# Shielding (atomic physics)

Shielding designated in an electron - atom the reduction of the attractive interaction between an electron and the nucleus by the action of other electrons. In the central field model of the atom, the energy of  an electron depends on the quantum numbers and : ${\ displaystyle \ varepsilon _ {n, l}}$ ${\ displaystyle n}$${\ displaystyle l}$

${\ displaystyle \ varepsilon _ {n, l} = - \ left ({\ frac {Z '} {n'}} \ right) ^ {2} \ cdot E_ {R}}$

With

• effective atomic number ${\ displaystyle Z '= Z _ {\ mathrm {eff}} = Z- \ sigma _ {n, l}}$
• Atomic number ${\ displaystyle Z}$
• Shielding constant  (see below)${\ displaystyle \ sigma _ {n, l}}$
• effective quantum number (see below) ${\ displaystyle n '= n- \ delta _ {n, l}}$
• Principal quantum number ${\ displaystyle n}$
• Quantum defect ${\ displaystyle \ delta _ {n, l}}$
• Rydberg energy (there also the formula for one-electron systems for comparison ).${\ displaystyle E _ {\ mathrm {R}}}$

For the radial parts of the associated one-electron wave functions , John C. Slater proposed the following analytical expression: ${\ displaystyle \ Psi _ {n, l, m} = R_ {n, l} (r) \ cdot Y_ {l, m} (\ theta, \ varphi)}$

${\ displaystyle R_ {n, l} (r) = N \ cdot r ^ {n'-1} \ cdot \ exp \ left (- {\ frac {Z '} {n'}} \ cdot {\ frac { r} {a_ {0}}} \ right)}$

with the normalization factor N.

One-electron wave functions with radial components determined in this way are called Slater orbitals .

## Slater rules

The shielding constant  and the effective quantum number  are determined as follows: ${\ displaystyle \ sigma _ {n, l}}$${\ displaystyle n '}$

1. Electron shells with principal quantum numbers greater than n are not taken into account.
2. Each additional electron with the same n contributes 0.35 (but only 0.3 for n = 1).${\ displaystyle \ sigma _ {n, l}}$
3. Each electron in shell n - 1 contributes to :${\ displaystyle \ sigma _ {n, l}}$
• for secondary quantum numbers l = 0 (s subshell) and l = 1 (p subshell): 0.85 each
• for secondary quantum numbers l = 2 (d subshell) and l = 3 (f subshell): 1.0 each.
4. All electrons from even deeper shells make a contribution of 1.0.

The following table follows:

n 1 2 3 4th 5 6th
n ' 1.0 2.0 3.0 3.7 4.0 4.2

To the quantum defect .

## impact

As part of the Sommerfeld atomic model, the orbital degeneration , i.e. the energy equality of states of the same principal quantum number n but different angular momentum quantum number l, is eliminated by the shielding, since the orbits of different angular momentum quantum numbers are subject to different shields.