Peter Pfeifer

Chemistry in noninteger dimensions between two and three. I. Fractal theory of heterogeneous surfaces

In this, the first of a series of papers, we lay the foundations for appreciation of chemical surfaces as D‐dimensional objects where 2≤D<3. Being a global measure of surface irregularity, this dimension labels an extremely heterogeneous surface by a value far from two. It implies, e.g., that any monolayer on such a surface resembles three‐dimensional bulk rather than a two‐dimensional film because the number of adsorption sites within distance l from any fixed site, grows as lD. Generally, a particular value of D means that any typical piece of the surface unfolds into mD similar pieces upon m‐fold magnification (self‐similarity). The underlying concept of fractal dimension D is reviewed and illustrated in a form adapted to surface‐chemical problems. From this, we derive three major methods to determine D of a given solid surface which establish powerful connections between several surface properties: (1) The surface area A depends on the cross‐section area σ of different molecules used for monolayer cov...

Surface geometric irregularity of particulate materials: The fractal approach

Abstract Surface geometric irregularities of a wide variety of particulate materials are determined by applying the fractal theory of surface science (P. Pfeifer and D. Avnir, J. Chem. Phys. 79, 3558 1983). Reanalysis and reinterpretation of previously published experimental data, revealed the following surface-fractal dimensions, all falling in the expected range 2.0 to 3.0, for the following materials: periclase—1.95; a number of quartzes—2.02 to 2.21; iron oxide—2.59; porous silica gel—3.0; coal dusts—2.33 and 2.53; six carbonate rocks—2.16 to 2.97; seven types of soils—2.19 to 2.99; a number of crushed rocks from nuclear test site—2.7 to 3.0. We describe in detail the method by which these fractal dimensions were determined, i.e., studying the dependence of monolayer values (n) on particle radii (R): n ∝ RD-3. Interestingly, several authors had presented their experimental results as log n vs log R, yet no explanation was offered to the straight lines obtained; the notion of self-similarity (fractal surfaces) fits nicely in all these studies. The various features of our approach are described in connection with the analyzed examples, e.g., we exemplify estimation of particle diameters by this approach. Attention is given to the question of range of self-similarity. It is suggested that materials with fractal surfaces were formed in general by an iterative mechanism. Based on the massive list of fractal surfaces found (D. Avnir, D. Farin, and P. Pfeifer, Nature (London) 308, 261, 1984), this is probably the operative mechanism for the majority of natural and synthetic materials.

Fractal dimension as working tool for surface-roughness problems

Abstract Surface analysis in terms of fractal (noninteger) dimension is designed to describe, by a single parameter, surface roughness over many orders of magnitude. It has been shown that this is possible, from the molecular level up. Several open problems, directed to the practical (routine) application of the method, are settled here. They include the question of minimal yardstick range and various aspects of sensitivity. For instance, we show that also for quite small yardstick ranges, i.e., when the pertinent power laws are established only over a narrow span of magnitudes, the full framework of fractal geometry is available; and that the method can identify, purely on roughness grounds, kaolinite and feldspar as concurring in an example studied here. The scope of fractal geometry is illustrated by an outlook on fractal analysis by means other than the present adsorption studies. It presents, among other things, the theoretical basis for measuring fractal dimension through Fourier analysis (Fourier series or power spectra) of surface profiles.

Ideally irregular surfaces, of dimension greater than two, in theory and practice

Abstract We apply Mandelbrots [1] concept of fractal (noninteger) dimension D to surfaces: We describe how surfaces can provide environments (e.g. for adsorbates) with 2⩽D 2 are “ideally irregular” (self-similar) surfaces; and D→3 obtains in the limit where a surface visits every point of a volume. We present three major methods to get D from experiment. Case studies give ample instances for well-defined D>2. Some eminent consequences are implied.

A stationary formulation of time‐dependent problems in quantum mechanics

A unifying framework is presented which treats time‐dependent problems in nonrelativistic quantum mechanics on equal footing with stationary ones. This is accomplished by elevating time t to the role of a dynamical variable and considering evolution in an extended space with respect to a progress variable τ. Among the new objects in the extended space are the time operator as well as operators, such as energy, that are not diagonal in time. The latter bear, e.g., on the question of ‘‘quantum chaos.’’ Only τ‐stationary states in the extended space are necessary to recover all of the Schrodinger time evolution. In particular, time‐dependent constants of the motion (and these include time‐evolved density matrices) elevate to stationary constants of the τ motion. Time‐dependent problems, which may also involve time‐dependent Hamiltonians, can then be solved by stationary‐state methods. Special attention is given to applications based on time‐dependent constants of the motion and to maximum‐entropy states subj...

Initial states of maximal entropy in formal scattering theory

For a precollision state which is not sharply defined, not all details of the dynamics can be obtained by examination of the final state. A version of formal collision theory which takes full advantage of the inherent loss of detail is provided. Only the relevant aspects of the dynamics need then be computed and an explicit identification of the relevant details is provided by a generalized intertwining theorem. Several explicit examples where this point of view can be implemented are worked out.

Hartree theory for matter in a photon or phonon field

Abstract For a Dicke-type Hamiltonian describing M two-level systems coupled to N modes of a Boson field, the expectation value is minimized with respect to unrestricted products of 2M-level states and field states. In the resulting Hartree ground state(s) ϕ ⊗ Π, ϕ is ground state of a cubic Schrodinger equation in C 2M and Π is the product of N coherent one-mode states depending on ϕ. It is proved that ϕ factorizes as well, into M two-level states determined by a nonlinear equation in R M; that for weak coupling the Hartree ground state is unique and independent of the parameters in H; and that for strong coupling there are consecutively 2,…,2L (and possibly even more) Hartree ground states where 1≤L≤min{M, 2N} counts certain reflection symmetries. Details of this symmetry-breaking bifurcation (such as structural stability of relaxed two-level systems) and connections to the true ground state(s) of H are worked out.