetry was key to these

results, it was suggested in the second of thesepapers, by Hoppe [12], that, given sufficientnumbers of projections, reconstruction should bepossible for fully asymmetrical systems. The last ofthe three early papers, by Hart [13], demonstrateda method of improving contrast in BF imagesusing an ‘average’ image, known as a polytropicmontage, calculated from a tilt series of micro-graphs. Although primarily used as a means tocombat the weak contrast in biological specimens,Hart acknowledged the 3D information generatedby such an approach without extending this to thepossibility of full 3D reconstruction. These earlypapers were followed by a number of theoreticalpapers discussing the theoretical limits of Fouriertechniques [14], approaches to real space recon-struction [15] and, at the time, somewhat con-troversial iterative reconstruction routines [16,17].

While theory advanced rapidly, experimentalresults were slow to appear because of a number ofimportant limiting factors such as beam damage,the poor performance of goniometers and the lackof processing power required for image processingand reconstruction. The last two factors are nolonger problems but beam damage is still alimiting factor in biological tomography even ifthe samples are cooled to liquid helium tempera-tures and examined at high voltages. Nevertheless,electron tomography in the biological sciences hasdeveloped to the point where the reconstruction ofunique objects is possible with a resolution of 2 nmin a volume of (150 nm)3 [18].

3. The Radon transform

Although the first practical formulation oftomography was achieved by Bracewell in 1956,in fact it was Radon who first outlined themathematical principles behind the technique in1917 [19]. The paper defines the Radon transform,R; as the mapping of a function f ðx; yÞ; describinga real space object D; by the projection, or lineintegral, through f along all possible lines L withunit length ds:

Rf ¼Z

L

f ðx; yÞ ds: ð1Þ

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P.A. Midgley, M. Weyland / Ultramicroscopy 96 (2003) 413–431 415

Radon J. (1917) Leipzig Math.-Phys.

Echigo et al. (2013) Am. Min.

Purely quantum mechanical considerations

Strong quantum confinement regime:

a < ae,ah

Intermediate quantum confinement regime:

Weak quantum confinement regime:

ae < a < ah

ae,ah < a

For hematite: 7.0 nm 3.8 nm

aB =εm0

µa0

written in terms of :aB a0

aB =(32)(9.11×10−31)(1.43×10−31)

(5.28×10−11) =10.8×10−9m =10.8nm

For PbS galena: aB =18nm

Quantum Confinement in Hematite