Simulation of angular and energy distributions for heavy evaporation residues using statistical model approximations and TRIM code

Nuclear Instruments and Methods in Physics Research A 700 (2013) 111–123

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Simulation of angular and energy distributions for heavy evaporation residues using statistical model approximations and TRIM code R.N. Sagaidak a,n, V.K. Utyonkov a,1, F. Scarlassara b,2 a b

Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, Dubna 141980, Moscow Region, Russian Federation ´ di Padova, Dipartimento di Fisica ‘‘Galileo Galilei’’, 35131 Padova, Italy INFN Sezione di Padova and Universita

a r t i c l e i n f o

abstract

Article history: Received 28 July 2012 Received in revised form 20 September 2012 Accepted 16 October 2012 Available online 23 October 2012

A Monte Carlo approach has been developed for simulations of the angular and energy distributions for heavy evaporation residues (ER) produced in heavy ion fusion-evaporation reactions. The approach uses statistical model approximations of the HIVAP code for the calculations of initial angular and energy distributions inside a target, which are determined by neutron evaporation from an excited compound nucleus. Further step in the simulation of transmission of ER heavy atoms through a target layer is performed with the TRIM code that gives final angle and energy distributions at the exit from the target. Both the simulations (neutron evaporation and transmission through solid media) have been separately considered and good agreement has been obtained between the results of simulations and available experimental data. Some applications of the approach have been also considered. & 2012 Elsevier B.V. All rights reserved.

Keywords: Monte Carlo simulation Fusion–evaporation reactions Statistical model Recoil separators Multiple scattering Angular and energy distributions Transmission efficiency

1. Introduction Angular, energy and charge state distributions of heavy evaporation residues (ERs) produced in complete fusion reactions induced by heavy ions (HI) are the main input parameters needed to evaluate the performance of kinematical recoil separators for low-energy HI. One can find a number of figures with calculated angular distributions for ERs in the literature. Thus, for example, calculations performed with the RECOIL code (written by Reisdorf from GSI and, unfortunately, unavailable from the literature) demonstrate good agreement with angular distributions measured for reactions leading to the 58Nin [1] and 111 n In [2] compound nuclei. This code was used for the calculations of ER angular distributions in reactions leading to heavier, Hgn compound nuclei. The calculated angular distributions were used for estimates of the performance of the recoil separator SHIP installed at the UNILAC accelerator in Darmstadt [3,4]. Later, simple formulae for estimates of parameters of angular and energy distributions for heavy ERs produced in compound nucleus (CN) reactions were published [5]. Those formulae imply a Gaussian form of angular and energy spreading for ERs due to

n

Corresponding author. Tel.: þ7 49621 64170; fax: þ7 49621 65083. E-mail addresses: [email protected] (R.N. Sagaidak), [email protected] (V.K. Utyonkov), [email protected] (F. Scarlassara). 1 Tel.: þ7 49621 64246; fax: þ7 49621 65083. 2 Tel.: þ39 049 827 5911; fax: þ 39 049 827 7102. 0168-9002/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2012.10.064

light-particle emission from the CN followed by multiple scattering (MS) and energy loss of the ER in a target of finite thickness. Numerical implementation of the formulae presented in Ref. [5] with the use of the systematics of nuclear temperature inherent in (HI, xn) reactions [6] for estimates of the kinetic energy of emitted neutrons and a Gaussian approximation to the theory of small-angle MS of ions [7] allowed to obtain quite satisfactory agreement of the calculated ER angular distributions with available experimental data at small deflection angles [8]. At the same time, the Gaussian approximation, as might be expected, does not allow to describe ER angular distributions measured at relatively large deflection angles [9,10]. Within recent years a number of Monte Carlo (MC) simulation codes incorporating calculations of energy and angular distributions for ERs have been mentioned in the literature. These codes are intended for calculations of ERs transportation through different recoil separation systems installed at HI low-energy accelerators. With the use of one of them, which was developed for the simulation of ion trajectories and optimisation of ERs transmission through the recoil separators VASSILISSA and SHIP, the ER angular and energy distributions were calculated for the specific reactions [11,12], however no details were given on the assumptions and approximations used in the simulation. The description of the ANAMARI code [13] applied to the calculations of ER collection efficiencies and position spectra of the Dubna gas-filled recoil separator (DGFRS) installed at the

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U400 cyclotron in Dubna implies an approach similar to the one used in Ref. [8] for the calculations of the MS contribution to ER angular distributions. At the same time, a contribution due to light particles evaporation from a CN is not considered in details. No ER angular and energy distributions calculated according to the description are presented in Ref. [13]. The TERS code [14], developed to calculate the transmission efficiency of the HIRA recoil separator, treats light-particles evaporation in great detail, whereas the treatment of MS is questionable. For the latter, the author [14] used the approximations [15] applicable for the transmission of relativistic particles through matter, which do not take into account the Coulomb screening of interacting atoms (see, for example [7]). Since the applicability of the approximations [15] was not discussed and no comparison of simulations with experimental data was provided, the angular distributions presented in Ref. [14] might be questionable. Recently the more sophisticated MOCADI-FUSION code was presented [16] for the description of transmission of ERs produced in fusion-evaporation reactions at near Coulomb-barrier energies through the recoil separator SHIP. The code includes the statistical code PACE2 [17] describing the nuclear reaction process leading to the production of ERs. It calculates probabilities for light particles emission up to the final ER and determines the angular distribution of ERs caused by this process. The resulting distributions are used as an input file for the ERs transmission through a target, which is simulated with the ATIMA [18] or TRIM [19] codes and thus the ER distributions at exit from a target are obtained. These distributions could be compared with the measured ones available from the literature to appreciate the validity of such simulations. However, one should mention that in this approach [16], as well as in those considered above [3–5,11–14], no comparisons of the simulated distributions with measured ones are presented. In the present work, the generation of the initial ER angular and energy distributions caused by neutron evaporation from a CN is considered in detail and compared with experimental data in Section 2. In Section 3, the suitability of the TRIM code [19] to the simulation of the angle and energy distributions for HI transmitted through solid targets is examined by the comparison of measured distributions available from literature with the simulated ones. In Section 4, some examples of the ER angular distributions resulted in simulations are compared with the corresponding ones obtained in experiments with thin targets. The use of the proposed approach to the study of ER charge state distributions and to the estimate of an optimal target thickness in experiments on synthesis of the heaviest nuclei is also considered in this section. Summary of the paper is drawn in Section 5.

