Terms Of Fundamental Constants Research Articles

Colloquium : Planckian dissipation in metals

Relaxational processes for conduction electrons in metals come with characteristic temperature dependences; classic examples at low temperatures are the $1/{T}^{2}$ law for the relaxation time in a Fermi liquid, and the Bloch $1/{T}^{5}$ law for electron-phonon scattering. The strange metal phase of cuprate superconductors is governed instead by the Planckian time, which is given in terms of fundamental constants only. This Colloquium discusses the concept of such a fundamental timescale and the degree to which it provides a lower bound for relaxation times, and illustrates these ideas with examples from conventional and unconventional metals.

A ‘Dynamic Kibble’ mass balance for the undergraduate physics teaching laboratory

Originally envisaged in 1975 to realise the SI unit of electrical current, the Ampere, the Kibble balance has since developed into a powerhouse of modern scientific measurement. By combining theoretical simplicity with precision of measurement, it has enabled the redefinition of Planck’s constant, and subsequently a practical method of defining the kilogram in terms of fundamental constants. This article introduces a novel version of this classic apparatus, the ‘Dynamic Kibble’ Balance. Dynamic in this case because the magnet velocity is now 3 orders of magnitude higher than the original, but the same theory applies. The apparatus is simple in approach, robust, easy to set up, and capable of a high level of precision using only electrical measurements (plus length and time). The importance of this measurement to metrology re-enforces the link between what is measured in the laboratory via calibration, measurement standards, and traceability. Using the apparatus and measurements described in this paper, the mass of the magnet assembly was measured as 19.4 ± 0.3 g, which lies within one SEM of the known value. This paper describes an uncomplicated method with a clear focus on the key physics and theory required. This experiment is intended for use in a first-year undergraduate physics laboratory. Further potential for both more advanced theory demonstration and experimental work is discussed.

Double Integral involving the Product of the Bessel Function of the First Kind and Modified Bessel Function of the Second Kind: Derivation and Evaluation

A double integral whose kernel involves the Bessel functions Kv(xβ) and Jv(yα) is derived. This integral is expressed in terms of the Hurwitz-Lerch zeta function and evaluated for various values of the parameters involved. Some examples are evaluated and expressed in terms of fundamental constants. All the results in this work are new.

Non-Zero Order of an Extended Temme Integral

A new three-dimensional integral containing f(x,y,z)Iv(xα) is derived where Iv(xα) is the Modified Bessel Function of the first kind and the integral is taken over the infinite cubic space 0<x<∞,0<y<∞,0<z<∞. The integral is not easily evaluated for complex ranges of the parameters. A representation in terms of the Hurwitz–Lerch zeta function, polylogarithm function and Riemann zeta functions are evaluated. This representation yields triple integral representations in terms of fundamental constants that can be derived. Almost all Lerch functions have an asymmetrical zero distribution.

A quadruple integral involving the product of generalized parabolic cylinder functions 𝐷ᵥ(𝛽𝑥)𝐷ᵤ(𝛼𝑧): Derivation and evaluation

The aim of the present document is to evaluate a quadruple integral involving the product of the generalized Parabolic Cylinder functions D v ( β x ) D u ( α z ) D_{v}(\beta x)D_{u}(\alpha z) expressed in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.

A Quadruple Integral Involving Chebyshev Polynomials Tn(x): Derivation and Evaluation

The aim of the current document is to evaluate a quadruple integral involving the Chebyshev polynomial of the first kind Tn(x) and derive in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. The zero distribution of almost all Hurwitz-Lerch zeta functions is asymmetrical. All the results in this work are new.

Quadruple Integral Involving the Logarithm and Product of Bessel Functions Expressed in Terms of the Lerch Function

In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s ζ(3) constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new.

Double Integral Involving Logarithmic and Quotient Function with Powers Expressed in terms of the Lerch Function

In this work the authors use their contour integral method to derive the double integral given by $\int_{0}^{\infty}\int_{0}^{\infty}\frac{x^{m-1} y^{m+\frac{q}{2}-1} \log ^k(a x y)}{\left(x^q+1\right)^2 \left(y^q+1\right)^2}dxdy$ in terms of the Lerch function. This integral formula is then used to derive closed solutions in terms of fundamental constants and special functions. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus a table of integral pairs is given for interested readers. All the results in this work are new.

