Statistical thermodynamics of an anharmonic oscillator

In the present study we investigate the statistical thermodynamics of the anharmonic oscillator, whose energies are characterized by the potential 1/2x ²+x ⁴. Employing the energies recently obtained by Hioe and Montroll, we compute the partition function and the thermodynamic quantities for the anharmonic and quartic oscillators. Low- and high-temperature formulas are presented for the thermodynamic quantities of the oscillators.     

Matrix moment methods in perturbation theory,boson quantum field models,and anharmonic oscillators

A matrix moment problem is considered in connection with anyx ^2m (m=2, 3, 4, ...) anharmonic oscillator as well as the (:^2m (x):g(x))₂ (m=2, 3) field theory models, whose Rayleigh-Schrödinger perturbation expansions for the ground state eigenvalue are known to diverge. The approximants related to such a problem are proven to converge to the eigenvalue, when applied to an expansion of the Brillouin-Wigner type. These approximants, whose construction involves only matrix elements occurring in the Rayleigh-Schrödinger expansion, are the approximants of aJ-type matrix continued fraction, i.e. the [N–1,N] matrix Padé approximants. The explicit analytical expression of matrix continued fraction is found in the anharmonic oscillators case.     

The energy levels for simplified anharmonic oscillators

The energy levels for quantum mechanical oscillators with interaction imitatingx (for integer >2) are found by perturbative methods in finite number of dimensions. It is argued that in the limit of infinite dimensional space the coefficients in the expansion for the energy of theith level are growing with the perturbation ordern like . For the ground state (i=1) this reproduces estimates established for anharmonic oscillators.     

No Zero Energy States for the Supersymmetric x 2 y 2 Potential

We show that the positive supersymmetric matrix-valued differential operator H = p _x ² + p _y ² + x ² y ² + x₃ + y₁ has no zero modes, i.e., H = 0 implies = 0. The result depends on a virial type argument for the corresponding supercharge. The model may be regarded as a simple relative of dimensional reductions of supersymmetric Yang–Mills theories.     

Z 3-Graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d³=0, but d²0. The entities dx ⁱ and d² x ^k generate an associative algebra; we shall suppose that the products dx ⁱ dx ^k are independent of dx ^k dx ⁱ , while theternary products will satisfy the relation: dx ⁱ dx ^k dx ^m =jdx ^k dx ^m dx ⁱ =j ²dx ^m dx ^m dx ⁱ dx ^k , complemented by the relation dx ⁱ d² x ^k =jd² x ^k dx ⁱ , withj:=e^2i/3.We shall attribute grade 1 to the differentials dx ⁱ and grade 2 to the second differentials d² x ^k ; under the associative multiplication law the grades add up modulo 3.We show how the notion ofcovariant derivation can be generalized with a 1-formA so thatD:=d+A, and we give the expression in local coordinates of thecurvature 3-form defined as :=d² A+d(A ²)+AdA+A ³.Finally, the introduction of notions of a scalar product and integration of theZ ₃-graded exterior forms enables us to define the variational principle and to derive the differential equations satisfied by the 3-form . The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensorF _ik and its covariant derivativesD _i F _km .     

Three-dimensional relativistic-covariant poisson brackets

Using the algebraic properties of Poisson brackets, we extend the three-dimensional brackets (for a single free particle) to conform to the demands of special relativity. This yields, in an essentially unique way, the manifestly covariant extension [x ,p _v]=+p p _v/m ² c ². Position and time then become fully dynamical variables expressible in terms of the canonical conjugateq _i andp _i and the time parameter asx _i =q _i +p _i(q ·p)/m ²c² andt = +E(q ·p)/m ²c⁴. In the quantized version, the length associated with a particle of massm is shown to be an integral multiple of the Compton wavelength _C =/mc.     

Magnetic Properties of Permalloy/Cu-Au Multilayers

The interlayer exchange coupling and GMR effect of (permalloy/Cu _x Au_1-x )30 (Py = Ni₈₃Fe₁₇; 0.29x0.75) sputtered multilayers (MLs) were investigated. The strength of the antiferromagnetic (AF) interlayer coupling J _AF was determined from M(H) and/or R(H) curves. GMR effect and AF coupling was found in entire investigated concentration range of Cu _x Au_1-x . For x<0.65 the J _AF values at the first maximum of AF coupling (1.3t _Cu-Au1.6 nm) were smaller than 3×10^-6 J/m² and for x>0.65 J _AF increased to a value characteristic of Py/Cu MLs (J _AF10^-5 J/m²). The second maximum of AF coupling (J _AF10^-7 J/m²) was only found for x0.75 at t _Cu-Au2.6 nm.     

