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Cluster impact fusion and cluster size distributions
Nuclear Instruments and Methods in Physics Research J3 88 (1994) 116-121 North-Holland
NOMB
Beam fnteraetions with Materials % Atoms
Cluster impact fusion and cluster size distributions R. Vandenbosch
*, J. Neubauer, D.I. Will, T.A. Trainor and D. Ye
University of Washington, Nuclear Physics Laboratory CL-IO, Seattle, WA 98195, USA
D + D fusion yields have been measured for small deuterated molecules incident on deuterated polyethylene and are one to two orders of magnitude smaller than previously reported by another group. They are, however, in good agreement with free-particle cross sections. Fusion has also been observed for pure carbon clusters incident on deuterated polyethylene. Comparison of fusion yields per carbon for clusters and single carbon atoms show no collective enhancements. The observed fusion for carbon clusters is shown to be due to a knock on mechanism. Cluster size distributions for graphite sputtered with cesium exhibit a rich periodic structure whose origin will be discussed in terms of molecular orbital theory.
1. Introduction
In 1987, Beuhler, Friedlander, and Friedman 111 reported a remarkable observation. When they bombarded deuterated titanium with a 300 keV cluster of about 100 deuterated water molecules, they observed the d + d fusion reaction to occur at a rate about 10Z times larger than expected from an extrapolation of the known d + d cross section to the 0.3 keV energy of each deuterium in the deuterated water cluster. This was very surprising, as generally one does not see any dependence of a nuclear reaction rate on the chemical environment. The energy and distance scales of chemical and nuclear reactions are so different that one does not expect any enhancement unless there are very large changes in the target density or velocity distributions of the projectile or target deuterons. Our interest in the problem was further stimulated by the report of Bae et al. [Z] who not only confirmed the observations of Beuhler et al. but also reported that an enhancement could be observed for clusters of as few as two molecules.
2. Cluster impact firsion studies
Our first experiment was designed to check the observations of Bae et al. for small deuterated water clusters. We used a direct extraction ion source and the injection accelerator of the University of Washington tandem-booster facility. The deuterated cluster anions were mass analyzed and a particular species was acceierated to typically
* Corresponding author. Old-583X/94/$07.00
325 keV. After acceleration a weak magnetic field was applied to deflect away any deuterons which might have been produced by collisions in the residual gas and accelerated to sufficient energy to produce a spurious signal. We used deuterated polyethylene targets on a thin ~~inurn backing foil. The protons from the d + d -+ t + p reaction passed through the aluminum backing and were detected in a semiconductor detector placed at 0”. This provided a large solid angle and also enabled us to exploit the kinematic dependence of the proton energy on the deuteron energy inducing the reaction. The proton energy at 0” varies from 3.86 MeV for 300 keV deuterons to 3.08 MeV for 2 keV deuterons. This provides additional discrimination against beam contamination, and potentially can be used to deduce the energy spectrum of deuterons inducing fusion. This diagnostic was not available in previous experiments [1,2], in which the detector was located at 90” where the kinematic shift vanishes and where the kinematic broadening is at a maximum. We first compare our yields with the data of Bae, Lorents and Young [2]. In Fig. 1 we show the total proton fusion yield per cluster as a function of cluster size n. Although our results for n = 1 are in reasonable agreement with those of Bae, Lorents and Young and also with the thick-target model of Carraro et al. [3], for larger IZ our yields fall much more rapidly with n. Our yields are approximately 1 order of magnitude lower than theirs at y1= 2, and our n = 3 yield is 2 orders of magnitude lower than the interpolation of their n = 2 and 4 points. We believe that our present results indicate the absence of any special enhancement due to cluster impact for n < 5 and can be understood simply in terms of the free-deuteron thick-target excitation function. An attractive feature of our experimental setup is
Q 1994 - Elsevier Science B.V. All rights reserved
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R. Vandenbosch et al. /MA.