A Series Representation for the Hurwitz–Lerch Zeta Function

We derive a new formula for the Hurwitz–Lerch zeta function in terms of the infinite sum of the incomplete gamma function. Special cases are derived in terms of fundamental constants.

The Double Laplace Transform Expressed in terms of the Lerch Transcendent

In this manuscript, the authors derive a formula for the double Laplace transform expressed in terms of the Lerch Transcendent. The log term mixes the variables so that the integral is not separable except for special values of k. The method of proof follows the method used by us to evaluate single integrals. This transform is then used to derive definite integrals in terms of fundamental constants, elementary and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

Definite Integrals Involving Combinations of Powers and Logarithmic Functions of Complicated Arguments Expressed in Terms of the Hurwitz Zeta Function

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta functions. These derivations are then expressed in terms of fundamental constants, elementary, and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

Chandrasekhar limit for rotating quark stars

The limiting mass is a significant characteristic for compact exotic stars. In the case of quark stars the limiting mass can be expressed in terms of fundamental constants and the bag constant. In the present paper, using bag model description, the maximum mass of a rotating quark star is found to depend on the rotational frequency apart from other fundamental parameters. The analytical results obtained agree with the results of several relevant numerical estimates as well as observational evidences.

Quantized electrochemical transport in Weyl semimetals

We show that under the effect of an external electric field and a gradient of chemical potential, a topological electric current can be induced in Weyl semimetals without inversion and mirror symmetries. We derive analytic expressions for the nonlinear conductivity tensor and show that it is nearly quantized for small tilting when the Fermi levels are close to the Weyl nodes. When the van Hove point is much larger than the largest Fermi level, the band structure is described by two linearly dispersing Weyl fermions with opposite chirality. In this case, the electrochemical response is fully quantized in terms of fundamental constants and the scattering time, and it can be used to measure directly the topological charge of Weyl points. We show that the electrochemical chiral current may be derived from an electromagnetic action similar to axion electrodynamics, where the position-dependent chiral Fermi level plays the role of the axion field. This posits our results as a direct consequence of the chiral anomaly.

40 years of the quantum Hall effect

The discovery of the quantum Hall effect (QHE) marked a turning point in condensed-matter physics. The measurement of the Hall resistance showed that electronic resistance could be defined precisely in terms of fundamental constants, even in a disordered and irregular sample. Over the past 40 years, the QHE has inspired new theories and led to experimental discoveries in a range of fields going beyond solid-state electronics to photonics and quantum entanglement. In this Viewpoint, physicists reflect on how the QHE has influenced their research.

The universal acceleration scale from stellar feedback

ABSTRACT It has been established for decades that rotation curves deviate from the Newtonian gravity expectation given baryons alone below a characteristic acceleration scale $g_{\dagger }\sim 10^{-8}\, \rm {cm\, s^{-2}}$, a scale promoted to a new fundamental constant in MOND. In recent years, theoretical and observational studies have shown that the star formation efficiency (SFE) of dense gas scales with surface density, SFE ∼ Σ/Σcrit with $\Sigma _{\rm crit} \sim \langle \dot{p}/m_{\ast }\rangle /(\pi \, G)\sim 1000\, \rm {M_{\odot }\, pc^{-2}}$ (where $\langle \dot{p}/m_{\ast }\rangle$ is the momentum flux output by stellar feedback per unit stellar mass in a young stellar population). We argue that the SFE, more generally, should scale with the local gravitational acceleration, i.e. that SFE ${\sim}g_{\rm tot}/g_{\rm crit}\equiv (G\, M_{\rm tot}/R^{2}) / \langle \dot{p}/m_{\ast }\rangle$, where Mtot is the total gravitating mass and $g_{\rm crit}=\langle \dot{p}/m_{\ast }\rangle = \pi \, G\, \Sigma _{\rm crit} \approx 10^{-8}\, \rm {cm\, s^{-2}} \approx \mathit{ g}_{\dagger }$. Hence, the observed g† may correspond to the characteristic acceleration scale above which stellar feedback cannot prevent efficient star formation, and baryons will eventually come to dominate. We further show how this may give rise to the observed acceleration scaling $g_{\rm obs}\sim (g_{\rm baryon}\, g_{\dagger })^{1/2}$ (where gbaryon is the acceleration due to baryons alone) and flat rotation curves. The derived characteristic acceleration g† can be expressed in terms of fundamental constants (gravitational constant, proton mass, and Thomson cross-section): $g_{\dagger }\sim 0.1\, G\, m_{\mathrm{ p}}/\sigma _{\rm T}$.

