Mass defect

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In nuclear physics, the mass equivalent of the binding energy of the atomic nucleus is called a mass defect (also known as mass loss) . It is expressed as the difference between the sum of the masses of all nucleons ( protons and neutrons ) and the actually measured (always smaller) mass of the nucleus.

The observable mass defect refutes the assumption of classical physics that the mass is retained in all processes .

The term mass defect was introduced in 1927 by Francis William Aston , who from 1919 had been the first to establish that atomic nuclei are lighter than their putative building blocks put together. The work Units for atomic weights and nuclide masses by Josef Mattauch contains u. a. also details on the history of the term mass defect and related quantities. This work contributed significantly to the agreement of chemists and physicists on a common atomic mass unit in 1960.

The mass defect ( English mass defect, mass deficiency ) is sometimes mistakenly equated with the mass excess ( English mass excess ), also mass excess . However, there are two differently defined quantities with clearly different value ranges. The mass defect is always positive (this expresses the fact that matter is stable against spontaneous decomposition into the relevant components); the excess of mass, an auxiliary quantity to facilitate calculations, can be negative or positive.

The measured atomic mass of a neutral atom is also smaller than the sum of the nuclear mass and the mass of the electrons in the atomic shell . This mass defect is, however, much smaller than the mass defect of atomic nuclei and is mostly neglected.

The mass defect that occurs when atoms form a chemical bond is even smaller . In practice, therefore, one can assume that the mass is retained in chemical reactions.

Connection with binding energy

The mass defect can be explained with the knowledge of relativistic physics that the energy of the resting particle can be read from the mass : the binding energy of the nucleons reduces the total mass, which would result from the sum of the masses of the individual core components. Thus, according to the equation , the binding energy of the nucleons released during the construction of an atomic nucleus is equal to the mass defect multiplied by the square of the speed of light. The larger the mass defect per nucleon, the more stable the atomic nucleus, since more energy has to be expended to break it down.

Mass defect at different mass numbers

The total mass defect of a nucleus increases with the number of nucleons ; H. the number of nucleons contained. He is measured z. B. by means of mass spectrometers . If one calculates the average mass defect per nucleon and thus the binding energy per nucleon (in the unit keV ), the relationship with the mass number shown in the figure results.

Average atomic nucleus
binding energy per nucleon as a function of the number of nucleons in the atomic nucleus for all known nuclides according to Atomic Mass Evaluation AME2016

The highest mass defects per nucleon are found in nuclides whose atomic nucleus consists of around 60 nucleons. A whole range of nuclides have almost identical values ​​here. The nuclide with the highest average mass defect per nucleon is 62 Ni , followed by the iron isotopes 58 Fe and 56 Fe.

Energy release in nuclear reactions

If light nuclides (located to the left of the binding energy maximum in the figure ) reach a higher number of nucleons through nuclear fusion, the mass defect per nucleon increases; this additional missing mass is converted into energy that can be used. Conversely, heavy nuclei (located to the right of the binding energy maximum) release energy when they are split into two nuclei of medium mass by nuclear fission . An energy-releasing conversion always takes place "in the direction of the maximum of the mass defect or the binding energy", that is to say with a rising curve.

However, the fusion reactions important in energy technology do not use the region of the highest mass defects at mass numbers around 60, but rather the strong local maximum at the helium isotope 4 He, because the relative increase in mass defects from the reactants deuterium and tritium to helium is particularly large, and at the same time the Coulomb barrier that has to be overcome for the fusion of the nuclei is comparatively low.


The mass defect of a nucleus of the mass is defined as

Here is the ordinal number (number of protons), the number of neutrons , the mass of a proton and the mass of a neutron.

To a good approximation, the mass defect can be calculated on a semi-empirical basis using the Bethe-Weizsäcker formula based on the droplet model.

In practice, the mass defect is not given for the isolated atomic nucleus, but for the entire, uncharged atom of the respective nuclide, i.e. the atomic mass . There are experimental reasons for this: Completely ionized , ie “naked” atomic nuclei are difficult to obtain and handle because their high positive electrical charge immediately catches electrons from the environment. The exact measurement of their mass would therefore hardly be possible, especially in the case of heavy elements (elements of high atomic number ) with their correspondingly particularly high charge.

Therefore, the mass defect of a neutral atom in the nuclear and electronic ground state with a mass is used and defined by

Here means the mass of a neutral atom of the light hydrogen atom. This definition of the mass defect via binding energies ( Total binding energy in keV ) is decisive today (2018).

