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1993

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Articles 31 - 60 of 404

Full-Text Articles in Physics

Spatially Resolved Transitions To Autoionizing States, J. Greg Story, D. I. Duncan, Thomas F. Gallagher Nov 1993

Spatially Resolved Transitions To Autoionizing States, J. Greg Story, D. I. Duncan, Thomas F. Gallagher

Physics Faculty Research & Creative Works

We have observed transitions to autoionizing states in Mg using two short optical pulses. Mg atoms are initially prepared in a high lying Rydberg wave packet with the first ps laser. A second ps laser is then used to excite the inner electron, producing an autoionizing state. The dependence of the transition probability on the delay between the two lasers shows that when the second laser is tuned away from the ionic resonance, the inner electron can make a transition only when the Rydberg wave packet is near the core.


Large-Cell Renormalization-Group Approach To Long-Range Hopping On Energetically Disordered Lattices, B. D. Bookout, Paul Ernest Parris Nov 1993

Large-Cell Renormalization-Group Approach To Long-Range Hopping On Energetically Disordered Lattices, B. D. Bookout, Paul Ernest Parris

Physics Faculty Research & Creative Works

We describe an approach for computing the conductivity associated with long-range hopping on energetically disordered lattices. Using a numerically exact supercell procedure we compute the distribution ρL(γ) of block conductances γL associated with conducting cubes of edge length L that are randomly chosen from the disordered system of interest. This distribution of block conductances is then used in a self-consistent numerical calculation to obtain the renormalized bulk conductivity. The approach displays a surprisingly fast approach to the infinite-system limit, allowing finite-size effects to be minimized. In this paper we use this approach to study transport in a series of ...


The Temperature Dependence Of Perpendicular Anisotropy In Co/Pt And Co/Au Multilayer Films, H.Y. Zhang, Y.J. Wang, G.G. Zheng, J.X. Shen, Z.S. Shan, David J. Sellmyer Nov 1993

The Temperature Dependence Of Perpendicular Anisotropy In Co/Pt And Co/Au Multilayer Films, H.Y. Zhang, Y.J. Wang, G.G. Zheng, J.X. Shen, Z.S. Shan, David J. Sellmyer

David Sellmyer Publications

We report the temperature dependence of the effective perpendicular anisotropy and interface anisotropy in the temperature region of 80K to 290K for Co/Pt and Co/Au multilayers. Different temperature dependence of the interface anisotropy have been observed for these two types of multilayers. We speculate that the strain due to the lattice misfit may not be the main reason responsible for the perpendicular anisotropy in Co/Pt and Co/Au multilayers.


Structural And Magnetic Properties Of Lafe13-XSiX Nitrides, Z.X. Tang, X.H. Deng, George C. Hadjipanayis, V. Papaefthymiou, David J. Sellmyer Nov 1993

Structural And Magnetic Properties Of Lafe13-XSiX Nitrides, Z.X. Tang, X.H. Deng, George C. Hadjipanayis, V. Papaefthymiou, David J. Sellmyer

David Sellmyer Publications

LaFe13-xSix compounds were found to have two types of crystal structure; the face-centered cubic Na13--type (x ≤2.4) and the body-centered tetragonal Ce2Ni17Si9-type (3.5≤x≤5.0). Nitrogenation has a strong effect on the cubic phase, including a large lattice expansion (6.2 to 10.8 %), an increase of Curie temperature (52 to 80 K) and spontaneous magnetization ( ~10 %). The tetragonal phase, however, was not significantly affected by the nitrogenation.


Nucleation Field And Energy Product Of Aligned Two-Phase Magnets - Progress Towards The '1 Mj/M3' Magnet, Ralph Skomski, J.M.D. Coey Nov 1993

Nucleation Field And Energy Product Of Aligned Two-Phase Magnets - Progress Towards The '1 Mj/M3' Magnet, Ralph Skomski, J.M.D. Coey

Ralph Skomski Publications

Exchange hardening of nanostructured two-phase systems composed of an aligned hard phase and a soft phase with high magnetization is investigated using a micromagnetic approach which accounts for interactions between the soft regions. For Sm2Fe17N3 (2.5 nm)/Fe65Co35 (9 nm) multilayers an energy product as high as 1 MJ/m3 (120 MCOe) is predicted, witb a rare-earth content of only 5 wt %. Giant energy products may also be achieved in suitable cellular and disordered structures.