Imtr. and Meth. in Phys. Res. B 88 (1994) 116-121
that we can measure the thick-target fusion yield as a function of energy for D- ions. The results of these measurements, taken interspersed with the cluster measurements, are shown in Fig. 2.If there is no special enhancement over the free-deuteron cross sections due to the other atoms in the cluster, the cluster yields should be intimately related to the free-deuteron thick-target yields. Thus we would expect our cluster yield per deuteron, when plotted as a function of deuteron energy, to fall on our experimental free-deuteron yield curve. We show in Fig. 2 that this simple expectation is satisfied within our uncertainties due to beam integration. We have also calculated the expected (C,DJ, thick-target proton yield following the method of Carraro et al. The results are shown by the full curves in Figs. 1 and 2. The agreement in shape is excellent. The absolute magnitude of the calculation is somewhat higher than our data, perhaps due to ~certainties in stopping powers in the calculation and in the absolute detection efficiency in the experiment. After our work was published [4], Beuhler et al. [5] withdrew their claim for having observed large fusion yields. They reported that they now believed the fusion yields were at least two orders of magnitude below their original claim. Very recently, Fallavier et al. [6] reported on a study with (CD,), clusters. For small clusters (pt = l-5) they find no evidence for collective effects, in agreement with our observation. For large
Deuterons
l
ttiii
Ti
M=?858
30 34 36
20
I
I
I
I
0
5
IO
15
z E E :D Mctuster cluster
1s
I
I
20
25
30
(keV)
Fig. 2. Proton fusion yield per cluster divided by the number of deuterons per cluster plotted as a function of the kinetic energy of each deuteron. Also plotted are measured yields for free deuterons. From ref. 141.
I
1
1 i
*
3 Eoe et 01. Beuhler et ai.
l
- This work
-
100
cluster
size n
Fig. 1. Proton fusion yield for 225 keV clusters incident on (C2D4jn targets. The full curve is the calculate< thick-target yield (Yrr> as discussed in the text. From ref. 141.
clusters they determined upper limits for fusion considerably lower than originally reported for water clusters. In the course of our water cluster study we also observed a small amount of d + d fusion when a water anion not containing deuterium was used as a projectile. It seemed likely this was due to a secondary reaction initiated by the heavy oxygen atom. Mizota et al [7] have reported evidence supporting this surmise. The Lyon group [8] has reported the absence of cluster impact fusion for large clusters of pure deuterium. We decided to investigate the mechanism involved using carbon clusters, so as not to have any confusion arising from the relative roles of H and 0 atoms in normal water clusters. Carbon clusters were obtained by sputtering graphite with Cs+ ions. We had noticed in related attemps to produce beams of deuterated organic molecules that a broad range of carbon clusters could be obtained. We measured fusion yields for clusters and also for single carbon atoms, the latter as a function of energy. Our fusion yields, expressed as protons per incident carbon, are plotted as a function of carbon atom energy in Fig. 3. On such a plot a collective enh~cement of the fusion yield would reveal
118
R. Vandenbosch et al. /Nucl. Ins@. and Meth. in Phys. Res. B 88 (1994) 116-121
per deuteron, and da/dU is the differential cross section for transfering energy V to a deuteron by a carbon of energy E in a heavy projectile atom-deuteron collision. We assume that d&J, E)/dU is given by Rutherford scattering. Below roughly 20 keV the true scattering becomes less than Rutherford [lo] and our use of Rutherford scattering leads to an overestimate of the knock-on yield. However, this overestimation is only significant for measurements at very low energy, due to the rapid decrease in the fusion yield
101
100
10-I
10-2
10-3
10-4
Y,(U). 10-5
ECclusterl/n
(keV1
Fig. 3. Comparison of fusion yields for carbon clusters with those for single carbon atoms. Collective effects would be revealed on this plot by a larger yield for clusters compared to single atoms at the same energy/atom.
itself as a larger yield at a given carbon atom energy when that atom was part of a cluster. One observes
that we have no evidence for cluster enhancement of the fusion yield for clusters up to n = 12. We had originally reported limits on yields for n = 1.5 and 19, but now believe that the species accelerated were predominantly CsC, and CsC, rather than C, and C,, as discussed later on in this report. The leading order process by which one might expect D + D nuclear fusion for a projectile which does not contain deuterons is scattering of the projectile atoms off of target deuterons, leading to recoil deuterons with sufficient energy to react with other stationary deuterons in the target. This mechanism was one of several considered by Carraro et al. [3] in their early exploration of processes which could account for the fusion yields resulting from the deuterated water clusters bombardments of Beuhler et al. 111. Carraro et al. calculated the knock-on contribution of the projectile oxygen atoms. In their calculation this contribution generally exceeded that from the projectile deuterons, except for the smallest clusters. The fusion yield from the knocked-on deuterons is given by rF
Y,, = n,n) -
dE
o IdE/dxl
where n, is the deuteron number density in the target, n is the number of atoms per cluster, E, is the initial laboratory energy of each atom in the cluster, dE/dx is the stopping power of each projectile atom in the target, U,,(E) = 4m,m,E,/(m, + rn,j2 is the maximum energy that a projectile atom can transfer to a struck deuteron, Y,(U) is the thick-target fusion yield
