U anium-238 decay chain
Ma ia A azo1, Ma c Ba oso1, Ósca De la To e1, Lau a Mo eno1, A iadna
Ribes1, Pa icia Ribes1, Ana Ven u a1, and Da id O ellana-Ma ín2
1Uni e si a de Ba celona
Email: {ma ia.a azo, ma c.ba oso4, osca .dela o e.pe ez, 95mo enolau a,
a ibesme idie i, ibesme idie i, a. en u aba oso}@gmail.com
2Resea ch G oup on Na u al Compu ing
Depa men o Compu e Science and A i icial In elligence
Uni e sidad de Se illa
E-mail: [email p o ec ed]
Summa y.
The main objec i e o his a icle is o modelize he p ocess o decay o
U anium 238 wi hin he amewo k o Memb ane Compu ing, so he e olu ion o g ea
numbe s o pa icles can be p og essi ely ollowed and he esul s o he desin eg a ions
(nuclei coming om αand β−decays) can be coun ed.
In o de o model he p ocess in an accu a e manne , exploi ing he p ope ies o
maximal pa allelism and non-de e minism o Memb ane Compu ing, a Popula ion Dynamic
P sys em (o PDP o sho ) es ic ed o one en i onmen and a P sys em con o med by
only he skin ha e been selec ed.
The di icul y in he cha ac e isa ion o his eac ions lays in he simul anei y o he
di e en decays, since he numbe o desin eg a ions o nucleous o each specie depend on
he numbe o a oms o he ini ial popula ion. In o de o sol e his p oblem and keep
hei a achmen , he cha ac e is ic ime o p oduc ion o each decay has been ansla ed
in o p obabili ies o dein eg a ion o a nucleous using he decay cons an λ.
1In oduc ion
In his pape we a e conside ing he U anium-238 decay, which will be explained
in he ollowing sec ions. One o he i s objec i es was o p o e ha making use
o Memb ane Compu ing and he P-lingua simula ion, we could ob ain he esul s
p e iously known, e.g. he ways ha in e media e p oduc s o he decay ook o
a i e o he inal p oduc o he amoun o di e en elemen s ha we e p oduced
du ing he chain. Ne e heless, du ing he de elopmen o his p ojec ano he
in e es ing p oblem, which will be explained and discussed in la e on, appea ed:
ime implemen a ion. A his poin , ou main goal was o look o di e en ways
o modeling he physical p ocess as close as possible o eali y. E en so, i is s ill
in e es ing o know which p oduc s we ob ain in each disin eg a ion, so we a e able
o p oo he mos p obable ways o decay.
114 A azo, Ba oso, De la To e, Mo eno, A. Ribes, P. Ribes, Ven u a, O ellana
The p oblem wi h hal -li e imes.
Du ing he modeling o he decay p ocesses we ound ou some p oblems when
implemen ing he ime in ol ing he eac ions, i.e. he disin eg a ion o one elemen
in o ano he one has an in insic hal -li e ime associa ed o i . This pa ame e
T1/2
,
ound in equa ions
(1)
and
(2)
, de e mines he ime ha akes o he elemen
o educe he numbe o i s nuclei o hal o he ini ial ones. Also, an impo an
cons an is
τ
, he decay cons an , de ined as:
τ=1
λ
, which ep esen s he p obabili y
o a nucleus o decay, pe uni o ime.
dN
d =−λN (1)
T1/2=ln 2
λ(2)
Being
N
he numbe o nuclei a a gi en ime
, and
λ
, he numbe o disin e-
g a ions pe second, which is a cons an o a gi en eac ion.
F om
(1)
i can be no iced ha he a e o disin eg a ion depends no only
on he cons an o disin eg a ion,
λ
, bu also on he popula ion o nuclei a he
ime we a e calcula ing he disin eg a ion a e. This is he eason why a he han
conside ing he a e as he pa ame e o cha ac e ize he eac ions, some imes is
be e o conside wha we de ine as hal -li e ime,
T1/2
, which is cons an because
i only depends on λ.
