Energy Levels of Helium Nucleus

doi:10.4236/jmp.2013.44064

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Journal of Modern Physics, 2013, 4, 459-462

http://dx.doi.org/10.4236/jmp.2013.44064 Published Online April 2013 (http://www.scirp.org/journal/jmp)

Energy Levels of Helium Nucleus

Cvavb Chandra Raju

Department of Physics, Osmania University, Hyderabad, India

Email: [email protected]

Received January 9, 2013; revised February 10, 2013; accepted February 22, 2013

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

The Helium-4 nucleus is more similar to the Hydrogen atom of atomic physics. In the case of hydrogen atom, there are

many energy levels which were experimentally seen and theoretically explained using non-relativistic quantum me-

chanics. In this note, we use a central potential to derive the energy levels of Helium-4 nucleus. The ground state and

the first few energy levels agree pretty well with experiment. The same potential can be used with nuclei like Oxy-

gen-17 and many more nuclei.

Keywords: Helium-4; Nuclear Energy Levels; Deuteron; Morphed Gravitational Potential

1. Introduction and Formulation of the

Problem

The Deuteron nucleus has no excited states. The ground

state energy of the Deuteron is experimentally found to

be −2.225 MeV and the measured radius of this nucleus

is 2.1 F. This is the distance between the center of mass

and either of the nucleon in the Deuteron nucleus [1].

There are no known solutions of the Schrödinger equa-

tion of this nucleus with Yukawa potential.

Are there any central potentials with which we can

solve the Schrödinger equation for many nuclei such that

their ground state wave functions and excited states can

be obtained? This question led us to a central potential

which is closely related to the gravitational potential en-

ergy.

There is no reason or experimental support to believe

that the universal constant of gravitation G is same for all

values of interacting masses. For interacting masses of

the order of nucleon masses G may not retain its univer-

sality. This led us to the following expression for the

gravitational potential energy of two particles whose

masses are and ,



012

GM mm











1eVr G , (1.1)

where in place of the usual constant of universal constant

G we have a modulating factor.

The constant g2 is a dimensionless real number whereas

has dimensions of mass. We believe that the expo-

nential goes to zero when the interacting masses are large

and the Universal law of Gravitation is restored. It is the

parameter M0 that causes the Universal Law of Gravita-

tion restored. An approximation to Equation (1.1) is giv-

en by,



11 mm

Vr Gr





















(1.2)

where “r” is the distance between the interacting parti-

cles. Simplifying Equation (1.2), we have,

Vr r

 

. (1.3)

The above potential energy is obtained from the gravi-

tational potential energy. It may be called “the morphed

gravitational potential energy”. There are two constants

g2 and

which we will obtain below.

2. Deuteron

The Deuteron is a bound system of a neutron and a pro-

ton with an orbital angular momentum of zero. The total

spin of the two nucleons is one. The deuteron nucleus has

no excited states. The experimentally measured ground

state energy of the Deuteron nucleus is −2.225 MeV and

its orbital angular quantum number . There is no

stable diproton. It is also known that the nuclear potential

depends on the spin orientation [2] of the nucleons inside

a nucleus. If the nuclear force is independent of their spin

orientation then the singlet (total spin = 0) state and the

triplet state (total spin = 1) will have the same energy.

0

C. C. RAJU

460













But this is not observed. This means that the singlet ten-

sor potential is weaker than the triplet potential. We con-

sider an extreme situation wherein the singlet potential of

the diproton is quite negligible. In that event the total

potential energy operating for a diproton nucleus is,

,, ,

rRrY



 





1,2,3, ,0,1, ,1and

,1,2,,

222

gcm ec



, (2.1)

where, the first term is the morphed gravitational poten-

tial energy and the second term is the Coulomb potential

energy between the two protons. Here 21

137

e

is the

fine structure constant. If, 0

m and 22



, the

total potential energy in the case of a diproton will be

zero and there will be no stable diproton nucleus! But

with these values for g2 and 2

the theoretically com-

puted binding energy for the deuteron nucleus turns out

to be quite small. Hence we chose the following values

for these parameters which are quite close to the values

mentioned above.

0.2254

g0.0323 84



10 gm



, (2.2)

and,

00.931826M . (2.3)

There are a few important points to note about the above

values of the parameters g2 and 0

 The parameter g2 is the weak interaction constant of

the electro-weak standard model [3] because it con-

tains e2 and the Weinberg mixing parameter sin2θW =

0.2254 as in the Standard model [4]. The other pa-

rameter is nearly equal to the square of the proton

mass and this is required to avoid a stable diproton

nucleus.

 The interaction parameter contains the square of mass

in the numerator and also in the denominator; Be-

cause of this reason the interaction constant of the

morphed gravitational potential has the same dimen-

sions as the electroweak constant. Hence it is renor-

malizable. This is an important point in favor of the

morphed gravitational potential.

