Presented and research paper by

Dr. M. Zafferullah Ghuman
 Department of Physics University of Gujrat Pakistan.

1. Introduction

In making a transition from one level to other the atomic nucleus can give up energy k angular momentum L and parity P in the form of  ray of a particle multipole; electric 2^L, EL , if the parity change is by ( -1)^L rule and magnetic 2^L , ML , if the parity change is by (-1)^L+1 rule. On the other hand, the nuclear transition can take place by transferring the energy, angular momentum and parity directly to one of the orbital K, L or M. The energy of the ejected electron is equal to the energy lose by nucleus, less than binding energy of the electron in the atom, while passing through the nuclear space i.e. r_e < r_n. This is known as internal conversion of electric and magnetic multiples, the rate of electron ejection W_e to the rate of rays emission W   is defined as the internal conversion co-efficient    [5, 6, 10]

        =  ,               (1)                                     

Where H_if is the interaction Hamiltonian which would be given in explicit from latter on and of conversion density states. The removal of the orbital particle takes place through interaction of electron and nuclear transition current and charges via electromagnetic field. Outside the region of the nuclear transition current and charges the form of the electromagnetic field depend only on gross properties of K, L and P and not on the detail of the nuclear current and charges distribution. The probability of the ejection of an electron depends only on the form and strength of the field and on the relevant election wave function.

2. Purpose of the work

In general an excited nucleus (radioactive element) can go to its ground state energy through two channels (a) particle emission such as  particles (b)  rays emission. Ecologically particle emissions are not so harmful since they can be absorbed by the surrounding easily. In the case of α-particle by acquiring two electrons, it can form helium atom, where as  particle is captured by other mater. The situation about -rays is different. Being powerful electromagnetic quanta can penetrate deep into matter.  It disturbs the life equilibrium and therefore is very harmful for any life. In artificial fission and fusion products; emission of  Rays is inevitable and consequently hazardous for health. At present nuclear waste damping is a major threat to our ecology and consequently to existing life.  The purpose of this paper is to find the effect of pressure on such atom, in anticipation to find the solution of the said problem. As mentioned in the abstract  conversion mode of nuclear de-excitation under compression increases and through the emission of -ray decrease , it means we can control the -ray emission in any radioactive material .Since through conversion process nucleus looses its extra energy rapidly and as a result we can find the initial element  neutralized in a controlled fashion. In other words in desired mode of de-excitation in a shorter span of time. In actual research our main work was about the cold fusion process therefore we made calculations of the transition probabilities for only light elements and this idea can be extended to heavy elements too. For muonic atoms the pressure range was very high∼10¹⁰ atms and in view of the significance, as a test case, we have converted our calculations for electronic atoms and found the scale of the pressure laboratorially manageable. To conclude this section it would be interesting to point out that many authors  have seriously studied radiation as well as conversion transition probabilities of free electronic atom and to some extent muonic atoms using uniform distribution of charge over the spherical nuclear volume i.e. with potential energy.

       ,                       (2)                  

Which is in good approximation for heavy elements, where as we have used the following nuclear charge distribution (to be denoted by RED-relatively exact distribution) which is a good approximation for relatively light elements, namely [11]

(x) = ,        (3)                                          

Potential energy calculated on the basis of (3) distribution can be given by the approximated expression [2].

       (4)

Where  =1, 20 10^-13 A^1/3 cm, A-nuclear number, in =m=c=I system of units  the square of the electron charge in coulombs.

To deal with the internal conversion process mathematically, we assume, the interaction between the muon and nuclear transition currents and charges occurs via the electromagnetic fields only. This interaction Hamiltonian of the system can be written in the following retarded potential form namely [5, 6]

                                                                            (5)

When the nucleus under goes a transition from state n_i to n_r giving up energy k=E_i – E _r that is muon is lifted from state  to  .The quantity  and  are the usual Dirac’s (muon) transition charge and current densities. In symbolic from they can be given by the expression [5, 12]

 ( ) (                  (6)

 ( ) (

Here  Dirac’s matrix, ’s are Dirac‘s muon orbital’s. The rate of muon ejection is proportional to the square of H_if . For a given L and M, the scalar, the electric and the longitudinal part of the interaction have the same parity and angular momentum selection rules. The magnetic part has the opposite parity behavior. Therefore it would be useful to split the interaction. Matrix element after having expanded in Multi poles, into magnetic and electric terms, as follow [5, 6] 

H_if =        

                                                                        (7)                

The explicit expressions for H_el (L, M) and H_el (L=0) would be given where they will be needed for further discussion. The last term in (7) usually known as electric monopole interaction matrix, can be given as [4, 5, 6]

Hel (L=O) =   ( )

                                                               (8)

These integral are from zero to r_0. EO- mode of nuclear de–excitation arises entirely from dynamical nuclear structure effect, usually called penetration effect.

