Image reconstruction techniques applied to nuclear mass models

PHYSICAL REVIEW C 81, 024304 (2010) Image econs uc ion echniques applied o nuclea mass models I ing O. Mo ales,1P. Van Isacke ,2V. Velazquez,3J. Ba ea,4J. Mendoza-Temis,1J. C. L´ opez Viey a,1 J. G. Hi sch,1and A. F ank1 1Ins i u o de Ciencias Nuclea es, Uni e sidad Nacional Au ´ onoma de M´ exico, 04510 M´ exico, D.F., Mexico 2G and Acc´ el´ e a eu Na ional d’Ions Lou ds, CEA/DSM–CNRS/IN2P3, Boˆ ı e Pos ale 55027, F-14076 Caen Cedex 5, F ance 3Facul ad de Ciencias, Uni e sidad Nacional Au ´ onoma de M´ exico, 04510 M´ exico, D.F., Mexico 4Ins i u o de Es uc u a de la Ma e ia, Consejo Supe io de In es igaciones Cien ´ ı icas, Unidad Asociada al Depa amen o de F´ ısica A ´ omica, Molecula y Nuclea , Facul ad de F´ ısica, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain (Recei ed 14 Ap il 2009; e ised manusc ip ecei ed 11 Janua y 2010; published 5 Feb ua y 2010) A new p ocedu e is p esen ed ha combines well-known nuclea models wi h image econs uc ion echniques. A colo -coded image is buil by aking he di e ences be ween measu ed masses and he p edic ions gi en by he di e en heo e ical models. This image is iewed as pa o a la ge a ay in he (N,Z) plane, whe e unknown nuclea masses a e hidden, co e ed by a “mask.” We apply a sui ably adap ed decon olu ion algo i hm, used in as onomical obse a ions, o “open he window” and see he es o he pa e n. We show ha i is possible o imp o e signi ican ly mass p edic ions in egions no oo a om measu ed nuclea masses. DOI: 10.1103/PhysRe C.81.024304 PACS numbe (s): 21.10.D , 42.30.Wb I. INTRODUCTION Nuclei a e pa icula ly complex sys ems, anging om a single p o on o mo e han h ee hund ed nucleons; oo la ge o a de ailed mic oscopic ea men bu oo small o s a is ical me hods. In e es in hem is no es ic ed o nuclea physicis s because he a omic nuclei ha cons i u e ou wo ld a e manu ac u ed inside s a s, de ining hei e olu ion and a e. A guably, he mos basic p ope y o a nucleus is i s mass. Unde s anding nuclea masses p o ides a es o ou basic knowledge o nuclea s uc u e and is an essen ial ing edien o he undamen al as ophysical p oblem o nucleosyn hesis, which o en akes place in a - om-s abili y condi ions, on ul asho ime scales [1]. Though g ea p og ess has been made in measu ing he mass o sho -li ed nuclei ha a e a om he egion o s able, na u ally occu ing iso opes, heo y is needed o p edic hei p ope ies and guide expe imen s ha sea ch, o example, o egions o inc eased s abili y [2]. Despi e he e o s in es ed in de eloping echniques ha can accu a ely desc ibe hese masses, he p edic ions made by di e en models o en u n ou o be disconce ingly di e en , e en in egions close o known masses. Reliable heo e ical models and me hodologies ha can p edic he mass and o he p ope ies o hese “exo ic” nuclei a e s ill missing [3]. The e o s o calcula e nuclea masses ha e been hampe ed by he absence o a ue e ec i e heo y o he nuclea in e ac ion and by he di icul ies inhe en o quan um many- body calcula ions. Ins ead, simpli ied app oaches o model he a omic nucleus ha e been de ised. The co ne s one o nuclea mass models is he semi-empi ical mass o mula p oposed by on Weizs¨ acke [4] and Be he and Bache [5] in he 1930s, which is based on a pic u e o he nucleus as a uni o m, e y dense, cha ged liquid d op— he so-called liquid-d op model (LDM)—which gi es an accu a e o e all desc ip ion o nuclea masses, bu lacks some quan um-mechanical e ec s due o shell closu es, pai ing, quad upole co ela ions, e c. Thus, la ge de ia ions om he measu ed masses a e obse ed in ce ain egions o he nuclea cha . The inclusion o hese e ec s usually equi es o he ea men s, like he ini e- ange d ople model [6], which combines mac oscopic e ec s wi h mic oscopic shell and pai ing co ec ions, he model o Du lo and Zuke [7], and he Ha ee-Fock app oach [8]. In gene al, howe e , i is di icul o ma ch heo y and expe imen ( o all known nuclei) wi h an a e age p ecision be e han abou 0.5 MeV [3]. Mo e oublesome is he ac ha di e en model p edic ions end o di e ge om each o he o egions o nuclei wi h unknown mass. II. SYSTEMATIC PATTERNS The s a ing poin in he p oposed app oach is he s iking colo pa e ns in he (N,Z) plane shown in Fig. 1, which esul a e aking he di e ence be ween he expe imen ally known