These p ocesses occu all a he same ime, so o say, om he i s momen
when we ob ain he second nucleus o he chain, ano he eac ion begins o ake
place: i does no wai o all he i s elemen s o eac . Taking in o accoun ha
P-Sys ems a e based in sys ems ha e ol e by s eps o ime we we e awa e we
had o ind a way o app oxima e as close as we could o he ac ha ime is
con inuous. To do so while ying no o di e a lo om wha happens in eali y
we wen h ough di e en models making some changes in he implemen a ion o
he ime. The wo me hods ha we selec ed, which will be u he explained in
ollowing sec ions, we e he ollowing:
•
S eps o ime: The i s app oach o he p oblem o U anium decay consis ed in
ansla ing he hal -li e imes o he di e en decays o he chain by a loga i hmic
scale so he conside ed ange o a ia ion was educed enough in o he o assign
a p opo ional and a bi a y amoun o ime o each s ep. The e i has been
conside ed ha one eac ion mus be applied o all nuclei be o e beginning
he ollowing eac ion o he chain. In his case, index no a ion was used o
ep esen he du a ion o each s ep. Al hough no being a model eally close
o he eal si ua ion, one could ob ain he expec ed esul s. So, o example,
he i s eac ion was assigned a coun e ha wen om 1 o 7. This coun e
ensu ed ha no eac ion could begin be o e ha ing ended p e iously he ea lie
s ep in he chain.
In a way, his p ocess oughly simula ed he di e en pe iods o ime equi ed
o each elemen o he chain o anish. Howe e i doesn’ allow ha di e en
U anium-238 decay chain 115
elemen s eac a he same ime. The nex s ep in he chain has o wai un il
he p e ious one inished. The e o e, al hough he simula ion ha implemen s
his ules app oaches easonably well he amoun o pa icles ga he ed a he
end o he p ocess, i was no a good app oach o eali y, as in a eal decay
se e al eac ions o he chain ake place a he same ime.
•
P obabilis ic model: in which i has been aken in o accoun ha once a nucleus
has decayed in o he nex one, he ollowing eac ion can ake place o ha
ecen ly gene a ed nucleus. This model is a use ul way o de e mine which
pa icles we e gene a ed a each momen , i.e. one could ho oughly examine
he in e media e s ages o he decay.
2U anium-238 decay chain
2.1Radioac i e se ies
Nuclea decays [
2
] a e ansi ions o less ene ge ic —and hus mo e s able— s a es.
An ini ial uns able nucleus can na u ally decay in o ano he nucleus, usually bu
no necessa ily ligh e , ollowing di e en modes cha ac e ized by he emi ed
pa icles and he esul an nuclei. The ones conce ning ou s udy a e he
α
decay
(3)
and he
β−
decay
(4)
, whe e he emi ed pa icles can be He nuclei (
α
pa icles)
o elec ons (along wi h hei co esponding an ineu ino).
A
ZX→A−4
Z−2Y+4
2He (3)
A
ZX→A
Z+1Y+e−+ ¯νe(4)
Whe e
A
is he mass numbe and
Z
he a omic numbe . O he possible decay
modes a e he
β+
decay (wi h emission o posi ons and elec onic neu inos), he
gamma emission, and he elec onic cap u e. O he h ee kinds o possible emi ed
pa icles (
α
,
β
and
γ
),
γ
pa icles ha e he la ges pene a ing powe , while
α
pa icles in e ac mo e wi h ma e .
The esul ing nuclei o a nuclea decay can s ill be uns able and he e o e decay
in o ano he nuclei and he co esponding pa icle. In his way, se e al decays may
ake place un il a s able nucleus is eached. This p ocess o chained decays ha
begins on a uns able pa en nucleus and end on a numbe o s able nuclei is called
a adioac i e se ies o decay chain.
The pa en nuclei o he adioac i e se ies usually ha e e y la ge li e imes (i.e.
he ime i akes o he ini ial popula ion o disappea en i ely). The e a e ou
main adioac i e se ies ( h ee o hem being na u al), and all o hem end in lead,
which is s able.
