ComputerPhysicsCommunications25 (1982) 297—309 297North-Holland PublishingCompany

ALPHA-DECAY HALF-LIFE SEMIEMPIRICAL RELATIONSHIPS WITHSELF-IMPROVING PARAMETERS

D.N. POENARU,M. IVASCU andD. MAZILU

CentralInstituteof Physics,P.O. Box MG-b, R-76900Magurele-Bucharest,Rornania

Received5 November1981

PROGRAM SUMMARY

Title of program: SEMIEMPIRICAL ALPHA HALF-LIFE tionedabove.Thereare threeoptions: one can useeither thepresentset of parametervalues,a new onegivenas input data,

Cataloguenumber:ABQQ or new valuescomputedby using a better set of experimentaldata (more accurateor more complete). For each group ofnuclides,up to 8 (or 9) families of curvescould be plotted,Program obtainablefrom: CPC ProgramLibrary, QueensUrn- . . .

versityof Belfast,N. Ireland(seeapplicationformin this issue) optionally,with the line pnnter.The calculatedlife-timecanbecomparedeitherwith theexperimentalone,if it is available,orwith the time of ref. [5].

Computer: IBM 370/135; Installation: Central Institute ofPhysics Methodof solution

Only the additive parametersof the log T formulae of refs.Operatingsystem:DOS/VS-310 [1—4]aremodified,by requestinga vanishingmeanvalueof the

absoluteerrors. The B-parametersdefined in the ref. [5] are

Programminglanguageused: FORTRAN IV obtainedby a parabolicleastsquarefit.

Restrictionon the complexityof the problem1-ugh speedstoragerequired: 95 kbyte .The maximum numberof alpha emittersin eachof the four

groupscan not be larger than 300 in a given run. Up to 40Numberof bits per b.vte: 8 curvesare acceptedby the plottingsubroutine.

Overlaystructure: none Typical running time

For 100 alpha emitters, the tupical CPU runningtime on anOtherperipherals us&I: cardreader,line printer IBM 370/135 rangesfrom 7 s (no plot) to 20 s (9 diagrams

plotted)in the self-improving modeof operationand from 6to 18 s with thepresentset of parameters.

Numberof cards in combinedprogram and testdeck: 1511unusualfeaturesof the program

Cardpunchingcode: EBCDIC The subroutinesMZDPLT and MZDPFT could be used inother programs to plot a family of curves,or to make a

• . . . . . bidimensionalparabolicleastsquaresfit respectively.Keywords: nuclearphysics,radioactivity, alpha decay,semiem-pirical formulae,half-life References

[If R. TaageperaandM. Nurmia,Ann;Aead~Sci;Fenn~Ser. ANatureof the Physical Problem 78 (1961).From thealpha decayQ-values,the partial half-life T is esti- [2] yE. Viola Jr. and G.T. Seaborg,J. Inorg. Nucl. Chem. 28mated by using five semiempirical relationships[1—5]. The (1966) 741.parametersof these formulaehave been obtained from a fit [3] K.A. Keller andH. MUnzel, Z. Phys.255 (1972) 419.with a given setof experimentaldata on four groupsof alpha [4] P. Hornshei),P.G. Hansen,B. Jonson,L. Ravn,L. West-emitters: even—even,even—odd,odd—evenand odd—odd. gaardandON. Nielsen,NucI. Phys.A230 (1974) 365.Foreachnuclideonly thestrongesttransitionis consideredand [5] D.N. Poenaru,M. Ivalcu and D. Mazilu, J. de Phys.Lett.the data are automaticallysorted into the four groupsmen- 41 (1980) L-589.

