Aniruddha Chakraborty

Nonadiabatic tunnelling in an ideal one-dimensional semi-infinite periodic potential system: an analytically solvable model

Nonadiabatic tunnelling in an ideal one-dimensional semi-infinite periodic potential system is analysed using our model. Using our simple analytically solvable model it is shown that an ideal one-dimensional semi-infinite periodic potential system can be thought as a model of molecular switch. The method is applicable to those systems where Greens function for the motion in the absence of coupling is known. The method has been applied to the problem where motion takes place on parabolic potentials, for which analytical expression of Greens function can be found.We propose an analytical method for understanding the problem of multi-channel electron transfer reaction in solution, modeled by a particle undergoing diffusive motion under the influence of one donor and several acceptor potentials. The coupling between the donor potential and acceptor potentials are assumed to be represented by Dirac Delta functions. The diffusive motion in this paper is represented by the Smoluchowskii equation. Our solution requires the knowledge of the Laplace transform of the Green’s function for the motion in all the uncoupled potentials. Understanding of electron transfer processes in condensed phase is very important in chemistry, physics, and biological sciences, for the experimentalists as well as theoreticians [1–20]. A large amount of research in this area has been dedicated in the understanding of the behavior of electron transfer reactions exhibited by donor-acceptor pairs in solutions. Multi-channel electron transfer in condensed phase is one of the very interesting problem to study. In general the quantum ”jumps” of a high frequency vibrational mode can open several new point reaction sinks for electron transfer [21], in contrast to the broadening effect by a low frequency mode [22]. The work of Jortner and Bixon [21] was based on a quantum mechanical treatment of the high frequency vibrational coordinate in place of classical one of Sumi and Marcus [15]. However, theoretical treatment of Jortner and Bixon [21] did not consider the dynamics of motion on the potential surfaces. In the following we propose a simple analytical method for understanding the problem of multi-channel electron transfer reaction in solution, modeled by a particle undergoing diffusive motion under the influence of one donor and several acceptor potentials explicitly. A molecule (donor - acceptors) immersed in a polar solvent can be put on an electronically excited potential (represents the free energy of the donor surface) by the absorption of radiation. The molecule executes a walk on that potential, which may be considered as random as it is immersed in the polar solvent. As the molecule moves it may undergo non-radiative decay from certain regions of that potential to several potentials (represents the free energy of acceptor potentials). So the problem is to calculate the probability that the molecule will still be on the electronically excited donor potential after a finite time t. We denote the probability that the molecule would survive on the donor potential by Pd(x,t). We also use P (i) a (x,t) to denote the probability that the molecule would be found on the i-th acceptor potential. It is very usual to assume the motion on all the potentials to be one dimensional and diffusive, the relevent coordinate being denoted by x. It is also common to assume that the motion on all the potential energy surface is overdamped. Thus all the probability Pd(x,t), P (i) a (x,t)s may be found at x at the time t obeys a modified Smoluchowskii equation.

Multi-channel scattering problems: an analytically solvable model

Nonadiabatic transition due to potential curve crossing is one of the most important mechanisms to effectively induce electronic transitions in collisions [1]. This is a very interdisciplinary concept and appears in various fields of physics and chemistry and even in biology [2–7]. The theory of non-adiabatic transitions dates back to 1932, when the pioneering works for curve-crossing and non-crossing were published by Landau [8], Zener [9] and Stueckelberg [10] and by Rosen and Zener [11] respectively. Osherov and Voronin solved the case where two diabatic potentials are constant with exponential coupling [12]. C. Zhu solved the case where two diabatic potentials are exponential with exponential coupling [13]. In this paper we consider the case of two or more diabatic potentials with Dirac Delta couplings. The Dirac Delta coupling model has the advantage that it can be exactly solved [14–19] if the uncoupled diabatic potential has an exact solution.We have proposed a general method for finding an exact analytical solution for the multi-channel scattering problem in the presence of a delta function coupling. Our solution is quite general and is valid for any set of potentials, if the uncoupled diabatic potential has an exact solution. We have also discussed a few examples, where our method can easily be applied.

Curve crossing problem with Gaussian type coupling: analytically solvable model

We give a general method for finding an exact analytical solution for the two state curve crossing problem. The solution requires the knowledge of the Greens function for the motion on the uncoupled potentials. We use the method to find the solution of the problem in the case of parabolic potentials coupled by Gaussian interaction. Our method is applied to this model system to calculate the effect of curve crossing on the electronic absorption spectrum and the resonance Raman excitation profile.

Transfer matrix approach to the curve crossing problems of two exponential diabatic potentials

In the present manuscript, we have presented a method of calculation of non-adiabatic transition probability using transfer matrix technique. As an example for the two-state curve crossing problem, we have considered two diabatic potentials (two exponential potentials in the present case) with opposite sign of slopes which crosses each other and there is a coupling between the two diabatic potentials. The coupling is chosen as a Gaussian coupling which is further expressed as a collection of Dirac Delta potentials and the transition probability from one diabatic potential to another is calculated.

