Helium atom

A helium atom is an atom of the chemical element helium . It is composed of two electrons that are bound to the atomic nucleus by the electromagnetic force , which consists of two protons and (depending on the isotope ) one or two neutrons , which are held together by the strong force .

In contrast to hydrogen, there is currently no closed solution of the Schrödinger equation for the helium atom. H. the energy levels cannot be calculated exactly. However, there are numerous approximations, such as the Hartree-Fock method , for estimating energy levels and the wave function .

Properties of the atomic shell

Helium is the second and last element of the first period in the periodic table . Since the atomic radii decrease with an increasing number of electrons within a period and hydrogen and helium are the only elements in which only the first electron shell is occupied, helium has the smallest atomic radius and the highest ionization energy of all elements .

Only van der Waals forces act between helium atoms . Due to the small size of the atoms, the electron shells can only be polarized very slightly. The van der Waals forces, which are already very weak, are therefore particularly small in the case of helium. This explains the extremely low boiling point of 4.2  K (−268.8 ° C).

Electron states

Ortho- and Parahelium

In the ground state the electrons are in state 1s ² , i.e. H. they occupy the only two possible states with the principal  quantum number 1 and the orbital angular momentum 0. The spins of the electrons are antiparallel and add up to the total spin S  = 0 ( singlet state ¹ S ₀ ).

The energetically lowest state with S  = 1 (triplet state ³ S ₁ ) has the electronic structure 1s2s and is 19.8 eV higher.

States with S  = 0 are called parahelium, those with S  = 1 orthohelium. Transitions between ortho- and parahelium are strongly suppressed (" forbidden "). The lowest Orthohelium state is therefore comparatively long-lived.

Quantum mechanical description

Term scheme for Para- and Orthohelium with one electron in the ground state 1s and one excited electron.

The quantum mechanical description of the helium atom is of particular interest because it is the simplest multi-electron atom and can also be used to understand quantum entanglement .

The Hamilton operator for helium, considered as a 3-body system (consisting of two electrons and the nucleus ), can be written in the center of gravity system as

${\displaystyle H({\vec {r}}_{1},\,{\vec {r}}_{2})=\sum _{i=1,2}{\Bigg (}-{\frac {\hbar ^{2}}{2\mu }}\nabla _{r_{i}}^{2}-{\frac {Ze^{2}}{4\pi \varepsilon _{0}r_{i}}}{\Bigg )}-{\frac {\hbar ^{2}}{M}}\nabla _{r_{1}}\cdot \nabla _{r_{2}}+{\frac {e^{2}}{4\pi \varepsilon _{0}r_{12}}},}$

where the electron mass, the nuclear mass, the reduced mass of an electron in relation to the nucleus, the electron-nucleus distance vectors are and is. The atomic number of helium is 2. In the approximation of an infinitely heavy nucleus one obtains and the mass polarization term disappears. In atomic units , the Hamilton operator is simplified to ${\displaystyle m}$ ${\displaystyle M}$ ${\displaystyle \mu ={\tfrac {mM}{m+M}}}$ ${\displaystyle {\vec {r}}_{1},\,{\vec {r}}_{2}}$ ${\displaystyle r_{12}=|{\vec {r_{1}}}-{\vec {r_{2}}}|}$ ${\displaystyle Z}$ ${\displaystyle M=\infty }$ ${\displaystyle \mu =m}$ ${\displaystyle {\tfrac {\hbar ^{2}}{M}}\nabla _{r_{1}}\cdot \nabla _{r_{2}}}$

${\displaystyle H({\vec {r}}_{1},\,{\vec {r}}_{2})=-{\frac {1}{2}}\nabla _{r_{1}}^{2}-{\frac {1}{2}}\nabla _{r_{2}}^{2}-{\frac {Z}{r_{1}}}-{\frac {Z}{r_{2}}}+{\frac {1}{r_{12}}}.}$