Phase Diagrams For The Blue Phases Of Highly Chiral Liquid Crystals, M. B. Bowling, Peter J. Collings, C. J. Booth, J. W. Goodby Nov 1993

Phase Diagrams For The Blue Phases Of Highly Chiral Liquid Crystals, M. B. Bowling, Peter J. Collings, C. J. Booth, J. W. Goodby

Physics & Astronomy Faculty Works

Polarizing microscopy and optical-activity measurements are used to determine the phase diagram for the blue phases of chiral-racemic mixtures of terephthaloyloxy-bis-4-(2'-methylbutyl) benzoate. Contrary to an earlier report, it is the second blue phase (BP II) rather than the first blue phase (BP 1) that is not stable relative to the other blue phases at high chirality. With this development, all phase diagrams for the blue phases reported to date have the same topology. Using similar data for two other highly chiral systems, it is found that a simple scaling of the temperature and chiral-fraction axes produces phase diagrams ...


Triply Differential (E,2e) Cross Sections For He From Threshold To 50 Ev Above, C. Pan, Anthony F. Starace Oct 1993

Triply Differential (E,2e) Cross Sections For He From Threshold To 50 Ev Above, C. Pan, Anthony F. Starace

Anthony F. Starace Publications

Distorted wave calculations of the triply differential cross sections for electron-impact ionization of He are presented for electrons sharing ≤ 50 eV excess energy (Ex) and having the coplanar, θ12 = π geometry. The results are in good agreement with absolute experimental measurements of T. Rösel et al. [Phys. Rev. A 46 (1992) 2539-2552] and of A. J. Murray, M. B. Woolf, and F. H. Read [J. Phys. B 25 (1992) 3021-3036]. For final state electrons sharing 2 eV excess energy equally, many-body perturbation theory corrections to the authors' first order (e,2e) triply differential cross sections [Phys. Rev. A ...


Morphology And Ordering In Thin Liquid-Crystal Films On Flat Glass Substrates, Yushan Shi, Brian Cull, Satyendra Kumar Oct 1993

Morphology And Ordering In Thin Liquid-Crystal Films On Flat Glass Substrates, Yushan Shi, Brian Cull, Satyendra Kumar

Physics Publications

Results of a high resolution x-ray reflectivity study of growth morphology and ordering of thin liquid-crystal films on flat glass substrate are reported. An adsorbed bilayer of crystalline order, under three dimensional clusters, was found to persist well beyond the nematic to isotropic transition. This adsorbed layer appears to be related to the surface memory effects in liquid-crystal devices. The morphology of these films can be described by the Stranski-Krastanov growth mode.


Inclusion Of Charge Correlations In Calculations Of The Energetics And Electronic Structure For Random Substitutional Alloys, Duane D. Johnson, F. J. Pinski Oct 1993

Inclusion Of Charge Correlations In Calculations Of The Energetics And Electronic Structure For Random Substitutional Alloys, Duane D. Johnson, F. J. Pinski

Duane D. Johnson

Currently, the coherent-potential approximation (CPA) implemented via the multiple-scattering theory of Korringa, Kohn, and Rostoker (KKR) gives the best first-principles description of the electronic structure for random substitutional alloys. However, the total energy has an important component of electrostatic energy missing, namely, that arising from the correlation of charges with varying atomic environments. We develop a ‘‘charge-correlated’’ CPA method (cc-CPA) which includes (some) local environmental charge correlations within the KKR-CPA method. We investigate the cc-CPA energetics for several alloys and show that the formation energies are in better agreement with experimental results. These calculations show that the excess charge on ...


Orientational Phases For M3C60, Taner Yildirim, Suklyun Hong, A. Brooks Harris, Eugene J. Mele Oct 1993

Orientational Phases For M3C60, Taner Yildirim, Suklyun Hong, A. Brooks Harris, Eugene J. Mele

Department of Physics Papers

The mechanism of the orientational ordering of C60 in alkali-metal-doped fullerenes M3C60 is studied. Since the M-C60 (M=K,Rb) interactions cause the C60 molecules to assume one of two standard orientations, this model is equivalent to a generalized Ising model on a fcc lattice. The Ising interactions depend on two type of energies: (1) the direct interaction, i.e., the orientationally dependent part of interactions between nearest-neighboring C60 molecules (each carrying charge -3e), and (2) the band energy of the electrons transferred from M+ ions to the C603- ...


