THE F18 NUCLEUS ENERGY LEVELS USING THE SERBER AND ROSENFELD FORCES ( Ph.D Thesis )

trusciencetrutechnology@blogspot.com

Volume 2013 Issue No.4, Dt.8 April 2013 Time: 8h22m A.M.

THE F¹⁸ NUCLEUS ENERGY LEVELS USING THE SERBER AND ROSENFELD FORCES

[A COMPARATIVE STUDY BASED ON DATA OF THE AUTHOR Ph. D 1964 THESIS.]

by

Dr. Kotcherlakota Lakshmi Narayana,

{Retd. Prof. of Phys, SU, Kolhapur} 17-11-10, Narasimha Ashram, Official Colony,

Maharanipeta. P. O, Visakhapatnam -530002.

Mobile No: +919491902867   and No.9594717723

ABSTRACT

                     The levels of the F¹⁸ nucleus are solved and the data listed with a comparative study of the level positions by using the Serber and Rosenfeld forces. 

The levels

1d²_5/2, 1d²_3/2, 1d_5/21d_3/2, 1d_5/2 2s_1/2, 1d_3/22s_1/2 and 2s²_1/2 were considered for the energy calculations.

INTRODUCTION

                          The author has made elaborate calculations during the years 1961 to 1964 using the seven figure log tables and the Swedish FACIT calculator.  The modern soft-wares used now to get the details of the wave functions and the associated energy level spacing of F¹⁸ nucleus. In O ¹⁸ the two extra neutrons are presumably in the d_5/2 with the d²_5/2neutron configuration has allowed states j=0, 2, 4. The ground state is a 0+ state and 2+ and 4+ are found to be at 2 and 3.5MeV respectively. Talmi (1964) states in the energy calculations in the Nuclear Shell Model   states that no consistency of the configuration assignment. It is interesting that Talmi (ref no.13  and p.125) asserts that two-body interactions in jj coupling of O¹⁸, O¹⁹ and as well as O²⁰ seem to be unchanged in the central potential. Energy level spacing of F¹⁸ nucleus offers a totally different aspect of the two nucleon interaction even in the case of central force consideration as evident of the calculations made by the author given in his Ph. D. Thesis. The desired constant with j=5/2 and n=2 particles yield the values a=(1/28)*(5*V2+23*V4), b=(1/14)*(V4-V2) and c= 1/6*V0 –(1/42)*(10*V2-3*V4) where b= 0.11 MeV.  Here V0, V2 and V4 are the interaction energies of j=0, j=2 and j=4 respectively. (Talmi. I,  p.125)

DIGRESSION

          The force between two nucleons is not just a central force and involves a contribution from the relative direction of spins and the radius vector between the neutron and the proton. The ground level of deuteron has nuclear angular momentum I=1 and is a ³S₁ state which is unstable by the order of about 50KeV. In central force potentials the potential energy is greater than the binding energy of deuteron and the radial wave function of deuteron decreases as exp(-r/ro ) where r0 is the range of the fore. The value of Vo used in the thesis (Ph. D kln AU 1964) is just –Uo of Evans book on the Atomic Nuleus(1955) on page 316 and b is just r0

1.      Yukawa Well :      Vo*exp(-r/r0)/ (r/r0).

2.      Exponential well:   Vo *exp(-r/r0)

3.      Square well:           =Vo  for r< r0

                               =0 for r>r0

4.      Gaussian well:         Vo * exp(-r^2/r0^2)

5.      Highly singulated potential:  Vo *exp(-r^2/r0^2) / (r/r0)^2

6.      Hard core potential:   Vo* r0^2 * exp(-r^2/r0^2)

Where Vo is potential depth, effective range of nuclear force is r0.

           The Serber force represented usually as ½ *(1+P_M) where P_M is the Majorana exchange operator with +1(attractive) for even angular momenta and -1 for the odd (repulsive). My Ph. D thesis 1964 of Andhra University presents the projection operators involving these nuclear exchange operators P w, P_M, P_H and P _B respectively for the Wigner, Majorana, Heisenberg and Bartlett forces. Serber force is therefore zero for odd states. The phase shift in the ³P state depends critically on the exchange character  of the n-p force. It is zro for Serber forces and has opposite sign for the pure Wigner and Majorana forces. (Theoretical Nuclear Physics, Blatt J. N and Weisskopf V.F 1952 p. 613. It is very interesting to note that there also exist anti-Serber forces in nature.