116 A azo, Ba oso, De la To e, Mo eno, A. Ribes, P. Ribes, Ven u a, O ellana
2.2U-238 decay chain
As we can see in Table 1, he U anium decay chain consis s o 15 main s eps (i.e.
decay eac ions). This able shows he mos p obable decay modes, bu he e a e
o he decays wi h an ex emely low p obabili y o occu ing, showed wi h mo e
de ail in he second pa o Table 1. Ne e heless, independen ly om he pa h
chosen, he inal p oduc is always lead (Pb-206), which is s able. A diag am o
he whole U-238 decay chain and i s less p obable decay modes can be ound in
Figu e 1.
U anium-238 decay chain 117
Pa en T1/2(s) λ(decays/s) τ(s) Decay modes Reac ion ,1
238
92 U (a)1.41 ×1017 4.92 ×10−18 2.03 ×1017 α(100%): 234
90 Th [a]0→x[b]09.68 ×10−1
234
90 Th (b)2.08 ×1063.33 ×10−73.00 ×106β−(100%): 234
91 Pa [b]0→z[c]05.33 ×10−1
234
91 Pa (c)2.41 ×1042.87 ×10−53.48 ×104β−(100%): 234
92 U[c]0→z[d]06.22 ×10−1
234
92 U (d)7.74 ×1012 8.95 ×10−14 1.12 ×1013 α(100%): 230
90 Th [d]0→x[e]02.29 ×10−1
230
90 Th (e)2.38 ×1012 2.92 ×10−13 3.43 ×1012 α(100%): 226
88 Ra [e]0→x[ ]02.53 ×10−1
226
88 Ra ( )5.05 ×1010 1.37 ×10−11 7.28 ×1010 α(100%): 222
86 Rn [ ]0→x[g]03.30 ×10−1
222
86 Rn (g)3.30 ×1052.10 ×10−64.77 ×105α(100%): 218
84 Po [g]0→x[h]05.70 ×10−1
218
84 Po (h)1.86 ×1023.73 ×10−32.68 ×102α(99.98%): 214
82 Pb
β−(0.02%): 218
85 A
[h]0→x[ ]0
[h]0→z[j]0
7.20 ×10−1
1.44 ×10−4
214
82 Pb ( )1.62 ×1034.27 ×10−42.34 ×103β−(100%): 214
83 Bi [ ]0→z[k]06.77 ×10−1
214
83 Bi (k)1.19 ×1035.81 ×10−41.72 ×103β−(99.979%): 214
84 Po
α(0.021%): 210
81 Tl
[k]0→x[n]0
[k]0→z[m]0
1.43 ×10−4
6.83 ×10−1
214
84 Po (m)1.64 ×10−44.22 ×1032.37 ×10−4α(100%): 210
82 Pb [m]0→x[p]01.00 ×100
210
82 Pb (p)7.00 ×1089.90 ×10−10 1.01 ×109β−(100%): 210
83 Bi
α(1.9×10−6%): 206
80 Hg
[p]0→x[o]0
[p]0→z[q]0
7.90 ×10−9
4.16 ×10−1
210
83 Bi (q)4.33 ×1051.60 ×10−66.25 ×105β−(100%): 210
84 Po
α(13.2×10−5%): 206
81 Tl
[q]0→x[s]0
[q]0→z[ ]0
7.45 ×10−7
5.64 ×10−1
210
84 Po ( )1.20 ×1075.80 ×10−81.72 ×107α(100%): 206
82 Pb [ ]0→x[ ]04.98 ×10−1
206
82 Pb ( ) S able — — — — —
O he (less p obable) decays
218
85 A (j)1.50 4.62 ×10−12.16 α(99.9%): 214
83 Bi
β−(0.1%): 218
86 Rn
[j]0→x[k]0
[j]0→z[l]0
8.16 ×10−1
8.17 ×10−4
218
86 Rn (l)3.50 ×10−21.98 ×1015.05 ×10−2α(100%): 214
84 Po [l]0→x[m]08.92 ×10−1
210
81 Tl (n)7.80 ×1018.89 ×10−31.13 ×102β−(100%): 210
82 Pb [n]0→z[p]07.38 ×10−1
206
80 Hg (o)4.99 ×1021.39 ×10−37.20 ×102β−(100%): 206
81 Tl [o]0→z[s]07.00 ×10−1
206
81 Tl (s)2.52 ×1022.75 ×10−33.64 ×102β−(100%): 206
82 Pb [ ]0→x[ ]04.98 ×10−1
Table 1: Hal -li e imes, decay cons an s, mean li e imes, decay modes, eac ion
and p obabili y unc ions o each eac ion in he U-238 decay chain. The le e
in b acke s co esponds o he le e assigned o each nucleus o implemen a ion.