Linear Dichroism In Photoemission From Oriented Molecules, N. A. Cherepkov, G. Schönhense Oct 1993

Linear Dichroism In Photoemission From Oriented Molecules, N. A. Cherepkov, G. Schönhense

Faculty Publications, Department of Physics and Astronomy

It is shown that oriented molecules reveal both Linear Dichroism (LD) and Linear Dichroism in the Angular Distribution (LDAD) of photoelectrons, which means a difference between photoelectron currents ejected at a definite angle by linearly polarized light of two mutually perpendicular polarizations. Measurements of LD and LDAD enable one to investigate the orientation of molecules and to make an essential step towards the complete quantum-mechanical experiment. As an example, LDAD in a model case of oriented heteronuclear diatomic molecules is presented.


Proposal For A Dod Combined Battlefield Electromagnetic Environmental Effects (E3) Initiative, George H. Baker Iii Oct 1993

Proposal For A Dod Combined Battlefield Electromagnetic Environmental Effects (E3) Initiative, George H. Baker Iii

Department of Integrated Science and Technology - Faculty Scholarship

The presentation emphasizes the growing importance of electromagnetic survivability and compatibility. Operation Desert Storm demonstrated the clear military advantage provided by sophisticated electronic weapon and communication systems. In addition, the offensive tactic of taking out the enemy's eyes and ears during the air war paid off, giving our military decisive air superiority. The lessons for the future are clear. High-tech electronics now so dominates the battlefield that the outcome of future conflicts could well be decided by electronics attrition rather than human casualties. Our Desert Storm experience thus accentuates the importance of guaranteeing that our electronic systems will not ...


Low-Field Magnetoresistance In Granular Pb Films Near The Insulator-Superconductor Transition, Richard P. Barber Jr., Robert C. Dynes Oct 1993

Low-Field Magnetoresistance In Granular Pb Films Near The Insulator-Superconductor Transition, Richard P. Barber Jr., Robert C. Dynes

Physics

We have studied the insulator-superconductor transition in quench-condensed granular Pb films from 0.1 to 10 K. Resistance measurements were made in zero magnetic field and low noise conditions. Magnetoresistance measurements were also performed for low magnetic fields (less than 100 G). The magnetoresistance results on the superconducting side of the transition suggest that we are directly probing the finite length scales associated with the range of phase coherence in these granular materials.


Spaceborne Radar Sensing Of Precipitation Above An Ocean Surface: Polarization Contrast Study, Alexander Kostinski, John Kwiatkowski, A. R. Jameson Oct 1993

Spaceborne Radar Sensing Of Precipitation Above An Ocean Surface: Polarization Contrast Study, Alexander Kostinski, John Kwiatkowski, A. R. Jameson

Department of Physics Publications

This feasibility study explores the potential benefits of polarization adjustment for spaceborne radar sensing of precipitation. More specifically, the role of the wave polarization in separating or “distinguishing” ocean surface return from the hydrometeor echoes of a “chirped” signal is examined.

To that end, experimental as well as computational data for the polarization scattering matrices of hydrometeors and ocean surfaces are obtained and used to calculate ocean and precipitation “response” to the transmitted pulse for various rain rates and incidence angles. The analysis is restricted to X and C bands, but simulations are performed for several signal-to-noise ratios, rain rates ...


Photoemission Study Of Ce-Catalyzed Oxidation Of W(110) And W(111), C. Gu, C. G. Olson, David W. Lynch Oct 1993

Photoemission Study Of Ce-Catalyzed Oxidation Of W(110) And W(111), C. Gu, C. G. Olson, David W. Lynch

Physics and Astronomy Publications

Different catalytic oxidation effects of Ce on W(110) and W(111) were observed with photo- emission spectroscopy. On Ce-covered W(110), the top W layer is quickly oxidized to a surface monoxide at room temperature. Oxidation is almost halted after the completion of one monolayer of WO. On Ce/W(111), instead of WO, WO3 grows on the surface under the same conditions. The monolayer suboxide formation found on Ce/W(110) and Ce/Ta(110) (as reported earlier) is a direct consequence of the most densely packed structure of the bcc (110) surfaces. The more effective oxidation promotion ...