masses [9] and p edic ions o se e al models: an imp o ed mac oscopic LDM [10], an upg ade o he LDM by he inclu- sion o schema ic mic oscopic shell co ec ions [10] (LDMM), he 31-pa ame e Du lo-Zuke (DZ) model [7], and he Ga ey- Kelson (GK) ela ions [11]. Fo each model, we obse e a esidual sys ema ic pa e n ela ed o he physics and ha is no included in he model. Thus, o example, he LDM does no con ain mic oscopic in o ma ion on independen -pa icle mo- ion, and consequen ly he pa e n shows sys ema ic de ia ions ela ed o shell closu es, nuclea de o ma ions, and esidual in e ac ions in a compelling g aphic o m [see Fig. 1(a)]. The mic oscopic beha io can be pa ame ized by he inclusion o a schema ic mic oscopic shell co ec ion [10]: BELDMM(N,Z)≡BELDM(N,Z) −a Fmax +a FF max +acons ,(1) whe e BELDM is he nuclea binding ene gy as ob ained in he LDM and Fmax =nν+nπ 2−nν+nπ 2, FF max =nν+nπ 22 −nν+nπ 22, 0556-2813/2010/81(2)/024304(9) 024304-1 ©2010 The Ame ican Physical Socie y IRVING O. MORALES e al. PHYSICAL REVIEW C 81, 024304 (2010) FIG. 1. (Colo online) Di e ences be ween expe imen al masses and hose calcula ed wi h he ou models: (a) he liquid-d op model (LDM), (b) he liquid-d op model wi h schema ic shell co ec ion (LDMM), (c) he Du lo-Zuke model (DZ), and (d) he Ga ey-Kelson ela ions (GK). a e linea and quad a ic e ms in he numbe s o alence neu on nν(p o on nπ) pa icles o holes, coun ed om he nea es closed shell, and he quan i ies in b acke s a e hei a e age alues. The cons an acons , used o e-cen e he p edic ion a e age (see Re . [10]), and a and a a e pa ame e s ound by i ing he expe imen al masses. Al hough his pa ame iza ion s ill leads o la ge e o s, i is able o schema ically ep oduce he main ends obse ed in he mass su ace nea he shell closu es. These e ms a e simila o he ones used in pa ame iza ions p oposed by R. F. Cas en [12]. The inclusion o hese e ms educes he ms de ia ion bu some sys ema ic pa e ns s ill appea , as is shown in Fig. 1(b) (no e he change o scale). Finally, i we ake a mo e complex app oach such as he DZ model, we s ill obse e small bu sys ema ically co ela ed e o s [Fig. 1(c)]. The emaining pa e ns sugges physics ha ha e no been inco po a ed in o he models. Figu e 1(d) displays he di e - ence be ween expe imen al masses and masses ob ained om he GK ela ions [11]. The emaining de ia ions a e, in his case, close o whi e noise, which shows ha i is in p inciple possible o p edic masses o a signi ican ly be e p ecision and, he e o e, ha i is wo hwhile o a emp o ake in o accoun he missing physical e ec s (see, howe e , Re . [13]). III. IMAGE RECONSTRUCTION The pa e ns shown in Fig. 1sugges an app oach o mass p edic ion based on image econs uc ion echniques. The basic idea is o conside ha he mo e han 2000 di e ences be ween he di e en models and he known nuclea masses ep esen pa ial iews o a la ge image a ay in he (N,Z) plane, and ha all o he mass di e ences (pe haps a ound 7000 in numbe ) ha exis be ween he neu on and p o on d ip lines emain hidden, co e ed by a “mask.” Thus, he ques ion is whe he we can “open he window” o un eil he emaining pa e n, o a leas pa o i . We show he e ha indeed his in o ma ion can be eliably unco e ed, a leas o egions no oo a om measu ed masses, wi h image econs uc ion echniques. The success o he app oach s ongly depends upon whe he he physical ea u es associa ed wi h he changing pa e ns in he nuclea landscape can be coded in e ms o (global) egula i ies in he (N,Z) plane, and whe he hey can in u n be modeled by a ini e numbe o ha monic componen s. An impo an assump ion in his app oach is ha he obse ed egula i y and ela i e smoo hness o he mass-di e ence landscape emains h oughou [14]. Al hough subs an ial changes in mass di e ences may occu in a a he sho in e al (e.g., because o shell s uc u e o shape ansi ions), he e a e no sudden jumps om one nucleus o ano he . Figu e 1(a) shows ha , o a good app oxima ion, he landscape can be conside ed o a y smoo hly as a unc ion o nucleon numbe . Mo e p ecise checks o his assump ion can be made, o example by es ing whe he he GK ela ions (which a e a measu e o smoo hness) become inc easingly inaccu a e o p edic ions u he away om s able nuclei. Ou es s indica e ha his is no he case— hey emain equally accu a e, e en when app oaching a eas close o he d ip lines [15]. We a e hus mo i a ed o ollow his app oach u he . 