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The present paper is based upon the fact that if an object is part of a highly stable oscillating system, it is possible to obtain an extremely precise measure for its mass in terms of the energy trapped in the system, rather than through a ratio between force and acceleration, provided such trapped energy can be properly measured. The subject is timely since there is great interest in Metrology on the establishment of a new electronic standard for the kilogram. Our contribution to such effort includes both the proposal of an alternative definition for mass, as well as the description of a realistic experimental system in which this new definition might actually be applied. The setup consists of an oscillating type-II superconducting loop subjected to the gravity and magnetic fields. The system is shown to be able to reach a dynamic equilibrium by trapping energy up to the point it levitates against the surrounding magnetic and gravitational fields, behaving as an extremely high-Q spring-load system. The proposed energy-mass equation applied to the electromechanical oscillating system eventually produces a new experimental relation between mass and the Planck constant.

Mass is traditionally interpreted as the inertia content of a body [

We shall be interested in systems consisting of a macroscopic object in some way attached to an environment or device capable of producing over it some restitution force. That is, if the object is initially at rest and is suddenly subjected to an external force there will be a reaction against such force tending to restore the object to its initial location. We will specialize in the well known Basic Physics example of one such system, which is the spring-load set. We develop the theory in full detail to facilitate the comprehension of the comparisons to be made with the electromechanical system to be discussed in Sections 2 and 3. The spring-load system will be taken initially at rest with no stored energy, until on time t = 0 a fixed external force F is imposed upon it and kept constant thereafter. Newton’s Law describes the displacement x(t) of the spring and load as:

Here k is the elastic constant of the spring. The initial conditions are. The solution of Equation (1) predicts a harmonic oscillation of the load around a position displaced of F/k from its initial position, with the amplitude of oscillation given by

with the frequency of the oscillations. Note that we are ideally assuming that no damping occurs, so that the total energy given by

is conserved. Equation (3) states that the total energy contained in the system is given by the work done upon the system by the force F along the range of oscillation. By taking the time derivative of Equation (2) we obtain the velocity of the oscillating load, and realize that Equation (3) can be rewritten in the concise form

Here v_{m} is the maximum velocity of the load, given by. We note the following. Firstly, the form of Equation (4) independs on the particular expression for F, as well as it independs on the details of the restitution force, which is simply taken as linear in the displacement. Provided the system is initially fully at rest the imposition of any constant force at t = 0 will result in Equation (4). Secondly, its applicability of course depends upon the ability of the system in conserving energy and upon the possibility of measuring E. In this resonating state the parameter mass is the ratio between the energy of the state and the characteristic speed squared. Therefore, we have derived an alternative definition for mass that can be very convenient provided a suitable system is devised to explore it.

If damping is negligible the system oscillates in a state of dynamic equilibrium with all the force fields, internal or external, around it. We will be interested in the particular case of F = mg, the weight of the load oscillating in the vertical direction. In this case v_{m} = g/ω, and the oscillations maximum amplitude is g/ω^{2}.

Equation (4) can be useful provided energy losses are vanishingly small. However, it is well known that real spring-load systems are highly dissipative [^{7} or more, so that the application of Equation (4) is justified. The present investigation is motivated by a particular (long-standing) Metrology problem, namely the determination of a new international standard for the kilogram in terms of an experimental expression involving well defined standardized parameters like the electronic charge and Planck’s constant. The current stage of such efforts has been described for instance in [3,4]. Our objective hereafter is to show that a relatively simple electromechanical system, in a single experiment, and adopting Equation (4) as the definition of mass might be able to produce an extremely precise measure for the mass of an object. The main result of this work is a new expression relating mass to other constants of Nature to be obtained by experiment (Section 2), followed in Section 3 by a full discussion of the principles behind the design of the experimental setup. Section 4 lists the Conclusions.

In order to obtain a real application for Equation (4), we need to find a system in which losses are practically eliminated. Air friction can be virtually eliminated by working under high vacuum and at very low temperatures, to markedly decrease the atmosphere viscosity. Heat dissipation in the restitution process responsible for the oscillations is much more difficult to eliminate. However, as we show hereafter it has been possible to design a system in which all dissipative processes can be diminished to the point that the application of Equation (4) becomes realistic. This system has been devised and previously analyzed in detail by Schilling [5,6] (although its equations of motion had independently been published much earlier by Saslow [

The superconducting oscillator (SEO) is shown in _{1 }and B_{2} normal to its surface. The need for two fields is explained in Section 3. These fields should be homogeneous upon the loop wire within a tolerance discussed in [

tridimensional equivalent of this system, comprising three mutually perpendicular rectangular oscillating loops subjected to the field of a magnetized sphere has recently been discussed in the Letter [_{m} will be compensated by flux Li generated by super currents i(t) around its perimeter, in such way that the application of Faraday’s Induction Law gives a null electromotive force ε:

Here L is the self-inductance of the loop. This is an example of the property of flux conservation displayed by superconducting loops and rings. The loop will move with speed v described by Newton’s Law:

