The calculation of binding energies for even-even Mg(A=20,22,28 and 30) isotopes The Calculation of Binding Energies for Even-Even Mg(A=20,22,28 And 30) Isotopes Ahmad M. Shweikh Dept. of physics/ College of Education Ibn- Al-Haitham for pure science (Ibn- Al-Haitham) / University of Baghdad Received in :3 February 2013, Accepted in :10 October 2013 Abstract The rotational model symmetry is a strong feature of 1d shell nuclei, where symmetry breaking spin-orbital force is rather weak. The binding energies and low-lying energy spectra of Mg (A=20,22,28 and 30) even-even isotopes have been calculated. The interaction used contains the monopole-monopole, quadrupole-quadrupole and isospin dependent terms. Interaction parameters are fixed so as to reproduce the binding of 8 nucleons in N=8 orbit for Z=12 isotope. Key words: Binding Energy, Even-Even Isotopes, rotational model symmetry, shell nuclei, spin. | Physics87 @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Introduction The spin and parity for the ground state for even-even Mg (A=20, 22, 28 and 30) isotopes are verified experimentally. The values of spin and parity for the ground states of these isotopes predicted by the shell model is − 2 1 and − 2 3 to be compared with experimental values of + 2 1 and − 2 1 respectively [1]. In the region of N=1 the single particle orbits, gives an explanation of experiment values of spin and parity. As a result, the intruder states are expected to be found in low lying spectra for Mg (A=20, 22, 28 and 30) isotopes [2,3]. The rotational model was proposed and used by Elliott to describe rotational bands in light nuclei [2,4]. The quantum members λµ are the appropriate repress notations of symmetry group are used in this model, to create many basis states for Mg nucleus. The four parameter residual interaction with monopole-monopole, quadrupole-quadrupole and isospin dependent interaction terms with strength parameters (po, p1), χ and β respectively are used in the calculations of binding energies and law-laying spectra for 20Mg, 22Mg, 28Mg and 30Mg even- even isotopes. The ground state band is described by the quantum number λµ that extended the ground state binding energy for a given strength χ of quadrupole-quadrupole interaction. Interaction parameters χ and po are fitted to experimental ground state binding energy and excitation energy of +12 state in Mg (A=20, 22, 28 and 30) even-even isotopes. Interaction parameters p1 and β are adjusted to give a best fit to experimental ground state binding energies of Mg isotopes with A=20,22,28 and 30. The calculated energy spectra are compared with available experimental data. The new mass formula [5], is also used to calculate the residual interaction. A comparison of our calculations and experimental data shows the reasonable agreement. Theory In this article we will write down the main equations of rotational model of Elliott [2], explaining the generators, subgroups and Casimir operators. The representation of rotational bands is characterized by quantum numbers ( λµ ), that which determine the eigenvalue of Casimir operator for symmetry group of rotational bands. In spherical basis, three components of orbital angular momentum operator L, and five components quadrupole moment operator Q constitute the eight generators of system group of rotational bands. A components of quadrupole moment, is defined as; ))p̂(Y p )r̂(Yr( 5 4 Q q,22 2 q,2 22 q α +α π = …(1) Where 2,1,0q ±±= ,  ω =α m2 and q,2Y are the spherical harmonics. The eight generators are satisfy the following commutation relations.        ′+′=′ ′+′−=′ ′+′−=′ ′+ ′+ ′+ )L)qq1qq22(103]Q,Q[ )Q)qq2qq21(6]L,Q[ )L)qq1qq11(2]L,L[ qqqq qqqq qqqq …(2) | Physics88 @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 The Casimir operator, Casimir operator eigenvalues and corresponding quantum numbers for symmetry group of rotational states are listed in the following table. SU(3) Casimir operators LL3QQ ⋅+⋅ Eigenvalues )33( 3 2 22 µ+λ+λµ+µ+λ Quantum numbers ( λµ ) The quantum numbers ( λµ ) characterize the permutation symmetry between harmonic oscillator quanta and Λ is proportional to the number of quanta in xy plane for case 0=µ [6]. The definition of the binding energy of isotope A in residual interaction zero state I(A) as: )8,21(S12A)9,21(S2)8,20(B)Z,A(B)A(I np −−−−= …(3) Where )Z,A(B is the binding energy of the nucleus with Z protons and ZA − neutrons, )9,21(Sp is one proton separation energy of Ne 21 and )8,21(Sn is one neutron separation energy of Al21 . An interaction Hamiltonian used here containing monopole-monopole, quadrupole- quadrupole and isospin dependent interaction terms that is; )1T(TQQ)n(FH o +β+⋅χ−−= …(4) Where n is the number of active nucleons (n=A-p). The isospin dependent interaction is repulsive while the monopole-monopole and quadrupole-quadrupole interaction are attractive. The strength of monopole-monopole interaction for nucleons in the same shell is po and for nucleons in different shells is p1. We write the monopole-monopole interaction for n1 and n2 ( 21 nnn += ) nucleons in oscillator shells N=1 and N=2, respectively,[7] as: 211 22 o 11 oo nnp2 )1n(n p 2 )1n(n p)n(F + − + − = The quadrupole-quadrupole interaction operator may be expressed as: )LLC2(3QQ )3(SU ⋅−χ−=⋅χ− Where )3(SUC is the Casimir operator of the rotational group. The ground state isospin is defined as 2 pn T − = , where n and p are the number of neutrons and protons respectively. A Fortran routine has been used in the calculations of C's and matrix elements of Q.Q operator between relevant good L states. To compare the mass formula of Bethe-Weizsacker [8] with the new mass formula, which has a new parameter )Z,N(∆ and has redefinition of the pairing term newδ , as: )Z,N( A )Z2A( a A )1Z(Z aAaAa)Z,A(B new 2 symcsnew 3 1 3 2 ∆+δ+ − − − −−= υ …(5) Where MeV85.15a =υ | Physics89 @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 MeV34.18a s = MeV71.0a c = MeV21.23a sym = δ−−=δ )) c Z exp(1(new 2ln 6c = nucleiAoddfor0 nucleioddoddforA12 ,nucleievenevenforA12 2 1 2 1 = −−= −=δ − − 45.0kwith) 3 Z exp(ZNZ 3 4 N)Z,N( k =−−=∆ Using the new mass formula to calculate the binding energy we obtain the residual interaction )A(Inew for Mg isotopes as: )n,1A(S)nA()p,1A(S2)n,p(B)p,A(B)A(I npnewnew +−−−−−= …(6) Results and Discussion Table 1, listed Mg(A=20,22,28 and 30) even-even isotopes, besides the isotopes of A-1 (Na) and A+1 (Al) nuclei. Half-life time, Q-value, spin ( π ) and decay mode for Mg isotopes are listed in the table also. The Mg(A=24,25 and 26) are stable. Figure 1, shows the variation of Q-value with Mg even-even isotopes for A=20,22,28 and 30. The valley between A=22 and A=28 isotopes, belongs to Mg(A=24 and 26), where they are stable isotopes and not including this study because they have a closed shell for neutron and proton. Table 2: lists the representations ( λµ ) that maximize the quadrupole-quadrupole interaction and the possible L values for the ground state band. In Figure 2, shows the plotting of the ground state binding energies in Table 3 for (A=20,22,28 and 30) isotopes, due to residual interaction active nucleons, the calculation based on the rotational band model and new mass formula (Eq. 3). The value of parameter χ is chosen to reproduce the excitation energy +12 state in Mg isotopes. The interaction parameters 1p and β are used to find the best fit with the experimental ground states binding energies of Mg isotopes with A=22,24,28 and 30. Table 4, lists the calculated and experimental excitation energies of +10 , + 12 and + 20 states for comparison. Figure 3, shows the fitting with experimental data, which is obtained with parameters 8.2p1 = MeV and 8.5=β MeV. The calculated binding energies show a reasonable agreement with the others. No intruder states are found in Mg(A=20,22,28 and 30) even-even isotopes from Na(A=19,21,27 and 29) and isotopes Al(A=21,23,29 and 31) isotopes. References 1. National Nuclear Data Center.http://www.nndc-bnl.gov/nndc/ensdf/. 