2. Neutron evaporation from CN As was mentioned earlier in a number of works (see, for example [8,13,20,21]), the differential ER angular distributions ds=dO as a function of the W angle in the lab system, which are obtained in fusion-evaporation reactions in thin target experiments, can be approximated by a Gaussian

s

ds=dO ¼ pffiffiffiffiffiffi exp½0:5ðW=sW Þ2  2psW

ð1Þ

where s is the integral production cross-section of ER and sW is the standard deviation of the distribution. As proposed in Ref. [5], in the case of an infinitely thin target, sW corresponding to light particles evaporation alone could be estimated as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sW ¼ xe=ðEER AER Þ ð2Þ where x is the number of evaporated particles from a CN with the average kinetic energy e in the c.m. system, EER and AER are the

energy and mass number of the ER, respectively. The ER kinetic energy is expressed via the projectile energy Ep as EER ¼ Ep Ap AER =ðAp þ At Þ2

ð3Þ

where Ap and At are the mass numbers of projectile and target nuclei, respectively. The ER energy distribution, corresponding to light particles evaporation, could be similarly [5] approximated by a Gaussian with the standard deviation sE pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sE ¼ 2 xeEER =ð3AER Þ: ð4Þ Earlier, in the consideration of the ER ds=dW angular distributions obtained in (HI, xn) reactions [22], it was shown that the 2 average square of angle /W S in the lab system could be estimated as 2

/W S ¼ 8T n AER =½3EER ðAp þ At þ AER Þ2 

ð5Þ

where Tn is the average total kinetic energy of evaporated neutrons in the c.m. system (the neutron mass corresponds to the 1 amu). 2 The relationship between /W S and sW could be obtained using the connection of ds=dW and ds=dO ds=dW ¼ 2p sin Wðds=dOÞ

ð6Þ

2

a definition of /W S and Eq. (1) for the ds=dO distribution that give us Z 1 Z 1 2 /W S ¼ W2 sin Wðds=dOÞ dW sin Wðds=dOÞ dW: ð7Þ 0

0

The integrals in Eq. (7) could be replaced by sums [23] with the accuracy sufficient for the present consideration (bearing in mind smallness of sW ) and thus a numerical relationship between 2 /W S and sW could be established (see, for example [8]). So, we obtain Eqs. (1)–(7) describing the ER angular and energy distributions for an infinitely thin target. For the estimates of e or T n one can use the values calculated with the statistical model PACE2 [17] used in Ref. [16] or similar. The problem is the only one of a choice of values of statistical model parameters that are in abundance. In this work, the wellknown and widely distributed HIVAP code [24] was used for the estimates of the e or T n values. Their ‘proper’ values were found with a fit of the calculated excitation functions to the measured ones for the reaction under investigation or for a similar one. An example of such fit to the measured excitation functions for (HI, xn) reactions resulting from the 156Dyn CN formation [25] are shown in Fig. 1, where the production cross-sections for Dy nuclei are presented as a function of the CN excitation energy calculated with the empirical masses [26] in the usual way. The data fit was obtained with a variation of nuclear potential parameters, a scaling factor at the liquid-drop fission barriers, parameters describing g emission and with a choice of an asymptotic leveldensity formula, i.e., the main HIVAP parameters described in Ref. [24] which determine (HI, xn) reaction cross-sections. Some examples of the model application to the description of the Po nuclei production in different fusion-evaporation reactions are presented in Ref. [27]. They are accompanied by the discussion on the parameter values selection. As the result of calculations HIVAP yields the kinetic energies of evaporated neutrons ein at each step of the CN de-excitation cascade along with the corresponding cross-section values for fission sifis and for intermediate nucleus production sitot starting with sCN . So, the average neutron kinetic energy over a cascade leading to the final ER, en , could be obtained as a mean-weighted value

en ¼

n X i¼1

, i i i n ð tot  fis Þ

e s

s

n X

ðsitot sifis Þ:

i¼1

ð8Þ

R.N. Sagaidak et al. / Nuclear Instruments and Methods in Physics Research A 700 (2013) 111–123

103

Nd(12C, xn)156-xDy

144

113

Simonoff & Alexander approach

0.20

146

Nd(103.7 MeV 11B,8n)149gTb Nd(122.8 MeV 12C,7n)149Dy 140 Ce(111.0 MeV 16O,7n)149Dy 144

7n

0.15

102 6n

0.10

5n

0.05 Dy (5n) Dy (6n) 149 Dy (7n) HIVAP (xn) 151 150

103

PER(θ)

Cross section (mb)

101

Dahlinger et al. approach

0.20 7n

102

0.10

6n

5n

101

0.15

0.05 Ce( O, xn)

100

140

16

156-x

Dy

0

40

60

80

100

E*CN (MeV) Fig. 1. The 144Nd(12C, xn) and 140Ce(16O, xn) cross-sections as a function of the 156 Dyn CN excitation energy. Different symbols are the results of measurements [25] (10% arbitrary values of error bars were added to the data). Solid lines are the results of the best fitted calculations obtained in the present work with HIVAP [24].

The en values calculated with HIVAP and Eqs. (1)–(8) allow to calculate the angular distributions for 149gTb and 149Dy which are caused by neutron evaporation from the 157Tbn and 156Dyn compound nuclei, respectively, produced in reactions with 11B, 12 C and 16O. The calculated distributions could be compared with the corresponding ones obtained in Ref. [22]. The latter were measured at different target thicknesses following with the reduction to the probability function PðWÞ ¼ ½dsðWÞ=dW=s corresponding to the infinitely thin target (W t ¼ 0 in Ref. [22]). This comparison is shown in Fig. 2 for the cases of the use of Eq. (2) for the standard deviation in the angular distribution described by Eq. (1) and of Eq. (5) for the average square of angle in the angular distribution described by Eqs. (1) and (6), respectively. Obviously, en and Tn are connected with the relation T n ¼ xen :

4

8

12

16

θlab (deg)

120

ð9Þ

As one can see, both Eqs. (2) and (5) give very similar results and reproduce the experimental distributions rather well with the mean kinetic energies of evaporated neutrons, which are estimated with the statistical model approximations used in HIVAP. At the same time, in spite of such agreement and bearing in mind future applications, it is desirable to extend the approach using the kinetic energy spectra generated by HIVAP for neutrons and light charged particles. It could be important for the angular distributions corresponding to proton and alpha particle evaporation, since the use of Eqs. (1)–(9) seems to be questionable in these cases.

Fig. 2. Angular distributions for 149gTb and 149Dy (probability of escape at the W angle), which correspond to neutron evaporation from the 157Tbn and 156Dyn compound nuclei, respectively. Symbols correspond to the data obtained in Ref. [22] for a zero target thickness, whereas lines are the results of calculations obtained with the mean kinetic energies of evaporated neutrons, which are obtained with HIVAP. In the HIVAP calculations the same parameter values were used as those for the calculated excitation functions best fitted to the data [25] shown in Fig. 1. Eqs. (2) and (5) are used for the estimates of the standard deviation sW in the calculations shown in the bottom and upper panel, respectively.

