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Nucleus mass should be equal to sum of mass of protons and neutrons, but after binding of these particles, its mass is reduced. Why? Due to reduced mass of protons or neurons? Then what's the purpose of calculated mass?

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marked as duplicate by Bill N, Jon Custer, sammy gerbil, Kyle Kanos, Cosmas Zachos Jun 10 '18 at 13:25

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The strong force is the one that keeps quarks and gluons in confinement in a proton or neutron, and the residual strong force or nuclear force is the force that keeps the protons and neutrons in a nucleus.

Now the nucleus has a lesser mass then the sum of its constituents. The reason is that you need to add energy to get the constituents back (separate them).

That energy that you need to add to the bound system to make it unbound again (to separate the nucleus's protons and neutrons) is exactly the same energy (or mass defect) that is missing from the bound system when you measure it compared to the sum of the separate constituents.

This is the reason why it is called mass defect and that is why we use E=m*c^2, because that missing mass that you ask about is exactly the same as the energy that is binding the nucleus together (protons and neutrons). And it is exactly the same energy that you need to add to the system to separate the neutrons and protons again, so the

Energy (binding)= Energy (that you need to separate again)=(mass defect)*c^2

So you can convert the energy into mass.

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One has to keep clear that there are two general frameworks where the term "mass" is used. One is the classical framework where $m$ is the inertial mass for $F=ma$, and Newton's laws dominate and one can use Archimedes principle to calculate the mass of alloys. In this system mass is an additive conserved quantity. This is also the low velocity frames, i.e. velocities very small compared with the velocity of light. The classical physics system.

For velocities near the velocity of light one cannot have Galilean transformations (necessary for Newton's mechanics) to go from one frame to another, but Lorenz transformations. One has to use four vectors, $(E,p_x,p_y,p_z)$ and the "length" of this four vector is called the invariant mass characterizing the system. Particles are characterized by this mass $m$ when measured individually, but when in a system, one has to add the four vectors and the sum can be different than the sum of the individual masses comprising the system.

The second frame is the frame of atoms molecules particles, a micro frame because distances are small, governed by quantum mechanics and special relativity.

Protons and neutrons within a nucleus are in a quantum mechanically bound state, i.e a state in a potential well. Their free particle four vectors when falling in the potential well, during the creation of the nucleus, lost energy and momentum by radiation $( α, β, γ )$ and the addition of the four vectors of the nuclei that make up the nucleus has a total mass less than the sum of the masses of the free nuclei.

With the above in mind lets see the question:

Nucleus mass should be equal to sum of mass of protons and neutrons,

Only in the classical physics frame, which is not the frame of protons and neutrons.

but after binding of these particles, its mass is reduced. Why?

Because of special relativity which induces a relationship between masses of systems that have constituents, through the four vectors of each constituent and the law of addition of four vectors.

Due to reduced mass of protons or neurons?

The protons and neutrons in the potential of the nucleus are described by virtual four vectors, and yes, those four vectors have instantaneously different masses , less than the free particle rest mass.

Then what's the purpose of calculated mass?

The measured rest( or invariant) mass of individual protons and neutrons is invariant when the particles are free. The measured rest mass of the composed nucleus is also invariant . The difference of the sum of individual masses to the mass of the nucleus tells how much energy will be needed to get the protons and neutrons free, outside their built up potential well. The potential well of a system of nucleons is built up by the strong force, a spill over of the quantum chormodynamic force between quarks which compose the neutrons and protons.

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The law of conservation of energy is one of the “building blocks” on which Classical Physics is fabricated so imagine the “surprise” when in 1932 a experiment was done by Cockcroft and Walton which showed that energy could be created.
Cockcroft and Walton used their accelerator to give protons (Hydrogen ions) sufficient kinetic energy so that when a proton hit a lithium nucleus two alpha particles (Helium nuclei) were produced.

$\rm ^1_1H+^7_3Li \rightarrow ^4_2He+ ^4_2He$

Knowing the kinetic energies of the protons and measuring the kinetic energy of the alpha particles showed that kinetic energy had been “created” in this reaction.

In 1905 Einstein had postulated the idea of the equivalence of energy and mass $E=mc^2$ and using the known masses of Hydrogen, Lithium and Helium atom/nuclei Cockcroft and Walton showed that in the reaction they were investigating the total mass of the products was less than the total mass of the reactants by exactly the same amount predicted from the Einstein formula.

You might now ask why it was that the law of conservation of mass was, and still is, used by chemists and other scientists in other fields.
The answer lies in the fact that in chemical reactions the energies involved are very much smaller (~eV, ~$1.6\times 10^{-19}\rm J$) than in nuclear reactions (~MeV) with corresponding much smaller (and non-measurable) reductions in mass.
This reduction in mass even occurs when you stretch a spring!

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The binding energy of protons and neutrons in the nucleus is large enough that a fraction of the mass of the bound nucleus (expressed simply as the sum of the individual nucleon masses) is converted into binding energy (by E = mc^2) and hence is "missing" in that sum.

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let's consider the simplest case: if a proton comes close to a neutron they start interacting through strong force, and in such system there is an excess of energy, which can be radiated or carried away by other particle, and if it happens then these proton and neutron would stay together, now they weigh less since some energy escaped, and to break them apart we now have to apply some energy to the system

combining protons and neutrons make nuclear of atoms, some combinations are stable and we have stable elements, other combinations are not and tend to change into a state with lower energy, and the process we observe we call radioactivity

deep inside of stars lighter elements get converted into heavier ones, and released energy lets stars shine for many millions and billions of years, however, this process stops at iron, which has the combination of neutrons and protons with the lowest energy

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