It is also necessary to parameterize the stopping of the projectile atoms in the CD, target. This was done as described in ref. [9]. The calculated knock-on yield is shown by the continuous curve in Fig. 3. The calculation provides a reasonable account of the energy dependence of the observed fusion yield, which varies more than four orders of magnitude over the energy range studied. Our results for both deuterated water molecules and carbon clusters do not show any indication of a collective enhancement. Does this mean that no collective enhancement is to be expected, or just that it occurs at a level below our experimental sensitivity? Collective enhancement can originate from a microscopic model by the Fermi shuttle mechanism. This mechanism has already been discussed previously in this conference in the context of producing high energy electrons. An example of this mechanism is when a projectile deuteron is backscattered off a heavy target atom (e.g. Ti) and then struck by a heavy projectile atom (e.g. 0). The resulting deuteron can obtain an energy up to six times its original energy by this mechanism. In order to assess the fusion enhancement which might arise from these and higher order processes, molecular dynamics calculations for deuterated water clusters have been performed by Hautala, Pan and Sigmund [ll], by Haftel [12], and by Valkealahti, Manninen and Hammare’n [13]. We show in Fig. 4 one of the results of the calculation of Haftel. One sees that, indeed, deuterons with energies several times that in the original clusters are produced. The fusion yield in both calculations [11,12] were obtained by taking the deuteron energy spectrum (or an exponential fit to the spectrum) at the end of the simulation (after the cluster has undergone significant stopping) and calculating the yield expected using the thick-target yield expression, mentioned previously, or its equivalent. The calculated results show a large enhancement (- 1015) over the yield expected from deuterons of the original projectile velocity. They are, however, still many orders of magnitude lower than the revised upper limit of the Brookhaven group [5]. We believe that the expected yield may be underestimated in two of these calculations [11,12] because of the neglect of fluctuations in the deuteron spectra
R. Vandenbosch et al. / Nucl. Instr. and Meth. in Phys. Res. B 88 (I994) 116-121
.”
10’
102
lol
E (eV)
10'
Fig. 4. Spectra of beam D atoms at 6 fs (short dashed line), 12 fs (long dashed line), and 36 fs (solid line) after the start of the simulation of 300 keV (D,O),,, bombardment. From ref. KY.
during the stopping process. Our concern can be seen in the results presented in Fig. 4. At the intermediate time of 12 fs the deuteron energy spectrum is seen to have a higher energy deuteron than does the later spectrum used in calculating the fusion yield. It seems to us that the proper way to calculate the yield is to calculate the fusion probability at the distance of closest approach for every d + d collision which occurs as the deuterons are being slowed down. This type of approach has been employed recently by Valkealahti et al. [13] (A related approximation has been made by Shapiro and Tombrello [14], who kept track of the highest energy reached by a particle in a study of mixed-metal cluster impacts.) We believe the incorporation of these fluctuations in the deuteron spectrum could lead to an additional enhancement of several orders of magnitude.
119
of 157, 181, 205 and 229. A more careful mass calibration has confirmed the latter assignments. The favoring of even-n clusters through n = 8 has been observed previously [17,18], as has the enhancement of odd-n clusters with n 2 13 [17]. The even-n anions through n = 8 are attributed to linear chain structures, and the larger odd-n anions are attributed to cyclic structures on the basis of molecular orbital theory calculations [19,20]. It is also known from other studies that the yield of Ci clusters inhibits the opposite odd-even effect, particularly for n < 10 [21]. We present here a simple molecular orbital (MO) theory explanation for these observations. Our starting point is the generally greater stability of chains compared to rings for n < 10, and of monocyclic rings over chains for larger n [22]. We consider first the molecular orbitals of straight chains. 2n + 2 electrons go into u orbitals along the axis of the chain (z-axis). The remaining 2n + 2 electrons go into the degenerate “X and n,, orbitals. The Hiickel MO rrX energy levels for n = 4 and n = 5 are shown at the left of Fig. 5. One can see that for even n an extra electron to make the C; anion can go into a lower-energy bonding orbital, whereas for odd n it has to go into a non-bonding orbital. This accounts for the favoring of even-n anions. The electron removed from the neutral cluster to make a positive ion comes from a bonding orbital for either odd or even n, and the favoring of odd cations reflects the greater stability of odd-n neutrals. Now consider the molecular orbitals for monocyclic rings. We assume there are 2n electrons in the u orbitals along the ring (taken to lie in the yz-plane). We neglect ring strain and assume that the rX and +rrY orbitals are degenerate. (Our qualitative conclusions will not be altered if the rX and rY orbitals are displaced slightly with respect to each other due to deviations from colinearity of the r,, orbital axes). The energy levels of the rX orbitals from Hiickel MO theory are shown at the right of Fig. 6. There are n
3. Cluster size distributions In the course of our study of the knock-on mechanism for fusion induced by carbon clusters we observed a striking odd-even effect in the yield distribution of C; clusters produced by Cs sputtering. A mass distribution is shown in Fig. 5. The yield of even-n clusters is enhanced over that for odd-n with n < 10. We had originally associated the strong peaks at A N 156, 180, 204 and 228 with C,,, C,,, Ci7, and C,,. It has been called to our attention [15,16] that these peaks were probably due to CsC,, CsC,, CsC,, CsC,, with masses
Fig. 5. Mass spectrum of negative ions from cesium bombardment of graphite.
