Theoretical Calculation of The Binding And Excitation Energies For 𝐍𝐢𝟐𝟖 𝟓𝟖 30 Using Shell Model And Perturbation Theory Hadi J. M. Naz T. Jarallah Sameera A. Ebrahiem Rajaa F. Rabee Dept. of Physics/ College of Education for Pure Science ( Ibn Al-Haitham), University of Baghdad Received in:13May 2013, Accepted in :24September2013 Abstract A theoretical calculation of the binding and excitation energies have been used at low – lying energies based on shell model and quantum theory. In this model, we evaluated the energies under assume Ni28 56 30 as inert core with two nucleon extra, nucleons in the 2P3/2 , 1f5/2 and 2P1/2 configuration. Modified Surface Delta Interaction (MSDI) and Reid's Potential (RP) theory for two body matrix elements are evaluated by using a Matlab program to calculate the energies of experimental and Reid single particle energies. Our results of the theoretical calculation have been compared with the experimental results, which show no good agreement with the experiment but have a good agreement with the theoretical studies of Non Zero Pairing Shell Model (NZPSM) and Energy Spectra Method (ESM). Key words: Modified Surface Delta Interaction, Reid's Potential, Binding and Excitation Energies, Shell Model, Ni28 58 30, Perturbation Theory. 225 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Introduction The nuclear shell model is based on the analogous model for the orbital structure of atomic electrons in atoms. In some areas, it gives more detailed predictions than the other model. In principle, the shell models energy level structure can be used to predict nuclear excited state [1]. The concept of symmetry in physics is a very powerful tool for the understanding of the behavior of nature. Symmetries are intimately related to conservation laws and to conserved quantities which, in quantum mechanics, is translated into good quantum numbers. In nuclear physics, several symmetries have been identified. In particular, isospin quantum number t, with symmetry is related to the identical behavior of protons tz = −1/2 and neutrons tz = 1/2 in the nuclear field [2]. In the study of nuclear structure properties, nuclear masses or binding energies (BE) and, more in particular, two-neutron separation energies (S2n), are interesting probes to find out about specific nuclear structure correlations that are present in the nuclear ground state[3]. In most cases, non-relativistic kinematics is used. The bare nucleon-nucleon (or nucleon- nucleon-nucleon) interactions are inspired by meson exchange theories or more recently by chiral perturbation theory, and must reproduce the nucleon-nucleon phase shifts, and the properties of the deuteron and other few body systems [4]. Maria G. Mayer’s discussion of the magic numbers in nuclei has clearly demonstrated the nuclear shell structure is associated with the independent-particle model for nuclei. In this model, each closed-shell configuration provides a convenient first approximation. In this approximation, one can assume that the system under consideration consists of a closed-shell core plus valence particles in a valence shell. This approach very successfully explains the ground state properties of nuclei [5]. Theory