New Exactly Solvable Hamiltonians - Shape Invariance And Self-Similarity, David T. Barclay, Ranabir Dutt, Asim Gangopadhyaya, Avinash Khare, A. Pagnamenta, Uday P. Sukhatne Oct 1993

New Exactly Solvable Hamiltonians - Shape Invariance And Self-Similarity, David T. Barclay, Ranabir Dutt, Asim Gangopadhyaya, Avinash Khare, A. Pagnamenta, Uday P. Sukhatne

Physics: Faculty Publications and Other Works

We discuss in some detail the self-similar potentials of Shabat and Spiridonov which are reflectionless and have an infinite number of bound states. We demonstrate that these self-similar potentials are in fact shape invariant potentials within the formalism of supersymmetric quantum mechanics. In particular, using a scaling ansatz for the change of parameters, we obtain a large class of new, reflectionless, shape invariant potentials of which the Shabat-Spiridonov ones are a special case. These new potentials can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for the energy eigenvalues, eigenfunctions and transmission ...


Energy-Projected Effective-Medium Theory Of Long-Range Hopping On Energetically Disordered Lattices, Paul Ernest Parris, B. D. Bookout Oct 1993

Energy-Projected Effective-Medium Theory Of Long-Range Hopping On Energetically Disordered Lattices, Paul Ernest Parris, B. D. Bookout

Physics Faculty Research & Creative Works

We introduce energy-projected equations of motion to treat the diffusive transport of charge carriers that undergo long-range (i.e., greater than nearest-neighbor) hopping among the sites of an energetically disordered lattice. This approach leads naturally to an energy-projected effective-medium theory for treating such systems. Exact expressions for the diffusion constant associated with the energy-projected effective medium theory are obtained. Using the formalism in conjunction with what is normally a rather poor approximation, i.e., the virtual-crystal approximation, we are able to obtain the exact diffusion constant for the long-range symmetric-random-well problem. Effective-medium calculations and numerical simulations are presented for nearest-neighbor ...


Renormalization Of Chiral Couplings In Tilted Bilayer Membranes, Philip C. Nelson, Thomas Powers Oct 1993

Renormalization Of Chiral Couplings In Tilted Bilayer Membranes, Philip C. Nelson, Thomas Powers

Department of Physics Papers

We study the effects of chiral constituent molecules on the macroscopic shapes attained by lipid bilayer membranes. Such fluid membranes are beautiful examples of statistical ensembles of random shapes~ sometimes coupled to in-plane order. We analyze them with methods of continuum elasticity theory, generalizing the well-known Canham-Helfrich model, and in particular incorporate the effects of thermal fluctuations. The condition that coordinate choice be immaterial greatly constrains the possible forms of the statistical weights in these systems, leading to very few independent couplings and hence physically simple models. Thermal fluctuations effectively reduce the chirality of a membrane at long scales leading ...


Electric-Field Effects On H- Photodetachment With Excitation Of H(N=2), Ning-Yi Du, Ilya I. Fabrikant, Anthony F. Starace Oct 1993

Electric-Field Effects On H- Photodetachment With Excitation Of H(N=2), Ning-Yi Du, Ilya I. Fabrikant, Anthony F. Starace

Anthony F. Starace Publications

A theory of the photodetachment of hydrogen negative ions in an external static electric field with the production of hydrogen in excited states is developed. We use a combination of the frame-transformation theory and the Green’s-function method. The cross section in nonzero fields is expressed in terms of the photodetachment matrix elements for zero field. The latter are calculated using the adiabatic hyperspherical representation. For energies close to the threshold we use a smooth extrapolation of the zero-field photodetachment matrix elements down to threshold taking care to enforce the proper threshold behavior of cross sections for nonzero fields. The ...