3. ER multiple scattering and stopping inside a target As mentioned in Section 1, MS and stopping of ERs inside a target layer were treated within a framework of the MC TRIM code [19] in our approach. A huge number of works demonstrates a general agreement between measured stopping powers and those obtained by SRIM/TRIM calculations/simulations for different atomic numbers and energies of HI passed solid media (see some discussion below). Unfortunately, there is a lack of data on MS of very heavy ions with energies common to ERs traversing heavy-elements materials. Generally, available data on measured HI angular distributions are mostly limited by half-widths of measured distributions. At the same time, the half-widths of MS angular distributions obtained in experiments (see, for example [28,29]) are in rather good agreement with the results of calculations [7]. Results are conveniently compared in reduced coordinates of a dimensionless target thickness t and half-width angle a1=2 . According to Ref. [7], these quantities are defined as

t ¼ pa2 W t NA =At

ð10Þ

where a is a screening radius, NA is the Avogadro constant and W t is a target thickness in g/cm2

a1=2 ¼ Ep aW1=2 =ð2e2 Z p Z t Þ

ð11Þ

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where W1=2 is the MS half-width angle measured in a lab system, Z p and Z t are the atomic numbers of a projectile and target, respectively, and e2 ¼1:44  1013 MeV cm. There is a question in a choice of the screening radius (see Refs. [28,29] and discussion in Ref. [30]) when experimental data are compared to the theory [7]. Some deviations from the theoretical dependence a1=2 ¼ f ðtÞ [7] are probably observed for very heavy U ions [28] and for large reduced thicknesses (t 4 102 ) [29]. In order to check the suitability of TRIM to the predictions of ER angular distributions, we have compared the reduced values of the half-width of angular distributions obtained with simulations to the dependence a1=2 ¼ f ðtÞ given by the theoretical calculations [7,30]. In our TRIM simulations for ERs we have chosen the projectile-target combinations mentioned in Refs. [1–5,9–12,14,16,25]. The ERs were interpreted as HI passing a whole target thickness. Data for these combinations and a1=2 values obtained in simulations are listed in Table 1. In Fig. 3 these values are compared with the numerical calculations [7,30] corresponding to the Lenz-Jensen potential. We also included the data derived from measured angular distributions shown in the figures of Refs. [28,31,32], which correspond to 1.2 MeV/amu of 238U passed through a Bi target and to 20Ne, 209Bi and 40Ar passed through thin carbon foils at the input velocity V ¼ 0:8V B (V B is the Bohr velocity). The corresponding a1=2 values obtained with TRIM simulations for these ion-target combinations are also included in Table 1. Using Eq. (11), the transformation of the W1=2 values derived from the measurements and simulations to the reduced a1=2 ones was performed with an empirical screening radius proposed in Ref. [33] aZBL ¼ 0:8853r B =ðZ 0:23 þ Z 0:23 Þ p t

ð12Þ

where r B is the Bohr radius. As one can see in Fig. 3, data obtained with TRIM simulations retrace in outline the theoretical curves [7,30].

In Fig. 4 the angular distributions for 20Ne, 209Bi [31] and 40Ar [32] at the initial velocity V ¼ 0:8V B , which are measured after passing thin carbon targets, are shown together with the corresponding TRIM simulations and analytical calculations [7]. The Lenz-Jensen potential together with the empirical screening radius aZBL calculated according to Eq. (12) were used in calculations. As one can see in the figure, good agreement at relatively small angles is obtained for the 20Ne and 40Ar experimental data compared to the results of calculations [7] and simulations [19]. At the same time, in the case of 209Bi, the simulation seems to follow the data while the analytical calculation [7] overestimates the data at angles larger than the maximal one determined as sin Wmax ¼ At =Ap , when Ap 4At (see the discussion in Ref. [30]). One can state that TRIM reproduces MS angular distributions of HI transmitted through the solid foils of a different thickness rather well and can be applied to simulations of ERs recoiling from a production target. Estimates of stopping of heavy atoms inside heavy targets seem not to be in question, since a large number of measurements demonstrates a good agreement of experimental data on stopping power with SRIM (analytical) calculations [19] (see, for example, such comparison in the database collected by Paul [34]). In Fig. 5 we compare stopping powers calculated by SRIM with the most significant data concerning the heaviest ions at energies above 0.15 MeV/amu, namely, Pb and U transmitted through heavy absorbers like Sn and Au [35,36]. Good agreement between the measured and calculated values allows to use SRIM for estimates of ER stopping in heavy targets, if fusion-evaporation reactions with rather heavy projectiles (heavier than Ar) are considered. With lighter projectiles, heavy ERs are produced with lower energies ( t0:15 MeV=amu) for which no data on stopping powers for corresponding ions, such as Au, Pb, Bi and U in Ta, Au, Pb and U, are available. We examined SRIM suitability for

Table 1 ER data on the projectile-target combinations mentioned in Refs. [1–5,9–12,14,16,25] and HI data [28,31,32] on transmission through the targets of a different thickness. The values of the reduced target thicknesses t corresponding to those listed in the table in mg/cm2 are obtained with Eqs. (10) and (12). The reduced half-widths a1=2 of MS angular distributions are obtained from the TRIM-simulated W1=2 values using Eqs. (11) and (12). Projectile

Projectile energy (MeV)

Target

Target thickness (mg/cm2)

ER

ER energy (MeV)

t

a1=2 (TRIM)

Reference

46

266.8 499 194 236 496 647 88 95 140 99

12

0.17 0.38 0.58 0.1

54

197.0 354 35.3 49.9 218 442 7.57 8.80 20.5 16.4 16.1 9.86 26.5 36.3 50.7 7.80 10.4 11.2 61.7 4.58 33.6 34.0 141.3 4.12 9.27 18.2 28.6

43.9 32.3 4.59 1.02 1.55 3.15 2.19 0.310 0.671 4.10 4.03 2.40 5.64 8.38 5.39 1.60 1.62 8.50 6.57 2.37 2.20 3.71 6.09 0.326 0.544 0.852 1.36 1.40 4.61 2.60 1.40

4.52 3.77 1.25 0.324 0.426 0.766 0.590 0.0769 0.193 1.03 1.08 0.680 1.49 2.04 1.39 0.434 0.452 2.13 1.62 0.688 0.631 0.915 1.49 0.0771 0.157 0.270 0.369 0.394 1.12 0.702 0.435

[1] [2] [3] [4]

Ti Kr 40 Ar 40 Ar 84 Kr 132 Xe 16 O 19 F 30 Si 22 Ne 84

22

Ne P 40 Ar 48 Ca 16 O 19 F 31

50

Ti C 40 Ar 12

84 12

22

Kr C Ne

238

U Ne 209 Bi 40 Ar 20

100 170 187 225 100 110 68 215 60 173 186.4 350 53.9 122.9 140.5 223.3 300 0.317 3.32 0.635

C Al 175 Lu 144 Sm 102 Ru 56 Fe 166 Er 181 Ta 170 Er 107 Ag 109 Ag 197 Au 164 Er 162 Dy 159 Tb 184 W 175 Lu 93 Nb 120 Sn 142 Nd 162 Dy 175 Lu 124 Sn 142 Nd 27

138

209 12

Ba

Bi C

0.25 0.04 0.08 0.26

Ni In Ac 179 Hg 181 Hg 183 Hg 178 Os 195 Pb 104 210

125

La La 215 Ac 191 Bi 198 Po 201 At 195 Pb 188 Hg 109 Sn 166 Hf 151 Dy 198 Po 211 Ac 207 Rn 151 Dy 149 Dy 151 Dy 149 Dy 127

0.35 0.65 0.95 0.6 0.21 0.2 0.44 0.5 0.218 0.25 0.469 0.5 0.03 0.05 0.075 0.12 0.22 0.0137 0.00483

[5] [9] [10] [11]

[12] [14]

[16]

[25]

[28] [31] [32]

R.N. Sagaidak et al. / Nuclear Instruments and Methods in Physics Research A 700 (2013) 111–123

100

VHI = 0.8VB

α1/2

Ne => C (13.7 μg/cm2) TRIM simulation Ar => C (4.83 μg/cm2) TRIM simulation Bi => C (13.7 μg/cm2) TRIM simulation

