_{1}

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Two problems in solid state physics and superconductivity are addressed by applications of dispersion dynamics. The first is the Hall effect. The dynamics of charges that yield positive Hall coefficients in material having no mobile positive charges have always been problematic The effect requires both electric and magnetic response, but magnetic deflection is only possible in mobile charges. In high temperature superconductors, these charges must be electrons. Contrary to Newton’s second law, their acceleration is reversed in crystal fields that dictate negative dispersion. This is evident in room temperature measurements, but a second problem arises in supercurrents at low temperatures. The charge dynamics in material having zero internal electric field because of zero resistivity; and zero magnetic field because of the Meissner-Ochsenfeld diamagnetism; while the supercurrents themselves have properties of zero net momentum; zero spin; and sometimes, zero charge; are so far from having been resolved that they may never have been addressed. Again, dispersion dynamics are developed to provide solutions given by reduction of the superconducting wave packet. The reduction is here physically analyzed, though it is usually treated as a quantized unobservable.

Dispersion dynamics grows immediately out of Special relativity by the joint introduction of Planck’s law and the de Broglie hypothesis. After first order differentiation of the standard formulae E 2 = p 2 c 2 + m o 2 c 4 (where E represents energy, p momentum, c the speed of light, and m_{o} rest mass) and the wave equivalent ω 2 = k 2 c 2 + m o 2 c 4 / ℏ 2 (where ω represents angular frequency, k wave vector, and ℏ the reduced Planck constant) [

d ω d k ⋅ ω k = v g ⋅ v p = c 2 (1)

i.e. the product of phase velocity v_{p} (angular frequency/wave vector) with group velocity v_{g}_{ }(the first derivative d ω / d k ) is equal to the square of the speed of light. Alternative normalization gives v ′ g ⋅ v ′ p = 1 , where v ′ g = v g / c and v ′ p = v p / c . The group velocity is the speed of the particle in special relativity; the phase velocity is faster than light, and is measured either as the inverse of the group velocity or as the ratio of energy to momentum^{1}. The phase velocity is singular in the rest frame: v p → ∞ when k → 0 . Massless particles in vacuo have v p = v g . More generally, v p = ( 1 + m o 2 / k 2 ) 1 / 2 , in units c = ℏ = 1 ; and v g = k / ( k 2 + m o 2 ) 1 / 2 ≃ k / ω when k ≪ m o , and m o ≃ ω .

Previously we have explored implications for Equation (1) in many fields including wave packets [

Mathematicians choose their axioms; physicists falsify them. Typically, we investigate a new problem with simplicity, expecting complexity to later resolve oversimplification. As an early example, Newton believed light is corpuscular, moving in straight lines, where the complexity of wave-particle duality, came to dominate the subsequent legacy of 19^{th }and 20^{th} Century physics. The corresponding theoretical postulate of point particles^{2}, however convenient for limited purposes, poses anomalies for the real world of physics. For example, the electron may be regarded variously as a propagating wave in the electron microscope; or as an atomic wave orbital; or as an intrinsic magnetic current loop [

Even at the relatively large scale of the atom, simplification is employed in time independent, wave functional solutions, where time is re-introduced through perturbations. The wave function then becomes a probability amplitude for locating an extreme point particle. This feature becomes more mysterious when it is reduced to reality during quantum measurement. Mystery and physics are antithetical. In particular, suppose a particle, say a photon, is emitted by an atom near point A so that it may contemporaneously arrive near B or C with appropriate probabilities on the same wave front. Two anomalous possibilities occur: on the one hand ask, “Can the particle be absorbed at both B and C by breaking energy conservation and doubling energy”; or on the other, “If absorption occurs at B, how does information arrive at C—without breaking the group velocity, v, constraints in special relativity (v ≤ c, the speed of light)—in order to prevent a second absorption and to conserve energy?” A new solution for the reduction of the wave packet is available to Dispersion Dynamics [

In the following discussion we suppose an objective reality in our mathematical framework that is not limited by Bohr’s usage of phenomena [^{3}.