With electron mass, the mass of a neutral atom of light hydrogen can be expressed as


It is the equivalent mass of the binding energy of the electron in a hydrogen atom. This binding energy, also called ionization energy , is precisely known (see Rydberg energy = 13.605 eV or ionization energy of hydrogen = 13.598 eV). There is no nuclear binding energy for light hydrogen as an element with only one nucleon, the proton.

The mass of a neutral atom is


with the mass defect of the electron shell, the mass equivalent of the binding energies of all electrons in the atom. As already pointed out at the beginning, this is much smaller than the mass defect caused by the nuclear bond and is often neglected or not yet covered by the measurement accuracy.

Since the masses of the electrons lift away, there is a connection between the mass defect of the atom and the mass defect of the atomic nucleus


If the nuclear mass defect is explicitly required, the mass defect of the electronic bond can be calculated approximately from formula 2 given in the article nuclear mass . In the case of mass defects, the index is usually left out, as is the case in the following section. When a mass defect is spoken of without an explanatory addition, this size is usually meant.

How to get to current mass defects

Similar to how CODATA ensures the reliability and accessibility of fundamental physical constants, the Atomic Mass Data Center (AMDC) does this for atomic masses and related quantities. Updated, estimated data was published by the AMDC at approximately 10-year intervals. The last update of Atomic Mass Evaluation 2016 (AME2016) was published in spring 2017. Explicitly published are not estimated values ​​of the mass defects, but rather the binding energy per nucleon , which can be taken from the atomic mass evaluation cited . For nuclear physics practice, several computer-readable ASCII files are published in parallel, of which the mass16 file contains the values ​​of the size under the heading BINDING ENERGY / A for the nuclear and electronic basic states of all known nuclides. The mass defect is obtained from the binding energy per nucleon by multiplying by the number of nucleons and dividing by the square of the speed of light :



The unit electron volt (eV) (and thus also keV) for current estimated excess masses, binding energies, Q values, etc. is no longer based on the unit volt of the international system of units , but on the slightly modified and more precisely measurable unit maintained volt V 90 ( Reproducible reference ) . Since 1990, the voltage unit volt in this definition has been determined using the Josephson effect and the Josephson constant.

To stay consistent, the CODATA recommendation

whose standard uncertainty is many times greater, can not be used in this context , although this only plays a role in the case of high demands on accuracy.


The CODATA table contains four nuclear masses, those of proton, deuteron, triton and alpha particles. For the atomic nucleus 4 He, the alpha particle, the mass defect can be calculated with these data.

Mass defect of the atomic nucleus of 4 He

According to CODATA2014 the mass of a neutron is 1.008665  u , that of a proton 1.007276 u. The nucleus of the helium isotope 4 He consists of two protons and two neutrons. The sum of the masses of the four free nucleons is 4.031883 u, but the masses of the nucleus 4 He is only 4.001506 u. This results in a mass defect of 0.030377 u. The mass of the core is 0.75% less than the sum of the masses of its (free) parts.

Particle / nucleus Nuclear mass (u)
n 2 × 1.00866491588 ± 0.00000000049
p + 2 × 1.00727646688 ± 0.00000000009
α - 4.00150617913 ± 0.00000000006
Mass defect core = 0.03037658639 ± 0.00000000071

The table also includes the standard uncertainties of the nuclear masses and the mass defect.

4 He atom mass defect

Let us now calculate the mass defect analogously, but with the atomic masses. The mass16 file contains not only mass defects and binding energies, but also explicitly atomic masses, in the unit µu, which are reproduced here in the unit u. The mass of the neutron according to CODATA 2014 and AME2016 only differs in the 11th decimal place, the mass of the light hydrogen atom is 1.007825 u. The masses according to the above formula multiplied by N = 2 or Z = 2 and added results in a mass of 4.032980 u. The mass of the atom of 4 He is 4.002603 u. This gives us 0.030377 u for the mass defect of the atom, which corresponds to that of the atomic nucleus with an accuracy of 6 decimal places.

n / atom Atomic mass (u)
n 2 × 1.00866491582 ± 0.00000000049
1 H. + 2 × 1.00782503224 ± 0.00000000009
4 He - 4.0026032541 3 ± 0.00000000006
Mass defect atom = 0.03037664199 ± 0.00000000071

Significantly faster we reach with those contained in the table binding energy per nucleon of 7,073.915 keV to the searched (atomic) mass defect . The result is an expected mass defect of 0.030377 u.