120
R. Vandenbosch et al. /Nucl. Instr. and Meth. in Phys. Res. B 88 (1994) 116-121
(a)
Chains
(b)
--
--
Rings ---
--
-
------
suggested that the neglect of fluctuations in the deuteron energy during the stopping process may lead to a significant underestimate of the final yield. A qualitative molecular orbital picture for understanding the striking odd-even patterns observed in carbon cluster size dis~butions has been presented.
Acknowledgement n=4
n=5
Fig. 6. Schematic molecular orbital energies for chains (a) and rings (b). Only the nx orbital energies are illustrated. The arrows indicate the electron occupancy for neutral C, clusters.
This work was supported in part by the US Department of Energy.
References
electrons each to go into the n, and n,, orbitals. One can see that one must add an electron to a non-bonding orbital to make the C, anion, whereas the C, anion can be made by adding an electron to a bonding orbital. Similarly for C, the extra electron has to go into a more antibonding orbital than does the extra electron in C,. These effects lead to a striking oddeven effect in the measured electron affinities [21], with the odd-n neutrals having 30-40% larger values. They are also consistent with the odd-n enhancement of Schauer et al. for yt > 13. This latter results has to be treated with caution as the possibility of mixed Cs and C clusters may have been a problem in this study also. The odd-n enhancement for large y1was not seen when an argon, rather than a cesium, beam was used. One can also understand from Fig. 6 why the n = 15 (and 11 and 19) positive ion yield is especially large in laser vaporization of graphite [23,24]. It can be seen that 2n f 3 cyclic neutral clusters have an unusually loosely bound electron. An easy although oversimplified way to understand the shift in anion stability from even to odd when going from chains to rings is to realize that two non-bonding electrons on one end of a chain are available to put into the 7r orbitals when one closes the chain into a ring. One of each goes into the 7~~ and n; orbitals, leading to a shift in stability from even to odd.
4. Summary We have demonstrated that the enhancement in the fusion yield for small deuterated water molecules reported by Bae et al. is in error. Our yields are in agreement with calculations which do not include any collective enhancement. We have, furthermore, shown that d + d fusion can be observed with projectiles which do not include deuterons and that this arises from a knock-on mechanism. We have discussed theoretical efforts to predict the collective enhancement, and have
w R.J. Beuhler, G. Friedlander and L. Friedman, Phys. Rev. Lett. 63 (1989) 1292. 121Y.K. Bae, DC. Lorents and S.E. Young, Phys. Rev. A 44 (1991) R4091. 131 C. Carraro, B.Q. Chen, S. Schramm and S.R. Koonin, Phys. Rev. A 42 (1990) 1379. [41 R. Vandenbosch, T.A. Trainor, D.I. Will, J. Neubaner and I. Brown, Phys. Rev. Lett. 67 (1991) 3567. 151 R.J. Beuhler, G. Friedlander and L. Friedman, Phys. Rev. Lett. 68 (1992) 2108. t61 M. Fallavier, R. Kirsch, J.-C. Poigat, J. Remilliew, I-I. Rothard and J.P. Thomas, Phys. Rev. Lett. 70 (1993) 1022. f71 T. Mizota, IS. Okushi, Y. Honjo, Y. Futami, Y.H. Pu, II. Toyokawa, SM. Lee, J. Kasagi, B. Heusch, M. Hodge-Ali and J. Stoquert, Proc. Nikko ‘91 Symp., AIP Conf. Proc. no. 250. 181 M. Fallavier, J. Kemmler, R. Kirsch, J.C. Poizat, J. Remillieux and J.P. Thomas, Phys. Rev. Lett. 6.5 (1990) 621. [91 R. Vandenbosch, D. Ye, J. Neubauer, D.I. Will and T. Trainor, Phys. Rev. A. 46 (1992) 5741. 1101P.D. Townsend, J.R. Kelly and N.E.W. Hartley, in: Ion Implantation, Sputtering, and their Application (Academic Press, 1976) p. 20. 1111M. Hautala, Z. Pan and P. Sigmund, Phys. Rev. A 44 (1991) 7428. WI MI. Haftel, Phys. D 24 (1992) 385. 1131 S. Va~ealahti, M. Mann~en and F. Hammare’n, Z. Phys. D 22 (1992) 547. D41 M.H. Shapiro and T.H. Tombrello, Nucl. Instr. and M&h. B 58 (1991) 161. fl51 R. Middleton, Nucl. Instr. and Meth. A 144 (19771 373, and private communication. P61 Y. Le Beyec and P. Hakansson, private communication. [17] R.E. Honig, J. Chem. Phys. 22 (1954); R.E. Honig, in: Advances in Mass Spectrometry, vol. 2, ed. R.M. Elliot (Pergamon, London, 19631 p. 25. [18] S.N. Schauer, P. Williams and R.N. Compton, Phys. Rev. Lett. 65 (1990) 625. [19] S.J. Striclder and K.S. Pitzer, in: Molecular Orbitals in Chemistry, Physics and Biology, eds. P.O. Lowdin and B. Pullman (Academic Press, New York, 1964) p. 281.