024304-2 IMAGE RECONSTRUCTION TECHNIQUES APPLIED TO ... PHYSICAL REVIEW C 81, 024304 (2010) We display he mass-di e ence able by a wo-dimensional a ay o pixels, wi h Nin he ho izon al posi ion and Zin he e ical one. Di e ences be ween expe imen al masses mexp (N,Z) and calcula ed masses m h(N,Z) de ine he colo image unc ion i(N,Z)≡m h(N,Z)−mexp (N,Z).(2) The ela ionship be ween i(N,Z) and he ull pa e n m(N,Z) is hen gi en by i(N,Z)=m(N,Z)·w(N,Z),(3) whe e w(N,Z) is a bina y mask unc ion, aking he alue 1 o posi ions (N,Z) whe e nuclea masses a e known, and 0 o he wise (al hough a “weigh ed” mask may be used ins ead— see Re . [16]). We hus need o ex ac m(N,Z) om Eq. (3). I I(kN,k Z), M(kN,k Z),and W(kN,k Z) a e he Fou ie ans o ms o i(N,Z), m(N,Z), and w(N,Z), espec i ely, hen I(kN,k Z)=M(kN,k Z)∗W(kN,K Z),(4) whe e M∗Wis he con olu ion o he unc ions Mand W. Since bo h i(N,Z) and w(N,Z) a e known o he en i e domain, hei Fou ie ans o ms I(kN,k Z) and W(kN,k Z) can be e alua ed di ec ly. The p oblem is na owed down o ob aining he unc ion M(kN,k Z), om which m(N,Z) can be eco e ed by applying an in e se Fou ie ans o m. Fo mally, his is a decon olu ion p oblem. Decon olu ion is non i ial and may lead o non-unique solu ions, bu he e exis se e al algo i hms, such as he CLEAN me hod, o en used in adio as onomy [17], and he maximum en opy me hod [18], which p o ide es able me hodologies. We ha e chosen a specially adap ed e sion o he CLEAN algo i hm used in he econs uc ion o ex u e pa ches [19]. The main assump ion o he CLEAN me hod is ha he emaining pa e ns shown in Fig. 1can be modeled by a ini e numbe o ha monic componen s. The Fou ie spec um o he known da a I(kN,k Z) is iewed as a co up e sion o he Fou ie spec um o he comple e da a M(kN,k Z). This co up ion is due o he mask and he main goal is o emo e his noise and cons uc a clean Fou ie spec um, choosing hose componen s ha bes explain he obse ed pa e ns in he image. The unco up ed e sion o he spec um is made o componen s such as T(kN,k Z)=aδ(kN−u, kZ− )+a∗δ(kN+u, kZ+ ), (5) whe e ais he complex ampli ude o he componen and (u, ) is he posi ion in Fou ie space. The e ec o he mask on his componen is I(kN,k Z)=T(kN,k Z)∗W(kN,K Z),(6) and by expanding he con olu ion we ob ain I(kN,k Z)=aW(kN−u, kZ− )+a∗W(kN+u, kZ+ ), (7) so ha his single componen becomes dis o ed. The o m o he mask gua an ees ha he dominan peak o i s Fou ie spec um is he DC o cons an e m (u, =0) and, because o his, he posi ion emains unal e ed, only he ampli ude is changed due o he mask, and he co up ion o he mask appea s as addi ional smalle peaks a ound he componen . To achie e he decon olu ion i is necessa y o de e mine he na u e o he o iginal componen , namely he ue ampli ude and posi ion (a,u, ) gi en he co up ed Fou ie spec um I(kN,k Z) and he Fou ie spec um o he mask M(kN,k Z). Because he posi ion o he ue componen s emains he same and he spu ious peaks a e smalle han he p incipal ones, we can ind he p incipal componen o he da a by loca ing he majo pai in I(kN,k Z). Subs i u ion o kN=u and kZ= in Eq. (7)gi es I(u, )=aW(0,0) +a∗W(2u, 2 ),(8) which we can ea ange using he conjuga e in o a=I(u, )W(0,0) −I∗(u, )W(2u, 2 ) W(0,0)2−W(2u, 2 )W∗(2u, 2 ).(9) Thus, we can eco e he ue ampli ude o he o iginal ha monic componen T(kN,k Z) and he ue na u e o he componen is hus ob ained o bo h he posi ion and ampli ude. The CLEAN algo i hm essen ially p oceeds by sequen ially inding he equency posi ion o he peak wi h maximal in ensi y in he co up ed Fou ie spec um. Once his peak is loca ed, he ue ampli ude is calcula ed wi h Eq. (9) and a clean e sion o he Fou ie spec um is upda ed. This clean spec um is cons uc ed placing he componen in he ue equency and wi h he co ec ampli ude be o e p oceeding o he nex i e a ion. A new e sion o he co up ed spec um is calcula ed elimina ing he emo ed componen and he e ec s p oduced on i by he mask. The new co up ed spec um is called he esidual spec um. I Ri(kN,k Z)is hei h esidual and R0(kN,k Z)= I(kN,k Z), he CLEAN algo i hm consis s o he ollowing s eps: (i) Loca e he posi ion (ui, i)o hei h clean componen om he maximum o Ri−1(kN,k Z). (ii) Calcula e he ue ampli ude wi h Eq. (9). (iii) Gene a e he i h esidual spec um Ri(kN,k Z)=Ri−1(kN,k Z) −(aiW(kN−ui,k Z− i) +a∗W(kN+ui,k Z+ i)).