Correlation Functions For Ionic Motion From Nmr Relaxation And Electrical Conductivity In The Glassy Fast-Ion Conductor (Li2s)0.56(Sis2)0.44, Ivar Svare, Ferdinando Borsa, D. R. Torgeson, Steve W. Martin Oct 1993

Correlation Functions For Ionic Motion From Nmr Relaxation And Electrical Conductivity In The Glassy Fast-Ion Conductor (Li2s)0.56(Sis2)0.44, Ivar Svare, Ferdinando Borsa, D. R. Torgeson, Steve W. Martin

Materials Science and Engineering Publications

The Li7 NMR spin-lattice relaxation and the electrical conductivity in the typical glassy fast-ion conductor (Li2S)0.56(SiS2)0.44 are discussed from models of Li+ionic motion with distributions of activation energies, as well as from stretched-exponential time-correlation functions. The measured correlation times from the two effects differ by two orders of magnitude, and the derived distributions are shifted greatly relative to each other. We relate the great differences to percolation around the high barriers in the distribution. We present a phenomenological theory that yields good quantitative fits to the observed NMR relaxation with a Gaussian distribution, and ...


Driven Diffusion Of Particles, First-Passage Front, And Interface Growth, R. Bidaux, Ras B. Pandey Oct 1993

Driven Diffusion Of Particles, First-Passage Front, And Interface Growth, R. Bidaux, Ras B. Pandey

Faculty Publications

We study a computer-simulation model for driven particles on a discrete lattice where a fraction p of the lattice sites is randomly occupied by frozen impurities (barriers), and an imposed bias governs the particles' hopping through the lattice. These particles (the carriers) are initially released from a source of wetting fluid from one end of the lattice in order to wet and the dry lattice on their trails. We study the transport of particles, frontier of their trail and the growth of the interface between the wet and dry regions as a function of the biased field and the number ...


Resonant And Non-Resonant Pieces Of The D\Rightarrow \Bar{K}Π Semileptonic Transition Amplitude, Joseph Schechter, A. Subbaraman, S. Surya Sep 1993

Resonant And Non-Resonant Pieces Of The D\Rightarrow \Bar{K}Π Semileptonic Transition Amplitude, Joseph Schechter, A. Subbaraman, S. Surya

Physics

We compare the resonant and non-resonant contributions in various regions of phase space for the D\rightarrow \bar{K}\p semileptonic transition amplitude, computed in a chiral model which incorporates the heavy quark symmetry. Remarks on the significance for experiment and for chiral perturbation theory are made. hep-ph/yymmnnn


Cholesteric Polymer-Dispersed Light Shutters, John L. West, R. B. Akins, J. Francl, J. W. Doane Sep 1993

Cholesteric Polymer-Dispersed Light Shutters, John L. West, R. B. Akins, J. Francl, J. W. Doane

Chemistry & Biochemistry Publications

We report cholesteric/polymer dispersed (CPD) materials that switch from a reflecting, planar state to a relatively transparent, focal conic state. The reflective wavelength of the CPD materials can be continuously adjusted by varying the cholesteric pitch length. The electro‐optic properties of these materials are similar to the recently reported polymer stabilized cholesteric materials; however, the polymer concentration is sufficient to provide a self‐sustaining and self‐adhering film, offering the potential of large area devices fabricated on flexible substrates.


Mirrorless All‐Optical Bistability In Bacteriorhodopsin, D.V.G.L.N. Rao, F. J. Aranda, B. J. Wiley, J. A. Akkar, D. L. Kaplan, J. F. Roach Sep 1993

Mirrorless All‐Optical Bistability In Bacteriorhodopsin, D.V.G.L.N. Rao, F. J. Aranda, B. J. Wiley, J. A. Akkar, D. L. Kaplan, J. F. Roach

Physics Faculty Publications

We report direct observations of all‐optical mirrorless bistability associated with saturable absorption in three kinds of bacteriorhodopsin (BR) samples: wild‐type BR in water solution and dispersed in thin films of a polymer matrix as well as water solution of the genetically engineered mutant BRD96N. The experiments are carried out with picosecond pulses at 532 nm. The values measured for the saturation intensity are explained in terms of the relaxation of the excited M state population to the B state of the BR photocycle for the three kinds of samples studied.


Evidence For Two Exponent Scaling In The Random Field Ising Model, Michael Gofman, Joan Adler, Amnon Aharony, A. Brooks Harris, Moshe Schwartz Sep 1993

Evidence For Two Exponent Scaling In The Random Field Ising Model, Michael Gofman, Joan Adler, Amnon Aharony, A. Brooks Harris, Moshe Schwartz

Department of Physics Papers

Novel methods were used to generate and analyze new 15 term high temperature series for both the (connected) susceptibility χ and the structure factor (disconnected susceptibility) χd for the random field Ising model with dimensionless coupling K=J/kT, in general dimension d. For both the bimodal and the Gaussian field distributions, with mean square field J2g, we find that (χd-χ)/K2gχ2=1 as TTc(g), for a range of [h2]=J2g and d=3,4,5. This confirms the exponent relation γ¯=2γ (where χ ...