             The energy levels for both the Serber Force and the Rosenfeld force now have been  explicitly calculated and data is presented herewith. 

DATA

                     Both the Serber force and the Rosenfeld Force  data of F¹⁸ is presented below giving the details of Energy level the Isotopic and Angular Momentum details.

Serber

Force

9.6262

T=1

j=2

-0.0654*1d2 5/2

0.9847*1d2 3/2

0.1117*1d 5/2 1d 3/2

-.0458*1d 5/2 2s 1/2

-.1072*1d 3/2 2s 1/2

8.5572

T=0

j=1

.1495*1d2 5/2

0.0652*1d 5/2 1d 3/2

0.0278*2s 2 1/2

0.9564*1d2 3/2

-.2408*1d 3/2 2s 1/2

8.062

T=1

j=0

0.86121*1d2 5/2

0.308*1d2 3/2

0.4044*2s 2 1/2

7.5921

T=0

j=3

0.1086*1d2 5/2

0.953*1d2 3/2

-.2797*1d 5/2 1d 3/2

0.0423*1d 5/2 2s 1/2

5.954

T=1

j=1

1.0*1d3/22s1/2

5.8099

T=1

j=2

-.0437*1d2 5/2

0.0566*1d2 3/2

0.3007*1d 5/2 1d 3/2

-.1762*1d 5/2 2s 1/2

0.9346*1d 3/2 2s 1/2

5.083

T=1

j=1

1.0*1d5/21d3/2

5.042

T=0

j=2

1d5/2

0.1095*1d2 3/2

1d 3/2 2s 1/2

4.5284

T=1

j=2

-.2143*1d2 5/2

0.4627*1d 5/2 1d 3/2

-0.9228*1d 5/2 1d 3/2

-.1698*1d 5/2 2s 1/2

0.2482*1d 3/2 2s 1/2

3.5134

T=0

j=1

-.366*1d2 5/2

1d23/2

0.0791*2s 2 1/2

-.1645*1d2 3/2

-.7866*1d 3/2 2s 1/2

3.4639

T=0

j=3

0.124*1d2 5/2

-0.2953*1d 5/2 1d 3/2

-.9354*1d 5/2 1d 3/2

0.15*1d 5/2 2s 1/2

2.8128

T=1

j=4

-.7113*1d2 5/2

1.422*1d 5/2 2s 1/2

2.3722

T=0

j=2

0.8104*1d5/21d3/2

-.5215*1d 5/2 2s 1/2

0.2671*1d 3/2 2s 1/2

1.528

T=0

j=4

1.0* 1d5/21d3/2

0.0631

T=0

j=1

-.419*1d2 5/2

-.0036*1d2 3/2

.2338*2s 2 1/2

0.1551*1d2 3/2

0.4785*1d 3/2 2s 1/2

-0.1998

T=1

j=2

-.6609*1d2 5/2

0.0421*1d 5/2 1d 3/2

0.7431*1d 5/2 2s 1/2

0.0959*1d 3/2 2s 1/2

-0.5418

T=0

j=3

.7303*1d2 5/2

-.0518*1d2 3/2

.0039*1d 5/2 1d 3/2

-.6811*1d 5/2 2s 1/2

-1.2851

T=1

j=4

0.71131*1d2 5/2

1.422*1d 5/2 2s 1/2

-1.637

T=0

j=2

-.4272*1d5/21d3/2

-.8379*1d 5/2 2s 1/2

-.3396*1d 3/2 2s 1/2

-2.0144

T=0

j=1

0.5149*1d2 5/2

.0571*1d2 3/2

-.8446*2s 2 1/2

-.0842*1d2 3/2

-0.0947*1d 3/2 2s 1/2

-2.1

T=1

j=0

-0.4076*1d2 5/2

0.123*1d2 3/2

.9114*2s 2 1/2

-3.1697

T=1

j=2

.7149*1d2 5/2

1d2 3/2

-0.2091*1d 5/2 1d 3/2

0.6211*1d 5/2 2s 1/2

.2104*1d 3/2 2s 1/2

-3.555

T=0

j=5

1.0*1d2 5/2

-4.0842

T=0

j=3

0.6629*1d2 5/2

0.0437*1d2 3/2

-.2165*1d 5/2 1d 3/2

-.7154*1d 5/2 2s 1/2

-7.174

T=0

j=1

0.6651*1d2 5/2

0.5607*1d 5/2 1d 3/2

0.4675*2s 2 1/2

-.1561*1d2 3/2

-.0172*1d 3/2 2s 1/2

-7.92

T=1

j=0

.8612*1d2 5/2

.308*1d2 3/2

0.4044*2s 2 1/2

ROSENFELD

FORCE

10.5788

T=1

j=2

.052*1d2 5/2

‘0.9985*1d2 3/2

-.0051*1d 5/2 1d 3/2

.0069*1d 5/2 2s 1/2

0.0144*1d 3/2 2s 1/2

9.8609

T=0

j=1

-.3093*1d2 5/2