In Equation (6) the last factor on the right is the result of the opposing magnetic forces F_{1} and F_{2} the fields B_{1} and B_{2} impose upon the currents in the two horizontal sides of the loop (see _{2}. We define B_{0} ≡ B_{1} − B_{2}. It must be stressed that none of the final results and conclusions depends on the precise determination of either a or of the magnetic fields, and neither on how sharp would be the boundary between these fields. From

which is the differential equation obeyed by the velocity of a harmonic oscillator. Assuming zero initial speed and an initial acceleration g, Equation (7) can be solved:

From Equation (7), the natural frequency of the oscillations is. As the loop is released from rest, the assumed perfect flux and energy conservations will make this initial position the uppermost point of its oscillating path (measured from the middle point of its oscillations range), with. The position is described by the equation

which is equivalent to (2) if F = mg. The maximum amplitude of the oscillating motion is g/Ω^{2}. It is clear that such equations are equivalent to the ones in Section 1 for the spring-load system, with ω replaced by Ω. It is possible then to combine (5) and (6) to obtain an equation for the current i(t), which flows within a very thin layer close to the wires surface [

whose solution is

for. We define.

In view of Equation (9), it is clear that Equation (3) also applies here, so that the same energy conservation is obeyed, with v_{m}= g/Ω from Equation (8), as in Section 1. We define Φ = Li_{0} as the flux threading the loop due to the average loop current value i_{0}, resulting after a simple manipulation in E = 2Φ^{2}/L. These two expressions for E lead to the following expression for the mass of the loop:

This is the main result of this work. A superconducting quantum interference device (SQUID) with a proper gradiometer as probe, with the oscillating loop between its coils as “specimen” will measure the flux Φ due to the average current in the loop within a fraction of the flux quantum Φ_{0} = h/(2e), with h the Planck constant and e the electronic charge. Optical interferometric measurements will accurately give the speed v_{m}, and L is a parameter for the loop, to be measured in advance in a separate experiment. Therefore, a simple experimental relation between mass and the Planck constant is obtained from this experiment (see [

In the next section, the actual design and implementation conditions necessary for this experiment are described in detail.

The design of the system has been discussed in [5,6], but we describe here the main details. However, this discussion does not include further details on the experimental techniques to be adopted in the measurement of the magnetic flux produced by the loop and of its position and speed, which lay beyond the scope of this paper and should be considered in due course. The design should start by the choice of a material to make the loop. Superconductors exist in the so-called types I and II [_{1} and B_{2} greater than the lower critical field, which are necessary in order to actually impose the full type-II superconducting conditions to the whole length of wire, avoiding the intermediate state of lower fields. The FLs carry a quantum of magnetic flux each and their cores contain electrons in the normal state. If these lines are not properly pinned, their motion under magnetic forces will generate heat. Fortunately, suitable working conditions can be devised to dramatically decrease such effects [_{0} of 0.3 T, will oscillate at 178 Hz frequency with an amplitude of oscillations of about 8 μm.

Although we can conclude that losses within the wire can be made very small indeed, a remaining important source of energy loss is the drag of the loop against the cryostat atmosphere, even under high-vacuum. Temperature should be kept at the lowest possible level, preferably below 0.1 K. It has been shown in [5,8] that if the above conditions alone dominate the energy dissipation process a quality factor Q on the order of 10^{10} might be reached. However, other possible sources of energy dissipation should be avoided. Any electric or magnetic coupling of the loop with its surroundings will incur in extra losses. Eddy currents might be induced in metallic parts like the cryostat itself and the magnets that produce the field. Electrically insulating materials should therefore be used. In addition, the loop will produce an external oscillating field on the order of some gauss, and if permanent magnets are adopted the material used in the magnets should be hard enough not to display hysteresis under this kind of field. If a superconducting magnet is used to produce the field imposed to the loop the mutual inductance effect would be part of the measured value for L and there would be no major dissipative terms added. Just to conclude, special care must be taken also to avoid the excitation of sidewise oscillations of the loop.

It might be possible to relax the requirement of ultralow temperatures, and work at, say, 4.2 K and highvacuum. The viscosity of the atmosphere will increase the drag effect upon the loop by a fator of about 1000 [5,12]. The theoretical quality factor will drop from 10^{10} to the 10^{7 }range, which is comparable to that for electronic clocks used commercially, but even so the system will still be very close to ideal conditions. Experiments at 77 K might even be tried, since the drag effect will increase by a factor of about 50 between 4.2 and 77 K, which might still be acceptable. In this case the loop might be made of a high-temperature superconductor, and the whole setup would require simpler refrigeration techniques.

This paper has discussed in detail how an alternative definition of mass can be proposed for a very high-Q oscillating spring-load system. Provided the system starts entirely at rest, the application of a constant force at t = 0+ will lead to a 3-parameter relation between total energy, mass and the square of the oscillations speed. Such relation, Equation (4), is the expression of the equilibrium achieved by the load in the presence of the surrounding force fields. The concept is applied to the extremely high-Q superconducting loop-oscillator, which is a system that traps energy from the surrounding magnetic and gravitational fields and levitates in a nonhysteretic motion. The entirely new Equation (12) involves only three parameters and relates the mass of the loop with the Planck constant and the electronic charge. Such experimental system should be of great interest for Metrologists since it provides an essentially magneto-dynamic technique for obtaining an extremely precise measure of the mass of an object.