(10) This p ocedu e is epea ed un il a s opping c i e ion is eached a i e a ion K, a e which he clean spec um C(kN,k Z)is cons uc ed: C(kN,k Z)= K  i=1 (aiδ(kN−ui,k Z− i) +ai∗δ(kN+ui,k Z+ i)).(11) Once he clean spec um is cons uc ed, he concealed image is ob ained by applying he in e se Fou ie ans o m o i . 024304-3 IRVING O. MORALES e al. PHYSICAL REVIEW C 81, 024304 (2010) IV. MASS PREDICTIONS Wi h he algo i hm desc ibed p e iously i is possible o ex apola e he emaining pa e ns shown by he mass di e ences m =mexp −m h. Once his ex apola ed pa e n is ob ained, i is hen possible o p edic he nuclea masses by adding he mass p edic ed by he model: m(N,Z)=mex apola ed(N,Z)+m h(N,Z).(12) I is in his sense ha he CLEAN algo i hm is able o imp o e on he model used o calcula e he mass di e ences. In o de o p o ide a measu e o he deg ee o imp o emen o he model used o he mass di e ences, wo es s ha e been applied. In he i s we ake he se o measu ed masses wi h N⩾28,Z⩾28 om he A omic Mass E alua ion 2003 (AME03) compila ion o Audi e al. [9] and di ide i in o wo subse s. We use he 1454 masses p esen in he p e ious AME95 compila ion [20] as inpu o gene a e he pa e n (by i ing he mass model’s pa ame e s using his se ), and hen we measu e he p edic abili y o he model by calcula ing mass di e ences o he o he 371 expe imen al masses [21]. These p edic ions can subsequen ly be compa ed wi h he esul s ob ained by applying CLEAN o he same inpu subse , as we explain below. We e e o his as he “AME95-03 es .” This es has been ex ensi ely used be o e o measu e he p edic i e powe o nuclea mass models [3]. The second es consis s o p edic ing he mass di e ences o 301 nuclei a he bo de o he AME03-measu ed landscape [21], ollowing he same p ocedu e as in he i s es . We ha e applied bo h es s o h ee models wi h di e en deg ees o accu acy. The i s is a mac oscopic LDM wi h 7 pa ame e s (Eq. (2) in Re . [10]) ha a e de e mined by i ing (a) he (upda ed) AME95 da a o he AME95-03 es , o (b) he AME03 da a wi hou he bo de o he AME03-bo de es . The second model is an upg aded e sion o his mac oscopic LDM, deno ed by LDMM, and desc ibed in Sec. II. Finally, he hi d mass model used o he econs uc ion is he 31-pa ame e DZ model [7]. In hese es s he CLEAN me hod is i e a ed un il a ms de ia ion o 100 keV is achie ed o he inpu da a i . Table I shows a compa ison o he ms de ia ions o he AME95-03 es , and hose ob ained wi h he CLEAN me hod o each model. Table II gi es a simila compa ison o he AME03- bo de es . In he AME95-03 es he la ges CLEAN imp o emen (∼62% ms educ ion) is ob ained o he LDMM o Eq. (1). The imp o ed ms, su p isingly, is compa able wi h he co esponding ms in he DZ model (see Table I). Fo he simple LDM (Eq. (2) in Re . [10]) we also ind a la ge ms educ ion o ∼54%, whe eas o he DZ model we ind i TABLE I. AME95-03 es : P edic abili y o he se o nuclei in AME03 bu no in AME95, es ic ed o N,Z ⩾28. Model RMS RMS (wi h CLEAN) LDM, Eq. (2)in[10] 1.9307 MeV 0.8763 MeV LDMM, Eq. (1) 0.9955 MeV 0.3718 MeV DZ model [7] 0.3348 MeV 0.2727 MeV TABLE II. AME03-bo de es : P edic abili y o he se o nuclei in he bo de o AME03, es ic ed o N,Z ⩾28. Model RMS RMS (wi h CLEAN) LDM, Eq. (2)in[10] 2.7763 MeV 0.9168 MeV LDMM, Eq. (1) 1.9804 MeV 0.9333 MeV DZ model [7] 0.4039 MeV 0.3133 MeV o be ∼20%. The la e , smalle imp o emen is expec ed, because his model is al eady in e y good ag eemen wi h he expe imen al masses. Ne e heless, his ms educ ion ep esen s a signi ican co ec ion. A simila si ua ion occu s o he AME03-bo de es . The la ges imp o emen (∼66% ms educ ion) occu s o he mac oscopic LDM, ollowed by LDMM o Eq. (1) wi h a ∼52% ms educ ion. Again, as in he AME95-03 es , he ms in he DZ model dec eases by ∼20%. These esul s sugges ha he CLEAN me hod can e icien ly inco po a e he esidual pa e ns obse ed in he expe imen al masses. The ms, howe e , being an a e age measu e, is no a su icien ly clea gauge. The e o e, we now u n o an ampli ied iew o ou esul s, using wo-neu on sepa a ion ene gies S2n. Two-neu on sepa a ion ene gies, S2n(N,Z)≡BE(N, Z)−BE(N−2,Z), con ain de ailed in o ma ion abou nu- clea s uc u e e ec s. In Figs. 2,3, and 4we plo S2n o iso ope se ies in he N∼78–128 egion o he AME95-03 es using he LDM, he LDMM o Eq. (1), and he DZ model, espec i ely. The p edic ions o