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. Phys. 8,82
(1936).
[6] P. M¨
olle , J. R. Nix, W. D. Mye s, and W. J. Swia ecki, A . Da a
Nucl. Da a Tables 59, 185 (1995).
[7] J. Du lo and A. P. Zuke , Phys. Re . C 52,R23
(1995).
[8] S. Go iely, F. Tondeu , and J. M. Pea son, A . Da a Nucl. Da a
Tables 77, 311 (2001).
[9]G.Audi,A.H.Waps a,andC.Thibaul ,Nucl.Phys.A729, 337
(2003).
[10] J. Mendoza-Temis, A. F ank, J. G. Hi sch, J. C. L´
opez Viey a,
I. Mo ales, J. Ba ea, P. Van Isacke , and V. Velazquez, Nucl.
Phys. A799, 84 (2008).
[11] G. T. Ga ey and I. Kelson, Phys. Re . Le . 16, 197
(1966); G. T. Ga ey, W. J. Ge ace, R. L. Ja e, I. Talmi, and
I. Kelson, Re . Mod. Phys. 41, S1 (1969).
[12] R. F. Cas en, Nuclea S uc u e om a Simple Pe spec i e
(Ox o d Uni e si y P ess, Ox o d, 1990).
[13] I. O. Mo ales, J. C. L´
opez Viey a, J. G. Hi sch, and A. F ank,
Nucl. Phys. A828, 113 (2009).
[14] G. Audi, p ep in a a Xi :nucl-ex/0302020 (2003).
[15] J. Ba ea, A. F ank, J. G. Hi sch, and P. Van Isacke , Phys. Re .
Le . 94, 102501 (2005).
[16] I. Mo ales, A. F ank, J. C. L´
opez Viey a, J. Ba ea, J. G. Hi sch,
V. Vel ´
azquez, and P. Van Isacke , In . J. Mod. Phys. E 15, 1855
(2006).
[17] J. A. H¨
ogbom, As on. As ophys. Suppl. 15, 417 (1974); B. G.
Cla k, As on. As ophys. 89, 377 (1980).
[18] R. Na ayan and R. Ni yananda, Annu. Re . As on. As ophys.
24, 127 (1986).
[19] A. A. Cla k, B. T. Thomas, N. W. Campbell, and P. G eenway,
in B i ish Machine Vision Con e ence (B i ish Machine Vision
Associa ion, 1999), p. 193.
[20] G. Audi and A. H. Waps a, Nucl. Phys. A595, 409 (1995).
[21] J. Mendoza-Temis e al., Nucl. Phys. A812, 28 (2008).
[22] K. Blaum, Phys. Rep. 425, 1 (2006).
[23] A. Kaup, K. Meisinge , and T. Aach, In . J. Elec on. Commun.
59, 147 (2005).
024304-9