R Vandenbosch et al. /Nucl. Instr. and Meth. in Phys. Res. B 88 (1994) 116-121
[ZO] Eb.+Pitzer and E. Clementi, J. Am. Chem. Sot. 81(1959) 1211 W. Geltner Jr. and R.J. Van Zee, Chem. Rev. 89 (1989) 1713. 1221 S. Yang, KJ. Taylor, M.J. Craycraft, J. Conceicao, C. Pettiette, 0. ~eshnovs~ and R.E. Smalley, Chem. Phys. Lett. 144 (1988) 431.
121
1231F.A. Rolfing, D.M. Cox and H. Kaldor, J. Chem. Phys. 81 (1984) 3322. [%I] J.W. McElvany, W.R. Cressy and A. O’Keefe, J. Chem. Phys. 85 (1986) 632.
NOMB
Beam fnteraetions with Materials % Atoms
Cluster impact fusion and cluster size distributions R. Vandenbosch
*, J. Neubauer, D.I. Will, T.A. Trainor and D. Ye
University of Washington, Nuclear Physics Laboratory CL-IO, Seattle, WA 98195, USA
D + D fusion yields have been measured for small deuterated molecules incident on deuterated polyethylene and are one to two orders of magnitude smaller than previously reported by another group. They are, however, in good agreement with free-particle cross sections. Fusion has also been observed for pure carbon clusters incident on deuterated polyethylene. Comparison of fusion yields per carbon for clusters and single carbon atoms show no collective enhancements. The observed fusion for carbon clusters is shown to be due to a knock on mechanism. Cluster size distributions for graphite sputtered with cesium exhibit a rich periodic structure whose origin will be discussed in terms of molecular orbital theory.
1. Introduction
In 1987, Beuhler, Friedlander, and Friedman 111 reported a remarkable observation. When they bombarded deuterated titanium with a 300 keV cluster of about 100 deuterated water molecules, they observed the d + d fusion reaction to occur at a rate about 10Z times larger than expected from an extrapolation of the known d + d cross section to the 0.3 keV energy of each deuterium in the deuterated water cluster. This was very surprising, as generally one does not see any dependence of a nuclear reaction rate on the chemical environment. The energy and distance scales of chemical and nuclear reactions are so different that one does not expect any enhancement unless there are very large changes in the target density or velocity distributions of the projectile or target deuterons. Our interest in the problem was further stimulated by the report of Bae et al. [Z] who not only confirmed the observations of Beuhler et al. but also reported that an enhancement could be observed for clusters of as few as two molecules.
2. Cluster impact firsion studies
Our first experiment was designed to check the observations of Bae et al. for small deuterated water clusters. We used a direct extraction ion source and the injection accelerator of the University of Washington tandem-booster facility. The deuterated cluster anions were mass analyzed and a particular species was acceierated to typically
* Corresponding author. Old-583X/94/$07.00
325 keV. After acceleration a weak magnetic field was applied to deflect away any deuterons which might have been produced by collisions in the residual gas and accelerated to sufficient energy to produce a spurious signal. We used deuterated polyethylene targets on a thin ~~inurn backing foil. The protons from the d + d -+ t + p reaction passed through the aluminum backing and were detected in a semiconductor detector placed at 0”. This provided a large solid angle and also enabled us to exploit the kinematic dependence of the proton energy on the deuteron energy inducing the reaction. The proton energy at 0” varies from 3.86 MeV for 300 keV deuterons to 3.08 MeV for 2 keV deuterons. This provides additional discrimination against beam contamination, and potentially can be used to deduce the energy spectrum of deuterons inducing fusion. This diagnostic was not available in previous experiments [1,2], in which the detector was located at 90” where the kinematic shift vanishes and where the kinematic broadening is at a maximum. We first compare our yields with the data of Bae, Lorents and Young [2]. In Fig. 1 we show the total proton fusion yield per cluster as a function of cluster size n. Although our results for n = 1 are in reasonable agreement with those of Bae, Lorents and Young and also with the thick-target model of Carraro et al. [3], for larger IZ our yields fall much more rapidly with n. Our yields are approximately 1 order of magnitude lower than theirs at y1= 2, and our n = 3 yield is 2 orders of magnitude lower than the interpolation of their n = 2 and 4 points. We believe that our present results indicate the absence of any special enhancement due to cluster impact for n < 5 and can be understood simply in terms of the free-deuteron thick-target excitation function. An attractive feature of our experimental setup is