To calculate the properties of nuclear ground and excited state depending on quantum mechanical and perturbation theories, one must have the available wave functions of these states. The wave function can satisfy of the schrodinger equation that is given by [6]: H |φ(r)〉 = E|φ(r)〉…………(1) Where H: is the non relativistic Hamiltonian operator can be formally by [6]: H = H0 + Hres…………(2) Where H0: is the Hamiltonian of one body potential, and Hres is the residual interaction substitute of Eq. (2) in Eq. (1) results. (𝐻0 + 𝐻𝑟𝑒𝑠) |𝜑(𝑟)〉 = 𝐻0|𝜑(𝑟)〉 +𝐻𝑟𝑒𝑠|𝜑(𝑟)〉…………(3) The wave function for the first order perturbation theory is given by [6]: |𝜑(𝑟)〉 = |𝜑0(𝑟)〉 + |�̀�(𝑟)〉…………(4) And energies 𝐸 = 𝐸0 + �́�…………(5) Inserting Eq. (4) in Eq.(3) we can find the zeros order quantity [7]: 𝐻° �𝜑°(𝑟)〉 = 𝐸°�𝜑°(𝑟)〉…………(6) And the first order quantity is given by [7]: 226 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 𝐻0|�̀�(𝑟)〉 +𝐻𝑟𝑒𝑠|𝜑0(𝑟)〉 = 𝐸0|�̀�(𝑟)〉 +�́� |𝜑0(𝑟)〉́ …………(7) Multiply Eq.(7) by |𝜑0(𝑟)〉on the left, results. �́��𝜑°(𝑟)�𝜑°(𝑟)� = �𝜑°(𝑟)�𝐻𝑟𝑒𝑠�𝜑°(𝑟)� + �𝜑°(𝑟)�𝐻° − 𝐸°��̀�(𝑟)�…………(8) According to Eq.(6) the second term of Eq.(8) vanishes: When H0 is Hermition operator, then �́� = �𝜑°(𝑟)�𝐻𝑟𝑒𝑠�𝜑°(𝑟)�…………(9) The energy of state in Eq. (5) 𝐸 = 𝐸0 + �́� = �𝜑°(𝑟)�𝐻𝜊�𝜑°(𝑟)� + �𝜑°(𝑟)�𝐻𝑟𝑒𝑠�𝜑°(𝑟)�…………(10) = ∑ 𝒞𝑎𝑘𝐴𝑘=1 +�𝜑°(𝑟)�𝐻𝑟𝑒𝑠�𝜑°(𝑟)�…………(11) Where ∑ 𝒞𝑎𝑘𝐴𝑘=1 : is the contribution of the single particle energies and �𝜑°(𝑟)�𝐻𝑟𝑒𝑠�𝜑°(𝑟)� is the residual interaction. For shell model calculation one assume that a meaning full description of a nucleus can be made in term of an inert core of closed shell and extra nucleons in the orbit S which can not be occupied by core nucleus. Then the total binding energies are given by[8]. 𝐸𝑏 (core +𝜆2) =2 𝒞𝜆 + 𝐸𝜆 ` (𝜆2) + 𝐸𝐵.𝐸(𝑐𝑜𝑟𝑒)………… (12) Where 𝒞𝜆 is the single particle energies, 𝐸𝜆 ` is the residual interaction energies and 𝐸𝐵.𝐸(𝑐𝑜𝑟𝑒) is the energy of the core that assume. When we assume the modified surface delta interaction, (MSDI) is the best potential interaction of the two bodies, the residual interaction Energy Ѐλ is given by [8]. 𝐸𝜆 ̀ = �jajb�𝑉1,2 𝑀𝑆𝐷𝐼�jajb�𝐽𝑇=1 = −𝐴 (2ja+ 1)(2jb+ 1) 2(2𝐽+1)(1+𝛿𝑎𝑏) �jb − 1 2 ja 1 2 �𝐽0� 2 [1 + (−1) 𝑙𝑎+𝑙𝑏+𝐽 +𝑇 ] + 𝐵 + 𝐶…………(13) And the two bodies, the residual interaction energy with Reid potential is given by [9]. 𝐸𝜆 ̀ = �jajb�𝑉1,2 𝑀𝑆𝐷𝐼�jajb�𝐽𝑇=1 × ( 18 𝐴 ) 1 3 ………… (14) The excitation energy 𝐸𝐸𝑥𝑡 (𝑘) of k th excited state is followed from the binding energy of the nucleus in k state which took the results in respect to the ground state k0 binding energy given by[6]: 𝐸𝐸𝑥𝑡 (𝑘) = 𝐸𝐵.𝐸 (𝑘) − 𝐸𝐵.𝐸 (𝑘0) ………… (15) Results 227 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Shell model developments have been applied to evaluate the binding and excitation energies for Ni28 58 30 nuclei that are assumed to be describes by an inert closed shell core and two nucleons. The two neutrons occupy in the 2P3/2 1f5/2 configuration orbit. The model space 〈2p3/21f5/2〉 describe in the representation (J π ,T ) combination (0+,1) and (2,1+) for │2p3/2〉 , (4+,1) and (2+,1) for│2p3/2 1f5/2 〉 and (0 +,1) , (2+,1) and(4+,1) for│1f5/2〉 and allowed for the two neutrons in configuration space. The matrix element of two particles interaction are calculated by using Modified Surface Delta Interaction (MSDI) Eq.