Hal -li es and p obabili ies o he chain decays ob ained om [
3
] [
5
]; p obabili ies
o he less p obable decays ob ained om [
4
]. The p obabili y unc ion associa ed
o each ans o ma ion ule depends on he decay cons an and he decay mode
p obabili y.

118 A azo, Ba oso, De la To e, Mo eno, A. Ribes, P. Ribes, Ven u a, O ellana
238
92 U
234
90 Th 234
91 Pa 234
92 U
230
90 Th
226
88 Ra
222
86 Rn
218
84 Po 218
85 A 218
86 Rn
214
82 Pb 214
83 Bi 214
84 Po
210
81 Tl 210
82 Pb 210
83 Bi 210
84 Po
206
80 Hg 206
81 Tl 206
82 Pb
α
β−β−
α
α
α
α
α
β−
β−β−
αα
β−
α
β−
αα
α
β−
α
β−
β−β−
Fig. 1: U anium decay chain.
U anium-238 decay chain 119
3P-sys em model
Popula ion Dynamics P sys ems (PDP sys ems) [
1
] a e a kind o P sys ems ha
combines he cha ac e is ics o bo h cell-like and issue-like models. A PDP sys em
is cons i u ed by (i) a se o connec ed en i onmen s placed in he nodes o a di ec ed
g aph (ii) iden ical cell-like s uc u es o hie a chically a anged memb anes placed
inside each en i onmen , (iii) a wo king alphabe o objec s and (i ) a se o ules
which desc ibe how objec s e ol e and mo e inside he P sys ems (
R
) and among
he en i onmen s (Rε).
Fo mally, a Popula ion Dynamics P sys em o deg ee (
q
,
m
) wi h
q, m ≥1
,
aking T ime uni s, T≥1, is a uple
(G, Γ, Σ, T, Rε, µ, R,{ ,j : ∈ R,1≤j≤m},{Mij : 1 ≤i≤q, 1≤j≤m})
(5)
whe e:
•G= (V, S)
is a di ec ed g aph and
V={e1, . . . , em}
a e he elemen s called
en i onmen s.
•Γ∪Σis he wo king alphabe .
•Tis a na u al numbe ha ep esen s he simula ion ime o he sys em.
•Rεis a se o communica ion ules be ween en i onmen s o he o m
(x)ej
p(x,j,j1,...,jh)
−−−−−−−−−→ (y1)ej1· · · (yh)ejh (6)
whe e
x, y1,· · · yh∈Γ, (ej, ejl)∈S(l= 1, . . . , h)
and
p(x,j,j1,...,jh)( )∈[0,1]
,
o each = 1,...T.
The p e ious de ini ion means ha , when a communica ion ule is applied,
objec
x
con ained in en i onmen
ej
passes o en i onmen s
ej1. . . ejh
, possibly
modi ied in o objec s
y1,...yh
. I mo e han one ule can be applied o
(x)ej
,
hen he ule execu ed is chosen andomly acco ding o he p obabili ies
p(x, j, j1, ..., jh).
•µ
is he memb ane s uc u e o he cells con ained in each o he
m
en i onmen s
and each consis ing on a se o
q
hie a chically a anged memb anes injec i ely
labeled by
1,...q
. The skin memb ane, o ou e memb ane is labeled by 1. The
memb anes can also ha e elec ical cha ges o pola iza ions, EC ={0,+,−}.
•R
is a se o e olu ion ules applied wi hin each cell. They a e o he o m
:u[ ]α
i→u0[ 0]α0
iwhe e u, , u0, 0∈M(Γ),i∈1,...q, and α0∈EC.
