Approximate Solution of Bohr-Mottelson Hamiltonian with Minimal Length Effect for Hulthen Potential Using Asymptotic Iteration Method

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1 pprxate Slut f Bhr-Mttels Halta wth Mal Legth Effect fr Hulthe Ptetal Usg sypttc Iterat Methd Isa Lls Elvyat, 1 Supar, 1, ad C Car 1, 1 Physcs Departet, Graduate Prgra, Sebelas Maret Uversty, Jl. Ir. Suta 36 Ketga, Surakarta 5716, Idesa Physcs Departet, Faculty f Matheatcs ad Fudaetal Scece, Sebelas Maret Uversty, Jl. Ir. Suta 36 Ketga, Surakarta 5716, Idesa Eal: sa.elvyat@gal.c bstract. The apprxate slut f Bhr-Mttels Halta rgd defred ucleus case fr Hulthe ptetal wth al legth effect was vestgated usg sypttc Iterat Methd. sypttc Iterat Methd was used t slve apprxately the Bhr-Mttels Halta t bta eergy spectru ad u-ralzed wave fuct. The eergy spectru was calculated uercally usg the Matlab sftware. The u-ralzed wave fuct was expressed the Hypergeetrc ter. The results shwed that the eergy spectru creased due t the creasg al legth paraeter. The eergy spectru als creased by the creasg rage f ptetal. 1. Itrduct The cllectve dels f the ucleus are a terestg tpc ucleus area f study. The cllectve dels whch are the cbat f lqud drp del ad shell del [1] are used t descrbe the quadruple dyac cllectve the eve-eve ucleus []. The quadruple s the fr f the defred ucleus that crrespdg t the exctat eergy, ad t s used t descrbe rtatal ad vbratal f the ucleus [3]. The ucleus whch s csdered t be the rtat at lw exctat eergy s called as rgd defred ucleus [3]. Bhr ad Mttels explaed abut the rtat ad vbrat f the ucleus the cllectve dels f the ucleus by usg Bhr-Mttels Halta [3].The Bhr-Mttels Halta has descrbed the ucleus dels wth tw teral varables ad, ad three Euler agles,,. The crrespd t ucleus defrat ad crrespd t agle syetrc [,4]. Fr s the axally syetrc case [3,5], crrespd t the prlate defred ucleus [6] ad ccurs t the rgd defred ucleus [3,5]. Fr s the traxal syetrc case ad crrespd 6 t the blate [6]. The three Euler agles,, shw agles f the ucleus the cllectve del ucleus. I 11, The Bhr-Mttels Halta has bee vestgated by Batss et al. wth Davds ptetal fr axal syetrc [4]. The ext years, the Bhr-Mttels Halta have bee slved cludg Eckart ptetal [], Kratzer ptetal [7] ad Hulthe plus Rg shape [8]. The ethds that are used t slve Bhr-Mttels Halta are sypttc Iterat Methd (IM) [8], Nkfrv- Uvarv [] ad SUSYQM [4]. Heseberg Ucertaty Prcple descrbes cutat relats betwee pst ad etu peratrs. Whe Heseberg Ucertaty Prcple s flueced by quatu gravty, the ths effect causes the rse f al bservable dstace the scale Plack legth [5,9], ad t s well-kw as Geeral Ucertaty Prcple (GUP) r al legth [3,5,9]. The Bhr-Mttels Halta wth al legth effect has bee studed by Chabab et al. [5]. By trducg the ew wave fuct Bhr-

2 Mttels Halta, t s reduced t the secd rde dfferetal equat the t s slvable fr the case fte square well ptetal wth V [5]. I addt, l Mhaad ad Hassaabad slved the Bhr-Mttels Halta wth a al legth effect a dfferet way wth respect t Chabab et al., they used tw steps slut, where the frst step f the slut s fr the case wth zer al legth paraeter that gves usual eergy. Fr the frst step f the slut, t was btaed the express f Laplaca as a varable that ly a fuct f pst ad usual eergy spectru [3]. The secd step f the slut, the quadratc f Laplaca that s btaed fr the frst step s serted t the Bhr- Mttels Halta wth a al legth such that ths equat beces slvable fr fte square well ptetal [3]. I ths paper, we slved Bhr-Mttels Halta wth the al legth effect wth Hulthe ptetal fuct fr the case f the rgd defred ucleus as Ref [5]. Fr Hulthe ptetal fuct, we eed Bal expas apprxat fr ptetal part f Bhr-Mttels Halta. It s slved aalytcally usg sypttc Iterat Methd. The wrk s rgazed as fllw. I sect, the apprxate slut f Bhr-Mttels Halta wth al legth effect s brefly trduced. sypttc Iterat Methd s revewed sect 3. The result ad dscuss abut the eergy spectru ad u-ralzed wave fuct are preseted sect 4. Fally sect 5 cclus s preseted.. The apprxate slut f Bhr- Mttels Halta wth al legth effect Heseberg Ucertaty Prcple s expressed by [1,11], X, P (1) By csderg the effect quatu gravty whch s trduced as sall paraeter fr cutat relats betwee pst ad etu peratrs (1), s (1)beces, [1,11], X P P, 1 () I () s the dfcat f cutat relats betwee pst ad etu peratrs, t s called as Geeral Ucertaty Prcple (GUP). Fr (), that lead t gettg, Fr (3) ad (4), Xˆ xˆ ˆ 1 ˆ ˆ P p p (3) (4) s a al legth paraeter that has very sall pstve values ad X s pst peratr. P ad p are etu peratrs at hgh ad lw eergy, respectvely. The agtude f the p s expressed by p [5]. I quatu echacs, squared etu peratr s gve by ˆP (5) where s a Laplaca peratr. We substtute (5) (4) [], that yeld, P 1 B (6) Metu peratr whch s flueced by al legth s shw by (6). The cllectve geetrcal del f ucleus s expressed by [3], ds g dx dx (7), j j j I (7) s curvlear crdates, where x s curved space ad g s a etrc tesr. The axal syetry j case, ucleus has three degrees f freed: q, q, q, s the etrc tesr s gve 1 3 by [3,5], 3 s g j 3 (8) 1 where s a varable crrespdg t ucleus defrat, ad are part f Euler agles. The Laplaca peratr as fllws [3,5], 1 1 gg (9) j g q q where g ad atrx 1 g j, j j are deterat ad verse f the g j, respectvely. By usg (8), t s btaed deterat f the atrx g j as, 4 g 9 s ad the verse f the atrx 1 g j (1) g j as fllws, 1 3 s (11)