Mass defect in the construction of the nucleon from quarks

The mass of the proton is significantly larger than the sum of the masses of the quarks

Occasionally, the term mass defect is also related to the structure of the nucleon from quarks. But it is not applicable there. The term mass defect assumes that a structure consists of a numerically precisely determined number of parts into which it can be broken down and whose masses are individually well-defined sizes. This idea is based in classical physics and also applies there and still applies to a very good approximation in non-relativistic quantum mechanics . The mass defect in the binding of nucleons to an atomic nucleus was discovered as soon as around 1920 the idea developed that nuclei are made up of building blocks. In the relativistic quantum mechanics and quantum field theory, however , this requirement of well-defined particle numbers does not apply in principle, at most to a good approximation in the non-relativistic limit case. The reason is the constant presence of virtual pairs of particles and antiparticles in an indeterminate number, as was already established around 1930 shortly after the Dirac equation, which applies here, was discovered. The relationships within the nucleon fall into the highly relativistic range, where this pair generation is the most important process and not only causes extremely small corrections to the quantities calculated using non-relativistic equations (see for example Lamb shift ). What is certain is not the total number of quarks and antiquarks, only that the quarks are present in an excess of 3. The determination of a mass defect is therefore impossible.

Individual evidence

  1. a b Klaus Bethge, Gertrud Walter, Bernhard Wiedemann: Kernphysik . 2nd, updated and exp. Edition. Springer, Berlin / Heidelberg 2001, ISBN 3-540-41444-4 , pp. 47 (XX, 402 p., Limited preview in Google Book search).
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  4. a b Josef Mattauch: Units of measurement for atomic weights and nuclide masses. In: Journal of Nature Research A . 13, 1958, pp. 572-596 ( online ). (PDF)
  5. Eric B. Paul: Nuclear and particle physics . North-Holland, Amsterdam 1969, ISBN 0-7204-0146-1 , pp. 5 (English, XIV, 494 p., Limited preview in Google Book Search). The difference between the exact atomic mass of an isotope and its mass number is called the mass excess or the mass defect .
  6. Harry Friedmann: Introduction to Nuclear Physics . Wiley-VCH, Weinheim, Bergstr 2014, ISBN 978-3-527-41248-8 , pp. 97 (XII, 481 p., Limited preview in Google Book search).
  7. Douglas C. Giancoli, Oliver Eibl: Physics: text and exercise book . 3. Edition. Pearson Studium, Munich 2010, ISBN 978-3-86894-023-7 , pp. 1251 (XXV, 1610 p., Limited preview in Google Book search).
  8. MP Fewell: The atomic nuclide with the highest mean binding energy . In: American Journal of Physics . Vol. 63, No. 7 , 1995, p. 653–658 , doi : 10.1119 / 1.17828 , bibcode : 1995AmJPh..63..653F (English).
  9. G. Audi, AH Wapstra: The 1993 atomic mass evaluation: (I) Atomic mass table . In: Nuclear Physics A . tape 565 , no. 1 , 1993, p. 1–65 , doi : 10.1016 / 0375-9474 (93) 90024-R ( [PDF; accessed on September 30, 2018] definition of total binding energy on p. 17).
  10. ^ Homepage of the Atomic Mass Data Center. (No longer available online.) Archived from the original on August 13, 2018 ; accessed on September 30, 2018 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  @1@ 2Template: Webachiv / IABot /
  11. ^ Mirror homepage of the Atomic Mass Data Center, the historical Web site of the AMDC. Retrieved September 30, 2018 .
  12. ^ Mirror homepage of the Atomic Mass Data Center, International Atomic Energy Agency, IAEA. Retrieved September 30, 2018 .
  13. a b c WJ Huang et al. : The AME2016 atomic mass evaluation (I). Evaluation of input data; and adjustment procedures . In: Chinese Physics C . tape 41 , no. 3 , 2017, p. 30002 ( [PDF; accessed September 30, 2018]).
  14. M. Wang et al. : The AME2016 atomic mass evaluation (II). Tables, graphs and references . In: Chinese Physics C . tape 41 , no. 3 , 2017, p. 30003 ( [PDF; accessed September 30, 2018]).
  15. a b c AME2016: Atomic Mass Adjustment, File mass16.txt. (ASCII; 418937 bytes) Retrieved September 30, 2018 .
  16. a b CODATA2014: Fundamental Physical Constants - Complete Listing. (ASCII; 38896 bytes) Retrieved September 30, 2018 .
  17. Abraham Pais: Inward Bound: Of Matter and Forces in the Physical World . Clarendon Press, Oxford 1986, pp. 350 .