(13) with Matlab version (6.5), on the other hand, we calculated the matrix element depending on Reid potential with same program using expression and result of our calculation are listed in table(1) and table(2) respectively. The core binding energy can be calculated from Eq.(12) with mass number are : M( Ni28 58 30) = (57.942116)amu ,M(0n1)=(1.008665) amu and M(1p1)=(1.007276) amu [ 10]. The binding energies of the nucleus of each case for configuration ( Jπ ,T )values have been calculated with Eq.(13) with single particle energies e2P3/2 =-10.2549, e1f5/2 =-9.4356 and e2P1/2 =-9.1562 MeV [11], he tresults are tabulated in table(3) for MSDI and table (4) for RP respectively . Therefore, the excitation energies follow directly from the different values and evaluated with Eq.(15 ), the results are summarized in tables(5) and(6) for MSDI and RP alternatively . Briefly the single particle energies were taken from the observed spectrum for A=57-56 with a least squares fit for 2P3/2 , 1f5/2 and 2P1/2 orbits which are equal to (-10.254, - 9.4356,and -9.1562)Mev respectively[ 12 ] . Discussion It is shown that the sequence of the lower levels is well, but the level spacing is somewhat tool the effect of variation of the strength parameter for calculation. It clear that for the MSDI or RP theory will agree quite well for the excitation energy comparing with experimental. These work quite for first or second excited state but for higher spectral, they become very complicated because several nucleons can be excited simultaneously into super position of many different configurations to produce a given nuclear spin and parity. Figure (1), shows the low- lying state that MSDI has an agreement with the experimental than RP and the variance with experimental that similar with NZPSM, but at lower. While figure (2), shows a good agreement with ESM method while not good with NZPSM. The ground state configuration indicates that all the proton sub shell filled, and all the neutrons are in filled sub shell except for the last two, which are in sub shell on there own. There are many possibilities to consider for the excited state. One neutron of the 2P3/2 promote to 1f5/2 or 2P1/2 gives a configuration 2P3/2 , 1f5/2 or 1P3/2 , 2P1/2 promote one of 1P3/2 to others. All these possibilities would correspond to the smallest energy shift, so it should be founded over the others, the next excited state might involves moving the last neutron up to a farther level to(1f5/2 )or putting it back where it was and adapting configuration option (2P3/2 , 1f5/2) which is favored over (1P3/2 , 2P1/2 )because it keeps the excited neutrons paired with another. This should have a slightly lower energy than creating two unpaired protons when comparing these predictions, with the observed excited levels, it is found that the expected excited do exist. The influence of the single particles energies are viewed by comparing the calculated values from the NZPSM, ESM model with those obtained from calculations. This shows that 228 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 the binding and excitation energies of the low –lying level