## Bindings in helium

#### Binding energy

Theory of atomic nucleuses has proved that binding energy
between particular nucleons does not depend on their charge,
i.e. that it approximately holds:

E(N,N)=E(N,P)=E(P,P), where N is neutron and P proton.

Often reported reason is, that nucleuses with inverse number
of protons and neutrons
(e.g. isotopes H(3,1),(p=1,n=2) and He(3,2),(p=2,n=1))
are similar and their bindings energy differs only by
Coulomb repulsion of protons.

Now let us do one test.

#### Example

Let us assume, we can compute bindings energy with this formula:

** E(p,n) = c(P,P)*E(P,P) + c(P,N)*E(P,N) + c(N,N)*E(N,N) **

where p,n is number of protons P and neutrons N and c(P,P), c(P,N), c(N,N) number
of bindings between them.

Bindings energy of the four simplest atomic nucleuses is:

X(A,Z) p n E(p,n) c(P,P) c(P,N) c(N,N)
---------------------------------------------
H (2,1) 1 1 2.224 0 1 0
H (3,1) 1 2 8.482 1 2 0
He(3,2) 2 1 7.718 0 2 1
He(4,2) 2 2 28.295 1 4 1

We get three equations:

E(1,1) = 0*E(P,P) + 1*E(P,N) + 0*E(N,N) = 2.224
E(1,2) = 0*E(P,P) + 2*E(P,N) + 1*E(N,N) = 8.482
E(2,1) = 1*E(P,P) + 2*E(P,N) + 0*E(N,N) = 7.718

Hence:
E(P,P) = 3.270
E(N,N) = 4.034
E(P,N) = 2.224
Check:
E(1,1) = 1*2.224 = 2.224
E(1,2) = 2*2.224 + 1*4.034 = 8.482
E(2,1) = 2*2.224 + 1*3.270 = 7.718

Binding energy of E(2,2) is then:

E(2,2) = 1*E(P,P) + 4*E(P,N) + 1*E(N,N)
E(2,2) = 1*3.270 + 4*2.224 + 1*4.034 = 16.200.

It is - in comparison with real energy of He(4,2) - very low

(the difference 28.295- 16.200= 12.095 is c. 3*E(N,N)).

#### Test

Let us complete the formula for E(p,n):

** E(p,n) = c(P,P)*E(P,P)+c(P,N)*E(P,N)+c(N,N)*E(N,N)+p*E(P)+n*E(N) **,

where

E(1,1) = 0*E(P,P) + 1*E(P,N) + 0*E(N,N) +1*E(P)+ 1*E(N) = 2.224
E(1,2) = 0*E(P,P) + 2*E(P,N) + 1*E(N,N) +1*E(P)+ 2*E(N) = 8.482
E(2,1) = 1*E(P,P) + 2*E(P,N) + 0*E(N,N) +2*E(P)+ 1*E(N) = 7.718
E(2,2) = 1*E(P,P) + 4*E(P,N) + 1*E(N,N) +2*E(P)+ 2*E(N) =28.295

If E(N)=E(P) we get:
E(P) = -6.048
E(N) = -6.048
E(P,P) = -2.778
E(N,N) = -2.014
E(P,N) = 14.319
Check:
E(1,1) = 14.319 -6.048 - 6.048 = 2.224
E(1,2) = 2*14.319 -2.014 -3* 6.048 = 8.482
E(2,1) = -2.778 + 2* 14.319 -3*6.048 = 7.718
E(2,2) = -2.778 + 4* 14.319 -2.014 -4*6.048 = 28.295

The binding energy E(P,P) is nearly the same as E(N,N),
but not the same as energy E(P,N).

Nucleuses with inverse number of protons and neutrons
confirm or disprove nothing - they have the same number of bindings
E(P,N).

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