University of Groningen Development of an atomic clock for ...æ 698;S 8EY 698;S 8E

University of Groningen

Development of an atomic clock for fission lifetime measurementsKravchuk, Vladimir Leonidovich

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Publication date:2004

Link to publication in University of Groningen/UMCG research database

Citation for published version (APA):Kravchuk, V. L. (2004). Development of an atomic clock for fission lifetime measurements. s.n.

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4He 25 MeV4He 40 MeV

4He 74 MeV

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with GBW the Bohr-Wheeler fission width, g the nuclear vis-cosity, t the time, and t the time at which the flux across thebarrier reaches 90% of the quasistationary value. We param-etrized t anew from the results of Grange et al.

t~b ,T !5~2g !21ln~10E f /T !10.0112gA/T @10221 s# . ~4!

Here, E f is the height of the fission barrier, T is the tempera-ture, and A is the mass number. t has been calculated forevery nuclear state in the cascade. Another expression for twas obtained by Bhatt et al. @16#, but the g-ray spectra cal-culated with this expression are hardly different.

In the original version of CASCADE the probability fordecay of a nucleus moving from the saddle point to the sciss-ion point is not calculated. To take this effect into account,the fission width has been treated analogously to the fissionflux. The flux at time t at the saddle point is ~almost! equal to

the flux at the scission point at time t1tssc , with tssc the timerequired to propagate from the saddle point to the scissionpoint @17#:

tssc5tssc0 ~A11g2

1g !. ~5!

From the graphs presented by Grange et al. @15#, a value fortssc0 the saddle-to-scission time in nonviscous nuclei, can beinferred: tssc

052.2310221 s. Assuming the same time depen-

dence for the fission width, one can approximate the latter asfollows:

G f~ t !50, t,tssc

5G fBW~A11g2

2g !

[email protected]~ t1td2tssc!/t#%, t.tssc . ~6!

The concept of time, which is unknown in the statistical codeCASCADE, was implemented as follows. The lifetime td of anuclear state with excitation energy E* and angular momen-tum J , which is used as a time step in the calculations, isgiven by

td5\

Gn~E*,J !1Gp~E*,J !1Ga~E*,J !1Gg~E*,J !1G f~E*,J ,t !~7!

in which Gn etc. are the decay widths for all decay channelstaken into account by CASCADE: i.e., neutron, proton, a par-ticle, g ray, and fission decay. The justification for the use ofG f in this equation is that the lifetime of a state is inverselyproportional to the total width including G f . The bookkeep-ing of these time increments is done in matrices t(E*,J)with the same dimensions as the matrices for the populationcross sections, i.e., every nuclear state is characterized by the

charge, mass, excitation energy, angular momentum, and thetime at which the decay to it took place. Note that Eqs. ~6!and ~7! are coupled; therefore, they are solved iteratively.Including G f in Eq. ~7! will influence through the coupling toEq. ~6! the fission probability and thereby the GDR yield.

In the original CASCADE code no distinction is made be-tween compound nuclear fission and quasifission. We imple-mented this distinction as follows. With the one-body dissi-

FIG. 2. Results of the fits of the GDR parameters at four differ-ent values of g, for the experiment with Eb5216 MeV. The solidcurve indicates the sum of the prefission spectrum ~short-dashedcurve! and the postfission spectrum ~long-dashed curve!. For com-parison, a calculation with g55 is also shown ~dotted curve!.

TABLE I. Results of the fission time scale for the differentvalues for the viscosity.

Eb g Bscale t fa

@MeV# ~10220 s)

216 0.01 1.0 4567216 0.1 1.2 1565216 1 1.0 4768216 4 0.7 49362238 0.01 0.8 4561238 0.1 1.0 861238 1 0.8 4761238 2 0.65 24061

aThe numbers used as error bars indicate the sensitivity to thethreshold on the fission cross sections that is taken into account inthe calculation. Here, it represents the difference in t f for thresholdof s f50.5 mb compared to 0 threshold.

54 R1517VISCOSITY AND FISSION TIME SCALE OF 156Dy

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assumption

v v

−v+2w −v+2w−v

v+2w

w w w w

−v+2w

v chaotisity

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ò Í °ÍzÏ Q ÿ U ¦ ¼GPxËû ¡Ô ¤ jû ¡R |0Ã x' bEE|

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ò Í °ÍÏ Q ÿ U ¦ ¼PxËû ¡Ô ¤³jû ¡R |¤ ÿH

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Additional information about caloric curves was reportedby the INDRA Collaboration @28–30#. It is our understand-ing that the excitation energy determinations and tempera-tures for those experiments are currently under review @31#;therefore, we have not included them in this work.