0010-4655/82/0000—0000/$02.75© 1982 North-Holland

298 D.N. Poenaruet al. / Semiempiricalalphahalf-life

LONG WRITE-UP

1. Introduction where1 is the total disintegrationperiod, ba is the

The alphapartial half-life, T, of a given parent alpha branchingratio in the presenceof othernucleus with a mass number A and an atomic competingmodesand~ir is the relativeintensityof

the alpha transitionconsidered(the strongestone,numberZ, canbe relativelyreliably estimated,if called usually favouredin caseof an odd numberthe releasedenergy Q [1] is known,by usingsome of neutronsor/and protons).One assumesthatsemiempirical relationships, the first of which both b~and ~ir are in percent.As the rangeof(1911)wasperhapsthefamousGeiger—Nuttallaw, life-times of different nuclidesextendsovermanyThe kinetic energy of the alphaparticle, Ea~is ordersof magnitudeit is morepracticalto usetherelatedto the Q-valueby the following equation: decimallogarithmof T. We haveselectedour set

E,~ QAd/A, (1) of experimentaldata containing: 123 even—even

where Ad = A — 4 is the mass number of the nuclides (group1); 111 even—odd(group2); 83daughternucleus, odd—even(group3) and59 odd—odd(group4),by

We havefocused our attention on four of the using mainly the table of Rytz [9], but also thesemiempirical formulae [2—5], more frequently refs. [10—18]for the life-times, branchingratiosused,allowing one to obtain smaller errors [6]. and transition intensities.The Q-valuesare thoseEachhassomeparametersdeterminedfor a group of WapstraandBos [1],exceptfor a small numberof even—even,even—odd,odd—evenand odd—odd of isotopesof Te, I, Xe, Cs, Hf, Ta, Os, Jr andNo.nuclei,by fitting a given set of experimentaldata. ThepresentparametervaluesCT, Cv, CK, CH,

The quality and the quantity of the data are C~,and {Bk}, of this program,havebeenobtainedincreasedcontinuouslyby making more accurate on the basis of this data. The program has threemeasurementsand by discovering new alpha options: to work with theseparameters,to findemitters,henceit is interestingto havefrom time new values of the parameters(self-improvingto time the possibility to changesomeparameters. mode) from a better set of experimentaldataOnecansimply require a vanishingmeanvalueof (more accurateand/or more complete) and tothe absoluteerrors for eachgroupof nuclidesand work with new valuesof the parametersas inputobtain, in this way, new improved values of the data.additive parametersCT, Cv, CK, CH and C~,of The nuclidesare automaticallysortedinto fourthe log T formulae, groupsmentionedabove and in éachgrOup Oñe

In orderto havea betteragreementwith experi- has the possibility to plot up to 8 (or 9 in themental data even in the neighbourhoodof the self-improving mode) diagrams with the linemagic neutron and proton numbers,where the printer: the systematicsof log Tret~vs. 250/vt~errors of the other relationshipsare usually as and of Q, log 7ef’ (Xexp)’ the errors of the fivelarge as someordersof magnitude,we havepro- formulaewith respectto Tef vs. N. Usually 7~ef=

• - posedanewfo~l~E,7jwitb8pa~icters~Bk, T~~~butif the experimentalhalf-life is not known,basedon the fission theory of alphadecay[8]. It the calculatedOne s h that of ref. [6]takes into considerationexplicitly not only the ‘1’~ef= T~(seeeq. (8)).dependenceon the proton numberZ, but also onthe neutronnumber,N, and their differencefrommagicity. A bidimensionalparabolic least square 2. Semiempiricalrelationshipsfit leadsto new valuesof the coefficientsBk in theself-improvingmodeof the presentprogram. The formula given by Taageperaand Nurmia

The experimentalpartial half-life, 7~,for the [2]alphadecay is expressedas

log TT= l.6l(Zd//~a~Z~/~)+CT, (3)100~2~~. (2)

‘~Xp = b~, ~ir where Zd = Z — 2 is the atomic number of the

D.N. Poenaruet al. / Semiempiricalalphahalf-life 299

daughter nucleus and only the parameter CT de- K~ 2.52956Zd(Ad/AQ)V2pendson the group of nuclei, was improved byKeller andMUnzel [4] X [arccos~ ~/x~(l—xv) 1~ (8)log TK = HK(Zd/~/~—z~/3)+ CK, (4) wherex~,= 0.4253Q(l.5874+A~(3)/Zd,it was as-

sumedthat the radius constantof the liquid drop

allowing also the parameterHK to takethe follow- model r0 = 1.2249 fm andthe zero pointvibration

ing valuesin different groups: HK = 1.61 for even energy EV~b 0.4 MeV. The quantity x is ap-—even; 1.65 for even—odd;1.66 for odd—evenand proximatedby the secondorder polynomial1,77 for odd—oddnuclei. Onehas to mentionthat

xB1+B2y+B3z+B4y2+B

5yz+B6z2 (9)

the authorsare usingan effectivevalueof Qin the reducedvariablesy andz:

Qeff = Q+ (65.3z7/5 — 80Z2/5)106MeV, (5)N-N N<N~<.N~

1,wherethe correctiondueto electronicscreeningis ~N —~

of the order of 15—30 keV. In the following thiscorrectionwill be ignored,in orderto usethe same Z — ZQ-valuesas Wapstraand Bos[1]. z ~~÷1 — ~ Z~<z ~ (10)

The equationpresentedin ref. [3] by Viola andSeaborgis of the form expressingthe distancefrom the closestmagic-plus

-one number of neutronsand protons: ~log Tv = (a1Z—a2)/1/Q—b1Z b2 + Cv, (6) 29, 51, 83, 127, 185, ...; Z~ ..., 29, 51, 83, 115,

where a1 = 2.42151; a2 = 62.3848; b1 = 0.59015; The quantityto be fitted in order to obtain the= 4.2109 for N = A — Z < 126 and a1 = parameters{Bk), given in table 1, is

2,11329;a2 = 48.9879; b1 = 0.39004;b2 = 16.9543forN> 126, Z>82, Xexp~1~l0(logT~~~+20.446)/K~. (11)

Hornsh.~cjet al. [5] haveproposedthe formula In the eqs.of this section the timesare expressed

in secondsand the energiesin MeV. The corre-

log TH = 0.80307( A~/3Z~) 1/2 spondingadditive parametersare given in table2,

both in initial form and after the improvement_____ using our set of experimentaldataand the condi-

~ (arcc~‘~ — ~/l— x + CH, (7) tion

‘I

~ log(T,/T1~~~)=0 (12)

in which x = 0.53S243QA~’3/Zd. i=1

In our [6,7] equationin eachof the four groupsof nuclides.Onecansee

log T1, = 0,434294K—20.446+ C~,K= xK~, that in the eqs.(6) and(8), (Cv — b2) and(C~—

Table IParametersBk

Groupof nuclei B1 B2 B1 84 B5 B6

I (even—even) 9.87722E—l 2.12269E—2 l.74075E—2 2.33424E—2 l.54586E—3 — l.65093E—2

2 (even—odd) I.0l462E0 — l.19094E—l 3.06064E—2 I.56507E—l 2.28$48E—l — l.09130E—l3 (odd—even) l.00366E0 2.l6260E—2 3.48049E—2 8.69919E—4 3.91902E—2 6.6l467E—24 (odd—odd) l.00807E0 — I.84468E—l 2.59041E—l 2.31988E—l 3.26171E—l 4.06200E—l

300 D.N. Poenaru et a!. / Semiempiricalalphahalf-life

Table 2Initial andimprovedvaluesof theadditive parameters(— Ck)

Groupof nuclei

I (even—even) 2 (even—odd) 3 (odd—even) 4 (odd—odd)

Eq. “k” old new old new old new old new

(3) T 21.02 20.789 20.86 20.346 20.64 20.470 20.33 19.758(6) V 0 0.043 — 1.066 —0.339 —0.772 —0.196 —1.114 —0.962(4) K 20.02 20.226 20.5 20.383 20.7 20.643 20.8 20.571(7) H 20.279 20.347 20.279 19.922 20.279 20.051 20.279 19.355(8) P 0 0.008 0 0.026 0 0.004 0 0.003

20.446), respectively,are the additive parameters, correspondingdataare ignored.A blank card (IAnot only Cv and Ci,. = 0) interrupts this sequenceandthe specific op-

erationsfor eachworking modegoesinto action,For IMPROV= 1, in eachgroupof nuclei the