10-1

Intensity ~ dσ/dΩ

100

10-1

SW-table ABL-table Ne, Ar, Bi => C, expt TRIM simulation U => Bi, experiment TRIM simulation ER => targ, TRIM sim

10-2

10-2

10-3

10-4

209

10

-1

10

0

τ

101

115

10-5 10

2

Fig. 3. The reduced half-width values of the angular distributions obtained with TRIM simulations (symbols) and theoretical dependence a1=2 ¼ f ðtÞ (lines) given by the calculations [7] (SW-table) and [30] (ABL-table) using the Lenz-Jensen potential. The data listed in Table 1 for ERs produced in the projectile-target combinations mentioned in Refs. [1–5,9–14,16,25] are shown by open triangles. The HI data derived from the simulated angular distributions (open symbols) and from measurements (full symbols) for 238U passed through the Bi target [28], for 20 Ne, 209Bi [31] and 40Ar [32] passed through thin carbon foils, are also shown (see Table 1 and the text for details).

such combinations using the available ER range data obtained in integral experiments (target thickness larger than the mean range) with the 12C and 10,11B projectiles [37,38]. The comparison of these data with SRIM calculations (see Fig. 6) shows that ranges of Po and At in Au [37] slightly exceed those obtained in the calculations, whereas similar data for the Os, Ir and Pt ranges in Pt [38] exceed the calculations by (20–30)% at energies above 4 MeV. In contrast, the Sn ranges measured in Ag [38] are below the calculations by the same value. Note that carbon stopping powers for very heavy ions (70 r Z p r 92) at V ¼ 0:8V B , which were measured within a narrow range of scattering angles W r 0:17J [32] (such measurements give us essentially the electron component of stopping power), show significantly lower values (by a factor of about 2) than those given by SRIM/TRIM calculations/simulations. One should keep these circumstances in mind applying SRIM (TRIM) calculations to the estimates of low energy ER stopping in heavy targets.

4. Application of simulations to fusion-evaporation reactions 4.1. Angular distributions for ERs escaping thin targets Simulations of angular distributions for ERs escaping thin targets can be realised assuming uniform random production of ERs along the target thickness. The ER produced at a random depth within the target thickness acquires a direction at a random

0

40

Bi 4

8

12

Ar

16

20

θlab (deg) Fig. 4. Comparison of the experimental angular distributions for 20Ne, 209Bi and 40 Ar scattered on carbon [31,32] (full symbols) with the results of TRIM simulations (open symbols) and theoretical calculations [7] using the Lenz-Jensen potential and empirical screening radius according to Eq. (12) [33] (solid lines). Arrows designate positions of Wmax for 209Bi and 40Ar (see the text for details).

polar angle W and energy EER determined by the evaporation of light particles (neutrons) from a CN, as described by Eqs. (1)–(6). A small correction to the projectile stopping inside a target can be taken into account with the corresponding stopping power calculated with the SRIM code. Thus, one has to perform preliminary MC simulation of ER distributions for producing a data file (TRIM.dat) to be used as input for TRIM execution. As in Ref. [16] this file contains a list with the fusion-evaporation events including: ER mass and atomic numbers (AER and Z ER , respectively), energy (EER ), lateral Y and Z positions, remaining target thickness for ER transmission (depth at which ER is produced) and three directional angles (cos X, cos Y and cos Z). The reference frame is chosen with the X-axis along the beam direction and with the Y, Z-axes oriented as in a clockwise system. The ER Y, Z coordinates could be simulated with a two-dimensional Gaussian distribution with adjustable widths according to the primary beam spot size on the target surface. In Fig. 7 the ER angular distributions, which are actually the probability values PðWÞ obtained with the a-counting activation technique using thin targets [22] in the same reactions as shown in Fig. 2, are compared to those obtained with the TRIM simulations as described above. As we see, the Gaussian approximations 2 corresponding to the experimental values of /W S (solid lines in Fig. 7) deviate from the experimental data and TRIM simulations at large deflection angles. Similar TRIM simulations of the ER angular distributions for the 19F þ 181Ta and 30Siþ 170Er reactions demonstrate a quite satisfactory agreement with the data obtained in a wide angular range using the time-of-flight technique [9] (see Fig. 8). In these simulations, the mean kinetic energies of the 5n evaporation

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U => Au

30

10,11

1.6

Geissel et al. Brown & Moak SRIM: (dE/dX)e

20

1.2

(dE/dX)n

10

Recoil Range (mg/cm2)

dE/dx (MeV/mg/cm2)

B+natAg =>113Sn 10 / B / 11B 10,11 B+109Ag =>113Sn 10 / B / 11B SRIM (113Sn)

50

Pb => Sn, Au Sn, Geissel et al. SRIM: (dE/dX)e

40

(dE/dX)n

30

Au, Geissel et al. SRIM: (dE/dX)e

20

0.8 10,11

0.4

1.2

12

B+181Ta =>191,192Pt* 188 Pt 186 Ir 185 Os SRIM (186Ir)

C+197Au =>201-205At,Po experiment SRIM (203At)

(dE/dX)n

0.8

10 0.4

10-2

10-1

100

E/A (MeV/amu) Fig. 5. Stopping powers of Sn and Au for the Pb and U ions as measured in Refs. [35,36] (symbols) in comparison with those calculated with the SRIM code (the electronic and nuclear components are shown with solid and dashed lines, respectively).

channel were calculated using the statistical model parameters providing the best fit of calculations to the measured ER and fission excitation functions as obtained in the corresponding data analysis [39]. The 5n evaporation channel contributes about 60% and 75% to the ER cross-sections measured at the 95 MeV energy of 19F and at the 140 MeV energy of 30Si, respectively. Minor underestimates in the yield of ERs at large angles are observed in the simulated distributions. They can be explained by the appropriate contribution of light charged particle evaporation channels, which was not taken into account in the present simulations (see also discussion in Ref. [9]). Nevertheless, we see that even for rather thin targets MS gives a significant contribution to the ER angular distributions observed at relatively large deflection angles. This contribution becomes dominating in the case of slower ERs escaping from thicker targets, as one can see in Fig. 9 for the angular distributions of 217,218Ac produced in the 12Cþ 209Bi reaction studied in Ref. [40]. In this reaction, unfortunately, only daughter/grand-daughter nuclei (213,214Fr/209,210At) were observed in experiments on their production cross-sections measurement [41–43]. In Fig. 10, these cross-section data are shown for the At and Fr isotopes produced in the 12Cþ 209Bi reaction leading to the 221 Acn CN. The ER data [41–43] together with fission cross-sections [43,44] are described with the statistical model calculations [24] implying evaporative xn and axn channels for the production of observed nuclei, as shown in the figure. Parameter values used in the calculations are the same as obtained in the analysis of ER and fission cross-sections measured in the 12CþPb reactions [39,45].

2

4

6

8

10

Recoil Energy (MeV) Fig. 6. Ranges of low energy heavy ERs (designated in the figure) in Ag, Ta and Au targets, which are obtained in integral experiments (symbols). Open and full symbols in the upper panel correspond to the reactions studied in Ref. [38] using 10 B and 11B, respectively. In the bottom panel 10% arbitrary values of error bars were added to the data [37]. SRIM calculations for the products designated in the figure are shown by corresponding lines.