•
Fo each
∈ R
and o each
j, 1≤j≤m
,
,j
is a compu able unc ion which
sa is ies ha , o each
u, ∈M(Γ)
all he ules
∈ R
whose le -hand side
is
(i, α, u, )
and he igh -hand side ha e a pola iza ion
α0
,
Pj=1 ,j( )=1
∀ ≤T.
•M1j,...,Mqj ∈M(Γ)
a e he ini ial mul ise s o objec s o en i onmen s
j= 1,...m placed inside he memb anes 1,...q o µ.
The uple o mul ise s o objec s p esen a any momen in he
m
en i onmen s
and a each o he egions o he P sys ems (cell-like s uc u es) cons i u es he a
120 A azo, Ba oso, De la To e, Mo eno, A. Ribes, P. Ribes, Ven u a, O ellana
con igu a ion o he sys em a any ime. A he ini ial con igu a ion o he sys em,
all en i onmen s a e assumed o be emp y and all he memb anes ha e neu al
pola iza ion.
The sys em e ol es om one con igu a ion o ano he a each ime s ep by
execu ing simul aneously all he applicable ules o he se
R=Rε∪Sm
i=1 RΠj
3
in a
maximal way. When he e a e ules ac ing on o e lapping le -hand sides, i.e.
u[ ]α
i
,
u0[ 0]α
i
whe e
u, u0, , 0∈M(Γ)
,
u6=u0∨ 6= 0
and
u∩u06=∅ ∨ ∩ 06=∅
, he
ule which is execu ed is selec ed andomly acco ding o he p obabili y associa ed
wi h each ule.
Finally, i is in e es ing o highligh he ac ha a global clock is conside ed in
he sys em, ma king he ime o he whole sys em, so he applica ion o all ules
(bo h om Rεand R) a e synch onized in all en i onmen s.
3Πj={Γ, µ, R,M1j,...,Mqj}
deno es he P sys em in en i onmen
ej
and
RΠj
, he
se o ules de ined on he conside ed P sys em.
U anium-238 decay chain 121
2,2
3,24,2
b5a2
a6
2,1
3,1
[ab]1→a2b[ ]1
[a]2
0,2
−−→ b[ ]2
[a]2
0,8
−−→ a[a]2
[ab]1→a2b[ ]1
[a]2
0,2
−−→ b[ ]2
[a]2
0,8
−−→ a[a]2
1,1
1,3
2,3
b7a5
[ab]1→a2b[ ]1
[a]2
0,2
−−→ b[ ]2
[a]2
0,8
−−→ a[a]2
b7a5
1,2
2,2
3,2
a5
a3b3
b4a2
(a)2
0,25
−−→ (b)1
(a)2
0,75
−−→ (a)1
(a)2
0,5
−−→ (a)3
(ab2)3→(a2)1
(ab)3→(b)2
(a)2
0,5
−−→ (b)3
3,3
Fig. 2: A g aphical example o a PDP sys em
4Implemen a ion
4.1Fi s model
The S ep o ime model is he simples possible modeliza ion o he U anium decay
p oblem, as i cha ac e izes he hal -li e ime, he cha ac e is ic ime pa ame e ,
o he nuclea eac ions using he clock s eps o ime de ined in he memb ane
compu ing model. This i s ough app oxima ion is based on wo assump ions:
•
The nuclea eac ion o a gi en elemen canno begin un il all he p ogeni o
nuclei o his elemen ha e eac ed.
•
The du a ion o each eac ion can be ep esen ed assigning di e en clock s eps
o e e y eac ion.
128 A azo, Ba oso, De la To e, Mo eno, A. Ribes, P. Ribes, Ven u a, O ellana
•
a e a la ge enough numbe o compu a ion s eps, all he emmi ed pa icles in
he eac ion (αand β−pa icles) ha e eached a s able s a e.
•
he decay chain akes place in a s agge ed way (we canno see he disc e e
inc emen s), as was sough in his second model.
•
he slope o he e olu ion o each p oduc ma ches he
λ
coe icien s we imple-
men ed, showing a co ec ela ionship be ween he p obabili y unc ions and
λ.
•
he p ocess is as e a gene a ing pa icles in he beginning and a he end o
he eac ion, whe eas he middle p oduc s las o a while.