0.0154*1d 5/2 1d 3/2

-0.0257*2s 2 1/2

‘-.9301*1d2 3/2

0.1957*1d 3/2 2s 1/2

7.8727

T=0

j=3

0.3175*1d2 5/2

0.8345*1d2 3/2

-.449*1d 5/2 1d 3/2

0.034*1d 5/2 2s 1/2

6.9967

T=1

j=2

.0071*1d2 5/2

.0141*1d2 3/2

.0479*1d 5/2 1d 3/2

.0546*1d 5/2 2s 1/2

.9172*1d 3/2 2s 1/2

6.689

T=1

j=1

1.0*1d 5/2 1d 3/2

6.686

T=1

j=1

1.0*1d 3/2 2s 1/2

6.0968

T=1

j=2

-.1386*1d2 5/2

-.0016*1d2 3/2

-.9892*1d 5/2 1d 3/2

-.0016*1d 5/2 2s 1/2

0.0469*1d 3/2 2s 1/2

5.4228

T=1

j=4

-.217*1d2 5/2

0.976*1d 5/2 1d 3/2

4.9648

T=0

j=2

-.4163*1d2 5/2

-.1283*1d2 3/2

.9001*2s 2 1/2

3.3474

T=0

j=1

-.5145*1d2 5/2

.4941*1d 5/2 1d 3/2

-.0454*2s 2 1/2

0.0335*1d2 3/2

-.6986*1d 3/2 2s 1/2

2.292

T=0

j=3

0.0017*1d2 5/2

-.4694*1d2 3/2

-.8543*1d 5/2 1d 3/2

.2234*1d 5/2 2s 1/2

2.0383

T=0

j=2

0.6547*1d2 5/2

-.7293*1d2 3/2

0.1988*2s 2 1/2

1.3815

T=1

j=0

-.1192*1d2 5/2

-.0021*1d2 3/22

0.9929*2s 2 1/2

1.0807

T=1

j=2

-.1598*1d2 5/2

‘0.0024*1d2 3/2

.0132*1d 5/2 1d 3/2

.9855*1d 5/2 2s 1/2

-.0557*1d 3/2 2s 1/2

0.4975

T=1

j=2

0.976*1d2 5/2

0.0527*1d2 3/2

-0.1377*1d 5/2 1d 3/2

0.1605*1d 5/2 2s 1/2

.0041*1d 3/2 2s 1/2

0.1821

T=1

j=4

-.976*1d2 5/2

-.217*1d 5/2 1d 3/2

0.0679

T=1

j=0

0.9872*1d2 5/2

0.1062*1d2 3/2

0.1187*2s 2 1/2

-0.3206

T=0

j=4

1.0*1d 5/2 1d 3/2

-1.3134

T=0

j=3

.6988*1d2 5/2

-.21562*1d2 3/2

.0325*1d 5/2 1d 3/2

-.6671*1d 5/2 2s 1/2

-1.6643

T=0

j=1

-.3852*1d2 5/2

-.5341*1d 5/2 1d 3/2

0.1329*2s21/2

-.2846*1d2 3/2

-.6838*1d 3/2 2s 1/2

-3.0914

T=0

J=2

0.631*1d2 5/2

0.6721*1d2 3/2

0.3876*2s 2 1/2

-3.7155

T=0

J=1

0.3986*1d2 5/2

-.2333*1d 5/2 1d 3/2

0.8752*2s 2 1/2

.1208*1d2 3/2

0.0775*1d 3/2 2s 1/2

-5.404

T=0

J=5

1.0*1d2 5/2

-6.7113

T=0

J=3

-.641*1d2 5/2

0.1329*1d2 3/2

-0.26*1d 5/2 1d 3/2

-.7098*1d 5/2 2s 1/2

-12.113

T=0

J=1

-0.5765*1d2 5/2

-.6449*1d 5/2 1d 3/2

-.4621*2s 2 1/2

0.1954*1d2 3/2

0.0079*1d 3/2 2s 1/2

                        The energy level with the specified T and J values and the explicit calculation made by the author of the F ¹⁸ nucleus using both the Serber and the Rosenfeld forces are presented. A detailed analysis of the corresponding wave functions is worth the effort.

CALCULATED LIST OF ENERGY LEVELS

                  The F¹⁸ data of levels are listed for both the Serber and the Rosenfeld Forces below giving the associated T and j values explicitly. A set of about a total of 26 levels are presented. Plots B refer to Rosenfeld force and Plots C refer to Serber force.