Production of 210At, the grand-daughter of 218Ac, at energies Elab 4 76 MeV could not be attributed to the 3n or a3n complete fusion-evaporation channels because the observed cross-sectionenergy dependence significantly differ from usual bell-shaped evaporation excitation functions calculated with HIVAP. The angular and range distributions measured off-line for long-lived 210At [41] confirm this statement. The angular distribution for 217Ac produced in the 4n evaporation channel is well reproduced in our simulation, as shown in Fig. 9. At the same time, similar distribution for 218Ac produced in the 3n reaction channel is not reproduced at large deflection angles, as one can see in the same figure. Bearing in mind that stopping and MS of the Ac atoms inside the target are about the same in both cases, one can assume that the initial angular distribution, due to neutron evaporation is wider than the one used in the simulation. One can remind that HIVAP calculates ER cross-sections, assuming neutron evaporation from an equilibrated CN. Such calculation strongly underestimates the 3n cross-section at the projectile energy of 80 MeV (see Fig. 10). In the framework of the pre-equilibrium statistical model [46], high energy tails in the xn excitation functions correspond to high energy spectra of the emitted pre-equilibrium neutrons. These spectra strongly differ from those obtained in standard statistical model calculations as with HIVAP. Obviously, Eqs. (1)–(9) are not suitable for the calculations of the initial ER angular and energy distributions corresponding to pre-equilibrium neutron emission.

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111 MeV 16O+140Ce (0.073 mg/cm2) =>149Dy (7n)

experiment approximation 7n evaporation TRIM simulation

10-2

103

dσ/dθ (arb. units)

10-3 123 MeV 12C+144Nd (0.077 mg/cm2) =>149Dy (7n)

PER(θ)

10-1

10-2

experiment approximation 7n evaporation TRIM simulation

104 MeV

10

10

-2

102

101

10-3 -1

95 MeV 19F+181Ta (0.04 mg/cm2) experiment 5n evaporation simulation

104

10-1

117

11

146

B+

2

Nd (0.089 mg/cm ) =>

149

Tb (8n)

100

experiment approximation 8n evaporation TRIM simulation

0

4

140 MeV 30Si+170Er (0.08 mg/cm2) experiment 5n evaporation simulation

0

8 12 θlab (deg)

16 149

20 149

Fig. 7. Angular distributions (probability functions) for Tb and Dy produced in (HI, xn) reactions with the targets of a finite thickness designated in the figure. The data obtained in the measurements [22] are shown by large symbols, whereas the results of simulations are presented by small ones. Solid curves are Gaussian 2 approximations obtained with /W S derived from the experimental data. Dashed curves are the calculated angular distributions corresponding to neutrons evaporation alone from compound nuclei. See the text for details.

Good reproduction of the measured angular distributions, which is generally obtained in the simulations considered in this sub-section, is preliminary to applying this approach to different studies such as the transmission and detection of heavy ERs produced in fusion-evaporation reactions. Some of applications are considered below. 4.2. Transmission through electrostatic deflector and charge states of ERs Estimating the transmission of ERs through electro-magnetic separators is one of the main applications in the use of angular and energy distributions simulated as the output from the target (see discussion in Section 1). In this sub-section, by way of example, we consider the simulation of the ER transmission through the electro-static deflector (ESD) [47] installed at the XTU-Tandem accelerator of LNL (Legnaro). Below we will see that these simulations allow evaluating a charge state distribution of energetic heavy ER atoms recoiling from a target. Usually, the setting of the ESD is done at some representative energy by varying the high voltage (electric-field strength) between the deflector plates: thus, a transmission (yield) curve for ERs is constructed and the field setting is chosen that guarantees maximal transmission. At the variation of the beam energy, the field is scaled according to the calculated electric

5

10

15

20

θER (deg) Fig. 8. The same as in Fig. 7, but for ERs produced in reactions leading to the 200 Pbn CN with the use of the 19F and 30Si projectiles. Large and small symbols correspond to the data obtained in the experiments [9] and to the TRIM simulations, respectively. For the latter initial angular distributions for the 5n evaporation channel alone were used. They contributes about 60% and 75% to the ER production for the 19F and 30Si induced reactions, respectively, as it follows from the data [9].

rigidity. This finding allows further effective measurements of the ER yield dependence upon the beam energy (see, for example, some results presented in review [48]). In the study of the Fr and Ra isotopes production with the use of fusion-evaporation reactions induced by 18O and 12C, respectively, two-humped curves of their yields as a function of the electric-field strength were observed [49,50]. Earlier, a similar character in the yield dependence had been observed with a similar technique for the Fr and Th ERs produced in reactions induced by 16O [51]. The twohumped character of the yield curves was explained by the presence of two components in the charge distribution of ER ions: equilibrium and non-equilibrium ones. The latter corresponds to much higher charge states due to relatively long-lived nuclear states decaying through Auger cascades following inner conversion. In the 18Oþ 197Au reaction study, the angular distribution of 210 Fr was measured using the ESD at the beam energy of 90 MeV [49]. In the present study, we averaged the yields corresponding to the same polar angle W (see Fig. 2 in Ref. [49]) and the angular distribution  ðds=dOÞ was transformed into the angular distribution  ðds=dWÞ, which is shown in Fig. 11. The results of the TRIM simulation reproduce rather well the measurements as shown in the same figure. Good agreement of the simulated angular distribution with the measured one suggests that the corresponding 210Fr energy distribution generated with TRIM could be used for the estimates of the ESD transmission. Note that

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80 MeV

12

C+

2

209

Bi (0.1 mg/cm ) =>

90 MeV 18O+197Au (0.15 mg/cm2) => 210Fr (5n)

Ac

200

Ac, experiment 4n-evaporation TRIM simulation 218 Ac, experiment 3n-evaporation TRIM simulation

101

dσ/dθ (arb. units)

217

102

dσ/dθ (arb. units)

217, 218

expt, Corradi et al. 5n evap. from 215Fr* simulation

150

100

50

100 0

5

10

15

20

25

θER (degree)

10-1

0

10

20 θlab (deg)

30

40

Fig. 9. The same as in Figs. 7 and 8, but for 217,218Ac produced in the 12C þ 209Bi reaction [40] (large open symbols for the data points) together with the corresponding TRIM simulations (small full symbols) using initial angular distributions for the 3n and 4n evaporation channels (dashed lines).

12 210

At, Bimbot et al. 210 At, Jin et al. 214g Fr, Le Beyec et al. 214 Fr, Jin et al.

3

Cross section (mb)

10

C+209Bi fission 4n+α4n

3n+α3n

102 α4n

4n

101 209

At, Bimbot et al. At, Jin et al. Fr, Le Beyec et al. 213 Fr, Jin et al. fis. (x2), Britt & Quinton fis. (x2), Jin et al.

