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Rutherford classical scattering theory, as its quantum mechanical analogue, is modified for scattering cross-section and the impact parameter by using quantum mechanical momentum,
(de Broglie hypothesis), energy relationship for matter oscillator (Einstein’s oscillator) and quantum mechanical wave vectors,
and
, respectively. It is observed that the quantum mechanical scattering cross-section and the impact parameter depended on inverse square law of quantum action (Planck’s constant). Born approximation is revisited for quantum mechanical scattering. Using Bessel and Neumann asymptotic functions and response of nuclear surface potential barrier, born approximations were modified. The coulombic fields inside the nucleus of the atom are studied for reflection and transmission with corresponding wave vectors, phase shifts and eigenfunctions Bulk quantum mechanical tunneling and reflection scattering, both for ruptured and unruptured nucleus of the atom, are deciphered with corresponding wave vectors, phase shifts and eigenfunction. Similar calculation ware accomplished for quantum surface tunneling and reflection scattering with corresponding wave vectors, phase shifts and eigenfunctions. Such diverse quantum mechanical scattering cross-section with corresponding wave vectors for tunneling and reflection, phase shifts and eigenfunctions will pave a new dimension to understanding the behavior of exchange fields in the nucleus of the atom with insides layers both ruptured and unruptured. Phase shifts,
*δ*
_{l} for each of the energy profile (partial) will be different and indeed their corresponding wave vectors for exchange energy eigenvalues.

It has not been successful, to our knowledge, to find such reported results anywhere in any literature the world over. Theoretical results were developed, the verification of which would, however, be needed with experimental results on beam physics. Rutherford classical scattering [

Quantum action deals with oscillatory behavior of matter waves (transverse waves) which configures a space of its own called a wave packet or quanta.

It is assumed that the parity of scattering remains conserved. The quantum mechanical scattering its self is an asymmetrical second order process. Diverse scattering cross sections can be determined for each of the described above cases with known phase shifts and the wave vectors for scattered particles. With scattering profiles (scattering eigenfunctions), the shape and size of scattering through modeling and simulation can be reproduced.

The scattering cross-section and indeed the differential scattering cross-section in Rutherford scattering depend on measurable entities like

Using de Broglie hypothesis,

where Ze is the charge of incident of particles,

Changing

With rigorous mathematical substitution of basic quantum mechanical entities, as mention in the above paragraph, a meaningful solution is obtained, i.e.,

where c is the velocity of light,

where

It is found that our both formulas (4) and (5), i.e., quantum mechanical scattering cross-section and the impact parameter follow the inverse square law of quantum action,

If highly energetic incident particles are considered which could tunnel through the Coulomb’s barrier then the incident particles will suffer nuclear surface barrier and the potential well which is envisaged as the interior of the nucleus. The incident particles should have sufficient energy to tunnel through the nuclear surface barrier to get accommodated in the interior of the nucleus. These incident particles will make the nucleus to undergo scattering, of course, by the brim of the potential well and indeed by tunneling the other side of the nuclear surface barrier. For r > R (range of classical scattering) the Coulombic field is encountered. Now using certain conditions of Born approximation [

For

where a is the width of the surface nuclear barrier and v the potential energy. For odd solution of

Reflection (scattering) from the nuclear surface barrier as well as transmission (quantum tunneling) of incident particles are evident. Some of the incident particles are pulled into the nucleus while some of them tunnel perfectly through the other side of the nuclear surface barrier. For reflection,

The first term in the first part of Equation (9), i.e.,

For

The incident particles having sufficient energy will tunnel through the other side of the nuclear surface potential barrier [

The incident particles which are captured by the nucleus

The wave vector in Equation (12) shows the quantum action of scattered particles either for complete tunneling or tunneling after capturing by the nucleus. The emission of partial waves, in either cases, is a manifestation of asymptotic dependence of Bessel and Neumann functions, respectively. The partial waves from inside (quantum well) of the nucleus differ from partial waves emitted, as a consequence of tunneling, through the nuclear surface potential barrier. The asymptotic conditions of Bessel and Neumann functions, respectively are written

The Neumann function

With asymptotic conditions for particles which tunneled through the surface potential barrier, captured within the nucleus and then scattered with phase shift,

and then only the Neumann function will work,

equation (12) will take the following shape:

Using second part of Equation (9), which is for transmission,the eigenfunctions(energy profile) for different values of

Equation (15) shows that scattered particles from inside various layers of the nucleus. For

Equation (16) represents the energy profile for particles which are scattered from the inside of the nucleus for any value of azimuthal quantum number. it is assumed that

For

For

For

For

the incident particles after tunneling the Coulombic field will encounter the nuclear surface potential barrier, where

With

where

With

where r_{l} = radius of the screened nuclear surface.

For

The diverse nuclear quantum mechanical scattering cross-sections for each of the described above cases can be determined by using a generally accepted universal formula available in reference books [

we infer the following conclusions from this study: Quantum theory of Rutherford Scattering is established. Born approximations for coulombic fields inside the nucleus of the atom are determined for reflection and transmission with corresponding wave vectors, phase shifts and eigenfunctions. Bulk Quantum mechanical tunneling and scattering with born approximations both for ruptured and unruptured nucleus of the atom are deciphered with corresponding wave vectors, phase shifts and eigenfunctions. Surface quantum mechanical tunneling and scattering with born approximations for ruptured and unruptured nucleus of the atom are studied with corresponding phase shifts and eigenfunctions. Diverse Quantum mechanical scattering cross-sections with corresponding phase shifts and eigenfunctions for bulk and surface behavior of layers inside the nucleus of the atom will help to resolve and understand the exchange fields inside the nucleus of the atom.

SaleemIqbal,FarhanaSarwar,Syed MohsinRaza, (2016) Quantum Mechanical Approach for Rutherford Scattering and Nuclear Scattering with Born Approximation. World Journal of Nuclear Science and Technology,06,71-78. doi: 10.4236/wjnst.2016.61007