3. Programdescription code1W for writing a table is read.If 1W = 1, atable will appearlater at the output. The inputdata are ordered by increasingvalues of Z, re-

The first information to be read,as input data, questedby the plotting subroutineMZDPLT. Thefrom a cardconcernsthe operationcodeIMPROV subroutineCHIEX computesthe quantityXexp —

(1, 2 or 3) and the number of the groups of eq. (11) — andthe subroutineOTHERTthe DN =

nuclidesNG (1, 2, 3 or 4) for which the program log(TT/TeXp), DV = log(Tv/Texp), DK =

has to run, Note: do not confuseNG with the log(T~/Texp) and DH = log(T~/T~~~),by usinggroupcodegiven above;for exampleif only odd— the parametersgivenin the subroutineADPAR.odd nuclei are considered(group code 4), one Then, from a bidimensional parabolic leasthasNG = 1. square fit madeby the subroutineMZDPFT, in

If IMPROV= 3, the computationsimplied in the reducedvariablesy andz definedby eqs.(10)the eqs.(3)—(8) are performedby usingthe param- and calculatedin the subroutineMZDNTS, oneeters {Bk} and (Ck}, improved values given in gets the new parameters{Bk} if the error codetables 1 and 2. JR =0.

If IMPROV= 1 (self-improving mode), new The subroutineMZDPFT attemptsto estimatevalues for {Bk} and (Ck) are found, and if IM- the parameters(Bk} in a functional relationshipPROV= 2, theseparametersare suppliedas input, by meansof experimentaldata. A standardproce-The value IMPROV= — I is used to stop the dureis to takemoremeasurementsthan the num-execution. bers of unknowns, so that n >6, and to find

Regardlessof the working mode,the main deck B = (Bk) minimizing thesumof the squaresof theof input data for IA A, IZ = Z, QEXP= Q deviations[Xexp(Yi’ z.) — x(y1,z,)]: that is oneseeks(MeV) and TEXP = log

1~~~(s)are storedby the to minimize the functional i~~:R6 —~R definedbysubroutineSTORin different groupsaccordingtothe parity of Z andN. Only if IMPROV 1, has ~ B~— ‘~‘ u — I \12 13the quantityTEXP to be suppliedasinput; in the X~ ‘ ~ ~Xexp\Y,,Z,) X\Y,,Z

1,j

other two modesone can use T~as a referencetime. When the maximum numberof nuclidesin The system to be solved is the set of so-calledanygroupattemptsto be surpassed,IH > 300, the normalequationsof theleast-squaresproblemand

D.N. Poenaru et al. / Semiempiricalalphahalf-life 301

it takes the form five for the errors DN, DV, DK, DH and DP

n versus N for various Z. The coefficient 250 of the~ [Xex (y,z) x(y,z)] Geiger—Nuttal plot was chosen as a convenient= i scaling constant. The factors SV and SH for the

vertical and horizontal scale of each family areX ~ (yj, z~) 0, (k = 1, 2,...,6). (14) also readin PLOTMA and then the plotting sub-

k routine MZDPLT is called. SV is the distance

To solve this systemone usesthe Gausselimina- betweentwo consecutiveprinterlines andSH is ation methodwith completepivot, multiplicativefactor for the abscissa.If, for exam-

If the numberof nuclei n in a group is smaller pie SH = 2, each output column with printedthan 6 (IR < 0), or a lossof significanceis detected quantitiesis followed by a blank one. Due to the(IR > 0), a messageis printed at the output and fact that usually the maximum numberof spacesthe following group of nuclei is analysed.When in aprinter line is 132,onehas(b — a)/SH~ 121,JR= 0, the calculationfor a given group is con- wherea andb are the maximumandthe minimumtinued andunity is addedto the counterLT. The valuesof the basevariable,subroutineESTVPcomputesthe errors of eq. (8): Every plot has a heading indicating the func-DP= log(T~/7~) and these are stored in the tion, the independentvariable andthe parameter.vector VP. The computednew parameters(Bk) For example LOG1O (TV/TREF)—VS.N FORare printed. The meanvalue, c, of the errors and VARIOUS Z meansthat the plotted function isthe standarddeviation log(T~/7~~)where 7~efcould be 7~or T~,the

independentvariableis the numberof neutronsNr -i’.’2I ~ / 2 / I and the parameteris the atomic numberZ, coded