he mac oscopic LDM, as shown in Fig. 2, a e comple ely la , wi h no s uc u e a all. In con as , he da a display s ong a ia ions a he magic numbe s N=82 and 126, and subs uc u es nea N∼90. A e econs uc ion, hose s uc u es a e well desc ibed (wi h an ms o ∼0.1MeV) in he i ed egion (AME95 da a). On he o he hand, he benchma k AME03 da a a e also su p isingly well p edic ed (wi h an ms o 0.8763 MeV—see Table I). Al hough he shell s uc u es a e well desc ibed by he CLEAN algo i hm, he econs uc ion is no su icien ly cons ained. This is illus a ed by he p esence o spu ious subs uc u es ha , al hough small in magni ude, a e no seen in he S2n da a. To minimize such spu ious e ec s, i is necessa y o impose addi ional cons ain s, and his can be achie ed, o example, by schema ically including shell co ec ion e ms as is done in he LDMM o Eq. (1). Figu e 3shows S2n in he N∼78–128 egion o he AME95-03 es , ob ained wi h he LDMM [iso opic lines in Fig. 3(a)], and he esul s ob ained a e applying he econs uc ion algo i hm [iso opic lines in Fig. 3(b)]. In he LDMM, he magic numbe s N=82 and 126 a e inco po a ed in o he model om he beginning. The expe imen al S2n in hose egions displays discon inui ies ha a e well desc ibed by he LDMM. Howe e , he subs uc u e obse ed in he da a a N∼90 is no p ope ly accoun ed o [see Fig. 3(a)]. A e applying he econs uc ion o LDMM, he subs uc u e is co ec ly desc ibed (wi h an ms o ∼0.1 MeV) in he i ed egion. I is ema kable ha he egion o he benchma k AME03 da a (g een bulle s in Fig. 3) is also accu a ely 024304-4 IMAGE RECONSTRUCTION TECHNIQUES APPLIED TO ... PHYSICAL REVIEW C 81, 024304 (2010) 80 90 100 110 120 12 14 16 18 20 22 N S2NMeV (a) 80 90 100 110 120 12 14 16 18 20 22 N S2NMeV (b) FIG. 2. (Colo online) AME95-03 es : Two-neu on sepa a ion ene gies S2n iso opic lines (in blue) p edic ed by (a) he LDM and (b) i s imp o emen using he CLEAN econs uc ion. Red bulle s indica e he inpu ( i ed) da a and g een bulle s indica e he benchma k (p edic ed) da a. p edic ed (wi h an ms o 0.3718 MeV—see Table I). The cons ain s imposed by he schema ic shell-co ec ion e ms in he LDMM o Eq. (1) a e hus su icien o emo e he spu ious subs uc u e obse ed in he LDM econs uc ion. The esul s o S2n ob ained wi h he DZ model in he N∼ 78–128 egion o he AME95-03 es , and i s imp o emen by he image econs uc ion me hod, a e shown in Figs. 4(a) and 4(b), espec i ely. In his case, he DZ model shows he p es- ence o he subs uc u e a N∼90,bu he expe imen al S2n a e no accu a ely desc ibed. Fu he mo e, he iso opic lines become la ou side he egion whe e measu emen s a e a ail- able. The DZ model has an ms o 0.3384 MeV (see Table I), gi ing an excellen o e all desc ip ion o S2n in he p edic ed egion. Ne e heless, he econs uc ion me hod can be used o imp o e he desc ip ion by a signi ican ∼20%. Figu es 2,3, and 4show ha he main e ec o applying he CLEAN algo i hm o he p edic ions o he a ious models is o add “ ex u e” o he mass su ace, which co esponds o some o he physical e ec s no included in he model. We ha e made simila es s in he N∼110–160 egion o he AME03-bo de case, using he S2n ob ained wi h he LDM, LDMM, and DZ models, and he co esponding calcula ions using he image econs uc ion me hod. The imp o emen s as a esul o he CLEAN p ocedu e a e o he same quali y as o he AME95-03 es . 80 90 100 110 120 12 14 16 18 20 22 S2N(MeV) 80 90 100 110 120 12 14 16 18 20 22 N N S2N(MeV) (a) (b) FIG. 3. (Colo online) AME95-03 es : Two-neu on sepa a ion ene gies S2n iso opic lines (in blue) p edic ed by (a) he LDMM and (b) i s imp o emen using he CLEAN econs uc ion. Red bulle s indica e he inpu ( i ed) da a and g een bulle s indica e he benchma k (p edic ed) da a. The mass p edic ions ob ained applying he CLEAN algo i hm o he DZ model a e compa able o he bes a ailable global mass p edic ions. Howe e , i is o in e es o also compa e hem wi h he masses ecommended by Audi e al. [20], which a e p edic ed using he sys ema ic ends o he mass su ace and i s de i a i es. This me hod p o ides he bes sho - ange mass ex apola ions [3], which ha e been published o se s o h ee o ou nuclides in he neighbo hood o hose wi h measu ed masses in he A omic Mass E alua ions [9,20]. These p edic ions a e pe o med nucleus by nucleus, combining a ( a he elabo a e) g aphical analysis wi h ele an physical