for A››30 are hardly affected by the inclusion of f5/2 orbit that the MSDI, RP, NZPSM and ESM which describes less than for 1p3/2 orbital and 2p3/2 orbital. Our results of the spectra that’s obtained from RP potential in figures (1-2), show that it is not good with experimental data but with a good agreement for the results that are obtained by using MSDI especially for low lying level. On the other hand, the expected spectra that gate from result is constructed from mixing state configuration. Conclusion The present contribution addresses the role of shell model calculation in low lying energy .We show that both the MSDI and RP are not present in experimental values exactly that lead to conclude MSDI and RP which are limited successfully describing for light nuclei ,and poor describe that mediate nucleus and heavy .Also the energies for binding and excitation spectra are produced from the cooperation of mixing state configuration ,and we expected that mixing calculation gave good results to describe these nucleus .On the other hand, the shell model calculation results depend on the single particles energies effect. Summarizing, we may conclude that for correlating the available experimental data, the present calculation with MSDI interaction is rather successful than RP. References 1.Martin, B.R.(2008)Nuclear and Particle Physics, John wiley &Sonc, Ltd published. 2.Lenzi1, S.M.and M.A.(2009), Test of Isospin Symmetry Alongthe N = Z Line ,Bentley2 Springer-Verlag Berlin Heidelberg,pp57-98. 3.Fossion R.C.Coster De. al, García-Ramos J.E.a,b, Werner T.a,c and Heyde,K.(2002),Nuclear Binding Energies: Global Collective Structure and local Shell- Model Correlations, Nuclear Physics A, 697 ,703–747. 4Alfredo, P.(2011),The Nuclear Shell Model: Past,Present and Future, Departamento de F´ısica Te´orica and IFT UAM-CSIC,Universidad Aut´onoma de Madrid, 28049, Madrid Spain. 5.Vesselin,G.G.(2002),Mixed-Symmetry Shell-Model Calculations in Nuclear Physics, Ph.D,thesis, Sofia University. 6. Kris, L. G. (1994), The Nuclear Shell Model , Book, Springer Verlag pub ,study edition. 7.Trun, B. (1999), Quantum Mechanics Concept and Application, Book , phys. Dep. State University New York . 8.Shatha,F.A. (2005), Calculation of Energy Levels for Nuclei (42Ca,42Ti,42Sc) by using Modified Surface Delta Interaction, M.Sc. Thesis, Kufa University. 9.Fiase,J.;Hamoudi,A.;Irvine,J.M. and Yazici, F.G. (1988),Nucl.Phys.,14. 10.Lilley, J. (2006 ),Nuclear Physics, John Wiley &Sons, Ltd. 11.Brussaard, P.J. and Blaudemans,P.W.M. (1977),Shell Model applications in nuclear Spectroscopy, North –Holland publishing Company. 12.Koops,J.E.and Gloudemans, P.W.M. (1977),Atoms and Nuclei, Z. phys. A280,181. Table No.(1): The calculated matrix element for modified surface delta interaction (MSDI) 229 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Ja Jb Jc Jd J T ‹ Ja Jb │VMSDI│ Ja Jb › 5/2 5/2 5/2 5/2 0 1 -3.0000 5/2 5/2 5/2 5/2 2 1 -0.6857 5/2 5/2 5/2 5/2 4 1 -0.2857 3/2 3/2 3/2 3/2 0 1 -2.0000 3/2 3/2 3/2 3/2 2 1 -0.4000 3/2 5/2 3/2 5/2 4 1 1.1429 3/2 5/2 3/2 5/2 2 1 0.3429 Table No. (2): The calculated matrix element for two bodies' matrix element using Raid potential (RP) Ja Jb Jc Jd J T ‹ Ja Jb │VMSDI│ Ja Jb › 5/2 5/2 5/2 5/2 0 1 -0.388 5/2 5/2 5/2 5/2 2 1 -0.0846 5/2 5/2 5/2 5/2 4 1 -1.307 3/2 3/2 3/2 3/2 0 1 -0.510 3/2 3/2 3/2 3/2 2 1 -0.136 3/2 5/2 3/2 5/2 4 1 1.123 3/2 5/2 3/2 5/2 2 1 0.0822 Table No.