~ ~IogT~ —logT ) i(.n—l)~ , .= ~ / j as is shown in the table which follows. For exam-

Il5\ ple 3 = 52 5 meansthat thereare five pointson the“ ~ curve Z = 52,everyonebeing markedwith 3. The

are calculatedby the subroutineMEARMS, both numberof points on a particularcurve is a usefulfor the valuesof the (Ck } given in table 2 and for indication especiallyat a crossing point, wherethe new onesobtainedfrom the condition e = 0. only one code is printed; the other is lost. TheAt this stage,the initial values of the deviations maximumnumberof distinct valuesof Z is 40.storedin the vectorsVN, VV, VK, VH andVP are After the repetitionof the cycle for eachof thelost; they are replacedby the new onessatisfying NG-l remaininggroups,the counter valueLT isthe eq.(12). TheparameterCk andthecorrespond- tested and if LT ~ NG, the executionstops, Ifing rms valuea are printed,in the first row before LT NG, a new valueof IMPROV is entered,the improvementand in the secondoneafter. It The input datacommon to all working modesmay happen,as in the test run output, that the (seetable3) are readandsorted.If IMPROV = 2,newvaluescoincidewith theold ones.If 1W = 1, a the parameterHK of eq. (4), given by the sub-tablecontainingthe input dataA, Z, Q, log(T~~~) routineADPAR, is storedin the vectorVHK, Theand the calculated quantities N, E~,~ newvaluesof Bk (k = 1,2,,..,6) and CT, Cv, CK,log(T~/T~~~),is printed for all the nuclei of this CH, C~are read for each of the NG groupsofgroup. nuclei andthesequantitiesarestoredin VB, VCT,

The next input data card, requestedby the VCV, VCK, VCH and VCP. Then a messagesubroutinePLOTMA, containsthe plotting codes PARAMETER VALUES ARE GIVEN AT THEIQ, IT, IG, IC, IN, IV, 1K, IX, IP for 9 curves. INPUT shows the working mode at the outputThe correspondingcurveis plottedonly if thecode and the transferto the label 1002 is made, mean-has a unit value.For IQ = 1, one has the family ing that the following statementsarecommonwithQ =f(N) at Z = constant.IT is the codefor the the caseIMPROV = 3.

family log 7~,= f(N), IG for the = As onecanexpect,the samestageof storageofIC for Xexp =f(N) andthe following the parametervalues is accomplishedin case of

302 D.N. Poenaruet a!. / Semiempiricalalphahalf-life

Table3Input datacards

Operationmode Variables Format Number of Groupofcards cardsrepeated

common to all IMPROV, NG 212 1 only if thereareworking modes IA, IZ, QEXP,TEXP 215, F25.3,F 15.3 ii variousoperation

blank I modesin a run

IMPROV I 1W 12 I NG timesIQ, IT, IG, IC, IN, IV, 1K. 912 I with p codes=IIX, IPSV, SH 2F10.5 p

IMPROV=2 BI, B2, B3, B4, B5, B6 6E12.5 I N(; timesCT, CV, CK, Cl-I, CP SF10.3

common to ITX, 1W 212 I NO timesIMPROV=2 and IQ, IT, 10, IC, IN, IV, 1K, 912 I with 1C0 andIMPROV=3 IX, IP p codes I

SV, SU 2F10.5 p

endof the run IMPROV — 1 12 1 —

IMPROV = 3, by calling the subroutinesADPAR cards is presented in table 3; the purposes of thefor (Ck} and {Hk} andBPAR for (Bk). variablesare given in table4.

