Kinematics (particle processes)

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In atomic, nuclear and particle physics, kinematics are the conclusions that result from the conservation of momentum and energy for the movement of the particles after decay or collision processes. The particles are viewed as point masses .

Decay processes

When a particle at rest decays into other particles, conservation of momentum means that the vector sum of all momentum is zero even after the decay. The figure shows schematic examples of the decay into two or into three particles.

Disintegration (schematic) into 2 or 3 product particles. The arrows represent the momentum vectors. If there are only 2 particles, their momentum amounts are the same and the energies are also fixed. With 3 particles all impulses and energies can assume continuous values; two special cases are indicated

In the simplest case, only two particles are present after disintegration, e.g. B. an alpha particle and the emitting atomic nucleus. Then the particles always move 180 degrees apart. Designate their masses, speeds and kinetic energies and if the speeds are small compared to the speed of light , conservation of momentum is required

.

The energies are

.

If one presses over the first equation z. B. by aus, we get for the ratio of the two energies

.

The kinetic energies are related to each other in the opposite way to the two masses. Since the sum of the two energies is fixed, one observes z. B. discrete, well-defined particle energies in alpha decay.

Even after a gamma decay there are only two particles and discrete energies occur. However, for the photon, the relationship between energy and momentum must be calculated relativistically .

If, on the other hand, continuous energy spectra appear as in a beta decay, this indicates that a total of three or more particles are present as a result of the decay. This consideration led Pauli to the neutrino hypothesis . (See also Dalitz diagram .)

Shock processes

Illustration: game of billiards

The elastic collision of a moving with a ball resting in the billiard game (without Effet -.., Zugstöße and the like) illustrates the point masses kinematics of shocks. If, as usual, the two spheres have the same masses, two extreme cases are easiest to describe:

  • Central impact: the impacting ball stays where it is, the hit ball completely takes over its speed according to amount and direction.
  • grazing impact: the impacting ball continues to roll (almost) unchanged, the hit ball moves at a speed of (almost) zero approximately at right angles to the side.

In all other cases in between, after the impact, both balls move in directions at right angles to each other at such speeds that the common center of gravity of the balls (which is always in the middle of the straight line connecting the ball centers) continues to move in a straight line. It does not happen that one of the balls rolls "backwards" (deviating more than 90 degrees from the direction of the impact) after the impact.

These elastic collisions are treated mathematically in impact (physics) .

Atoms, atomic nuclei, elementary particles

The conservation of momentum and energy also determine the kinematics of processes that have to be described by quantum mechanics . For example, in scattering processes, such as the Compton scattering of a photon on an electron, this shows how the energy of the scattered photon is related to the scattering angle. Compared to the classic mechanical case of billiards, however, there are complications in some cases:

  • The particles can have different masses. If a lighter particle meets a heavier particle, it can be deflected by more than 90 degrees, ie it can be “scattered back”. The backscattering of alpha particles on a gold foil observed by Ernest Rutherford was therefore a clear indication of the existence of the heavy atomic nucleus .
  • There may be more than 2 particles after the impact.
  • The kinetic energy of the colliding particle ( projectile ) can partly be converted into other forms (excitation energy, mass of new particles).
  • Conversely, in addition to the projectile energy, further energy can be converted into kinetic form (see exothermic reaction ). This happens in particular with the nuclear fission and nuclear fusion reactions carried out for technical energy generation .
  • If the speeds that occur are no longer negligible compared to the speed of light, the relativistic energy and the relativistic momentum must be used in the calculations (see tests of the relativistic energy-momentum relationship, for example ).

Center of gravity coordinate system

The description of the kinematic conditions usually facilitates by selecting a coordinate system in which the common center of gravity of the particle is at rest ( center of mass system , Eng. Center-of-mass system, CMS ). In this system the two particles present before the collision move 180 degrees in opposite directions towards each other; after the collision, the above considerations apply as for a decay.

In the laboratory system - the "normal" system in which only the projectile particle moves before the impact - the speed of each particle is generally the vector sum of its speed calculated in the center of gravity system and the speed of the center of gravity in the laboratory system ( Galileo transformation ). In the relativistic case, instead of the Galileo transformation, the Lorentz transformation is necessary to convert between the laboratory and center of gravity system .

Colliding beam experiments

The kinetic energy that is contained in the movement of the center of gravity (seen in the laboratory system) is not available for conversion into other forms, such as new particles, because the common center of gravity “must” move on with constant momentum after the impact. This is the reason to conduct experiments in high-energy physics as colliding beam experiments . Here (with oppositely equal momentum vectors of the colliding particles) the center of gravity system coincides with the laboratory system, so that the entire energy of both particles can be “consumed”.

Strongly exothermic reactions

Nuclear reactions with a high energy release, such as neutron-induced nuclear fission or nuclear fusion, can be viewed as decays if the kinetic energy of the triggering impact can be neglected, because then the laboratory and center of gravity system coincide here too. Example: The fusion of a triton with a deuteron results in an alpha particle and a neutron and releases 17.6 MeV of energy. The alpha particle has about four times the mass of the neutron; If the impact energy was only a few keV (as in a fusion reactor , for example ), neutrons with four fifths of the energy released, i.e. 14.1 MeV, and alpha particles with 3.5 MeV are found in the laboratory system.

literature

  • P. Marmier, E. Sheldon: Physics of Nuclei and Particles . Vol I. New York, London: Academic Press, 1969. Contains a detailed description of the disintegration, scattering and reaction kinematics
  • K. Bethge, G. Walter, B. Wiedemann: Kernphysik , 3rd edition, Berlin etc. 2007, ISBN 978-3-540-74566-2 , pp. 169-182
  • R. Hagedorn: Relativistic kinematics: a guide to the kinematic problems of high-energy physics , New York 1964
  • W. Greiner, J. Rafelski: Spezial Relativitätstheorie , 3rd edition, Frankfurt 1992, ISBN 3-8171-1205-X , pp. 136-185

Individual evidence

  1. Bethge / Walter / Wiedemann (see list of literature) p. 169; Greiner / Rafelski (see literature list) p. 136