3. Electric monopole (EO) Transition Probabilities         in Muonic Atoms.

A case of special interest arises in the transition of nucleus between two states of the same spin. The whole EO mode of nuclear de excitation occurs only through muonc penetration into nuclear space same. EO transition occurs between same nuclear spin and parity by transferring the energy to an orbital muon.  The monopole mode implies a zero angular dependence of the operator and therefore EO operator is purely radial .If the wave function of a conversion muon is normalized to unit energy, then general formula for the probability of EO conversion for any atomic shell can be given as.

W(EO)=2πα (1/ ²                               (9)                                                     

In getting (9) we have replaced ρ and j by their explicit expressions given by (6). Relation (9) is obtained supposing that muonic Electric monopole transition takes place under coulomb’s interaction   between orbital muon and the protons in the nucleus having Dirac‘s initial and final bound state function ��_i , ��_f respectively with radial components given in the form of power series i.e. : (r)=  And (r) = , where  is constant which Must be found fulfilling the following conditions: =λ +1/2; =λ + 1/2 here λ=-1/2 for  <0 and  λ=1/2 for >0 and equating Dirac’s radial function for inside and outside nuclear space at r=  the nuclear boundary ,  in the power of r is relativistic quantum number of muonic bound states. Where as for Free State it will be denoted by . In expression (9), after having integrated with respect to r(r, θ, ф) probability for muon EO-conversion can be brought in the following form i.e. the rate of EO-conversion may be written in the usual form of a muon factor times the square of a nuclear matrix element (6)

W_µ (EO) = (EO)  (EO),             (10)

Where as  (EO) is usually called relative nuclear matrix element of monopole and is given by:

(EO)= ( (11)

Where as Ω_μ(EO)-relative probability of EO-conversion given by [6]

(EO)=           (12)

Here  and   constants of normalization for initial and final muonic states. In obtaining (11) and (12) it was considered that the nuclear quantum number  and  defining initial and final muonic states must be equal to each other.

4. Concept of Pressure:

In quantum mechanic we usually considered infinite atomic volume which in practice is extremely not acceptable. This is idealized views which in its ultimate application give relatively more acceptable results. For free atomic volume we considered radius of the volume extended from 0 to infinity. In other words our wave function turns to be zero at r→∞. If we apply external pressure on this atomic volume the wave function must reduce to zero at r= r_o where r_o is called radius of the compressed atomic volume. In solving Dirac’s equation we find two component solution i.e. f(r) and g(r). Form the study of volume boundary conditions, in classical quantum mechanic radial probability density = ��*�� must have extreme value at the desired pressure point or volume boundary condition namely

  = (d /dr           (13)

From here we have a number of possibilities either (r_o) =0 or =0, or both are equal to zero. In relativistic quantum mechanics we have Dirac’s radial function, which for probability and current densities are different from Schrödinger’s radial function, which for both current as well as probability densities at r= r_o are identically reduce to zero. ; Where as in Dirac’s case such coincidence does not exit. Since in Dirac’s case current density for the boundary condition j(r=r_o)=0, we have f(r_o) g(r_o)=0 and it does not mean that ρ= f²(r_o)+ g²(r_o)=0 .

Condition ρ= 0 for relativistic probability cannot be used as Dirac’s equation for infinite deep potential well is not applicable. From the condition f(r_o) g(r_o)=0, again we have a number  of possibilities. For example either f(r_o)=0; or g(r_o)=0, which cannot be implied that ��’(r_o)=0 given non relativistic case. Further in the third possibility simultaneously if one of them become zero, then other cannot be zero either f(r_o)=0, g(r_o) ≠0 or g(r_o)=0, f(r_o) ≠0. In this work we shall use only one of them particularly g(r=r_o) =0. As we have already mentioned about the pressure techniques we further assume that due to the atomic compression nuclear volume almost remain unchanged although some perturbations take place and consequently nuclear volume must expand as a reaction. This was noticed in our work while calculating effects of nuclear size in compressed atoms. This was pointed out as stabilization factor against the external aggression; while comparing the difference of atomic orbital energy level shifts for both free and compressed atoms. In going from free to compressed atomic states EO-conversion probability calculations, we have to renormalize the muon function in accordance with the required normalization of each initial and final state. If C_po is the compressed bound normalization constant than C_p can be found by the expression.