Fig. 5: Nuclei popula ion. I is ep esen ed he numbe o pa icles a each s ep
o ime, in loga i hmic scale. I can be seen ha he numbe o ini ial pa icles
(U anium-238) dec eases whe eas he inal p oduc inc eses un il i eaches an
s a iona y alue, he same as he ini ial numbe o pa icles, as expec ed. As
eac ions ake place, he new elemen s a e c ea ed, showing an imp essi e inc ease
ha slowly dec eases hen as eac ions con inue. As we each mo e ad anced s ages
(S eps o ime
≈
250 and mo e) a noise in he numbe o elemen s appea s: his
is because less eac ions o he same elemen s ake place simul aneously and so
he numbe o pa icles changes sha ply a each s ep, depending on i he eac ion
in ol ing ha speci ic elemen has aken place o no .

U anium-238 decay chain 129
The luc ua ions, which can be app ecia ed in he las s eps o ime in Figu e 5,
a e p obably due o he o e lapping o di e en eac ions in he ad anced s eps o
he u anium chain which simul aneously p oduce and consume a ce ain nucleus,
i.e., as he p oduc s e ol e, mo e eac i es a e gene a ed a he same ime ha
hey a e e ol ing o he nex p oduc . Tha way, a dependence on he decay a e
on he ela i e abundances o each eac i e is app ecia ed.
7Conclusions
Decay chains a e based in a sys em o di e en ial equa ions ha once sol ed allow
o ob ain he p oduc s a each ime,
. Howe e , he solu ion is eached a e
sol ing a coupled sys em o nume ous di e en ial i s o de equa ions, which is
compu a ionally cos ly. The MC ools allow o ep oduce he p ocess and o ob ain
he expec ed inal p oduc s jus by making some sligh app oxima ions.
This means ha compe i ion ules which appea na u ally in MC can assume he
ole o he bounds be ween di e en ial equa ions almos i ially, so he men ioned
sys em o di e en ial equa ions does no need o be sol ed in o de o simula e he
eal si ua ion.
In addi ion, his a icle a emp s di e en ways o implemen ing ime in a decay
p ocess. Ins ead o modeling ime as an independen pa ame e , which would be
he model whe e indexes a e used (conside ing s eps o compu a ion as ime), i
has inally been in oduced as a pa o he p obabili y, gi en by he decay cons an
λ
. The i s me hod is eally une icien because he sys em was es a lo o ime
jus skipping p ocesses (while indexes change) and so by his ime, he p og am is
no eally wo king on he chain eac ion i sel .
Acknowledgmen s.
We would like o hank: Agus ín Riscos-Núñez, Ca men G aciani and Ma io J.
Pé ez-Jiménez om Uni e sidad de Se illa and Rica do G aciani om Uni e si a
de Ba celona o hei aluable commen s and sugges ions, as well as o hei
e iews o he se e al e sions o his pape ; he 14 h BWMC a endan s o hei
pa ience and ad ice on ou i s ideas; and also F ancesc Sal a , Assump a Pa eño,
Rica do G aciani and B uno Juliá-Díaz om Uni e si a de Ba celona o o e ing
us he oppo uni y o a end he B ains o ming.
Re e ences
1.
M. Colome , A. Ma galida, and M. J. Pé ez-Jiménez. Popula ion Dynamics P Sys em
(PDP) Models: A S anda dized P o ocol o Desc ibing and Applying No el Bio-Inspi ed
Compu ing Tools. PLOS ONE,8(4):1–13,2013.
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K. S. K ane. In oduc o y nuclea physics. John Wiley and Sons Inc., New Yo k, NY,
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130 A azo, Ba oso, De la To e, Mo eno, A. Ribes, P. Ribes, Ven u a, O ellana
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Li e Cha o Nuclides: nuclea s uc u e and decay da a.
h ps://www-nds.iaea.o g/ elnsd/ cha h ml/VCha HTML.h ml.
4. Na ional Nuclea Da a Cen e : Cha o Nuclides. h p://www.nndc.bnl.go /cha /.
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N. Radionuclide Hal -Li e Measu emen s.
h p://www.nis .go /pml/da a/hal li e-h ml.c m
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