Q 1994 - Elsevier Science B.V. All rights reserved
SSDZ 016%583X(93)E0925-7
R. Vandenbosch et al. /MA.
Imtr. and Meth. in Phys. Res. B 88 (1994) 116-121
that we can measure the thick-target fusion yield as a function of energy for D- ions. The results of these measurements, taken interspersed with the cluster measurements, are shown in Fig. 2.If there is no special enhancement over the free-deuteron cross sections due to the other atoms in the cluster, the cluster yields should be intimately related to the free-deuteron thick-target yields. Thus we would expect our cluster yield per deuteron, when plotted as a function of deuteron energy, to fall on our experimental free-deuteron yield curve. We show in Fig. 2 that this simple expectation is satisfied within our uncertainties due to beam integration. We have also calculated the expected (C,DJ, thick-target proton yield following the method of Carraro et al. The results are shown by the full curves in Figs. 1 and 2. The agreement in shape is excellent. The absolute magnitude of the calculation is somewhat higher than our data, perhaps due to ~certainties in stopping powers in the calculation and in the absolute detection efficiency in the experiment. After our work was published [4], Beuhler et al. [5] withdrew their claim for having observed large fusion yields. They reported that they now believed the fusion yields were at least two orders of magnitude below their original claim. Very recently, Fallavier et al. [6] reported on a study with (CD,), clusters. For small clusters (pt = l-5) they find no evidence for collective effects, in agreement with our observation. For large
Deuterons
l
ttiii
Ti
M=?858
30 34 36
20
I
I
I
I
0
5
IO
15
z E E :D Mctuster cluster
1s
I
I
20
25
30
(keV)
Fig. 2. Proton fusion yield per cluster divided by the number of deuterons per cluster plotted as a function of the kinetic energy of each deuteron. Also plotted are measured yields for free deuterons. From ref. 141.
I
1
1 i
*
3 Eoe et 01. Beuhler et ai.
l
- This work
-
100
cluster
size n
Fig. 1. Proton fusion yield for 225 keV clusters incident on (C2D4jn targets. The full curve is the calculate< thick-target yield (Yrr> as discussed in the text. From ref. 141.
clusters they determined upper limits for fusion considerably lower than originally reported for water clusters. In the course of our water cluster study we also observed a small amount of d + d fusion when a water anion not containing deuterium was used as a projectile. It seemed likely this was due to a secondary reaction initiated by the heavy oxygen atom. Mizota et al [7] have reported evidence supporting this surmise. The Lyon group [8] has reported the absence of cluster impact fusion for large clusters of pure deuterium. We decided to investigate the mechanism involved using carbon clusters, so as not to have any confusion arising from the relative roles of H and 0 atoms in normal water clusters. Carbon clusters were obtained by sputtering graphite with Cs+ ions. We had noticed in related attemps to produce beams of deuterated organic molecules that a broad range of carbon clusters could be obtained. We measured fusion yields for clusters and also for single carbon atoms, the latter as a function of energy. Our fusion yields, expressed as protons per incident carbon, are plotted as a function of carbon atom energy in Fig. 3. On such a plot a collective enh~cement of the fusion yield would reveal
118
R. Vandenbosch et al. /Nucl. Ins@. and Meth. in Phys. Res. B 88 (1994) 116-121
per deuteron, and da/dU is the differential cross section for transfering energy V to a deuteron by a carbon of energy E in a heavy projectile atom-deuteron collision. We assume that d&J, E)/dU is given by Rutherford scattering. Below roughly 20 keV the true scattering becomes less than Rutherford [lo] and our use of Rutherford scattering leads to an overestimate of the knock-on yield. However, this overestimation is only significant for measurements at very low energy, due to the rapid decrease in the fusion yield
101
100
10-I
10-2
10-3
10-4
Y,(U). 10-5
ECclusterl/n
(keV1
Fig. 3. Comparison of fusion yields for carbon clusters with those for single carbon atoms. Collective effects would be revealed on this plot by a larger yield for clusters compared to single atoms at the same energy/atom.