B1: plot(x^3 - 12.4123* x^2 +15.9826* x -1.0278 ,x);     B2:  p =  1.0e+003*[0.0010*x^5+0.0043*x^4-0.1267*x^3   -0.2901*x^2+1.3657*x+2.4723]; No B3 plot since they are individual values. B3=[6.689                     0; 0                         6.686]        B4 : plot(x^5-25.2505*x^4+ 219.0681*x^3 -749.9132*x^2 +809.5767*x-242.6108,x); B5: plot(x^3-3.9117*x^2-11.5295*x+31.2839,x); B6: plot(x^4 - 2.08* x^3 -55.2413* x^2 +55.4855*x +160.4777,x);  Rosenfeld Force F 18 nucleus 1964 KLN B7: plot(x^2-5.6049*x+0.9876,x); B8    T=0    J=4    1d5/21d3/2   Eig=  -0.3206 B9     T=0     J=5    1d25/2      Eig =  -5.404 SERBER FORCE DATA C1: plot(x^3+1.951*X^2-64.132*x-133.8162,x);  C2: plot(x^5 -2.9452*x^4 -66.212*x^3 +106.0021*x^2 +428.0445*x -27.416,x);  C3: Eigen values are 5.083 and 5.954 only two. No C3 plot since they are individual values. C4 : plot(x^5-16.595*x^4 + 59.1912*x^3  + 158.0714*x^2 -773.6768*x -160.3725,x);  C5 : plot(x^3 -5.7777 *x^2 – 0.1747*x   +  19.5843,x); C6 : plot(x^4 - 6.43* x^3 -22.634* x^2 +97.189*x +58.1947,x);  C7 : plot(x^2 -1.5277*x – 3.6147,x);  C8 : T=0  J=4  1d5/21d3/2       Eig=  1.528 C9 : T=0  J=5    1d2 5/2          Eig =  -3.555 The Nuclear levels calculated for F 18 nucleus  with Serber Force and Rosenfeld Force

DISCUSSION
                             Rosenfeld force yields the ground state as T=1 j=0; while the Serber force shifts it to 0.0679 level. The three levels T0 j=1; T=0 j=3; and T=0 j=5; happened to occur sequentially upward above the ground state levels. But the Serber force shifts away the T=1 j=2; level to a very high state at 0.4975. This is fascinating result. Similarly the sixth level of Rosenfeld force shifts the level to 1.3815 level Serber force.  The seventh and eighth levels of Rosenfeld force lie down at -3.0914 and -3.7155 of the Serber force. Rosenfeld T=0 j=3; goes up considerably by Serber force at 2.292. The fifteenth, fourteenth and eleven numbered levels of  Rosenfeld force fall back to lower levels of Serber force at -1.663, -1.3134 and 0.3206 levels. T=1 j=2; level of the Serber force is slightly higher relative to -1.998 of Rosenfeld force. T=0 j=3; seems to well balanced of the two forces around the level 7.8721 and7.5921 of the Serber and Rosenfeld forces. Similar is the case with the T=0 j=2; but Serber gives lower value 2.0383 while Rosenfeld sets it at 2.3722. This seems to be a nice contrast with the previous levels of T=0 j=3. For T=1 j=1 the Serber force 6.686 gives a very high value relative to the Rosenfeld force at 5.083. Next the T=1 j=0; brings the Serber force 10.963 level to a lower value of Rosenfeld force 8.062. Main conclusion is that the Rosenfeld force sets the levels at slightly lower values compared to the Serber force.

