Spin–orbit interaction
The spin -orbit coupling and spin- orbit interaction is occurring in atomic, nuclear and elementary particle physics interaction, whose strength depends on the position of the spin of the particle relative to its orbital angular momentum. For bound particles, the spin-orbit interaction leads to a splitting of energy levels, which contributes to the fine structure of the level scheme. For the electrons of the atomic shell, these effects are relatively minor, but have important effects on the atomic structure.
The spin -orbit interaction is expressed within the framework of nonrelativistic quantum mechanics by a separate term in the Schrödinger equation that contains the scalar product of orbital and spin angular momentum of the particle in relativistic quantum mechanics, a corresponding energy contribution is automatically obtained.
Bound particles
The spin -orbit interaction was observed with the electrons in the atomic shell first. Here it causes a splitting of the spectral lines and thus contributes (in addition to relativistic effects and the Darwin term ) for the fine structure of atomic spectra. A well-known case is the splitting of the yellow D line of sodium, which has already been observed with a good prism.
Much stronger spin -orbit coupling is in the nucleus, the protons and neutrons ( see shell model ( Nuclear physics ) ).
In atomic bound electrons
Assuming intrinsic angular momentum (spin ) and magnetic moment of the electron as given, the spin -orbit coupling can be clearly justified already in the Bohr model of the atom: From the Maxwell theory and the theory of special relativity follows that an electron, when the electric field of an atomic nucleus circles, affects a magnetic field. In the rest frame of the electron, a circular motion of the core is perceived namely. This movement is due to the charge of the nucleus is a circular current, which according to the law of Biot- Savart magnetic field generated parallel to the orbital angular momentum vector. Since the magnetic moment of the electron spin is anti-parallel to its concerns a spin direction parallel to the field and have a higher energy for the opposite lower. As for a spin 1/2 only these two options exist, a single energy level is split into two levels, and there is in the optical spectra of two compared to the original position slightly shifted lines, but appear to be a case of gross consideration.
In nonrelativistic quantum mechanics, a corresponding summand is added to the Schrödinger equation for each electron in relativistic quantum mechanics, spin, magnetic moment and spin -orbit interaction arising automatically from the Dirac equation.
Spin-orbit coupling energy of an electron
The operator for the spin-orbit interaction of an electron in the electrostatic field is central
This is the spin-orbit coupling constant
Me is the electron mass, e is the electron charge, μ0 is the magnetic field constant and the reduced Planck constant, the distance r from the center of the electron.
The result for states with next power shift:
J is the quantum number of the total angular momentum of the particle, which is quantized in half-integer multiples of. In the Bohr model r is the orbital radius of the electron ( n principal quantum number, Bohr radius). Therefore, is greatest for the innermost track Bohr (n = 1). Overall, the splitting due to spin -orbit coupling with increasing atomic number that is growing like. In quantum mechanical treatment of the factor is to replace the withdrawn through the respective orbital mean. Neglecting the influence of other electrons results
The distance between the split levels. It occurs for example in the X-ray photoelectron spectroscopy (XPS), with the absorption of X-radiation and the emission of characteristic X-rays experimentally in appearance, because these processes depend directly on the binding energy of individual electrons in the inner shells of the atom.
Jj - coupling with several electrons
Because of the magnitude of the spin -orbit interaction at large Z (eg lead with Z = 82) plays with heavy atoms, the interference of the electrons is not so important. Therefore, every electron is in a state with a "good quantum number " ji for its total angular momentum. The angular momenta ji all the electrons are made up to the total angular momentum J of the atom. Only the electrons in non fully occupied shells are taken into account, because the total angular momentum of a closed shell always evaluates to zero. The total angular momenta of the individual electrons with quantum numbers so well defined and coupled to the total angular momentum of the atomic shell with a quantum number. This coupling scheme is called the jj - coupling.
LS- coupling with several electrons
The LS- coupling prevails for the lighter atoms. Erroneously, it is easily accommodated because of their names with the spin -orbit interaction in context, although this is currently being neglected in bringing about the LS- coupling. The dependence of the energy of each individual electron from the dot product described above is in smaller nuclear charge numbers Z namely so weak that the electrons are in a non- closed shell primarily influenced by their mutual Coulomb repulsion, which is independent of the spins. The total wave function of an energy eigenstate is therefore to be put to a good approximation as a product of a spatial wave function of all electrons with a spin function of all electrons. In such conditions has (except for ) no electron hold a state which is characterized by a quantum number on its total angular momentum. However, the total orbital angular momentum
A fixed size ( quantum number, the eigenvalue for the operator). These states are formally broken down further according to the quantum number for the total spin of the electrons:
(. Indeed, one does not need to be considered closed shells, as their bodies have automatically ) If at least two electrons in the same subshell, we can and each have several different values that belong to the same energy, if other energy contributions - are neglected - yet. Only the combinations of and, which correspond to the Pauli principle, that is, an antisymmetric wave function arise when two electrons are interchanged come. Now the spatial wave function of two electrons to itself in a given permutation are (within a sub- shell ) on whether is always either symmetric or antisymmetric, depending on even or odd. The spin wave function for a given total spin is either symmetric or antisymmetric, only in the opposite direction. Thus, a total antisymmetric function arises position and spin function of a level must have opposite symmetry.
The next step is given by the mutual electrostatic repulsion of the electrons, the energy of the state is increased. The energy contribution is different for the spatial wave functions at different total orbital angular momenta, in particular the rejection of a symmetric spatial wave function is larger than for antisymmetric. The energy thus depends on the symmetry character of the spatial wave function, which - as shown flat - suitable for the symmetry character of the particular spin function must. Thus, the final result for each value of a different energy, although the spins of the electrons have been calculated not yet involved in the interactions. For light atoms ( up to about the atomic number ) that is a good approximation. The finally observable levels of light atoms can ensure that the quantum numbers and are assigned. This is the LS- coupling scheme. For jj - coupling it is opposite in a sense.
In the following step, the still existent spin-orbit coupling is taken into account for each electron. It is noticeable by a further fine splitting, is assigned by each possible eigenvalue of the total angular momentum of a slightly different energy. The result is a multiplet with ( in general) closely adjacent levels that match their quantum numbers and all.
LS- coupling that is, each electron still quantum numbers, but not. A level around the atomic shell, the three quantum numbers, which are summarized in the term symbol.
With increasing atomic number, the description by the LS- coupling is an increasingly poor approximation to average atomic numbers from the spin -orbit interaction of the individual electrons is so large that the jj - coupling scheme applies increasingly better. They say that the LS- coupling is broken.
Occasionally, the LS- coupling is referred to by physicists Henry Norris Russell and Frederick Albert Saunders as Russell - Saunders coupling.
Splitting in the magnetic field
A level with specific and contains individual states with different. Without a magnetic field, they are energetically degenerated to form a single level. In a finite magnetic field is no longer true:
- If this splitting so large that it is no longer negligible compared to the energy difference to the levels with neighboring values of the coupling and at a fixed value is increasingly broken up. The energy eigenstates have to be the quantum numbers and, but no fixed value more. Their energies do not vary linearly with the magnetic field, to the extreme case of strong field ( Paschen - Back effect ) the states at fixed values and energy eigenstates and their energies are linearly dependent on the magnetic field again.
Unbound particles
When a particle is, for example, scattered and distracted from his direction of flight, calls the spin -orbit interaction in general a dependence of the differential cross section on the azimuthal angle out (see also spin polarization, Mott scattering ). Even in nuclear reactions and for all elementary particles with strong interaction ( hadrons ) it plays an equivalent role.