It must be emphasized that the underlying space is not the ordinary space, but a 6-dimensional configuration space . In this approximation ( Pauli approximation ) the wave function is a spinor of the second level with 4 components , whereby the indices represent the spin projection of the electrons in the selected coordinate system ( direction up or down). It must meet the usual standardization condition. This general spinor can be represented as a 2 × 2 matrix and consequently also as a linear combination of an arbitrary basis of four (in the vector space of the 2 × 2 matrices) orthogonal constant matrices with scalar coefficient functions as . A suitable basis is formed from an antisymmetric matrix (with total spin , corresponding to a singlet state ) and three symmetric matrices (with total spin , corresponding to a triplet state ) , ${\displaystyle ({\vec {r}}_{1},\,{\vec {r}}_{2})}$ ${\displaystyle \psi _{ij}({\vec {r}}_{1},\,{\vec {r}}_{2})}$ ${\displaystyle i,j=\,\uparrow ,\downarrow }$ ${\displaystyle z}$ ${\displaystyle \sum _{ij}\int d{\vec {r}}_{1}d{\vec {r}}_{2}|\psi _{ij}|^{2}=1}$ ${\displaystyle {\boldsymbol {\psi }}={\bigl (}{\begin{smallmatrix}\psi _{\uparrow \uparrow }&\psi _{\uparrow \downarrow }\\\psi _{\downarrow \uparrow }&\psi _{\downarrow \downarrow }\end{smallmatrix}}{\bigr )}}$ ${\displaystyle {\boldsymbol {\sigma }}_{k}^{i}}$ ${\displaystyle \phi _{i}^{k}({\vec {r}}_{1},\,{\vec {r}}_{2})}$ ${\displaystyle {\boldsymbol {\psi }}=\sum _{ik}\phi _{i}^{k}({\vec {r}}_{1},\,{\vec {r}}_{2}){\boldsymbol {\sigma }}_{k}^{i}}$ ${\displaystyle S=0}$ ${\displaystyle {\boldsymbol {\sigma }}_{0}^{0}={\tfrac {1}{\sqrt {2}}}{\bigl (}{\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}}{\bigr )}={\tfrac {1}{\sqrt {2}}}(\uparrow \downarrow -\downarrow \uparrow )}$ ${\displaystyle S=1}$ ${\displaystyle {\boldsymbol {\sigma }}_{0}^{1}={\tfrac {1}{\sqrt {2}}}{\bigl (}{\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}{\bigr )}={\tfrac {1}{\sqrt {2}}}(\uparrow \downarrow +\downarrow \uparrow )}$ ${\displaystyle {\boldsymbol {\sigma }}_{1}^{1}={\bigl (}{\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}{\bigr )}=\;\uparrow \uparrow ,}$ ${\displaystyle {\boldsymbol {\sigma }}_{-1}^{1}={\bigl (}{\begin{smallmatrix}0&0\\0&1\end{smallmatrix}}{\bigr )}=\;\downarrow \downarrow .}$

It is easy to show that the singlet state is invariant under all rotations (a scalar quantity), while the triplet can be mapped to an ordinary space vector with the three components , and . ${\displaystyle (\sigma _{x},\sigma _{y},\sigma _{z})}$ ${\displaystyle \sigma _{x}={\tfrac {1}{\sqrt {2}}}{\bigl (}{\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}}{\bigr )}}$ ${\displaystyle \sigma _{y}={\tfrac {i}{\sqrt {2}}}{\bigl (}{\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}{\bigr )}}$ ${\displaystyle \sigma _{z}={\tfrac {1}{\sqrt {2}}}{\bigl (}{\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}{\bigr )}}$

Since in the above (scalar) Hamiltonian all spin interaction terms between the four components of are neglected (e.g. an external magnetic field , or relativistic effects such as spin-orbit interactions ), the four Schrödinger equations can be solved independently of each other. ${\displaystyle {\boldsymbol {\psi }}}$

The spin only comes into play here because of the Pauli principle , which requires antisymmetry for fermions (like electrons) with simultaneous exchange of spin and coordinates , i.e.:

${\displaystyle {\boldsymbol {\psi }}_{ij}({\vec {r}}_{1},\,{\vec {r}}_{2})=-{\boldsymbol {\psi }}_{ji}({\vec {r}}_{2},\,{\vec {r}}_{1})}$