B. Correlation of temperature, excitation energy, and mass

In Fig. 1 we plot the correlated values of the temperaturesand excitation energies per nucleon, which have been deter-mined in the experiments represented in Table I. For com-parison, we plot curves corresponding to Fermi-gas modelpredictions with inverse level density parameters K58 and13. Also included, for further reference, is a ‘‘total vaporiza-tion’’ line representing the ~purely hypothetical! two-stagescenario of separation into constituent nucleons, at a cost of8 MeV/nucleon followed by thermal heating. Although thereis a significant increase in the divergence of the results athigher excitation energies, the combined data still exhibit thequalitative features observed previously in many of the indi-vidual experiments, i.e., an apparent Fermi-gas-like rise atexcitation energies per nucleon below 3–4 MeV/nucleon, avery slow rate of temperature increase at higher energies(;4 –9 MeV/nucleon), and some indication of an increaseagain at higher energy, ;9 MeV/nucleon.

In some of the experiments considered, the more centralcollisions were selected by an appropriate experimental filter,and the excitation energy was varied by changing the projec-tile energy @9,10,18,19#. In such cases the masses of the ex-cited systems ~following pre-equilibrium emission! were rea-sonably well constrained. In other experiments a single

projectile energy was used, and a range of impact parametersfrom peripheral to central was investigated @6–8,11–14,16,17#. In those cases the initial masses and excitationenergies of the excited systems studied may vary signifi-cantly with impact parameter. That the results included inFig. 1 include experiments which span a broad range of massis illustrated in Figs. 2 and 3, which present plots of theprimary excitation energies and temperatures as a function ofthe derived values of the primary mass. The wide mass varia-tion inherent in many of the individual experiments is clear,as is the fact that different experiments may sample the samemass range at significantly different excitation energies. Wepreviously suggested that mass variation is an important fac-

FIG. 1. Caloric curve data from Refs. @6–20#. Measurements oftemperature vs excitation energy per nucleon are represented bysymbols. Reference @6#, open circles; Ref. @7#, solid black circles;Ref. @8#, solid black diamonds; Ref. @9#, open triangles; Ref. @10#,open square; Refs. @11,12#, solid black squares; Ref. @13#, solidblack diamonds (84Kr), open diamonds (134La), and shaded dia-monds (197Au); Ref. @14#, shaded triangles; Ref. @15#, cross inshaded squares; Refs. @16,17#, shaded circles ~Ag!, and shadedsquares (197Au); Refs. @18,19#, 3 in shaded squares; Ref. @20#, 3’s.Fermi-gas model lines for K58 ~dashed line! and K513 ~solidline!, and a ‘‘total vaporization’’ line ~see the text!—connectedsmall open circles are shown for comparison.

FIG. 2. Excitation energy per nucleon as a function of A. Ref-erence @6#, open circles; Ref. @7#, solid black circles; Ref. @8#, solidblack diamonds; Ref. @9#, open triangles; Ref. @10#, open square;Ref. @11,12#, solid black squares; Ref. @13#, solid black diamonds(84Kr), open diamonds (134La), and shaded diamonds (197Au); Ref.@14#, shaded triangles; Ref. @15# cross in shaded squares; Refs.@16,17#, shaded circles ~Ag! and shaded squares (197Au); Refs.@18,19#, 3 in shaded squares; Ref. @20#, 3’s.

FIG. 3. Temperature as a function of A, the mass number of theprimary de-exciting nucleus. Reference @6#, open circles; Ref. @7#,solid black circles; Ref. @8#, solid black diamonds; Ref. @9#, opentriangles; Ref. @10#, open square; Refs. @11,12#, solid black squares;Ref. @13#, solid black diamonds (84Kr), open diamonds (134La), andshaded diamonds (197Au); Ref. @14#, shaded triangles; Ref. @15#

cross in shaded squares; Refs. @16,17#, shaded circles ~Ag!, andshaded squares (197Au); Refs. @18,19#, 3 in shaded squares; Ref.@20#, 3’s.

CALORIC CURVES AND CRITICAL BEHAVIOR IN NUCLEI PHYSICAL REVIEW C 65 034618

034618-3

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University of Groningen Development of an atomic clock for ...æ 698;S 8EY 698;S 8E