Before proceeding to a treatment of reduction, it will be necessary to clarify the notion of uncertainty in all its variations through multiple dimensions. The noteworthy example in superconductivity requires prior treatment of the Hall effect. In particular, high temperature superconductors, besides being ionic, have critical temperatures T_{c} that are typically an order of magnitude greater than classical, metallic, low temperature superconductors. Above respective critical temperatures T > T_{c}, the former group displays positive Hall coefficients, R_{H} > 0; the latter negative. The Hall voltage is generated in two stages: by a transient acceleration of electrons, across an applied magnetic field; in which magnetic deflection in the Lorentz force builds a charge potential in the steady state. The consequent field is normal to the two applied fields and produces a corresponding Hall voltage. When charge carriers are positive ions the Hall coefficient, R_{H} > 0; typically, negative charges yield R_{H} < 0. In high temperature superconductors, the charge carriers produce R_{H} > 0, though the only positive charges are stationary ions. These charges are subject to negative dispersion in ionic crystal fields [

A second and worse anomaly is the common description of superconducting currents. These are due to Cooper pairs of electrons that co-exist, in the superconducting state, with normal electrons [

In mathematical quantum mechanics the wave function is a probability amplitude for finding a point particle that is bounded by limits given by the Heisenberg uncertainty principle. By contrast, classical wave mechanics contains many techniques for predicting wave behaviors that depend on particular circumstances: the image plane; or near field; or far field; whether longitudinal or transverse; whether 1-dimensional; or 2-dimensional; or spherically symmetric; or cylindrical etc. Physical uncertainty describes properties of wave groups [

The stable wave packet ϕ is defined by mean angular frequency ω ¯ and mean wave vector k ¯ , which together stabilize the packet through conservation laws in energy and momentum (

ϕ = A ⋅ exp ( X 2 2 σ 2 + X ) , with X = i ( k ¯ x − ω ¯ t ) (2)

where σ is the coherence, A is the wave amplitude, and x, t are coordinates in space and time. We can define the uncertainty in time by the full width at height 1/e ( [

Δ t = 2 σ ω ¯ (3)

and uncertainty of frequency by Fourier transform:

Δ ω = 4 ω ¯ σ (4)

The product is a dual uncertainty in which σ cancels! By substituting for ω and using Planck’s law:

Δ E ⋅ Δ t = 8 ℏ , (5)

and by similar operations on de Broglie’s hypothesis:

Δ p x ⋅ Δ x = 8 ℏ . (6)

Both values are more than an order of magnitude greater than Heisenberg’s limit, ( ℏ / 2 ) (

Next consider diffraction in the transverse plane that is caused by a horizontal slit, width d. The diffracted beam intensity tends to zero when the angle away from axis θ = tan − 1 ( λ / d ) where λ represents wavelength, and θ is sufficiently small. Then, using the de Broglie relationship, p x = h / λ and Δ p y = 2 p x sin ( θ ) = 2 h / d ( [

Δ p y ⋅ Δ y = 2 h . (7)

This value is again greater than Heisenberg’s limit, here by the factor 8π.

It is well known that in Fresnel diffraction a narrow slit pinches a transmitted

beam in the near field, which is contrary to general effects elsewhere [^{2}/(3λ) downstream from the slit, the beam width is ~d/3, From this it follows that Δ p y = h / λ ⋅ d / 3 ⋅ 3 λ / d 2 = h / d , and:

Δ p y ⋅ Δ y ~ h / 3 . (8)

The dual uncertainty is 6 times smaller than in Equation (7) and over 4 times larger than Heisenberg’s limit. However, at gaps away from Critical, the contracted beam dimension varies rapidly and is illustrated elsewhere [

When a circular opaque screen of diameter Δs, is placed transverse to a parallel beam and viewed from downstream, an illuminated spot is seen on the axis. The intensity of the spot is greatly amplified when the circle is divided into optical half zones, of dimensions such that only zones of either odd or even sequence transmit. This zone plate acts like a lens [

The problem we address is this: how is a hole deflected in a magnetic field? This is supposed to occur in the Hall effect measured in a p-type semiconductor, P doped Si for example. The charge carrier has a positive Hall coefficient like a positive ion such as a proton or Li^{+} ion in whatever matrix. In the two step process, the charge is first accelerated in an electric field and then deflected in a crossed magnetic field. The force F on the deflection depends on the charge and is proportional to its velocity, v_{g}. Including the first stage electric acceleration. Instantaneously:

F = q E + q v g × B (9)

Where q represents the carrier charge, now presumed electronic, negative e; E represents the applied Coulomb field; and B the crossed magnetic intensity. Since v_{g} is parallel to E, the magnetic part F_{m} is normal to both E and B (

F = − e E − e v g × B (10)

So that the acceleration a can be written approximately:

a ≃ − e E m − e E t ′ m × B . (11)

where t' is a kind of mean time between collisions or the inverse of the number of collisions per unit time (^{4}.