3 By applyg(1) ad (11) t (9), we get Laplaca peratr, s gve as s 3 s s Halta peratr s expressed by[3], B (1) P H T V V (13) where P s etu peratr, V s ptetal eergy fuct ad B s a ass paraeter. We serted (6) ad (1) (13), s t btaed, 4 V,, E,, (14) B B Bhr-Mttels Halta al legth effect s expressed (14). T slve (14) s used the ew wave fuct [5] s gve by,,, 1,, (15) By substtuted (15)ad 1(atural ut) [3] (14),t s get, B E V,,,, (16) 14 B E V,, The slut f (16) [5] wll be btaed aalytcally by usg bal expas t apprxat the deteratr f the secd ter f (16), s (16) beces, E V,, B,, 14 B E V,, I (17), we have set (17). The separat varable ethd s used t slve (17) by settg,, R, s we have Euler agles part f Bhr-Mttels Halta wth al legth, (18) s s s ad part f Bhr-Mttels Halta wth al legth, 1 R (19) B E V R R 3 8B E EV V ( ) R Fr the case ptetal fuct rgd defred ucleus, we use Halta wth al legth. part f Bhr-Mttels s cstat f separat varable whch s crrespdg t agular etu quatu uber. By applyg R ad LL 1 U LL 1, s we have, d U U d 3 B E V U () 8B E EV V U The apprxate equat f Bhr-Mttels Halta fr a part a al legth effect fr rgddefred ucleuscase s shw by (). 3. sypttc Iterat Methd sypttc Iterat Methd s ethd t slve the secd rder dfferetal equat ter [1,13], where, y t t y t st y t t ad s (1) t are the ceffcet f a dfferetal equat ad s a quatu uber. T bta slut, we dervatve (1), s we bta, where, y t t y t s t y t z1 z1 z1 t t s t t t z z1 z1 z1 s t s t s t t z z1 z1 () (3) (4) z 1,,3,... (5) The egevalue s btaed fr the quatzat cdt whch s gve by, t ts t ts t z z z1 z1 z (6) T bta the wave fuct, (1) s reduced t the frals, as fllws, N 1 at t 1 y t 1 N bt t y t (7) N wt y N t 1 bt I (7) s e-desal Schrdger lke equat that s reduced t a hypergeetrc type dfferetal equat. The asscated egefuct s btaed fr the slut f (7), s gve as, N 1 1, ; ; y t C N F bt (8) where t N 3, N, N t 1 b a (9) b C s ralzat cstat ad F 1 s a hypergeetrc fuct. The u-ralzed wave