Beginning with the label 1002, for each of NGgroups of nuclei, the following operations aremade. The parameters (Bk) are printed and the 4. Test run outputcomponentsof the vector VVS, VVN, VVQ andVVT- containing--the-input--dataZ, N, -Q---and-7~- --The--three--workiiigiiiudes of the programare~are ordered by increasing values of Z by the tested by using our set of experimental datasubroutine MZDORD. The codes ITX, 1W are selectedas mentionedabove. In this way everygiven as input. If ITX 1 one has ~ 7~,; user can either utilize this set, or improve it. Inotherwise 7e1 = T~and this is specified at the order to get the testrun output to be as short asoutput. possible,while showingat the sametime its versa-

The time TP log T~is calculated in the sub- tility, the data for various nuclides were sortedroutinePOTIM andthe errors,DN, DV, DK, DH into four groupsaccordingto the parity of Z andwith respectto are obtainedin the subroutine N, an operationwhich is usually madeautomati-OTHERT. The standarddeviation a~are calcu- cally.lated only if n ~ 6. Then a table similar to that I. As one can seein table4, at the beginningwritten for IMPROV 1 is printed at the output the self-improvingmodeis illustratedfor the odd—if 1W 1. Of course,this time the quantityXexp is even(group3) nuclei. It is not surprising that theno longerpresent,and T = T~,if ITX ~ 1. Onehas new parameterscoincidewith the old ones,as thealso to put IC = 0 in the following data card experimentaldata were not changed.From thecontainingthe plotting codes1W, IT, IG, IC, IN, tableof the absoluteerrorsandfrom the valuesofIV, 1K, IP, IX of which p havethe value1. Corre- the standarddeviationsRMS, one can seethat aspondingly the numberof SV, SH cardsmust be satisfactoryagreementwith experimentalresultsisp ~ 8. obtained with all eqs., except from some light

The executionstopsif IMPROV = — 1, the last alphaemitterswere the errors are larger than aninput data card. A summary of the input data order of magnitude.

D.N. Poenaruet al. / Semiempiricalalphahalf-life 303

iN ‘~1 C’ ~‘0 C

I I~0

C ~fl (N ~ I ~ ~ — ~ o ~ — I I C

‘0 0’ —‘a C

I C Cr~i r- (N ‘0 I C — C C C C C C C — — I I C

Sc C

‘~~1 C (N(N —

I I I I — r~i~.°i ~ ,t~

(N — SC (N — r— I C C C C C — C C — C —, .0 0 .0 0~i~b

V SC ©

-~ ‘110’ ~a ~0C —(N —

~ c-!. — C C C C C C C C C I I I

— ——

‘11

B 11’11 c.i ~

LI~ .L, ‘a 11 (N

-;~

~ .~ ~ ~~ ‘~ .B ~

-~ i ~ Uv ~ k” k’ k” k~ ~— 0,’C ‘~ “-‘~ .0 ~ ~ I o

C.~ ,~ ~ .-~ i~ ~‘ a -“-‘V .>kk.-.-kkkE~,k .,a ‘ ‘t .0~ ,~ ~ ~5OC~5S5 ‘°.B °‘ .‘

,~ ~ ~ ~(‘1 8~v ~ .~ ~Q~~QOO ~ ~

C .0. B ~ o. ~‘a-a’a’~~-a ~ ‘~ ~-

.3E~ ~ ~ ~ ~ ~

~, ~ ~EB~’.~C V VVVVVVVVV ~ ~ :~:: i

~ .~

0

“0(N

o ~‘ ‘0’ Sc

304 D. N. Poenaruet a!. / Semiempiricalalphahalf.life

II, The IMPROV 2 mode(parametersBk and [3] V.E. Viola Jr.and G.T. Seaborg,J. Inorg. NucI. Chem.28

Ck given on input) is testedfor the group1 (even— (1966) 741.

even nuclei). In this case the RMS values are of [4] K.A. Keller and H. Munzel,Nucl. Phys.Al48 (1970) 615;Z. Phys. 255 (1972) 419.the same order of magnitude as the preceding [5] P. Hornsbe~j.PG. Hansen,B Jonson,L. Ravn,L. West-

ones,Thetableof 123 nuclides is not printed. The gaardandON. Nielsen,Nucl. Phys.A230 (1974) 365.first plot show the good agreementof eq. (6) with [6] D.N. PoenaruandM. Iva~cu,Invited lecturepresentedat

experimentaldata in the region of heavy alpha the InternationalSummerSchool on Critical Phenomena

emittersanda systematic increase of the absolute in Heavy Ion Physics (25 Aug.—10 Sept. 1980) PoianaBra~ov,Romania.

valueof errorsin the regionof the magicneutron [7] D N. Poenaru,M. Ivalcuand D Mazilu, J. dePhys.Lett

number N = 126 and of the light alphaemitters 41(1980)L-589.