2. Elliott,J.P.(1958), Collective Motion in the Nuclear Shell Model. I. Classification Schemes for States of Mixed Configurations, Proc. R. Soc., (London) A245, 128. | Physics90 @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 3. Hammermesh ,M.(1962) Group Theory and its applications to physical problems, Addison- Wesley Publishing Company , Reading MA. 4. Bohr,A. and Mottelson,B. R.(1969), Nuclear Structure, V.I, W.A, Benjamin, Inc, New York,. 5. Samanta ,C. and Adhikari ,S. (2002), Extension of the Bethe-Weizsäcker mass formula to light nuclei and some new shell closures, Phys. Rev. C, 65, 03708. 6. Guillemaud-Mueller ,D.; Detroz ,C.; Langerin ,M.; Naulin ,F.; Desaint-Simon, M.; Thiboult, .C.; Touchard, F.and Epherre ,M. (1984), β-Decay schemes of very neutron-rich sodium isotopes and their descendants, Nucl. Phys. A, 426. 37. 7. Firestone ,R. B. and Shirley ,V. S. (1998), Table of Isotopes, Wiley-Interscience; 8th edition. 8. Fayache. M. S., Moya de Gura .E., P., Sarriguren .Y., Sharon Y., and Zamick. L., (2000), Erratum: Reply to Comment on Question of low-lying intruder states in 8Be and neighboring nuclei, Phys. Rev. C, 61, 059901. Table (1): The Mg(A=20-30) even-even isotopes with their neighbor nuclei No. Isotopes isotopes [7] A-1 Mgeveneven A+1 21T Q-value MeV Spin ( π ) Decay mode 1 Na19 Mg20 Al21 95 ms 10.730 +0 EC 2 Na21 Mg22 Al23 3.857 s 4.725 +0 EC 3 Na27 Mg28 Al29 20.910 h 1.832 +0 −β 4 Na29 Mg30 Al31 0.335 6.990 +0 −β Table No.( 2): Values of ( 111 Lµλ ) for active nucleons No. Ground state band isotopes 11µλ 1L 1 Mg20 20 0,2 2 Mg22 30 0,2,4 3 Mg28 22 0,2,3,4 4 Mg30 22 0,2 Table No.(3): Values of Binding energies B(A,Z) and I(A) MeV No. isotopes B(A,Z) I(A) MeV 1 Mg20 130.066 11.404 2 Mg22 162.487 12.747 3 Mg28 245.447 15.666 4 Mg30 244.274 15.629 Table No.( 4): Excitation energies of +10 , +12 and +20 states in Mg20 , Mg22 , Mg28 and Mg30 nuclei in MeV Spin Mg20 Mg22 Mg28 Mg30 | Physics91 @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Calc. Exp. Calc. Exp. Calc. Exp. Calc. Exp. + 10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 + 12 3.26 3.26 3.21 3.67 3.14 3.48 3.22 2.89 + 20 17.88 18.75 16.91 6.78 9.87 13.78 18.81 14.27 Figure No.(1): The variation of Q-value with Mg (A=20,22,28 and 30) isotopes. | Physics92 @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Figure No.( 2): The comparison between results of the present calculations, new mass formula and experimental data Figure No.( 3): Calculated of rotational model and new mass formula and experimental ground state binding energy in Mg isotopes | Physics93 @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 30و 28، 22، 20حساب طاقات الربط لنظائر المغنیسیوم ذات األعداد الكتلیة الزوجیة-الزوجیة أحمد موسى أشویخ قسم الفیزیاء/ كلیة التربیة ابن الھیثم للعلوم الصرفة/جامعة بغداد 2013الثاني تشرین 10، قبل البحث 2013شباط 3استلم البحث في الخالصة ب�رم ض�عیف، –یكون قوة تناظر تكسر المدار إذ، 1dنموذج التناظر الدوراني وصفا جیدا لالنویة ذات الغالف إیعتبر –الزوجی�ة Mg (A=20,22,28 and 30)طاقات الربط واطیاف طاق�ات المس�تویات الواطئ�ة لنظ�ائر المغنس�یوم حسبت رباعي القط�ب واالیزوس�بن –احادي القطب، رباعي القطب –الزوجیة تم حسابھا. استخدم التفاعل المتضمن احادي القطب .Z=12و N=8لھذه النظائر، معامالت التفاعل تم حلھا العادة حساب طاقة الربط لثمان نیكولیونات وعدد ،غالف النواة ،برم. جیة ، نموذج التناظر الدورانيزو–الكلمات المفتاحیة : طاقة الترابط ، النظائر زوجیة | Physics94 @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014