Larmor precession

Under the Larmor precession ( after the Irish physicist Joseph Larmor ) is the precession of the angular momentum of a particle with a magnetic dipole moment to the direction of an externally applied magnetic field. It is particularly due to the splitting of the spectral lines in the magnetic field, the Zeeman effect, observable.

The frequency of precession is called the Larmor frequency. It differs from the cyclotron frequency of the particle in the corresponding magnetic field, the electron to ( almost exactly ) a factor of 2

• 3.1 Zeeman effect
• 3.2 precession

Larmor precession

Precession (Example of Severe gyro )

At a toy gyroscope, which is not supported at its center of gravity and the axis of rotation is not perpendicular, gravity acts with a torque that is perpendicular to the gravity and to the gyro axis and can therefore fall over the top, if it does not rotate. Wherein (not too slow ) rotation on the other hand, the torque causes the precession, which leads the rotor axis and thus the angular momentum vector in a circle around the solder around. The angle to the perpendicular remains constant, and the angular velocity of precession is the same for all angles of attack.

Precession in the magnetic field

The Larmor precession based on the fact that each of charged particles with angular momentum is also a magnetic dipole. This also applies to overall neutral particles (eg neutron, neutral atom with an odd number of electrons ), which are composed of charged particles whose magnetic moments do not add up to zero. In a magnetic field acting on the particles, a torque that tends to place the dipole parallel to the field direction. It is. Magnitude and direction of the dipole are given by the angular momentum vector. This is the gyromagnetic ratio is a constant which can be calculated using the Landé formula depending on the particle type for each energy level. From the equation of motion of the gyro, follows the precession at the Larmor frequency. This is to be and flux density of the magnetic field proportional

Or as angular frequency ( with the Landé factor of the charge and the mass of the particle )

Macroscopic effect

The above description applies equally in the classical and quantum physics. If, for example, a water droplet by a strong magnetic magnetized something that make it (partially) aligned magnetic moments of protons ( nuclei of hydrogen ) along a weak macroscopic dipole magnet, which is connected via the same gyromagnetic factor with a small total angular momentum. When the magnetic field is sufficiently rapidly replaced by one in another direction, this dipole holds for a short time in its original orientation and performs the Larmor precession. He generated in an antenna coil, an easily observable induced alternating voltage whose frequency is the Larmor frequency. The amplitude of the AC voltage decreases to the extent on how the perpendicular to the field direction strength of the rotating dipole is reduced because the macroscopic magnetization to the new field direction adapts ( longitudinal relaxation ), and because the individual protons get out of sync due to small disturbances ( transverse relaxation ). Both the accurate measurement of the frequency and the observation of relaxation include in materials research to the most important tools in the study of structures and reactions. In geophysics, this method is applied in the proton magnetometer to measure the magnetic field of the earth and its interference accurate.

Quantum mechanical description

Zeeman effect

Quantum mechanically causes the magnetic moment in a magnetic field splitting of the energy level with the angular momentum quantum number in equidistant levels to the different possible magnetic quantum numbers. The level spacing is always ( it is the reduced quantum of action ). This splitting was observed at optical spectral lines for the first time in 1896 and was one of the first approaches to the study of processes in atoms and thus to the development of quantum mechanics.

In formulas: From the O.A. Torque results, that the particle in the magnetic field an additional energy

, said to -parallel component of the vector and the field direction is selected as the z- axis. As to the quantum numbers are (see directional quantization ), splits the level into as many Zeeman levels. Their energies are

These energies are equidistant, the distance between them is straight.

Precession

At a single Zeeman state can be read no movement of quantum mechanics, no rotation about the rotor axis or the precession of the spin axis about the z- axis. As an eigenstate of the energy of the state is stationary, ie with the passage of time does not change its shape, but only the quantum mechanical phase of its state vector by means of the phase factor. States of various energy change its phase at different speeds. In the split depending on the magnetic quantum number Zeeman states with energy of the phase factor is accordingly. Just as is the eigenvalue for the z component of the angular momentum of the Zeeman condition concerned, the phase factor is the same as a rotation by an angle about the z- axis. For a Zeeman state alone, this phase or rotation expresses in no observable fact, but just in according to quantum mechanics in principle arbitrary phase factor of the associated state vector.

A rotational movement around the z- axis can be seen only from a condition that is characterized at each instant a certain direction transverse to the z- axis in a measurable manner. To this end, he must be a superposition of several Zeeman states. The axis to be awarded perpendicular to the z axis, then depends on the relative phase of its Zeeman components. For example, a particle may be in the spinning conditions and Zeeman, and directed to the x-axis is given by the state of superposition (except for a common factor, see also the spinning properties ). When the phases of both components but have developed for any reason to 90 ° apart, aligned with the spinning means according to the Y axis out (up to a common factor) and has state. After a further 90 ° phase difference is, the state and is oriented to x -, etc.

Since just change with the passage of time, the individual state vectors as if they were all rotated by the same angle around the z- axis now describes the same superposition of a state that has really carried this rotation. He showed at the beginning of a polarization that was not parallel to the z - axis, then later shows the same shape and thickness of the polarization, but in an appropriately rotated direction.

In other words, the system rotates very described with the angular velocity, in complete accordance with the intuition. Here it is clear that the energy splitting of the angular momentum eigenstates as in the Zeeman effect allows such a simple spatial illustration because it is equidistant. A splitting proportional to the square of the magnetic quantum number, such as by the interaction of the electric quadrupole moment with an inhomogeneous electric field, can not be interpreted in this way.

Magnetic Resonance

By radiating an alternating magnetic field transitions between the split in the Zeeman levels are excited as the frequency of the alternating field coincides with the Larmor frequency ( resonance). By varying the frequency produced an absorption spectrum with a visible absorption line. This method is called depending on the observed object electron spin resonance or nuclear magnetic resonance and allows measurements of extreme accuracy. For example, the influence of the chemical bonding of the atom and its wider area can be measured by nuclear magnetic resonance, the magnetic field as it acts on the core to millionths fractions modified (chemical shift ).

Also, this absorption of energy can be understood macroscopically, because a linearly polarized alternating field contains a circularly polarized component which on the precessing dipole exerts a constant ( in its rest frame ) moment at the right frequency. Does it have the direction, " as if it wanted to speed up the precession ", is fed to the centrifugal energy meanwhile. But he can not save in the form of a faster precession, but must respond ( with constant Larmor frequency ) with magnification of the pitch away from the constant field. At a large toy top precesses in the gravitational field, one can observe this behavior directly, if you try with your finger to accelerate the precession ( or slow down ).