Strong coupling expansion for classical statistical dynamics

We discuss the simple, randomly driven systemdx/dt = –x –x³ +f(t), wheref(t) is a Gaussian random function or stirring force with f(t)f(t) = (t – t). We show how to obtain approximately the coefficients of the expansion of the equal-time Green's functions as power series in (1/R)ⁿ, whereR is the internal Reynolds number ()^1/2/, by using a new expansion for the path integral representation of the generating functional for the correlation functions. Exploiting the fact that the action for the randomly driven system is related to that of a quantum mechanical anharmonic oscillator with Hamiltonianp ²/2 +m ² x ²/2 +vx ⁴ +x ⁶/2, we evaluate the path integral on a lattice by assuming that thex ⁶ term dominates the action. This gives an expansion of the lattice theory Green's functions as power series in 1/(a)^1/3, wherea is the lattice spacing. Using Padé approximants to extrapolate toa = 0, we obtain the desired large-Reynolds-number expansion of the two-point function.Supported financially by the National Science Foundation and the U.S. Department of Energy.     

Decay of helicity in homogeneous turbulence

The self-similar relaxation of helicity in homogeneous turbulence has been considered taking into account integral invariants ∫ ₀ ^∞ r ^m 〈u(x)ω(x + r)〉 dr = I _m ^h (where ω = curlu and r = |r|). It has been shown that integral invariants with m = 3 for both helicity and energy are possible in addition to helical analogs of Loitsyanskii (m = 4) and Birkhoff-Saffman (m = 2) invariants associated with the conservation laws of momentum and angular momentum, respectively. Helicity always relaxes more rapidly than the energy. Its decay exponent is in the interval from ?3/2 to ?5/2 versus the interval from ?6/5 to ?10/7 for the energy.     

A class of exact solutions for doubly anharmonic oscillators

The solution of a difference equation in the form of an infinite continued fraction is used to obtain a class of exact solutions for the eigenfunctions and eigenvalues of doubly anharmonic oscillators described by potentials of the type (1/2)²x²+(1/4)x⁴+(1/6)x⁶, n>0, provided certain constraints on the couplings are satisfied. The class is denumerably infinite but not complete.     

Supertransferred hyperfine magnetic fields at111Cd(←111In) in ferrimagnetic oxides with the spinel structure

The supertransferred hyperfine magnetic fields,H _STHF, at¹¹¹Cd(¹¹¹In) on the tetrahedral sites of the spinel oxides Fe_3-xM_xO₄ (M = Ni or Co) were measured by means of the time differential perturbed angular correlation. The observedH _STHF were analyzed as the sum of the fields transferred from individual B site magnetic ions: 11 kOe/Fe³⁺, 8 kOe/Fe²⁺, 4 kOe/Ni²⁺ and 4 kOe/Co²⁺. A brief discussion is presented on these results in comparison with previous ones onH _STHF at¹¹¹Cd(^111mCd) in rocksalt-type oxides and the one at¹¹¹in in NiFe₂O₄.     

Magnetic Study of Doped Ba Ferrites Prepared by Citrate Method

The M-type barium ferrite is doped with either Co²⁺-Ti⁴⁺ or Co²⁺-Zr⁴⁺ pairs to reduce its intrinsically high uniaxial magnetic anisotropy in order to make fine particles for magnetic recording. The magnetic parameters were investigated by magnetic measurements and the Mössbauer spectroscopy. Compounds (BaF) obtained from BaCo _x (Ti,Zr) _x Fe_12-2x O₁₉ with 0x1.0 have been prepared by the modified citrate method with the initial ratio of Ba:Fe equal to 1:10.8. The substitution of Co - Ti or Co - Zr for Fe³⁺ ions affects mainly the positions 2a and 4f ₂. Ba ferrite with the substitution x 0.8 is promising for perpendicular recording media applications.     

119Sn-mössbauer,IR, and DSC study of tin oxyfluoride glasses

The¹¹⁹Sn-Mössbauer spectra ofxSnO·(70–x)SnF₂·30P₂O₅ glasses (0 x70) measured at 78 K comprised a doublet due to Sn²⁺ (=3.30–3.36 mm s^–1, = 1.70–1.72 mm s^–1) and a weak singlet due to Sn⁴⁺ located at –0.23 mm s^–1 with respect to BaSnO₃. The and of Sn²⁺ were comparable to those of Sn₂P₂O₇. Small Debye temperatures (146 and 155 K) were obtained from the low-temperature measurements. These results indicate that Sn²⁺ and Sn₄₊ occupied interstitial sites, being loosely and ionically bonded to distorted PO_4–x F _x tetrahedra.     