itself as a larger yield at a given carbon atom energy when that atom was part of a cluster. One observes
that we have no evidence for cluster enhancement of the fusion yield for clusters up to n = 12. We had originally reported limits on yields for n = 1.5 and 19, but now believe that the species accelerated were predominantly CsC, and CsC, rather than C, and C,, as discussed later on in this report. The leading order process by which one might expect D + D nuclear fusion for a projectile which does not contain deuterons is scattering of the projectile atoms off of target deuterons, leading to recoil deuterons with sufficient energy to react with other stationary deuterons in the target. This mechanism was one of several considered by Carraro et al. [3] in their early exploration of processes which could account for the fusion yields resulting from the deuterated water clusters bombardments of Beuhler et al. 111. Carraro et al. calculated the knock-on contribution of the projectile oxygen atoms. In their calculation this contribution generally exceeded that from the projectile deuterons, except for the smallest clusters. The fusion yield from the knocked-on deuterons is given by rF
Y,, = n,n) -
dE
o IdE/dxl
where n, is the deuteron number density in the target, n is the number of atoms per cluster, E, is the initial laboratory energy of each atom in the cluster, dE/dx is the stopping power of each projectile atom in the target, U,,(E) = 4m,m,E,/(m, + rn,j2 is the maximum energy that a projectile atom can transfer to a struck deuteron, Y,(U) is the thick-target fusion yield
It is also necessary to parameterize the stopping of the projectile atoms in the CD, target. This was done as described in ref. [9]. The calculated knock-on yield is shown by the continuous curve in Fig. 3. The calculation provides a reasonable account of the energy dependence of the observed fusion yield, which varies more than four orders of magnitude over the energy range studied. Our results for both deuterated water molecules and carbon clusters do not show any indication of a collective enhancement. Does this mean that no collective enhancement is to be expected, or just that it occurs at a level below our experimental sensitivity? Collective enhancement can originate from a microscopic model by the Fermi shuttle mechanism. This mechanism has already been discussed previously in this conference in the context of producing high energy electrons. An example of this mechanism is when a projectile deuteron is backscattered off a heavy target atom (e.g. Ti) and then struck by a heavy projectile atom (e.g. 0). The resulting deuteron can obtain an energy up to six times its original energy by this mechanism. In order to assess the fusion enhancement which might arise from these and higher order processes, molecular dynamics calculations for deuterated water clusters have been performed by Hautala, Pan and Sigmund [ll], by Haftel [12], and by Valkealahti, Manninen and Hammare’n [13]. We show in Fig. 4 one of the results of the calculation of Haftel. One sees that, indeed, deuterons with energies several times that in the original clusters are produced. The fusion yield in both calculations [11,12] were obtained by taking the deuteron energy spectrum (or an exponential fit to the spectrum) at the end of the simulation (after the cluster has undergone significant stopping) and calculating the yield expected using the thick-target yield expression, mentioned previously, or its equivalent. The calculated results show a large enhancement (- 1015) over the yield expected from deuterons of the original projectile velocity. They are, however, still many orders of magnitude lower than the revised upper limit of the Brookhaven group [5]. We believe that the expected yield may be underestimated in two of these calculations [11,12] because of the neglect of fluctuations in the deuteron spectra
R. Vandenbosch et al. / Nucl. Instr. and Meth. in Phys. Res. B 88 (I994) 116-121
.”
10’
102
lol
E (eV)
10'
Fig. 4. Spectra of beam D atoms at 6 fs (short dashed line), 12 fs (long dashed line), and 36 fs (solid line) after the start of the simulation of 300 keV (D,O),,, bombardment. From ref. KY.
during the stopping process. Our concern can be seen in the results presented in Fig. 4. At the intermediate time of 12 fs the deuteron energy spectrum is seen to have a higher energy deuteron than does the later spectrum used in calculating the fusion yield. It seems to us that the proper way to calculate the yield is to calculate the fusion probability at the distance of closest approach for every d + d collision which occurs as the deuterons are being slowed down. This type of approach has been employed recently by Valkealahti et al. [13] (A related approximation has been made by Shapiro and Tombrello [14], who kept track of the highest energy reached by a particle in a study of mixed-metal cluster impacts.) We believe the incorporation of these fluctuations in the deuteron spectrum could lead to an additional enhancement of several orders of magnitude.