(Z < 64). The advantageof usingeq. (8) is clearly [8] D.N. Poenaru,M. Iva~cuand A. Sàndulescu,J. de Phys.

seen both from the second plot and from the Lett. 44 (1979) L-465; Rev. Roum.Phys.24(1979)917: J.

reducedvalueof the standarddeviationa~for all Phys.G 5 (1979) L-169.[9] A. Rytz, At. DataNucl. Data Tables23 (1979) 507.

the analysedgroupsof nuclei. [10] H. Gauvin, Y. Le Beyec,J. Livet andJ.L. Reyss,Ann. deIII. The even—odd (group 2) and odd—odd Phys. 9 (1975) 241.

(group4) set of dataare usedto testthe IMPROV [II] C. Cabot,S. Della Negra,C. Deprun,H. Gauvin andY.

= 3 operationmode (unmodifiedparameters).In Le Beyec,Z. Phys.A 287 (1978) 71.

the former case, the referencetime is the experi- [12]5. Hofman, W. Faust, G. MUnzenberg,W. Reisdorf, P.Armbruster,K. Güttner and H. Ewald, Z. Phys. A 291

mental one both for computingRMS values and (1979) 53.

for the plot showingthe systematicsof the alpha [13] E. Hagberg,PH. Hansen,P. Hornsh~j,B. Jonson. S

decaylife time. In the latter one,as is mentioned Mattson and P. Tidemand-Peterson, NucI Phys. A 318

at the output, the referencetime is T~. (1979) 29.By analysingthe diagramand the RMS values [14] K.H. Schmidt,W. Faust,G. Mtinzenberg,HG. Clerc, W.Lang, K. Pielenz,D. Vermeulen,K. Wollfarth, N. Ewald

of this groupandthoseof the precedingones,one andK. Guttner,NucI. PhysA 318 (1979) 253.

can concludethat the Keller—MUnzel eq. (4) gives [IS] D. Schardt,R. Kirchner, 0. Klepper, W. Reisdorf, E.

the closestresults to eq. (8). But even in this case Roeckl, P. Tidemand-Peterson,G.T. Ewan,E. Hagberg,B.

one can have rather large differences in the Jonson,S. Mattson and G. Nyman, NucI. Phys. A 326

neighbourhoodof the magic numberof neutrons. (1979) 65.[16] V.M. Gorbachev, Yu.S. Zamiatnin and A.A. Lbov,

Osnovnye haracteristiki izotopov tiajelyh elementov(Atomizdat, Moscow, 1975).

References [17]CM. Ledererand VS. Shirley, Table of isotopes(JohnWiley, New York, 1978).

[I] AU. WapstraandK. Bos,At. DataNucl. DataTables19 [18] W. Seelmann-Eggebert,G. Phennig and H. Munzel,(1977) 215. Nuklidkarte,KernforschungszentrumKarlsruhe,4 Auflage

[2] R. TaageperaandH. Nurmia, Ann. Acad. Sci. Fenn. Ser. (1974).A 78(1961).

D.N. Poenaruet al. / Semiempiricalalphahalf-life 305

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c_i “. ..............s.... •••.....•••••Z~ O~ 00000000000000000000000000000000LU I- I III I I i 11111 p ii

ci-J

-, .or’-,-imtcim.tc-40’m1nI’- ‘O0’Ocnr’-1-.0 P-U’it’-LflQs i~1’4Lu0Sm~~ci 1’ (“&OD-’WiI”V ‘0U~t’i4”O4’ 4fl0”tO 00 0~0) .400Q(-wfl’0~J

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