Mössbauer studies on the series La1-xSrxFeO3

Mössbauer spectra have been recorded at 4.2 and 300 K on the series La_1–x Sr _x FeO₃, wherex varies from 0 to 1.0 in steps of 0.1. Neutron diffraction experiments have shown that the crystal structure is orthorhombic for 0x<0.3, rhombohedral for 0.4x0.7, and cubic for 0.8<x1.0. Mössbauer spectra at 4.2 K are composed of magnetic sextet components arising from different charge states of iron ions. In the orthorhombic and rhombohedral phases, the charge states Fe³⁺ and Fe⁵⁺ coexist. In the cubic phase, iron is present as Fe³⁺ and Fe⁴⁺ states. At 300 K, the samples are magnetically ordered in the range 0 x0.3 and the coexistence of Fe³⁺ and Fe⁵⁺ remains. For samples 0.4x1.0, the samples are paramagnetic. Fits to these spectra require two components, one corresponding to an Fe⁴⁺ state, the other being best described as an Fe³⁺ ion forx0.7 but forx>0.7 having a mean charge state which increases to 3.5 forx=1.0.     

Sol–gel deposition and luminescence properties of lanthanide ion-doped Y2(1-x)Gd2xSiWO8 (0≤x ≤1) phosphor films

Y_2(1-x)Gd_2xSiWO₈:A (0x1; A=Eu³⁺, Dy³⁺, Sm³⁺, Er³⁺) phosphor films have been prepared on silica glass substrates through the sol–gel dip-coating process. X-ray diffraction (XRD), Fourier transform infrared spectroscopy (FT-IR), thermogravimetric and differential thermal analysis (TG-DTA), atomic force microscope (AFM), scanning electron microscopy (SEM) and photoluminescence spectra as well as lifetimes were used to characterize the resulting films. The results of the XRD indicated that the films began to crystallize at 800 °C and crystallized completely at 1000 °C. The AFM and SEM study revealed that the phosphor films, which mainly consisted of closely packed grains with an average size of 90–120 nm with a thickness of 660 nm, were uniform and crack free. Owing to an efficient energy transfer from the WO₄^2- groups to the activators, the doped lanthanide ion (A) showed its characteristic f–f transition emissions in crystalline Y_2(1-x)Gd_2xSiWO₈ (0x1) films. The optimum concentrations for Eu³⁺, Dy³⁺, Sm³⁺, Er³⁺ were determined to be 21, 5, 3 and 7 mol % of Y³⁺ in Y₂SiWO₈ films, respectively. The above lanthanide ions showed higher emission intensity for 02(1-x)Gd_2xSiWO₈ films. PACS 73.63.Bd; 78.55.Hx; 78.66.Nk; 81.15.Lm; 81.20.Fw     

Approximate vacuums in (:φ2m :g(x))2 field theories and perturbation series

A class of perturbation problems is considered, in which the Rayleigh-Schrödinger perturbation series for the ground state eigenvalue and eigenvector are presumed to diverge. This class includes the (:^2m :g(x))₂, (m=2, 3) quantum field theory models and the quantum mechanical anharmonic oscillator. It is shown that, using matrix elements and vectors which occur in the series coefficients, one may construct convergent approximants to the eigenvalue and eigenvector. Results of a calculation of the ground state energy of thex ⁴ anharmonic oscillator are given.Supported in part by the National Research Council of Canada.     

A variational problem for a system of magnetic monopoles joined by Abrikosov vortices

An action functional, related to the Higgs model to field theory, depending on a complex scalar field and aU(1) connection is defined. The complex scalar field is a section of a line bundle associated to a principalU(1)-bundle with base space ³\{x ₁,...,x _n }. The pointsx ₁,...,x _n are the positions ofn magnetic monopoles of magnetic chargesm ₁,...,m _n, with . The existence of minimizers of the action functional is proven using direct methods of the calculus of variation. Regularity and decay properties of the minimizers are obtained. By constructing explicit comparison field configurations, we establish accurate upper and lower bounds for the action of the minimizers in a variety of special situations, e.g.n=2 andm ₁=–m ₂.     

The hole concentration on oxygen sites in the highT c superconductor Y1−Ba2−Cu3−O7−x

From XPS core level spectroscopy the average copper charge on the Cu sites in the high temperature superconductor Y₁Ba₂Cu₃O_7–x is determined as function of the oxygen vacancy concentrationx. Analysis of these data leads to the suggestion that there are holes on the oxygen sites in the basal plane of the crystal structure. The probability for holes on these oxygen ions is rather constant for 0x0.3 with a value of 0.64 and decreases to zero forx=0.5. The dependence of the superconducting transition temperature on the hole concentration is discussed. An energy level diagram for Cu²⁺ and Cu³⁺ in YBa₂Cu₃O_7–x is constructed.     

Vacuum Solutions Which Cannot Be Written in Diagonal Form

The vacuum solution ds² = dx² + x² dy² + 2 dz dt + x dt² of the Einstein gravitational field equation follows from the general ansatz ds² = dx² + (x) dx ^dx but fails to follow from it if the symmetric matrix g (x) is assumed to be in diagonal form.