The molecular beam magnetic resonant (MBMR) technique has significantly contributed, as is well known, to the development of atomic and molecular physics (1). And it makes possible to measure de Larmor frequency of an atom or molecule in the presence of a magnetic field. In the original technique, developed by I.I. Rabi and others [1] -[3] the molecular beam is forced to pass through four different fields:

• A non-homogeneous polarizer field (A) where the molecules are prepared.

• A resonant unit (C) that consists of two, a static and an oscillating, fields.

• A non-homogeneous analyzer field (B). Only molecules in the prepared state reach the detector.

• The two non-homogeneous magnetic fields A and B have opposite directions.

The molecular beam describes a sigmoidal trajectory and, finally, is collected in a detector (see Figure 1).

Rabi explained this effect in terms of spatial reorientation of the angular moment due to a change of state when the transition occurs.

In this case the depletion explanation is based in the interaction between the molecular magnetic dipole moment and the non-homogeneous fields.

The force is provided by the field gradient interacting with the molecular dipolar moment (electric or magnetic). On the resonant unit the molecular dipole interact with both, homogeneous and oscillating, fields. When the oscillating field is tuned to a transition resonant frequency between two sub states, a fraction of the molecular beam molecules is removed from the initial prepared state. As a consequence, the dipolar moment changes as well as the interaction force with the nonhomogeneous analyzer field (B). As only molecules in the initial prepared state reach the detector the signal in the detector diminishes.

During the last years some interesting experimental results have been reported for N₂O, NO, NO dimer, H₂ and BaFCH₃ cluster [4] -[7] . The main result consists in the observation of molecular beam depletion when the molecules of a pulsed beam interact with a static electric or magnetic field and an oscillating field (RF) as in the Rabi’s experiments. But, in these cases, instead of using four fields, only two fields those which configure the resonant unit (C), are used, that is, without using the non-homogeneous magnetic, A and B, fields. See Figure 2.

In a similar way, when the oscillating field is tuned to a transition resonant frequency between two sub states,

the fraction of the molecular beam that is removed from the initial prepared state does not reach the detector. But the important thing is: differently to the previous method, it happens without using non-homogeneous fields. Obviously, the trajectory change has to be explained without considering the force provided by the field gradient. There must be another molecular feature that explains the depletion. It looks as though the linear momentum conservation principle were not satisfied. These experiments suggest that a force depending on other fundamental magnitude of the particle, different from mass and charge must be taken into account.

In order to find out an explanation, let’s consider the following case:

An electron is moving, with speed constant in modulus, in a homogeneous magnetic field B where is perpendicular to B.

Its kinetic energy will be:

The electron, as is well known, describes a circular trajectory (in general case a helix) with a radius r, being:

and

due to the Lorentz force:

On the other hand, as the electron has a magnetic moment, , and spin, S, the presence of the magnetic field B produces a torque when interacting with the electron magnetic moment. The angle between S and (the direction of the magnetic field B) remains constant but the spin S revolves about with angular velocity. This phenomenon bears the name of Larmor precession.

The electron kinetic energy must increase with the energy due to spin precession. But it should be considered that the forces producing the torque are perpendicular to the precession motion and, as a consequence, do not modify the energy of the system. It looks like if the principle of energy conservation be violated.

We will find, in Quantum Mechanics, the phenomenon equivalent to that described for a particle with classic magnetic moment and spin when moving in a uniform magnetic field B and which bears the name of Larmor precession.

Let us assume that, at time, the spin is in the state

To calculate the state in an arbitrary state and as is already expanded in terms of the eigenstates of the Hamiltonian we will obtain

Or, using the values of and:

The presence of the magnetic field B therefore introduces a phase shift, proportional to time, between the coefficients of the kets and.

Comparing the equation (14) for with that for the eingenket for the observable

We see that the direction along which the component is, with certainty, is defined by the polar angles:

that coincides with the direction along which the classic spin should be pointing out.