α3n

209 213

3n

100 60

70

80

60

70

80

90

Elab (MeV) Fig. 10. Cross-section data obtained in the 12C þ 209Bi reaction for the 209,210At and 213,214 Fr nuclei production [41–43] and fission [43,44] (symbols). The lines represent the evaporative xn, axn and fission excitation functions calculated by HIVAP with parameter values providing the best fit to the data (see Refs. [39,45]).

the measured angular distribution was obtained with the rotation of the ESD around a target position. The ESD was operated at the same high voltage during the measurement of the angular distribution. Therefore, one may ask if the transmission of the ER is really independent of the angle. In Fig. 12, the 210Fr energy

Fig. 11. The angular distribution for 210Fr produced in the 18Oþ 197Au reaction as was measured in Ref. [49] using the electro-static deflector [47] (large open circles) in comparison with the results of the TRIM simulation (small full circles). The initial angular and energy distributions corresponding to neutron evaporation alone were obtained with the mean neutron kinetic energy given by the HIVAP calculations using parameter values mentioned in Ref. [49]. The former is shown by the dashed line.

distributions obtained with the TRIM simulation, which correspond to all emitted angles and to the angles of 21, 51, and 81, are shown. In the last cases of the definite angles, the full symbols correspond to the range of ER angles of 70.51. The open symbols correspond to the simulations with a real geometry inherent in the ESD experiments. The geometry takes into account the size of the beam spot on the target, the position and shape of the entrance diaphragm. As we see, the energy distributions corresponding to the definite angular region differ little from each other, as may be inferred from the Gaussian fits shown in the figure. So, one can use the ER energy normally distributed around an average value for the simulation of the ER transmission through the ESD. Further simplification in the simulation is the assignment of an effective size to the entrance diaphragm. This size is chosen to reproduce the solid angle subtended by the stop detector of the ESD (  2:1 102 msr, the contemporary value measured with a-source). It implies some angular spread of ERs in further simulations, which is determined by small sizes of the beam spot taking into account and ‘effective’ diaphragm (about several mm in both cases) together with a relatively large distance between them (881 mm). Obviously, one could neglect the ER angular distribution within a small range of angles cut out by such geometry. Fig. 13 shows the experimental transmission curve for the 210 Fr ions as measured in Refs. [49,50]. The data are compared to the simulations that use the energy distribution of 210Fr which is calculated as explained above. In the transmission simulation, the trajectories of the charged ions were calculated step by step taking into account the deflector geometry with the assumption of uniform electric field between the deflector plates and null outside (see, for example [52]). The fringe effects were taken into account by the use of effective lengths for the plates. To initialise transmission simulations, we used a ‘seed’ charge distribution parameterised with the systematics of ER equilibrium charges

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105

90 MeV 18O+197Au (0.15 mg/cm2) => 210Fr (5n)

90 MeV 18O+197Au (0.15 mg/cm2) => 210Fr (5n)

experiment SIMULATIONS: eq Weq=100%, Qeq m , σQ (syst.)

0.20

10

4

all angles ° θm ER=2

eq Weq=15%, 1.1Qeq m , 1.05σQ

Gauss fit ° θm ER=5

Wneq=85%, non-eq. set-1 summarised eq Weq=16%, 1.1Qeq m , 1.05σQ

0.16

Gauss fit ° θm ER=8

Wneq=84%, non-eq. set-2 summarised eq Weq=13%, 1.1Qeq m , 1.05σQ

Gauss fit

103

Transmission

Counts / 0.5 MeV

119

102

0.12

Wneq=87%, non-eq. set-3 summarised

0.08 x1/3

101

100

0.04

2

4

6

8

10

EER (MeV) Fig. 12. The 210Fr energy distribution integrated over all angles (corresponding to angular distribution shown in Fig. 11) as obtained with the TRIM simulation (large spheres). Small open symbols represent the energy distributions corresponding to the mean ER angles of 21, 51, and 81 with the selection within 70.51 around these values, whereas small full symbols correspond to the selection taking into account a real geometry used in the measurements with the ESD. See details in the text.

[53]. To adjust the simulated transmission curve to the measured one we have to use three-component charge state distribution for the 210Fr ions, as proposed in the similar study of reactions leading to the Fr and Th ERs [51]. We have to scale slightly main parameters of the ‘seed’ charge distribution (standard deviation and mean values) for the equilibrium component corresponding to the ‘rigid’ part of the yield curve. The non-equilibrium component could not be described by a single Gaussian, but it requires two modes described by two Gaussians. The measured transmission values allow to vary the relative contribution of the equilibrium and non-equilibrium components within Weq ¼(13–16)% and Wneq ¼(87–84)%, respectively. For the equilibrium component, the best fit to the data corresponds to the mean charge and standard deviation values exceeding those given by the systemaeq tics [53] by 10% and 5%, i.e., they are equal to 1:1Q eq m and 1:05sQ , respectively. The non-equilibrium charge component consists of two modes with the mean charge values exceeding the equilieq brium one and corresponding to ð2:322:4ÞQ eq m and ð3:623:7ÞQ m , respectively. Standard deviations of these components exceed the one given by the systematics [53] and correspond to ð2:923:2Þseq Q . The relative contributions of the non-equilibrium modes with ‘smaller’ and ‘larger’ charges are varied within wneq 1 ¼ ð60270Þ% and wneq ¼ ð40230Þ%, respectively. In Fig. 13 these variations are 2 designated by three sets of the parameter values (set-1–set-3) allowing the description of the transmission curve with about the same quality.

1

2

3

ε (kV/cm) Fig. 13. Transmission efficiency for 210Fr produced in the 18Oþ 197Au reaction as a function of the electric-field strength of the ESD, as was measured in Refs. [49,50] (large spheres), in comparison with the results of MC simulations using different sets of charge distributions (small symbols connected with lines). These simulations correspond to (a) the equilibrium distribution alone (Weq ¼ 100%) detereq mined by the Q eq m and sQ parameter values given by the systematics [53] and (b) the reduced contribution of the equilibrium component, Weq ¼(13–16)%, with scaled values of the mean charge and standard deviation corresponding to 1.1 Q eq m and 1:05seq Q , respectively, and the significant contribution of the non-equilibrium neq component, W ¼(87–84)%. The latter consists of two modes described with three different sets of parameter values (set-1–set-3) reproducing the measured transmission curve much the same as shown by the summarised curves which resulted from the simulations. The parameter values for the non-equilibrium charge distributions are given in the text.

In Fig. 14 the charge distribution for the 210Fr ions is shown. It consists of three components (equilibrium and two nonequilibrium ones) and is the average of three charge distributions corresponding to the three sets used in our simulation to reproduce the measured transmission curve shown in Fig. 13. In the bottom panel of Fig. 14, the charge-state distribution for 192Pb produced in the 16Oþ 182W reaction [54] is also shown for comparison. These data were obtained with the use of a magnetic spectrograph and off-line g-activation technique for the detection of long-lived reaction products collected in the focal plane of the spectrograph. The Gauss fits to the charge distributions shown in Fig. 14 allow to compare the fitted parameter values with some results of the analysis [55] of similar data [54]. The fitted parameters for the equilibrium charge distributions could be also compared to the estimates of corresponding values from different systematics (see some Ref. [53]), whereas high charges of the non-equilibrium component might be compared to the predictions of charge distributions obtained by vacancies created in inner shells of atoms (see, for example [56,57]). Such quantitative comparison is presented in Table 2.