Parahelium is thus the singlet state with a symmetrical function and Orthohelium is the triplet state with an antisymmetrical function . If the electron-electron interaction term is ignored (as a first approximation), both functions can be written as linear combinations of any two ( orthogonal and normalized ) one-electron eigenfunctions : or for the special cases (both electrons have identical quantum numbers, only for parahelium ) . The total energy (as the eigenvalue of ) is then the same in all cases (regardless of the symmetry). ${\displaystyle {\boldsymbol {\psi }}=\phi _{0}({\vec {r}}_{1},\,{\vec {r}}_{2}){\boldsymbol {\sigma }}_{0}^{0}}$ ${\displaystyle \phi _{0}({\vec {r}}_{1},\,{\vec {r}}_{2})=\phi _{0}({\vec {r}}_{2},\,{\vec {r}}_{1})}$ ${\displaystyle {\boldsymbol {\psi }}_{m}=\phi _{1}({\vec {r}}_{1},\,{\vec {r}}_{2}){\boldsymbol {\sigma }}_{m}^{1},\;m=-1,0,1}$ ${\displaystyle \phi _{1}({\vec {r}}_{1},\,{\vec {r}}_{2})=-\phi _{1}({\vec {r}}_{2},\,{\vec {r}}_{1})}$ ${\displaystyle \phi _{x},\;x=0,1}$ ${\displaystyle \varphi _{a},\varphi _{b}}$ ${\displaystyle \phi _{x}={\tfrac {1}{\sqrt {2}}}(\varphi _{a}({\vec {r}}_{1})\varphi _{b}({\vec {r}}_{2})\pm \varphi _{a}({\vec {r}}_{2})\varphi _{b}({\vec {r}}_{1}))}$ ${\displaystyle \varphi _{a}=\varphi _{b}}$ ${\displaystyle \phi _{0}=\varphi _{a}({\vec {r}}_{1})\varphi _{a}({\vec {r}}_{2})}$ ${\displaystyle H}$ ${\displaystyle E=E_{a}+E_{b}}$

This explains the absence of the - state (with ) for Orthohelium, where consequently (with ) the metastable ground state is. (A state with the quantum numbers : principal quantum number , total spin , angular momentum quantum number and total angular momentum is denoted by.) ${\displaystyle 1^{3}S_{1}}$ ${\displaystyle \varphi _{a}=\varphi _{b}=\varphi _{1s}}$ ${\displaystyle 2^{3}S_{1}}$ ${\displaystyle \varphi _{a}=\varphi _{1s},\varphi _{b}=\varphi _{2s}}$ ${\displaystyle n}$ ${\displaystyle S}$ ${\displaystyle L}$ ${\displaystyle J=|L-S|\dots L+S}$ ${\displaystyle n^{2S+1}L_{J}}$

When the electron-electron interaction term is included, the Schrödinger equation is not separable . But even if it is neglected, all the states described above (even with two identical quantum numbers, as with ) cannot be written as a product of one-electron wave functions:  - the wave function is entangled . One cannot say that particle 1 is in state 1 and the other in state 2; and one cannot take measurements on one particle without influencing the other. ${\displaystyle {\tfrac {1}{r_{12}}}}$ ${\displaystyle 1^{1}S_{0}}$ ${\displaystyle {\boldsymbol {\psi }}=\varphi _{1s}({\vec {r}}_{1})\varphi _{1s}({\vec {r}}_{2}){\boldsymbol {\sigma }}_{0}^{0}}$ ${\displaystyle \psi _{ik}({\vec {r}}_{1},\,{\vec {r}}_{2})\neq \chi _{i}({\vec {r}}_{1})\xi _{k}({\vec {r}}_{2})}$

See also

• Dihydrogen cation
• Ortho and para hydrogen

Web links

Wiktionary: Helium atom  - explanations of meanings, word origins, synonyms, translations

Wikiversity: Helium - Periodic Table  - Course Materials

Individual evidence

1. ↑ John Stewart Bell : Quantum Mechanics, Six Possible Worlds and other articles . de Gruyter, Berlin 2015, ISBN 978-3-11-044790-3 , pp. 42, 65.
2. ↑ P. Rennert, H. Schmiedel, C. Weißmantel: Small Encyclopedia Physics. VEB Bibliographisches Institut, Leipzig 1988, pp. 192–194.
3. ↑ LD Landau, EM Lifschitz: Textbook of Theoretical Physics. Volume III: Quantum Mechanics. Akademie-Verlag, Berlin 1971, chap. IX, p. 218.