 [ (r) dr]=1,    (14)

Where f(r) and g(r) are the Dirac’s radial functions for the discrete atomic spectrum. Function inside nuclear region are integrated from zero to R_o and outside nuclear region from R_o to r_o. if we replace discrete atomic spectrum in [14] by continuous atomic spectrum function than our C_po replaced by C_p will be considered as normalization constant for the compressed atom muon final states. The compute compressed atom EO-conversion probability we have to replace free atom normalization constant by the newly computed atom normalization constants C_po and C_p respectively. All other factors of formula remain unchanged. In our calculation compressed atom volume radius r_o is expressed through the nuclear radius by the relation r_o=νR_o; where v can be any positive real integral value. The choice of the relation between r_o and Ω_μ (EO) from r=r_o to r=R_o is not very wide.

5. Conclusions and Comments

To analyze the effects of pressure on EO-conversion, which is the major mode of nuclear de-excitation in all sorts of atoms? For example according to our calculation at r_o=10R i.e. ten times the nuclear radius Ω EO-conversion transition probability of compressed mu-He i.e. (Z=2, A=4) with transition energy k=0, 2 almost 12 times increase. As we know muonic Bohr’s radius for µHe i.e. a_μ (Z=2, A=4)=1, 28x10^-11cm and in this case r_o (Z=2, A=4)=1, 90x10^-11cm this means that when a_μ/ r_o=7, Ω_μ EO in comparison with that of free 12 times increase this means on the introduction external pressure muonic binding energy decreases and possibility of its remaining inside the atomic spaces reduces. In other words EO-conversion becomes faster in compressed atoms. That is muon takes away extra nuclear energy quickly and therefore possibility of gamma emission becomes a lesser. The choice of transition energy has been taken in ħ=m_μ=c=1 system of units equal to those of major workers in this field so that our result could be compared with others [7, 8, 9].

Considered the table 1 in which K –shell pressure increment coefficient (PIC) of Ω_μ (EO) for μ-Helium with UD, RED- nuclear models and energy values k=0, 2, 0 , 4 have been given these PIC values of Ω_μ (EO) given in the table must be independent of the orbital particle and can be applicable in general. If we look into the same table we can see for the same parameters PIC (UD) > PIC (RED) for k=0, 2 where as for k =0, 4 the relation is reversed. This mean in some where between k (0,2 & 0,4) both models give the identical results. It may be said that at lower exited energies UD model predominates where as shell model aspect is less prominent. At higher excited energies shell model aspect becomes more prominent. If we draw a graph between v and PIC we get a usual curve for exponential decay. Actually in this case decay process is speeded up so that extra nuclear could be emitted rapidly.  Consequently radiation mode is suppressed. In other words we can artificially neutralize the radio activity well before and in the desired fashion than it could have cased naturally. At v=10 and k=0, 4 the PIC values for the both nuclear model come down to ~2. This means that at r= 10R the compressed atom Ω_μ (EO) values become equal to those fo free atoms Ω_μ (EO). This mean variation range from r= r_o to r_o = R_o is almost ten times. This was indicated at end of previous section.

According to our calculation  for mu-He K-shell (n=1, = -1) critically ionization radius r_o  10^-10cm for which we need 8x10¹⁰ atm pressure in case of point nucleus; where as 5x10¹⁰atm in case of finite nucleus. In order to define pressure unit in rmu system we have λ= /mc and therefore P=m⁴ c⁵/ ³. From this we can have one rmu pressure ~ 2, 5.10²⁸ atm. This mean in case of electronic the pressure factor would reduce ~ (206,8)⁴ times or ~ 10⁹ times. For the sake of rough estimation of pressure to electronic atoms we can assume that for a given temperature spherical atomic volume decrease uniformly. In this case Boyle-Marriott’s law of ideal gases i.e. p₁ r₁³= p₂r₂³ may be used to defined the pressure volume state.

6. Conversion of EO-Conversion Calculations to     Electronic Atoms

Table1. K-Shell pressure PIC Ω (EO) for mu-Helium

        (Z=2,A=4,        

More to come..