Since the only positive charges in p-type Si, for example, are on immobile nuclei, it is obvious that the carriers cannot be ions; they must be electrons. To understand how the charges react like positive ions to produce positive Hall coefficients, we have to return to Einstein’s special relativity and derive the second derivative following Equation (1):

d 2 ω d k 2 = d v g d k = d v g d p = ( 1 m ′ − k 2 m ′ 3 ) = 1 m e f f = a F (12)

where m' is the relativistic mass m o / ( 1 − v g 2 / c 2 ) 1 / 2 ; effective mass m_{eff} is as defined in the brackets; a is acceleration in Newton’s second law of motion corresponding to applied force F, such as the electric force (Equation (9)). in electricity and magnetism. Notice that a negative second derivative, or curvature (in p-Si, Al etc.), causes negative effective mass and negative acceleration under the Coulomb force. By contrast, the magnetic force is normal to the velocity and applied magnetic intensity so that its potential is structured differently from those in electrostatics and gravity. The result is that in magnetism, δ ω = 0 and the effective mass responding to the magnetic part of the electromagnetic Lorentz force is conventional (see

Charge q | D^{±} curvature | m_{eff} | F z e l | a z e l = F z / m | s | 〈 v g z 〉 ≃ a t ′ / 2 | F x m | a x m = F x / m | q ⋅ a x m ≃ E x H a l l | |
---|---|---|---|---|---|---|---|---|---|---|

+ve ion | e | >0 | m_{i} | eE | eE/m | >0 | eEt'/2m | eB(eEt'/2m) | eB(eEt'/m^{2}) | >0 |

Electron | -e | >0 | m | -eE | -eE/m | <0 | -eEt'/2m | eB(eEt'/2m) | eB(eEt'/m^{2}) | <0 |

Holey electron | -e | <0 | -m | -eE | eE/m | >0 | eEt'/2m | -eB(eEt'/2m) | -eB(eEt'/m^{2}) | >0 |

Glossary: m_{i} ionic mass; a z e l electric acceleration; t' time between collisions: expected velocity is half this time x acceleration; 〈 v g z 〉 mean z-component velocity.

d v g d k = 1 m = a x F x m . (13)

These formulae describe the motion of a single electron in an electron gas; they should be multiplied by carrier density to find measured values.

In summary:

1) The Hall effect occurs in two moments: a linear Coulomb acceleration; and a magnetically deflected charge. The Hall voltage results from the steady state deflection of charges.

2) As for free electrons, in typical metals (e.g.Cu) the normal dispersion band curvature D + = d 2 ω / d k 2 > 0 at Fermi level intercepts. R_{H} < 0 when the carriers have charge -e (Equation (11)).

3) By contrast, because all positive charges in p-Si and Al are immobile and since R_{H} > 0, the charge carriers in the magnetic field must be electrons.

4) Dispersion dynamics predicts that in crystal fields with negative dispersion D^{−}, the effective mass and acceleration are negative (Equation (12)).

5) In consequence, the velocity of electrons due to the Coulomb force F^{el} is reversed in D^{−}, compared with D^{+}.

6) In further consequence, the backward velocity reverses the otherwise conventional magnetic force F^{m} and deflection (^{−} crystal field dispersion, the Hall coefficient R_{H} > 0.

This gives the sign of R_{H} for “holey” electrons. Dispersion dynamics gives the reasons how and why those electrons behave like positive ions in the Hall effect. The argument is significant in high temperature superconductivity, where, because all positive charges are nuclear and immobile, the charge carriers can only be electrons [

Physically, how does it happen that a supercurrent is measured outside a superconductor, when its internal measurables, E, B, pair momentum K, pair spin S, resistivity r, etc., are all nominally zero? Reduction is mysterious when introduced as a change between time-independent wave functions: by hypothesis, the change may be represented as instantaneous at some point in space. In reality it is not so: in modern physics, resonant wave packets are the typical instruments for the force of change, and the packet is extended in both space and time. The space is divided between a direction of propagation and the corresponding transverse plane. Each of these differentiates as in Equation (1), where the group propagates with properties that are distinct from its internal transverse phase. In massive particles the group travels slower than light; the phase faster. The phase velocity v_{p}, in a particle with non-zero rest mass, is singular in the rest frame. Transmission of information is therefore theoretically instantaneous and Newtonian, i.e. within the transverse spatial coherence of the wave packet.

In massless photons travelling in vacuo, by contrast, the group and phase velocities share common speed c. In fact of course, the transmission is spread over time that is determined by the coherence 2σ, of the wave packet. This coherence determines likewise the spatial transmission, consistently with the methods of Huygen’s wavelet construction and with the methods of Fresnel, Fraunhoferetc. [

Transmission of energy or momentum by massive particles, like the electron, is most easily considered in the rest frame: energy and momentum are conserved when a wave front encounters two targets simultaneously. The transverse phase velocity is singular and time is Newtonian, so that energy supplied to one target cannot be given to the other owing to released wave amplitude.