119
of 157, 181, 205 and 229. A more careful mass calibration has confirmed the latter assignments. The favoring of even-n clusters through n = 8 has been observed previously [17,18], as has the enhancement of odd-n clusters with n 2 13 [17]. The even-n anions through n = 8 are attributed to linear chain structures, and the larger odd-n anions are attributed to cyclic structures on the basis of molecular orbital theory calculations [19,20]. It is also known from other studies that the yield of Ci clusters inhibits the opposite odd-even effect, particularly for n < 10 [21]. We present here a simple molecular orbital (MO) theory explanation for these observations. Our starting point is the generally greater stability of chains compared to rings for n < 10, and of monocyclic rings over chains for larger n [22]. We consider first the molecular orbitals of straight chains. 2n + 2 electrons go into u orbitals along the axis of the chain (z-axis). The remaining 2n + 2 electrons go into the degenerate “X and n,, orbitals. The Hiickel MO rrX energy levels for n = 4 and n = 5 are shown at the left of Fig. 5. One can see that for even n an extra electron to make the C; anion can go into a lower-energy bonding orbital, whereas for odd n it has to go into a non-bonding orbital. This accounts for the favoring of even-n anions. The electron removed from the neutral cluster to make a positive ion comes from a bonding orbital for either odd or even n, and the favoring of odd cations reflects the greater stability of odd-n neutrals. Now consider the molecular orbitals for monocyclic rings. We assume there are 2n electrons in the u orbitals along the ring (taken to lie in the yz-plane). We neglect ring strain and assume that the rX and +rrY orbitals are degenerate. (Our qualitative conclusions will not be altered if the rX and rY orbitals are displaced slightly with respect to each other due to deviations from colinearity of the r,, orbital axes). The energy levels of the rX orbitals from Hiickel MO theory are shown at the right of Fig. 6. There are n
3. Cluster size distributions In the course of our study of the knock-on mechanism for fusion induced by carbon clusters we observed a striking odd-even effect in the yield distribution of C; clusters produced by Cs sputtering. A mass distribution is shown in Fig. 5. The yield of even-n clusters is enhanced over that for odd-n with n < 10. We had originally associated the strong peaks at A N 156, 180, 204 and 228 with C,,, C,,, Ci7, and C,,. It has been called to our attention [15,16] that these peaks were probably due to CsC,, CsC,, CsC,, CsC,, with masses
Fig. 5. Mass spectrum of negative ions from cesium bombardment of graphite.
120
R. Vandenbosch et al. /Nucl. Instr. and Meth. in Phys. Res. B 88 (1994) 116-121
(a)
Chains
(b)
--
--
Rings ---
--
-
------
suggested that the neglect of fluctuations in the deuteron energy during the stopping process may lead to a significant underestimate of the final yield. A qualitative molecular orbital picture for understanding the striking odd-even patterns observed in carbon cluster size dis~butions has been presented.
Acknowledgement n=4
n=5
Fig. 6. Schematic molecular orbital energies for chains (a) and rings (b). Only the nx orbital energies are illustrated. The arrows indicate the electron occupancy for neutral C, clusters.
This work was supported in part by the US Department of Energy.
References
electrons each to go into the n, and n,, orbitals. One can see that one must add an electron to a non-bonding orbital to make the C, anion, whereas the C, anion can be made by adding an electron to a bonding orbital. Similarly for C, the extra electron has to go into a more antibonding orbital than does the extra electron in C,. These effects lead to a striking oddeven effect in the measured electron affinities [21], with the odd-n neutrals having 30-40% larger values. They are also consistent with the odd-n enhancement of Schauer et al. for yt > 13. This latter results has to be treated with caution as the possibility of mixed Cs and C clusters may have been a problem in this study also. The odd-n enhancement for large y1was not seen when an argon, rather than a cesium, beam was used. One can also understand from Fig. 6 why the n = 15 (and 11 and 19) positive ion yield is especially large in laser vaporization of graphite [23,24]. It can be seen that 2n f 3 cyclic neutral clusters have an unusually loosely bound electron. An easy although oversimplified way to understand the shift in anion stability from even to odd when going from chains to rings is to realize that two non-bonding electrons on one end of a chain are available to put into the 7r orbitals when one closes the chain into a ring. One of each goes into the 7~~ and n; orbitals, leading to a shift in stability from even to odd.
4. Summary We have demonstrated that the enhancement in the fusion yield for small deuterated water molecules reported by Bae et al. is in error. Our yields are in agreement with calculations which do not include any collective enhancement. We have, furthermore, shown that d + d fusion can be observed with projectiles which do not include deuterons and that this arises from a knock-on mechanism. We have discussed theoretical efforts to predict the collective enhancement, and have
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