Core mass

The nuclear mass describes the mass of a naked atomic nucleus freed from all electrons of the shell . ${\displaystyle m_{\mathrm {K} }}$

It differs from the atomic mass by the masses of the electrons bound in the atom and the mass equivalent of the binding energy of all electrons: ${\displaystyle m_{\mathrm {A} }}$ ${\displaystyle E_{\mathrm {b} }}$

${\displaystyle m_{\mathrm {K} }=m_{\mathrm {A} }-Z\cdot m_{\mathrm {e} }+{\frac {E_{\mathrm {b} }}{c^{2}}}}$

It mean

• ${\displaystyle Z}$the ordinal number
• ${\displaystyle m_{\mathrm {e} }}$the electron mass
• ${\displaystyle c}$the speed of light .

The mass equivalent of the electronic binding energy ${\displaystyle \Delta m_{\mathrm {e} }}$

${\displaystyle \Delta m_{\mathrm {e} }={\frac {E_{\mathrm {b} }}{c^{2}}}}$

can be referred to as an electronic mass defect in analogy to the nuclear mass defect .

The binding energy of the electron shell cannot be determined directly experimentally, instead one has to rely on theoretical estimates . The calculation according to the Thomas-Fermi model provides  the approximate value for an atom with electrons ${\displaystyle E_{\mathrm {b} }}$ ${\displaystyle Z}$

${\displaystyle E_{\mathrm {b} }=15{,}7\cdot Z^{\frac {7}{3}}\ \mathrm {eV} }$ (Formula 1).

Binding energies calculated according to the Hartree-Fock-Slater method for all electrons of the atom can be taken from tables from 1976. Another numerical equation , which approximates the binding energy of all electrons in an atom better than Formula 1, is given in a journal article from 2003:

${\displaystyle E_{\mathrm {b} }=(14{,}4381\cdot Z^{2{,}39}+1{,}55468\cdot 10^{-6}\cdot Z^{5{,}35})\ \mathrm {eV} }$ (Formula 2).

Binding energy of the electrons bound in the atom depending on the atomic number

The figure shows the curves for the binding energy of the electrons bound in the atom as a function of the atomic number for both formulas. For uranium isotopes ₉₂ U, that is, a total electronic binding energy according to formula 1 of 600 keV or according to formula 2 of 763 keV. For comparison, remember that the energy equivalent of the mass of an electron is 511 keV. ${\displaystyle Z}$ ${\displaystyle Z=92}$

The accuracy of the calculated values ​​for the size is unknown. Presumably the uncertainty for uranium isotopes in the tables from 1976 is less than 2 keV. ${\displaystyle E_{\mathrm {b} }}$

In practical calculations (nuclear masses with 6 to 7 valid digits in the case of light nuclides), the electronic mass defect can sometimes be neglected. Then the approximation applies: ${\displaystyle \Delta m_{\mathrm {e} }}$

${\displaystyle m_{\mathrm {K} }\approx m_{\mathrm {A} }-Z\cdot m_{\mathrm {e} }.}$

Nuclear masses can be determined very precisely in mass spectrometers if the corresponding atom is completely ionized . However, this is only possible with a reasonable amount of effort if the ordinal numbers are low. Nowadays there are some measurements of the mass of completely (i.e. bare atomic nuclei ) or almost completely ionized atoms.

Atomic nuclei are lighter than the sum of their components ( proton and neutron masses):

${\displaystyle m_{\mathrm {K} }=Z\cdot m_{\mathrm {P} }+N\cdot m_{\mathrm {N} }-{\frac {E_{\mathrm {bK} }}{c^{2}}}}$

It mean

• ${\displaystyle Z}$the ordinal number
• ${\displaystyle m_{\mathrm {P} }}$ the mass of a proton
• ${\displaystyle N}$the number of neutrons
• ${\displaystyle m_{\mathrm {N} }}$ the mass of a neutron.

The nuclear mass defect ${\displaystyle \Delta m_{\mathrm {K} }}$

${\displaystyle \Delta m_{\mathrm {K} }={\frac {E_{\mathrm {bK} }}{c^{2}}}}$,

the mass equivalent of the binding energy of the nucleons of the atomic nucleus is orders of magnitude larger than the electronic mass defect. ${\displaystyle E_{\mathrm {bK} }}$

Individual evidence

1. ↑ ^{a b} Keh-Ning Huang et al. : Neutral-atom electron binding energies from relaxed-orbital relativistic Hartree-Fock-Slater calculations 2 ≤ Z ≤ 106 . In: Atomic Data and Nuclear Data Tables . tape 18 , no. 3 , 1976, p. 243-291 , doi : 10.1016 / 0092-640X (76) 90027-9 .
2. ↑ D. Lunney, JM Pearson, C. Thibault: Recent trends in the determination of nuclear masses . In: Rev. Mod. Phys. tape 75 , 2003, p. 1021 , doi : 10.1103 / RevModPhys.75.1021 .
3. ^ ^{A b} Georges Audi: A Lecture on the Evaluation of Atomic Masses . 2004, p. 11 (31 pp., In2p3.fr [PDF; accessed January 10, 2017]).