The angle between and (the direction of the magnetic field B) therefore remains constant, but revolves around with angular velocity proportional to the magnetic field. And the mean values of

, and behave like the components of a classical angular momentum of modulus constant un-

dergoing Larmor precession.

As is well known, helicity is not, in general, a constant of motion. The reason is that helicity operator does not commute, in general, with the Hamiltonian H. Nevertheless, it will be proven that, at least for the interaction here considered (Larmor), the helicity eigenvalue is conserved along the electron’s (classical) trajectory.

We redefine now the helicity, , in order that its eigenvalues be ±1, as, where and. The initial velocity of the electron is, and we assume the initial spin state of the electron to be an eigenstate of the helicity with eigenvalue +1, which is given in equation (12), with that is:

At the time t the velocity of the electron is, as it is known,

where is given in Equation (7). According to Equations (15) and (16), with, at time t the spin state is:

and the helicity at time, ,

Now, taking into account that,

We easily obtain:

This shows that is an eigenstate of the helicity of eigenvalue +1; in other words, helicity is conserved along the electron’s (classical) trajectory.

It has been proven, for cases here considered, that helicity is a constant of motion. As a consequence of this result, the linear momentum must have the same precession angular velocity (Larmor angular velocity) than the spin S. The equation of motion describing the linear momentum evolution must be then equivalent of the equation of motion which describe the evolution of the spin S. This means that:

It is concluded the particle will be under a central acceleration, perpendicular to. The particle is then under a central force:

This kind of forces related with the spin will be designed as Lorentz-like forces. In this case, the trajectory will be a circular one. The radius will be:

And its kinetic energy:

which is equal to the initial one shown in Equation (2). The force (Equation (24)) is the responsible of the electron circular trajectory inside the field B and should be related to the spin S of the electron.

In conclusion, as helicity is, in those cases, a constant of motion, the particle is under a Lorentz-like force and the principles of conservation are not violated.

In 1939 Alvarez and Bloch [11] measured the neutron magnetic moment by using a neutron beam passing through a resonant unit. Neutrons from the Be + D reactions were slowed to thermal velocities and diffused down a Cadmium lined tube through the water tank to the polarizer magnet, B_A. After passing through the reso-

nant unit that consists of two, a static and an oscillating, fields and the analyser magnet, B_B they were detected in a BF₃ chamber. The polarizer B_A and analyser B_B are strongly magnetized iron pieces. A neutron resonant dip is observed in the signal of the neutron beam when the oscillating resonant frequency corresponding to the transition between the two states up and down is reached. According to the previous theoretical description and recently results obtained for NO₂, NO, NO dimer, H₂ and BaFCH₃ cluster, if the Álvarez and Bloch experiment is carried out without using analyser magnet B_B, we anticipate that the experimental results will be the same as those obtained by Álvarez and Bloch in the experiment of 1939.

According to the new explanation, the trajectory change takes place when neutrons pass through the resonant unit and the oscillating field is tuned to a transition resonant frequency between two states, up and down, of the spinof the neutron. In case of Álvarez and Bloch experiment, they used a magnetic field for the neutron resonance of 622 Gauss and a resonant frequency of oscillator of 1843 kilocycles.

• Some recent experimental results have reported the observation of molecular beam depletion when molecules of a pulse interact with a homogeneous static electric or magnetic field and an oscillating field (RF).

• In absence of non-homogeneous fields it is not possible to use the force provided by the field gradient interaction with the molecular dipole in order to explain this depletion.

• A unknown force depending on other fundamental magnitude of the particle, different of mass and charge must be considered.

To the best of our knowledge, it seems that existence of forces described in this paper, related with the spin of the particles, is the more adequate way to explain, from a theoretical point of view, the experimental result here considered. These forces are called Lorentz-like forces.

However, more experimental works are needed to support this conclusion. In this sense the experiment with neutrons, suggested in our proposal, is a very good example of a relevant experiment to be carried out.

The author is very grateful to prof. José L. Sánchez Gómez, Universidad Autónoma de Madrid and to Prof. Ramón Fernández Álvarez-Estrada, Universidad Complutense de Madrid, for useful discussions.