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8

90 MeV 18O +197Au (0.15 mg/cm2) => 210Fr (5n) averaged sim. distrib. three-Gauss fit equilibr. component non-equilibr. comp.-1 non-equilibr. comp.-2 equilibr. distr. (syst.)

6

x1/4

Charge state fraction (%)

4

2

12

16

O (102 MeV) +182W => 192Pb (6n)

9

x1/2

expt, Maidikov et al. two-Gauss fit equilibr. component non-equil. component equil. distrib. (system.)

6

3

0

10

20

30

40

Charge state Fig. 14. The charge state distribution for 210Fr produced in the 18O þ 197Au reaction (large full circles) is shown in the upper panel. This is the result of averaging the charge distributions used in the MC simulations of the transmission curve measured in the experiments [49,50] (see Fig. 13). The results of the threecomponent Gauss fit are shown by the solid line for the summarised distribution and by the dashed, dash-dotted and dash-double-dotted lines for the equilibrium, the first non-equilibrium and the second non-equilibrium components, respectively. The fitted parameter values for all the components are presented in Table 2. The ‘seed’ equilibrium charge distribution [53] corresponding to the energy distribution resulting from neutron evaporation and stopping of 210Fr inside the target is shown for comparison (small full squares connected with the dotted line). In the bottom panel, the charge state distribution for 192Pb produced in the 16 O þ 182W reaction (the histogram) [54] is shown for comparison. The results of the two-component Gauss fit and the charge distribution for the 192Pb equilibrium charges alone [53] are also shown in a manner like the 210Fr case.

So, we believe that the present approach, as exemplified by the experiment on the production of 210Fr, provides a useful instrument for estimates of performance of recoil separators, in particular, their transmission efficiency for ERs. It can be also used for evaluation of the charge state distribution of ER energetic atoms produced in HI fusion-evaporation reactions. 4.3. Optimal target thickness in experiments with recoil separators Experiments aimed at production of new heavy nuclei at the borders of the nuclides chart, which are usually carried out with recoil separators, face the choice of an optimal target thickness providing the highest yield of the desired nucleus that, as a rule, has an extremely low production cross-section. Obviously, using a thick target, along with increasing ER yield, widens the ER angular and energy distributions and leads to the loss of transmission and to the degradation of background conditions, since the yield of undesired products (recoil target atoms, transfer-reaction

products etc.) increases proportionally to the target thickness. Thus, in the experiments with the DGFRS [13], on the 48Caþ 206Pb - 252No reaction study, the layer thickness of the metallic 206 Pb-target was limited to  0:5 mg=cm2 [58,59]. Below we propose the way to estimate the optimal target thickness by the example of the 252No yield simulation in the 48Caþ 206Pb reaction, which is applied to the DGFRS geometry. We consider the optimal target thickness as a value, exceeding of which does not increase the yield of desired products in specified conditions of their production and detection. The measurement of the 252No yield versus target thickness in the experiment with the DGFRS was carried out at the 48Ca input energy of 217 MeV, the same one for six targets of metallic 206Pb of the thicknesses of 0.15–0.64 mg/cm2. In the experiment the targets were installed on a rotation wheel whose rotation was synchronised with a precise time recording during the run (accuracy of timing was about 1 ms). The time of flight of 252No from the target to a focal plane detector is about 1 ms. Thus, one could assign the time of implantation of 252No into the detector and its decay time, which are recorded in a list mode by the DGFRS acquisition system, to a definite phase of the wheel and thus obtain the yields corresponding to different target thicknesses in a single run, without any additional normalisation. In our study of the 252No yield versus target thickness, we based on the cross-section data obtained with the DGFRS in the 48 Ca þ 206Pb reaction [59]. These data are shown in Fig. 15 together with the calculated excitation functions obtained with HIVAP, which are the ones best fitted to the data. As in the previous cases, we used mean neutron kinetic energies given by these calculations. In the figure we show the ranges of the 48Ca energy absorbed by the target layers of 0.1–1.0 mg/cm2 of thickness, which were used in our simulations. Note that the energy losses in the targets thicker than 0.4 mg/cm2 correspond to a steep fall-off of the 252No excitation function. This circumstance should be taken into account in simulations. We generated the ER production along the target thickness according to the energy dependence of the 252No cross-section, as given by the calculations below 217 MeV (see Fig. 15). Further TRIM simulations could be performed in principle for the transmission of the No ions through the Pb target layers, but U is the heaviest ion that TRIM can deal with. However, the general dependence of the deflection (scattering) angle on the atomic numbers [7,30] allows extrapolations to scattered ions heavier than U. Using previous notations (see Section 3), one can introduce a correction factor kZ for the deflection angles generated by TRIM for U (Z p ¼ 92) and obtain corresponding angles for ions with Z p 4 92 þZ 0:23 Þ=92=ð920:23 þZ 0:23 Þ: kZ ¼ Z p ðZ 0:23 p t t

ð13Þ

Obviously, Eq. (13) implies the use of the same values of other parameters in transformation of the angles obtained for U into the angles for desired ions with Z p 4 92. We tested the selfconsistency of this correction with the transformation of the angular distributions generated by TRIM for 252Pb, 252Fr into the 252 U ones and by their comparison with the angular distributions simulated directly for 252U. All simulations were performed at the energy corresponding to the recoil energy of 252No that is produced at the 48Ca energy of 217 MeV and transmitted through the 206Pb target of thicknesses of 0.1, 0.4 and 1.0 mg/cm2. The results of these simulations are shown in Fig. 16. A correction, analogous to the one given by Eq. (13), applied to the Pb and Fr angles, makes the transformed angular distributions for 252Pb and 252 Fr indistinguishable from the one for 252U, as one can see in the figure. The really important value for estimates of transmission through the DGFRS is the fraction of ions transmitted through

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121

Table 2 Mean values Qm and standard deviations sQ for the equilibrium (eq) and non-equilibrium (neq) components of the charge distribution for the 210Fr ions produced in the 18 O þ 197Au reaction [49,50]. The data obtained in the present analysis are compared to the results of the analysis of charge distributions for some ERs produced in the reactions studied in Ref. [54]. The analysis of the data [54] was performed in Ref. [55]. Relative velocities V=V B of ER ions, predictions of the equilibrium charge distribution parameters according to the systematics [53] and of the mean charge values for charge distributions formed by a sudden vacancy created in the K-shell of ions [56,57] are listed for reference. V =V B

Qeq m

seq Q

Q neq m1

sneq Q1

2.18

13.77 0.1 13.6

2.87 7 0.13 2.04

20.7 70.2

4.35 70.23

9.2 9.1 15.3 7 0.5

3.64 70.53

10.5 24.9 7 0.6

3.1

34.3 71.1

4.4

10.5 25.4 7 1.1

3.8

34.4 73.3

3.5

10.8 16.6 7 0.7

6.8 70.2

26.3 71.3

6.77 0.3

Reaction

ER

22

Neþ 138Ba

153

16

182

192

Pb

1.30

9.297 0.20 7.53

1.13 7 0.28 1.75

40

Arþ 158Gd

192

Pb

2.71

20.17 0.1 18.9

2.46 7 0.10 2.21

40

Arþ 164Dy

199

Po

2.64

20.47 0.1 18.6

1.99 7 0.07 2.22

Fr

1.19

7.627 0.03

1.87 7 0.03

7.05

1.72

18

Oþ

W

Oþ 197Au

210

Dy

11.2

Fig. 15. The No isotopes production cross-sections obtained in the 48Ca þ 206Pb reaction [59] (symbols) and the best fitted excitation functions calculated with HIVAP (solid lines). The ranges of the 48Ca energy absorbed by the 206Pb target layers of 0.1–1.0 mg/cm2 of thickness are shown by horizontal segments with the points corresponding to the mean energy values.

the entrance diaphragm of the separator. Such fractions are (89.7270.92)%, (54.5470.65)% and (20.3170.35)%, respectively, as calculated for U ions transmitted through the 0.1, 0.4 and 1.0 mg/cm2 thick targets in the angular range (0–41) that is approximately cut by the entrance diaphragm of the DGFRS.