On The Escape Of Oxygen And Hydrogen From Mars, Jane L. Fox Sep 1993

On The Escape Of Oxygen And Hydrogen From Mars, Jane L. Fox

Physics Faculty Publications

Escape rates of oxygen atoms from dissociative recombination of O2+ above the Martian exobase are computed in light of new information from ab initio calculations of the dissociative recombination process, and our recently revised understanding of the Martian dayside ionosphere. Only about 60% of the dissociative recombinations occur in channels in which the O atoms are released with energies in excess of the escape velocity. Futhermore, we find that the computed escape fluxes for O depend greatly on the nature of the ion loss process that has been found necessary to reproduce the topside ion density profiles measured by ...


Numerical Study Of A High-Order Quasiconserved Quantity In The Henon-Heiles Problem, Paul Finkler, C. Edward Jones, Glenn A. Sowell Sep 1993

Numerical Study Of A High-Order Quasiconserved Quantity In The Henon-Heiles Problem, Paul Finkler, C. Edward Jones, Glenn A. Sowell

Physics Faculty Publications

Recent efforts to derive and study a quasiconserved quantity K in the Henon-Heiles problem in terms of a single set of variables are discussed. Numerical results are given, showing how the value of such a quantity varies with time and order in a power-series expansion for K in terms of monomials of the coordinates and velocities. The lowest order in the power series for K corresponds to n =4 and the highest order to n =27, so that 24 orders are included in the series. The results are compared with an earlier study by the authors [Phys. Rev. A 42 ...


Review Of Genius In The Shadows: A Biography Of Leo Szilard, The Man Behind The Bomb By William Lanouette And Bela Silard, David W. Hafemeister Sep 1993

Review Of Genius In The Shadows: A Biography Of Leo Szilard, The Man Behind The Bomb By William Lanouette And Bela Silard, David W. Hafemeister

Physics

No abstract provided.


Solar Cycle And Seasonal Variationsin F Region Electrodynamics At Millstone Hill, M. J. Buonsanto, M. E. Hagan, J. E. Salah, Bela G. Fejer Sep 1993

Solar Cycle And Seasonal Variationsin F Region Electrodynamics At Millstone Hill, M. J. Buonsanto, M. E. Hagan, J. E. Salah, Bela G. Fejer

Bela G. Fejer

Incoherent scatter radar observations of ion drifts taken at Millstone Hill (42.6°N, 288.5°E) during 73 experiments in the period February 1984 to February 1992 are used to construct, for the first time at this station, average quiet-time E×B drift patterns for both solar cycle maximum and minimum, for the summer, winter, and equinox seasons. The daily variation of V⊥N shows a reversal from northward to southward drifts near noon, and a return to northward drifts in the premidnight hours. The weaker southward drift in the afternoon in summer noted by Wand and Evans (1981 ...


Semiclassical Approach To Quantum-Mechanical Problems With Broken Supersymmetry, Ranabir Dutt, Asim Gangopadhyaya, Avinash Khare, A. Pagnamenta, Uday P. Sukhatne Sep 1993

Semiclassical Approach To Quantum-Mechanical Problems With Broken Supersymmetry, Ranabir Dutt, Asim Gangopadhyaya, Avinash Khare, A. Pagnamenta, Uday P. Sukhatne

Physics: Faculty Publications and Other Works

The semiclassical WKB approximation method is reexamined in the context of nonrelativistic quantum-mechanical bound-state problems with broken supersymmetry (SUSY). This gives rise to an alternative quantization condition (denoted by BSWKB) which is different from the standard WKB formula and also different from the previously studied supersymmetric (SWKB) formula for unbroken SUSY. It is shown that to leading order in ħ, the BSWKB condition yields exact energy eigenvalues for shape-invariant potentials with broken SUSY (harmonic oscillator, Pöschl-Teller I and II) which are known to be analytically solvable. Further, we show explicitly that the higher-order corrections to these energy eigenvalues, up to ...