in o ma ion [9]. This p ocedu e leads o an ms o 0.1615 MeV o he p edic ions co e- sponding o he AME95-03 es . Since hese ex apola ions a e mo e accu a e han he DZ+CLEAN p edic ions, we expec ha he masses ob ained applying he CLEAN algo i hm o he DZ model should be close o he p edic ions o Audi e al. han he DZ masses. In o de o es his we ha e calcula ed he ms de ia ion be ween he masses ex apola ed by Audi e al. and he p edic ions o he DZ model wi h and wi hou CLEAN. Fo he DZ model we ob ain an ms o 0.3305 MeV while o he DZ+CLEAN p edic ions we ge an ms o 0.2694 MeV, which is signi ican ly smalle . This shows ha he DZ p edic ions a e indeed imp o ed using he CLEAN algo i hm. 024304-5 IRVING O. MORALES e al. PHYSICAL REVIEW C 81, 024304 (2010) 80 90 100 110 120 12 14 16 18 20 22 N S2N(MeV) 80 90 100 110 120 12 14 16 18 20 22 N S2N(MeV) (a) (b) FIG. 4. (Colo online) AME95-03 es : Two-neu on sepa a ion ene gies S2n iso opic lines (in blue) p edic ed by (a) he DZ and (b) i s imp o emen using he CLEAN econs uc ion. Red bulle s indica e he inpu ( i ed) da a and g een bulle s indica e he benchma k (p edic ed) da a. V. THE ACCURACY OF CLEAN The ms de ia ions ob ained in he p e ious es s show ha he CLEAN me hod imp o es he mass p edic ions o he models used as inpu . Howe e , hese esul s e lec he global beha io o CLEAN. I would be desi able o measu e he accu acy o he CLEAN ex apola ion as a unc ion o he dis ance o he known egion. Howe e , de ining such a dis ance is no simple. A eliable me hod o ob ain accu a e p edic ions o nuclea masses in he neighbo hood o a known egion is based on he GK ela ions [11] used in an i e a i e p ocedu e [13]. These ela ions a e equa ions in ol ing six neighbo ing masses; om he knowledge o i e masses i is possible o ob ain a p edic ion o he six h (al hough he e a e six ways in which his can be done o each mass). The expe imen ally known masses accu a ely sa is y he GK ela ions and, as men ioned be o e, he e is no e idence ha his accu acy dec eases away om s abili y. Un o una ely, because o he o m o hese ela ions i is only possible o make p edic ions nea known egions o he nuclea cha . Howe e , i is possible o use p edic ed masses as known masses and epea he p ocess in an i e a i e ashion [13]. A each i e a ion new masses a bi u he away om he p e ious masses a e p edic ed. We can use his i e a ion p ocedu e as a means o de ining a FIG. 5. (Colo online) The i e a ion in he GK p ocess a which each nucleus is p edic ed in he AME95-03 es . dis ance. Fo example, he nuclei p edic ed in he i s i e a ion a e de ined o be a dis ance one, he nuclei p edic ed in he second i e a ion a e wo uni s away, and so on. Figu e 5 shows he i e a ion o he GK p ocess a which each nucleus is p edic ed in he AME95-03 es , showing his o be a consis en measu e o dis ance om he known egion. Fo he bo de es his de ini ion o dis ance is no use ul because all nuclei in he p edic ion a e as close as possible o he known egion. Two measu emen s o de ia ion can be de ined ha a e a unc ion o dis ance: σi, which is he ms o nuclei a dis ance i, and Aσ i, which is he ms o nuclei wi h dis ance alues up o i( i.e., an accumula i e ms up o dis ance i). Figu e 6and FIG. 6. (Colo online) (a) σiand (b) Aσ ias a unc ion o he numbe o nuclei o each uni o dis ance p edic ed in he AME95-03 es o he GK i e a i e p ocedu e (black ci cles), LDM (blue s a s), LDM imp o ed by CLEAN (blue s a s dashed), LDMM (g een iangles), LDMM imp o ed by CLEAN (g een iangles dashed), DZ ( ed squa es), and DZ imp o ed by CLEAN ( ed squa es dashed). 024304-6 IMAGE RECONSTRUCTION TECHNIQUES APPLIED TO ... PHYSICAL REVIEW C 81, 024304 (2010) TABLE III. σiand Aσ ide ia ions in MeV o he GK i e a i e p ocedu e, LDM, LDM imp o ed by CLEAN, LDMM, LDMM imp o ed by CLEAN, DZ, and DZ imp o ed by CLEAN o he AME95-03 es . The second column shows he numbe o nuclei used o calcula ing he ms de ia ions. A g aphical compa ison o hese da a is shown in Fig. 6. Dis ance Nuclei GK LDM LDM+LDMM LDMM+DZ DZ+ CLEAN CLEAN CLEAN σ185 0.144 1.866 0.208 1.095 0.202 0.299 0.168 Aσ 185 0.144 1.866 0.208 1.095 0.202 0.299 0.168 σ256 0.205 2.186 0.471 1.055 0.407 0.329 0.224 Aσ 2141 0.161 1.999 0.338 1.079 0.301 0.311 0.192 σ340 0.256 1.698 0.489 0.990 0.332 0.392 0.260 Aσ 3181 0.186 1.937 0.376 1.060 0.308 0.331 0.209 σ432 