The Schrödinger equation for the deuteron is given by,



2,,

2Vr r



,,E r

















 





Vr 1

, (2.4)

where, is given by Equation (1.3) with



and . The reduced mass

mm



is given by,







. (2.5)

Since is a central potential, from the methods

of quantum mechanics, it just follows that [5,6],

. (2.6)

As in the case of the Hydrogenatom, the quantum

numbers, n, and m have their values,





 

 

  (2.7)

The radial function is given by,



RA L









, (2.8)

where,



, (2.9)

and,



cmm





. (2.10)

These results are obtained by a simple transcription of

the Hydrogenatom calculations. The energy of the deu-

teron is given by,



22 2

EMn





 







. (2.11)

The normalization constant A in Equation (2.8) is

given by,



Anan n





 



 

 







10 2.2251 MeV.E

. (2.12)

Using the above results we can now estimate the

ground state energy, ground state wave function and the

radius of the Deuteron nucleus.

(2.13)





100 3







04.31734 10cma



. (2.14)

. (2.15)

here a0 is the distance between the two nucleons. The

radius of the Deuteron is the distance of either of the nu-

cleon from the center of mass. The center of mass lies

halfway between the line joining the two nucleons be-

cause the two nucleons have almost equal mass. We can-

not avoid the excited states of the deuteron nucleus

however small these may be.

3. The Helium-4 Nucleus

The Helium-4 nucleus is a strongly bound system of two

neutrons and two protons. It used to be known as an



particle. In fact using this



-particle itself the very ex-

C. C. RAJU 461

istence of a nucleus was inferred. Our morphed potential

energy can be put to test using the estimated energy lev-

els for this nucleus. A rough picture of this nucleus is

like this. Any one nucleon experiences a total morphed

gravitational potential due to the remaining three nucle-

ons. This outer nucleon must be a proton because two

protons of this nucleus cannot be together because of

their Coulomb repulsion. This rough picture indicates

that,



npp

2mm

cec





Vr M



 , (3.1)

where the second term is due to the Coulomb repulsion,

and the first term is the morphed gravitational potential

energy. We will now apply the methods of quantum me-

chanics in an effort to obtain a theoretical description of

Helium-4 nucleus or alpha-particle.

The Schrödinger equation with the above potential is

given by



2,,

2Vr r



,,E r

















 

, (3.2)

where the reduced mass is now given by,



749 10gm







21.254

npp

mmm







. (3.3)

The central potential



enables us to find a closed

solution for this nucleus. As in the case of the H-atom the

radial function is given by,















, (3.4)

where,



, (3.5)

here,



2np

gcm m











2

mec













. (3.6)

In Equations (3.4) and (3.5), n is the principal quantum

number as in Equation (2.7). The energy Eigen values for

the Helium nucleus are given by,



mec











gcm m















. (3.7)

The normalization constant A in Equation (3.4) is giv-

en by,



Anan n





 



 

 







. (3.8)

For the Helium nucleus the ground state orbital angu-

lar momentum quantum number is zero. Hence the

ground state wave function of the Helium-4 nucleus is

given by,

100 3







10 28.5176 MeVE

. (3.9)

The ground state energy of Helium-4 nucleus is given

by,

. (3.10)





This value should be compared with the binding en-

ergy of this nucleus which is about 28.3 MeV; we ob-

tained this value through ordinary quantum mechanics

and by using masses and the charge as shown above. In

general the principal energy levels of the tightly bound

Helium-4 nucleus are given by,

28.5176MeV



28.5176 MeV,7.1294 MeV,

3.168622 MeV,1.78235 MeV,

1.140704 MeV,0.792156 MeV.

. (3.11)

The possible energies for this nucleus are listed below:





 



(3.12)

These energy levels are obtained ignoring the tensor

forces. These energy levels can be put to test to find out

the viability of the morphed gravitational potential. Our

efforts to find the principal energy levels were not very

successful. In reference [7], only we could obtain the

measured energy spectrum and this contains only two

energy levels. The energy diagram is arranged by setting

the ground state energy zero, this amount to adding

+28.5176 MeV to all the energies above. Our results

agree pretty well with the two principal levels given in

Ref [7]. The general wave function for the Helium-4 nu-

cleus for any orbital angular momentum is given by,













,, ,

rRrY



 

 . (3.13)

The wave function given here for the ground state of

the Helium-4 nucleus can be used in alpha-decay or in

reactions involving the alpha particle.

4. Conclusion

In this note we assumed that the universal constant G is

not universal for all values of the interacting masses.

Motivated by this idea we changed G and used an ap-

proximated constant for interacting nucleons. When this

potential is applied along with the methods of ordinary

C. C. RAJU

462

quantum mechanics the estimated results are all in agree-

ment with experiment. This very procedure can be appli-

ed to such nuclei Oxygen-17, F-17 and many more nuclei.

The constant 0

is not universal. It is an adjustable

parameter. The functional dependence of this parameter

on the product of the interacting masses is unclear as of

now. But the idea cannot be dismissed just like that. But

certainly for larger masses the exponential in Equation

(1.1) goes to zero.

5. Acknowledgements

The author is very grateful to Prudhvi Rchintalapati. But

for his help the work could not have seen the light of the

day.

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