CONCLUSIONS

           The main contents of the paper the solutions of the energy matrices of F¹⁸ nucleus that have been solved now and presented. The nuclear level characteristic spacing elaborated. The energy levels for both the Serber Force and the Rosenfeld force now have been  explicitly calculated and data is presented herewith.

ACKNOWLEDGMENT

                 These matrices forms a part of my Ph. D (Andhra) Thesis during the years 1961-1964 of Andhra University,  Waltair from the laboratories of Late Prof. K. R. Rao, D.Sc. (Madras), D.Sc. (London)  to whom I am deeply indebted. I am greatly indebted to Late Prof .K. R. Rao D.Sc. (Madras), D.Sc. (London) for enumerating me on the nature of Nuclear Forces and the Nuclear Spectrum of F ¹⁸ nucleus using the Serber and the Rosenfeld forces.

REFERENCES

1. T. Berggren, T. Erickson, K. L. Narayana,” A single Particle excitable

    core model for Li⁶  nucleus“

    Arkiv Fur Fys (Sweden), Vol.30, p. 545 , 1965

2. K. L. Narayana and C. I. Patil, “On the Energy level structure of

   the ¹⁰B nucleus”,

   Acta Physica Polonica, Vol. B8, No.4, p.559-572, 1977. &

   Paper No.78, Nucl. & Solid State Physics Sympsium, Vol.17-B, Bombay,    1974.

3. K. L. Narayana, “ Dual core Model for He³  ”,

     Current Science, Vol.38, p.487, 1969.

4. K. L. Narayana, “ Dual Core model for ¹⁰B nucleus”,

     Shivaji University Journal, Vol.4, No.8, p.73-75, 1971.

5. K. L.Narayana, “On the low lying energy levels of O¹⁸ nucleus”,

     Current Science, Vol.37, p.132, 1968.

6. K. L.Narayana, “On the low lying even parity levels of F¹⁸  nucleus”,

     Current Science, Vol.38, p.38-40, Jan 1966.

7. K. L. Narayana, “The Uranium qudrupole moment based on a dual core  Fissionable Model”,

       Current Science, Vol.38, p.261, 1966.

8. K. L. Narayana, “On The unification of gravitation And the quantum theory”,

      (invited Talk),    

     Presented at GRG, 8th National Symposium at Bhavanagar, 1978 &

     Shivaji University J., Vol.17. p.13-21. 1977

9. K. L. Narayana, “ Resonating Magic Cores Model for B¹⁰ nucleus”,

    Proc. Nucl. & Solid State Physics, Roorke, Paper No. N-93,1969.

10. K. L. Narayana, ” On the binding energy of ¹⁰B “,

      Proc. Nucl. & Solid State Physics, Madurai, 1970.

11. K. L. Narayana, “The structure of mass 19 nuclei”,

      Proc. Nucl. & Solid State Physics, Madras, 1968

12. K. L. Narayana, “ Quantal functional Representation and Estimate

      of Actinides Nuclear Properties”, Nuclear Science Symposium, No.164,

      Omni-Shoreham Hotel, Washington D. C., Oct-Nov.1, 1986.

13. Talmi . I, Book on “Selected Topics in Nuclear Spectroscopy”, compiled

      B J  Verhaar, p.107 and article “on energy calculations in the Nuclear

      Shell  Model” p.106-132, 1964 references in the Ph. D Thesis of the

      author 1964 of Andhra University, Waltair.

14. See references listed in the trusciencetrutechnology@blogspot.com,

       Volume 2013 Issue No.3, Dt.23 March 2013 Time:1h08m P.M. 
       CALCULATIONS ON THE ENERGY LEVELS SPACINGS OF F ¹⁸ NUCLEUS

15. J. M. Blatt and V. F. Weisskopf, p.613 etc., 1958 reprint

16  R. D Evans, “The Atomic Nucleus”, p.296, Chap.10, p.316-318, 1955