0.323 1.741 0.624 0.978 0.352 0.334 0.307 Aσ 4213 0.211 1.909 0.423 1.048 0.315 0.331 0.226 σ525 0.543 1.706 0.783 0.951 0.493 0.388 0.370 Aσ 5238 0.265 1.888 0.474 1.038 0.338 0.338 0.245 σ620 0.615 1.743 0.747 0.819 0.449 0.388 0.387 Aσ 6258 0.306 1.878 0.500 1.023 0.348 0.342 0.259 σ718 0.587 1.571 0.767 0.594 0.346 0.292 0.250 Aσ 7276 0.332 1.859 0.522 1.001 0.348 0.339 0.259 σ815 0.841 1.566 0.945 0.579 0.348 0.268 0.279 Aσ 8291 0.375 1.845 0.552 0.983 0.348 0.336 0.260 σ913 1.326 1.879 1.278 0.751 0.390 0.268 0.281 Aσ 9304 0.456 1.847 0.601 0.974 0.350 0.333 0.261 σ10 15 1.206 1.850 1.186 0.822 0.389 0.253 0.230 Aσ 10 319 0.515 1.847 0.641 0.968 0.352 0.330 0.259 σ11 11 1.567 1.891 1.367 0.744 0.287 0.305 0.239 Aσ 11 330 0.582 1.848 0.677 0.961 0.350 0.329 0.259 σ12 9 1.054 2.067 1.766 1.022 0.386 0.213 0.178 Aσ 12 339 0.583 1.854 0.728 0.963 0.351 0.326 0.257 σ13 6 1.822 1.890 2.120 0.672 0.485 0.382 0.413 Aσ 13 345 0.624 1.855 0.774 0.959 0.353 0.327 0.260 Table III show a compa ison o he esul s ob ained o each model in he AME95-03 es . The ms de ia ions ob ained wi h he GK i e a i e p ocedu e a e also p esen ed. Figu e 6 shows ha he CLEAN algo i hm imp o es he accu acy o all he models es ed and also displays he deg ee o imp o e- men achie ed as a unc ion o dis ance. The LDM (blue) p edic ion is imp o ed by CLEAN a small dis ances bu his imp o emen disappea s a e which he co ec ion oscilla es a ound he alue o he model. The imp o emen o he LDMM (g een) is mo e s able and has a longe ange, whe eas in he DZ case ( ed) i is appa en ha beyond ou uni s he ms o he CLEAN me hod ceases o imp o e he model p edic ions and oscilla es a ound he ms de ia ion o he model. The nuclei egion whe e he imp o emen on he p edic ion can be judged as eliable depends on he model and is de ined as he dis ance whe e he CLEAN ms oscilla es a ound he ms o he model, so o he models conside ed he e he anges a e 8 uni s o dis ance o he LDM, 4 uni s o he DZ case, and 12 o he LDMM. I is qui e imp essi e o obse e how he CLEAN me hod imp o es he LDM p edic ions, making hem as accu a e as he ones om he DZ model o he i s uni s o dis ance. This is because he pa e n ob ained wi h he di e ence o he LDM is e y clea and i epea s i sel ac oss he nuclea landscape, only changing i s scale [see Fig. 1(a)]. On he o he hand i is clea ha he DZ model is signi ican ly mo e di icul o imp o e wi h he CLEAN me hod, gi en i s al eady accu a e p edic ions. One o he main disad an ages o he CLEAN algo i hm is ha i s p edic ions depend on he model used o ob ain he mass-di e ence pa e n. Because i is known ha di e en models can ha e wildly di e en p edic ions, e en when hey apply o masses close o he expe imen ally known egion [22], i is impo an o discuss he unce ain ies due o he use o di - e en models. A consis en desc ip ion o he nuclea masses would be expec ed in he egion whe e he imp o emen o he models is conside ed eliable. I he p edic ions o he CLEAN algo i hm using di e en models a e consis en wi h each o he hen he di e ence be ween he models mus be educed a e using CLEAN. Table IV shows he ms de ia ions be ween he p edic ions o he di e en models as a unc ion o he dis ance de ined p e iously bo h be o e and a e using he CLEAN algo i hm, which indica e ha hese p edic ions a e consis en . As shown in he able, he mean de ia ion be ween he p edic ions o he di e en models is d as ically educed a e using CLEAN o he h ee possible combina ions, LDM 024304-7 IRVING O. MORALES e al. PHYSICAL REVIEW C 81, 024304 (2010) TABLE IV. σide ia ions in MeV as a unc ion o dis ance be ween he p edic ions o he di e en models conside ed (LDM, LDMM, and DZ) be o e and a e applying he algo i hm CLEAN. The second column shows he numbe o nuclei used o calcula e he ms de ia ions. Dis ance P edic ed LDM LDM+CLEAN LDM LDM+CLEAN LDMM LDMM+CLEAN Nuclei s. s. s. s. s. s. LDMM LDMM+CLEAN DZ DZ+CLEAN DZ DZ+CLEAN 1 165 2.15376 0.197661 2.53302 0.318165 1.44478 0.294493 2 122 2.14802 0.347624 2.57365 0.568731 1.60051 0.52335 3 121 2.20684 0.505986 2.67907 0.877885 1.69461 0.845331 4 120 2.07145 0.665372 2.71753 1.25192 1.99971 1.21614 5 111 1.99655 0.791645 2.94985 1.62756 2.25071 1.58315 6 111 1.97813 0.889578 3.18726 2.02946 2.52272 1.97478 7 112 1.98505 0.962786 3.4904 2.42825 2.76453 2.35319 8 114 1.98924 1.09671 3.84949 