4 fuct f Bhr-Mttels Halta s btaed by usg (7)-(9) [14,15]. 4. Result ad Dscuss The Hulthe ptetal s shrt rage ptetal physcs, t s used uclear, partcle physcs, ad atc physcs [16,17]. The Hulthe ptetal s gve by [18,19], e V( ) V 1 e where ad (3) V are a rage ad cstat ptetal, respectvely. Fr the case, atc ucleus V s defed as Z e. The Z s a atc uber ad e s a charge f the electr []. T get sple slut, (3) was chaged hyperblc trgetrc ter [1], s gve as, Ze V cth 1 (31) By settg the cetrfugal apprxate 1 [4] ad serted (31)t (), sh we btaed, L L1 1 B V 3 sh du 8B V E 4B V cth U d BV BE BV 8B E 8B V E 4B V By settg, L L1 vv 1 B V 3 (3) (33) q 8B V E 4B V B V (34) B E B V 8B E k 8B V E 4B V (3), the we gt, d U vv 1 (35) q cth k U (36) d sh The dfferetal equat lke Schrdger equat fr Mag Rse ptetal [] was shw by (36). I (36) ust be reduced t hypergeetrc type by usg the sutable varable chage cth 1 z, yeld, d U du z 1 z 1 z dz dz 4 4 v v 1 U 4z 41 z By settg, (37) q k 4 ad qk 4 (38) vv 1 v v 1 3 (39) where, ad v are the hypergeetrc paraeter. I (37) s teredate f the hypergeetrc dfferetal equat ad by trducg the ew wave fuct as, we btaed, 1 U z z g z (4) z g z1 z dz g z 1 z dz v v 1 1g z By dvdg (41) wth z1 z IM type equat, as 1 1 (41), the t was reduced t dz z z dz v v 1 1 g z z1 z By cparg ()ad (4), we had g z z g z 1 1 z 1 z 1 v v 1 (4) (43) z (44) s 1 v v 1 1 z The egevalue was btaed by usg (6)-(7) ad(41)-(44), s gve as, 1 1 v v (45) By usg (38)-(39) ad (45), we btaed the eergy spectru equat f Bhr-Mttels Halta wth the al legth effect s gve as, 1 E B 1 v v 1 E v v 1 wth, 1 1 L L1 1 v B V 3 4 8B V E 4B V B V (46) (47) (48) V E 4B 4 E B V E B V (49) The eergy spectru f Bhr-Mttels Halta wth al legth effect whch was btaed by

5 usg apprxate slut was expressed by (46). The s the quatu uber f the ucleus. The = was eergy level f grud state bad ad =1 was eergy level f frst exted. I the case, we used = ad =1 fr rtat cdt f the ucleus. O the ther had, fr the level eergy wth >1, the ucleus was the vbrat cdt [5]. Ths paper, B was defed as the su f prts ass ad eutrs ass. T get the eergy spectru uercally, Matlab sftware was used. The easureets were preseted usg atural uts. The uercal result f eergy spectru was shw Table 1 fr = ad L= wthut a al legth paraeter ad the presece f al legth paraeter fr se stpes. TBLE 1. The eergy spectru fr se stpes wth e=.8544 ad.1. Istpes E (MeV) Ru Pd Cd Xe Ba Nd S Dy Os Pt Fr the Table 1, t was shw that the eergy spectru was egatve [17-19] wthut the al legth effect, whle by the presece f a al legth effect, the eergy spectru becae pstve due t rasg the V factr (46). The stpe Pt had the hghest eergy spectru value wth ad wthut al legth paraeters. It was caused, Pt had greater stpes ass tha the thers stpes. The uercal result f eergy spectru was shw Table fr Hulthe ptetal as a fuct f rage ptetal fr se stpes. TBLE. The eergy spectru fr se stpes wth varus f rage ptetal, L=, ad e= Istpes E (MeV) Ru Pd Cd Xe Ba Nd S Dy Os Pt Table shwed the eergy spectru creased by the crease f the rage ptetal value. The stpe Pt had the hghest eergy spectru value wth the varus rage ptetal. It was caused, Pt had greater stpes ass tha the thers stpes. The uercal result f eergy spectru level fr = wth the varus stpes ad al legth paraeter was shw Fgure 1.

6 FIGURE 1. The eergy spectru fr L= wth varus al legth paraeter fr se stpes. Fgure 1 shwed eergy spectru f se stpes fr = wth varus al legth paraeter. The eergy spectru f se stpes creases fr the creasg f a al legth paraeter. Fr cstat al legth paraeter, the eergy spectru creased, t s caused by the creasg the stpes. The stpe whch had the greatest ass had the hghest eergy spectru. It was shwed by Pt whch had hghest eergy spectru. Ths result was agreeet wth the result Ref [3,5]. The result ref [3,5] shwed the al legth effect creased, t caused the creasg spectra eergy [3,5]. The geeral u-ralzed wave fuct f Bhr- Mttels Halta was btaed by applyg (6)- (8)ad (4), we get 1 C 1 1 U z (5) F1, 1, 1, z By substtutg (5) t (4)ad tgether applyg trasfrat varable f cth 1 z btaed, U 1cth 1cth C, we (51) 1cth 1cth C 1 (5) 1 cth U The u-ralzed wave fuct f = ad =1 are shw by (51) ad (5), respectvely. The wave fuct apltude fr se stpes depeds the value ad. 5. Cclus We vestgated apprxate slut f Bhr- Mttels Halta fr Hulthe ptetal wth al legth effect the case f the rgd defred ucleus. The apprxate slut f Bhr-Mttels Halta wth al legth effect was slved by usg sypttc Iterat Methd t bta eergy spectru ad u-ralzed wave fuct. The results shwed that the eergy spectru creased by the creasg f al legth effect value fr se stpes. The eergy spectru creased due t the creasg the rage ptetal fr se stpes. The greatest ass stpes had the hghest eergy spectru. ckwledgeet Ths research was partly supprted by Sebelas Maret Uversty Hgher Educat Prject Grat Hbah Peelta Berbass Kpetes 18. Refereces [1] W. Greer ad.j. Maruh, Nuclear Mdel, Sprger, New Yrk, 1996.

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