(3): The binding energy B.E (MeV) for ( 𝑵𝒊𝟐𝟖 𝟓𝟖 30) that is calculated with (MSDI) 230 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Configuration Jπ T Binding Energy (B.E)(MeV) 2P3/2 0+ 1 -504.9516345 2+ 1 -504.2621793 2P3/2 1f 5/2 4+ 1 -502.7778362 2+ 1 -503.1226638 1f 5/2 0+ 1 -503.744269 2+ 1 -502.7467259 4+ 1 -502.5743121 Table No. (4): The binding energy B.E(MeV) for ( 𝑵𝒊𝟐𝟖 𝟓𝟖 30) that is calculated with Raid potential (RP) Configuration Jπ T Binding Energy (B.E)(MeV) 2P3/2 0+ 1 -504.9088 2+ 1 -504.6054 2P3/2 1f 5/2 4+ 1 -502.5785 2+ 1 -503.6193 1f 5/2 0+ 1 -504.1892 2+ 1 -503.3922 4+ 1 -503.0182 Table No.(5): Results of the excitation energies EExt. (MeV) for 𝑵𝒊𝟐𝟖 𝟓𝟖 30 that are calculated with (MSDI) 231 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Configuration Jπ T Excitation Energies (Ex.E)(MeV) 2P3/2 0+ 1 0 2+ 1 -0.6896552 2P3/2 1f 5/2 4+ 1 -2.1739983 2+ 1 -1.88291707 1f 5/2 0+ 1 -1.2075655 2+ 1 -2.2051086 4+ 1 -2.377522431 Table No. (6): Results of the excitation energies EExt.(MeV) for 𝑵𝒊𝟐𝟖 𝟓𝟖 30 that are calculated by using Raid potential (RP) Configuration Jπ T Excitation Energies (Ex.E)(MeV) 2P3/2 0+ 1 0 2+ 1 -0.3034 2P3/2 1f 5/2 4+ 1 -2.3303 2+ 1 -1.2895 1f 5/2 0+ 1 -0.7196 2+ 1 -1.5166 4+ 1 -1.8906 232 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 0.5 1.5 2.5 3.5 0.0 1.0 2.0 3.0 4.0 En er gy (M eV ) 0+ 2+ 4+ 0+ EXP. MSDI RP NZPSM 0+ 2+4+ 0+ 2+4+ 2+ 4+ 0+ 0+ 4+ 2+ 2+ 2+ 2+ 4+ 0+ 0+ 2+ 4+ 2+ 0+ Fig.(1):Our results for spectra level energy for Ni-58 using MSDI and Rp. compared with experimental and NZPSM method. 233 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 0.5 1.5 2.5 3.5 0.0 1.0 2.0 3.0 4.0 En erg y ( Me V) EXP. MSDI RP ESM 0+0+ 0+ 0+ 2+ 2+ 2+ 2+ 2+ 2+ 0+ 0+ 4+ 4+ 4+ 4+ 4+4+ 2+ 2+ 2+ 0+ 0+ 2+ 2+ 2+ 4+ 4+ Fig.(2):The calculation of spectra energy level for Ni-58 compared with experimental and ESM method. 234 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 )لنواة واالثارةطاقات الربط ل ات النظریةحسابال 𝑵𝒊𝟐𝟖 𝟓𝟖 نموذج ا عمالستاب (30 ضطرابالقشرة ونظریة اال ھادي جبار مجبل ناز طلب جارهللا سمیرة أحمد أبراھیم رجاء فیصل ربیع جامعة بغداد /)أبن الھیثم(كلیة التربیة للعلوم الصرفة /قسم الفیزیاء 2013ایلول 24حث في قبل الب 2013ایار 13ستلم البحث في ا الخالصة ونظریة النووي موذج القشرةانللطاقات بأعتماد ات الدنیالمستویل نظریة لطاقات الربط واالثارةالحسابات ال عملتأست ) على أفتراض نواةالطاقات تمقی نموذجالكم. في ھذا األ 𝑁𝑖28 58 ، وھذه داخلي مع أثنین من النیوكلیونات الخارجیة قلب (30 . 2P3/2 , 1f5/2 and 2P1/2 شكیلضمن تالنیوكلیونات ھي لجسیمتیین من (RP) جھد ریدو (MSDI)تفاعل سطح دالتا المطور حساب قیم مصفوفةل بتالالمابرنامج عمالتم أست الحسابات النظریة مع النتائج العملیة، ورنتقجسیمة رید المنفردة. ةقعناصر المصفوفة وذلك لحساب الطاقات العملیة وطا ألنموذج القشرة المزدوج مع الدراسات النظریة آجید كان توافقال ولكن ،معھا آضعیف آالتي تبین توافق .) ESM طریقة طیف الطاقة(و، NZPSM)الالصفري( 𝑁𝑖28القشرة، نموذجأ، طاقات الربط واالثارة، تفاعل سطح دالتا المطور،جھد رید مفتاحیة:الكلمات ال 58 ،نظریة 30 .ضطراباال 235 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013