However, creation and annihilation of a massless particle, like the photon, is not so simple because c is the constant velocity for both group and phase velocities. By ignoring diffraction, the transverse spatial coherence then becomes c multiplied by the temporal coherence. This photon is described in Maxwell’s theory by two real, sinusoidal, wave functions on mutually normal planes, one lagging in phase by π/2. Reduction of the packet thus occurs within its spatio-temporal coherence. From Equation (2), the longitudinal coherence length is found to be 2cσ; the coherence time 2σ/c. Through σ, the coherence values depend on initial conditions for the transmitting wave packet.

What can be understood about the transverse spatial coherence? Can a wave front become wider than the coherence? Does the coherence limit beam spread? Compare the temporal coherence from section 2.1 (2σ/c) with the transverse spatial coherence from Section 2.2 (tan^{−1}(λ/d)). It is clear that when, typically λ ≪ d , the ratio c Δ t / ( λ / d ) ~ 2 σ d / λ . Thus, when σ > d / λ , the transverse spatial coherence is determined by longitudinal temporal coherence; otherwise wave diffraction, d/λ, determines the transverse coherence. These features are in principle demonstrated by comparing diffraction of light generated by a laser source or point source. Supercurrents are massive, so we can now consider their reduction in their rest frame, i.e. in Newtonian time.

The solution for superconductivity was given by Bardeen, Cooper and Schrieffer [_{c}, and applied magnetic field H < H_{c}. The superconducting state consists of a pair of electrons bound together by a lattice distortion and energy gap ΔE. This Cooper pair has zero net momentum K = 0; zero net spin S = 0^{5}; and electronic charge 2e. The wave is superconductive, with zero resistivity ρ = 0 and therefore zero internal electric field E = 0. Meanwhile zero internal magnetic field induction B = 0 is demonstrated by the Meissner-Ochsenfeld effect. This is the basic state, though minor complexity arises in type II superconductors where electrons propagate quantized magnetic flux lines, and also at normal state boundaries as in the Josephson dc and ac tunneling effects. Since the normal and superconducting states are, by supposition, time independent, anomalies naturally arise in their dynamics. In high temperature superconductors, we understand the Cooper pair to be Wannier excitons with charge Q = 0 [

Consider for contrast, the more comprehensible oscillator strength in spectral transitions. This is a calculation of the relative transition rate between two stable electronic states, initial and final, excited by an operator O: | 〈 y f | O | y i 〉 | 2 , typically integrated over space, time, and density of states. In a dipole transition for example, O = er, where e represents electronic charge. Apply such a transition to measurement of resistivity of a superconductor by the four probe technique: at T < T_{c} and H < H_{c}. When a voltage is applied between two outer terminals the supercurrent is measured at the inner sensor terminals. This current consists of normal electric charges that emerge from the positive terminal of a superconductor containing no electromagnetic fields. In dispersion dynamics this feature is not surprising: the terminals supply energy that breaks the Cooper pairs which release their normal charges to metallic terminals in Newtonian time, i.e. when considered in the rest frame. The dynamics will be governed by the transfer of thermal or electrical energy to the Cooper pairs. Each pair has zero net momentum, while any dynamic momentum that is transferred to the normal electrons at terminals, can be obtained by the pairs from the massive lattice as in X-ray diffraction and in the Mossbauer effect. Between the terminals, a supercurrent flows in absence from any internal electric or magnetic fields.

In dispersion dynamics, the product of the group velocity and phase velocity in a wave packet is equal to the square of the speed of light. In consequence, uncertainty is not an arbitrary limit, but a calculated expectation that has varied effects in reduction. Moreover, since immobile positive ions are not deflected by the magnetic Lorentz force, the positive Hall coefficients that are measured in certain metals and doped semiconductors must be generated by (negatively charged) “holey” electrons that exist in states having peculiar dispersive dynamics. Furthermore, in ionic high temperature superconductors, those electrons are supposedly contained in bosonic pairs of Wannier excitons. How then do supercurrents flow in the absence of internal electromagnetic fields and with zero momentum, zero spin, and zero net charge? In dispersion dynamics, the flow occurs by the reduction of time-independent waves that are consistent with packet decay, i.e. in Newtonian time between terminals. This reduction is a new solution for an unanswered problem. Following the analysis, the reduction is in principle verifiable by time-dependent measurements. Superconductivity is a physical paradigm for Boson statistics, for condensation, and for related phenomena that occur throughout the broad scope of physic. These further applications of dispersion dynamics will be objects for further study.

The author declares no conflicts of interest regarding the publication of this paper.

Bourdillon, A.J. (2020) Reduction of Superconducting Wave Packets in Dispersion Dynamics. Journal of Modern Physics, 11, 365-377. https://doi.org/10.4236/jmp.2020.113023