Buckled nano rod – a two state system and quantum effects on its dynamics using system plus reservoir model

We consider a suspended elastic rod under longitudinal compression. The compression can be used to adjust potential energy for transverse displacements from harmonic to double well regime. As compressional strain is increased to the buckling instability, the frequency of the fundamental vibrational mode drops continuously to zero (first buckling instability). As one tunes the separation between the ends of a rod, the system remains stable beyond the instability and develops a double-well potential for transverse motion. The two minima in the potential energy curve describe two possible buckled states at a particular strain. From one buckled state it can go over to the other by thermal fluctuations or quantum tunnelling. Using a continuum approach and transition state theory (TST) one can calculate the rate of conversion from one state to the other. The saddle point for the change from one state to the other is the straight rod configuration. The rate, however, diverges at the second buckling instability. At this point, the straight rod configuration, which was a saddle until then, becomes a hill top and two new saddles are generated. The new saddles have bent configurations and as the rod goes through further instabilities, they remain stable and the rate calculated according to harmonic approximation around a saddle point remains finite. In our earlier paper, a classical rate calculation including friction was carried out [J. Comput. Theor. Nanosci. 4, 1 (2007)], by assuming that each segment of the rod is coupled to its own collection of harmonic oscillators – our rate expression is well behaved through the second buckling instability. In this paper we have extended our method to calculate quantum rate using the same system plus reservoir model. We find that friction lowers the rate of conversion.

EXACT RESULTS ON DIFFUSION IN A PIECEWISE LINEAR POTENTIAL WITH A TIME-DEPENDENT SINK

The Smoluchowski equation with a time-dependent sink term is solved exactly. In this method, knowing the probability distribution P(0, s) at the origin, allows deriving the probability distribution P(x, s) at all positions. Exact solutions of the Smoluchowski equation are also provided in different cases where the sink term has linear, constant, inverse, and exponential variation in time.

Effect of curve crossing induced dissociation on absorption and resonance Raman spectra: An analytically solvable model

An analytically solvable model for the crossing of a harmonic and a Morse potential coupled by Dirac Delta function has been proposed. Further we explore the electronic absorption spectra and resonance Raman excitation profile using this model and found that curve crossing had significant effect on the resonance Raman excitation profile.

Exact solution to the curve crossing problems of two linear diabatic potentials by transfer matrix method

In the present work we have reported a simple exact analytical solution to the curve crossing problem of two linear diabatic potentials by transfer matrix method. Our problem assumes the crossing of two linear diabatic potentials which are coupled to each other by an arbitrary coupling (in contrast to linear potentials in the vicinity of crossing points) and for numerical calculation purposes this arbitrary coupling is taken as Gaussian coupling which is further expressed as a collection of Dirac delta functions. Further we calculated the transition probability from one diabatic potential to another by the use of this method.

Simple Models of Magnetism, Oxford Graduate Texts, by Ralph Skomski

at which electronic devices can be included in siliconintegrated circuits. Faced with this dichotomy, an approach immediately presents itself: use silicon for OEICS and then high-density integration will follow. One major drawback to that approach is that, due to the indirect band gap of silicon, it is not possible to construct efficient light emitters or lasers in that material. By the same token, high-speed light modulators cannot easily be made in silicon and also silicon’s band gap does not allow an easy interface with the near infra-red wavelengths used by optical communications system. There is arguably a ‘third way’ where III–V materials are used as a base for developing customised OEICS. The trick here would be to standardise the design of components, such as lasers, optical waveguides and detectors which are used in such circuits. Then a foundry approach may be developed to enhance capabilities of III–V OEICs. It is suggested that gaining agreement on which designs to utilise may be more challenging than pressing ahead with the development of OEICs – whether in III–Vs or in silicon. The viewpoint of this volume, written by acknowledged experts in the field, is that the obstacles preventing the adoption of silicon in OEICs can be overcome. In that context, the authors have assembled a highly readable volume which offers both the basic physics and the technology needed to develop silicon OEICs. The material is well-structured with basic properties of silicon being treated carefully in the opening chapters. Then, in a logical sequence, attention is given to quantum structures, optical processes, including optical processes in quantum. This approach leads up to an exposition of light emitters in silicon, silicon light modulators and photo-detectors. Raman lasers occupy Chapter 9, and then Chapters 10–12 treat a number of aspects of silicon optical waveguides, including silicon waveguides for dense wavelength division multiplexing. The final chapter of the book captures fabrication techniques and material systems. The target audience for the book is postgraduate students, researchers and technologists ‘engaged in research and development and the study of materials in electronics and photonics’ as well as industrial scientists. Students will appreciate the problems which appear at the end of the chapters whilst the many references appended to the chapters will particularly assist researchers. In a fast moving field, the authors have endeavoured to produce a book which captures the state of the art in this field. In a word, one may say that they have succeeded in that respect. More generally the authors have managed to produce an accessible text which will benefit the target audience. The authors are, therefore, congratulated for the sustained effort over a decade, which has led to this volume.

Essentials of Electromagnetics for Engineering, by David A. De Wolf

Cambridge, Cambridge University Press, 2012, 524 pp., £35.00 (1st paperback edition), ISBN 978-0-52-166444-8. Scope: review. Level: undergraduate. This textbook is the paperback edition (with corre...