2.93075 3.11124 2.79128 9 112 1.98101 1.18803 4.25154 3.47196 3.41349 3.25024 10 112 1.95507 1.30185 4.69261 4.01583 3.82372 3.70363 e sus LDMM, LDM e sus DZ, and LDMM e sus DZ. This educ ion is p opo ional o he dis ance o he known egion, as expec ed. We conclude ha he p edic ions o he CLEAN algo i hm a e consis en wi h each o he , and ha all o hem ge co ec ed in he same di ec ion, a leas a sho ange. The esul s shown in his sec ion indica e ha he CLEAN ex apola ions a e eliable o a he sho (bu s ill sizable) anges. I is necessa y o imp o e he decon olu ion algo i hm in o de o achie e mo e accu a e p edic ions a la ge dis ances. One o he basic assump ions o he CLEAN me hod is ha he pa e ns o ex apola e can be modeled by a ini e numbe o pe iodic ha monic componen s. Howe e , he emaining pa e ns ob ained wi h he di e ence be ween expe imen al nuclea masses and he models es ed he e a e no pe iodic, bu a he ha e a well-de ined change in scale ha is speci ied by he shell sizes (i.e., he magic numbe s), so i is a he di icul o model i wi h ha monic componen s. I has been demons a ed in Re . [23] ha he CLEAN algo i hm is a special case o a mo e gene al echnique ha is no limi ed by he use o a Fou ie basis. Resea ch in his di ec ion is in p og ess. VI. CONCLUSIONS A me hod o imp o e p edic ions o a bi a y nuclea mass models was p esen ed, based on he de ec ion and ex apola ion o egula i ies in he pa e n o di e ences be ween expe imen al and heo e ical nuclea masses. The CLEAN algo i hm was shown o be capable o de ec ing ea u es in he mass-di e ence landscape associa ed wi h physical cha ac e is ics no included in he model. The model conside s hese pa e ns as he isible pa o a concealed image. The CLEAN me hod is essen ially a decon olu ion algo i hm ha applies a pa icula me hodology in o de o e eal he “ ue” image behind a mask. The CLEAN image econs uc ion echnique was applied o imp o e he heo e ical p edic ions gi en by h ee di e en models each ha ing a di e en deg ee o accu acy in hei p edic ions. The models analyzed a e: (a) a mac oscopic LDM, (b) a mac oscopic LDM wi h he inclusion o shell- co ec ion e ms (LDMM), and (c) he Du lo-Zuke model. The econs uc ion me hod has been applied o wo subse s o he known nuclea landscape. In all cases impo an imp o emen s in he p edic ions gi en by he di e en models we e ob ained wi h he CLEAN econs uc ion. We analyzed he wo-neu on sepa a ion ene gies S2n and ound ha he CLEAN econs uc ion me hod seems o adequa ely encode he physics con en o he da a and o ep oduce in a eliable way—in bo h p edic ion es s— he p esence o shell and de o ma ion s uc u es. We ha e also es ed he accu acy o he CLEAN me hod as a unc ion o dis ance o known masses, whe e a special de ini ion o dis ance based on he Ga ey-Kelson i e a i e p ocedu e has been de ined. This analysis measu es how he imp o emen o he di e en models by he CLEAN algo i hm a ies wi h dis ance o he known egion. We a e cu en ly s udying he use o al e na i e bases ha may imp o e accu acy o egions u he emo ed om measu ed masses o be able o make mo e eliable p edic ions. We belie e ha he CLEAN me hod p esen ed he e is a ela i ely simple me hod ha imp o es p edic ions by nuclea mass models and ha can be cons an ly imp o ed by he inco po a ion o new measu emen s. I may also be suscep ible o u he imp o emen s by he inco po a ion o p ocedu es ha can be e disce n he pa e ns p esen in he g owing se o nuclea mass da a. Finally, he same echniques can be applied o o he physical obse ables, including nuclea adii, ission ba ie s, and exci a ion spec a, as long as enough measu emen s ha e al eady been made o display a pa e n. Some o hese ques ions a e cu en ly unde in es iga ion. ACKNOWLEDGMENTS Con e sa ions wi h R. Cas en and A. Zuke a e g a e- ully acknowledged. This wo k was suppo ed in pa by PAPIIT-UNAM and Conacy -Mexico. 024304-8 IMAGE RECONSTRUCTION TECHNIQUES APPLIED TO ... PHYSICAL REVIEW C 81, 024304 (2010) [1] C. E. Rol s and W. S. Rodney, Cauld ons in he Cosmos (Uni e si y o Chicago P ess, Chicago, 1988). [2] Y. Oganessian, Na u e (London) 413, 122 (2001). [3] D. Lunney, J. M. Pea son, and C. Thibaul , Re . Mod. Phys. 75, 1021 (2003). [4]C.F. onWeizs ¨ acke , Z. Phys. 96, 431 (1935). [5] H. A. Be he and R. F. Bache , Re . Mod. 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