Relativistic quantum backflow

In this paper we discuss relativistic quantum backflow. The general theory of relativistic backflow is presented and it is shown that the backflow can be written as a function of a simple parameter ϵ, which is defined in terms of fundamental constants and the backflow period. Backflow eigenfunctions are determined numerically for a range of values of ϵ, and an explicit expression for the relativistic backflow eigenvalue in terms of the non-relativistic backflow constant is presented. Backflow eigenvectors are then fitted with some standard functions, which lead to substantially higher backflow than has been found previously with fitting procedures, for some values of ϵ. In analysing the non-relativistic limit of the theory we show that this problem is one of those rare cases where the relativistic theory is intrinsically more simple than the non-relativistic theory.

Future trends in I&M: Connections with the universe

On 16 November 2018, the member states of the Metre Convention, at the 26th General Conference on Weights and Measures (CGPM) in Versailles, France voted to revise the International System of units (SI) and thus change the definition of the kilogram, the ampere, the kelvin and the mole [1]. All SI units are now defined in terms of fundamental constants: specific physical quantities that we believe to be perfectly stable in time and across the universe.

Solutions for Series of Exponential Equations in Terms of Lambert-W Function and Fundamental Constants

Series of exponential equations in the form of were solved graphically, numerically and analytically. The analytical solution was derived in terms of Lambert-W function. A general numerical solution for any y is found in terms of n or in base y. A solution is close to the fine structure constant. The equation which provided the solution as the fine structure constant was derived in terms of the fundamental constants.

A brief history of the unit of mass: continuity of successive definitions of the kilogram

The very first definition of the kilogram was in terms of a constant of nature, although this idea could not be fully realized at the end of the 18th century. Instead the kilogram was defined by an artefact whose mass was made to approximate as closely as possible a physical constant with unit kg m−3—the maximum density of distilled water at atmospheric pressure. For the next two centuries, mass comparators improved greatly as did the materials from which artefacts could be constructed. These improvements put tighter constraints on the realization of a non-artefact definition of the kilogram. However, it is now expected that the goal of redefining the kilogram in terms of fundamental constants will be achieved in 2018. We present a history of the kilogram with emphasis on continuity of this unit of mass each time it has been redefined and the stability of a unit defined by the mass of an artefact.

The minimum mass of a charged spherically symmetric object in D dimensions, its implications for fundamental particles, and holography

We obtain bounds for the minimum and maximum mass/radius ratio of a stable, charged, spherically symmetric compact object in a $D$-dimensional space-time in the framework of general relativity, and in the presence of dark energy. The total energy, including the gravitational component, and the stability of objects with minimum mass/radius ratio is also investigated. The minimum energy condition leads to a representation of the mass and radius of the charged objects with minimum mass/radius ratio in terms of the charge and vacuum energy only. As applied to the electron in the four-dimensional case, this procedure allows one to re-obtain the classical electron radius from purely general relativistic considerations. By combining the lower mass bound, in four space-time dimensions, with minimum length uncertainty relations (MLUR) motivated by quantum gravity, we obtain an alternative bound for the maximum charge/mass ratio of a stable, gravitating, charged quantum mechanical object, expressed in terms of fundamental constants. Evaluating this limit numerically, we obtain again the correct order of magnitude value for the charge/mass ratio of the electron, as required by the stability conditions. This suggests that, if the electron were either less massive (with the same charge) or if its charge were any higher (for fixed mass), a combination of electrostatic and dark energy repulsion would destabilize the Compton radius. In other words, the electron would blow itself apart. Our results suggest the existence of a deep connection between gravity, the presence of the cosmological constant, and the stability of fundamental particles.