Q neq m2

sneq Q2

Reference [54] [53] [56] [57] [54] [53] [56] [54] [53] [56] [54] [53] [56] This data [53] [56]

The same fractions obtained with the correction applied to the Pb deflection angles correspond to the values of (89.5070.92)%, (52.9870.64)% and (19.8970.35)%, and with the correction applied to the Fr angles these values are (89.46 7 0.92)%, (53.9570.64)% and (20.4970.35)%, respectively. As we see, such scaling gives quite acceptable results even in the case of ‘far extrapolations’ of the Pb angles into the U ones for rather thick targets. In Fig. 17 the angular distributions (AD) for 252No produced in the 206Pb(48Ca, 2n) reaction at the input energy of 217 MeV are shown for the different target thicknesses. These angular distributions were obtained from the angles generated by TRIM for 252U and scaling with Eq. (13). As we mentioned above, our simulation of the initial distribution of 252U (252No) atoms along the target thickness corresponded to the excitation function shown in Fig. 15. In the simulation shown in Fig. 17, the number of generated 252No events is a function of the target thickness, whereas in Fig. 16, the same number of events was generated for each thickness, for the sake of comparison. In Fig. 17 the angular distributions cut out by the entrance diaphragm of the DGFRS (AD/D1) are also shown for the corresponding target thicknesses. These distributions correspond to the real geometry of an entry section of the DGFRS (diameter of the beam spot on the target surface, distance between the target and entrance diaphragm, the form and sizes of the diaphragm). The fractions of the 252No atoms transmitted through the diaphragm (D1-tr) for the corresponding target thicknesses are also presented in the figure. In Fig. 18 we show the calculated yield of the 252No atoms behind the diaphragm as a function of the target thickness, which is compared to the yield curve obtained in the DGFRS experiment [58,59]. The curves are normalised to unity corresponding to the maximal yield observed in the experiment for the target thickness of 0.49 mg/cm2 and to the one obtained in our simulations for the target thickness of 0.5 mg/cm2. As we see both results are in close agreement with each other. Such agreement seems to be surprising at first sight, since the simulations do not take into account the 252No atoms transmission through the dipole magnet and their focusing by the quadrupole lenses of the DGFRS. At the same time, the gas pressure and field setting were the same for all targets. They were not optimised for the different target thicknesses (the different ER energies/velocities). It is thought that the used settings were actually optimal for the thickness of 0.5 mg/cm2.

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10

41 MeV 252[U,U(Pb),U(Fr)] =>206Pb

3

Wt

10

252

U

252

U(Pb)

252

U(Fr)

0.1

2

101

Counts / 0.4 deg

100 103

Wt

252

U

252

U(Pb)

252

U(Fr)

0.4

102

101

Wt 1.0

252

U

252

U(Pb)

252

U(Fr)

102

0

4

8

12

16

20

24

θER (deg) lab Fig. 16. Angular distributions obtained with TRIM simulations for the 252U ions (open symbols) transmitted through the 206Pb target of thicknesses Wt ¼ 0.1 (upper panel), 0.4 (middle) and 1.0 (bottom panel) mg/cm2. Full symbols correspond to angular distributions originally simulated for Pb and Fr followed by the transformation to those for U with the correction factor, analogous to the one given by Eq. (13). These distributions are designated as 252U(Pb) and 252U(Fr) in the panels. The energy of the ions corresponds to the recoil energy of 252No produced at the 48Ca beam energy of 217 MeV.

In concluding this sub-section, one can state that entrance geometry and a target thickness might be mainly responsible for the efficiency of separating devices such as recoil separators for ERs produced in fusion-evaporation reactions. MC simulations similar to those described above allow estimating the optimal target thickness needed for the effective work of these devices.

5. Summary The Monte Carlo approach has been developed for simulations of the angular and energy distributions for heavy ERs produced in HI fusion-evaporation reactions. The approach uses statistical model approximations of the HIVAP code for the calculations of initial angular and energy distributions inside a target, which are determined by neutron evaporation from an excited compound nucleus. Further step in the simulation of transmission of ER heavy atoms through a target layer is performed with the TRIM code that gives final angle and energy distributions at the exit from the target. Both the simulations (neutron evaporation and transmission through solid media) have been separately considered and good agreement has been obtained between the results of simulations and available experimental data.

Fig. 17. The angular distributions for 252No produced in the 206Pb(48Ca, 2n) reaction at the input energy of 217 MeV for the target thicknesses of Wt ¼ 0.1, 0.2, 0.4 and 0.8 mg/cm2 (full symbols corresponding to the acronym AD). The data were transformed from the angular distributions generated by TRIM for 252U following with the scaling according to Eq. (13). The angular distributions cut out by the entrance diaphragm of the DGFRS (AD/D1) are shown by open symbols connected with lines. The fractions of the 252No atoms (in %) transmitted through the diaphragm (D1-tr) are indicated. See details in the text.

1.2

206Pb(48 Ca, 2n)252No

1.0 Yield (relative units)

103

0.8 0.6 0.4 DGFRS-experiment polynomial fit D1-transmission

0.2

0.2

0.4 0.6 0.8 Target thickness (mg/cm2)

1.0

Fig. 18. The 252No atoms yield behind the entrance diaphragm of the DGFRS as a function of the target thickness as obtained in our simulations is compared to the yield curve measured in the DGFRS experiment [58,59]. The curves are normalised to unity corresponding to the maximal yield observed in the experiment for the target thickness of 0.49 mg/cm2 and to the one obtained in our simulation for the target thickness of 0.5 mg/cm2.

Some applications of the approach have been also considered. Thus, it reproduces rather well ER angular distributions obtained in experiments with thin and moderately thick targets and can be used in different applications where such simulations are needed. One of applications is concerned with the estimates of

R.N. Sagaidak et al. / Nuclear Instruments and Methods in Physics Research A 700 (2013) 111–123

the transmission efficiency of different (electro-magnetic) separating systems. As an example, the simulation of ER transmission through the electrostatic deflector has been considered. Such simulations allow the study of charge state distributions of heavy ER atoms. Varying the charges of atoms passing electric field in simulations, one can reproduce the ER yield as a function of the strength of electric field, which is observed in experiments with the deflector. Our approach could be useful for the estimates of the optimal target thickness in experiments with recoil separators. A knowledge of the optimal target thickness is especially important for the effective work of these devices in experiments aimed at production of ERs having very low production crosssections.

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