**4.3.1 Magic number sequence of nana-clusters for liquid metal Al**

For liquid metal Al, for simplicity, we only analyze ten groups in the system in turn by the numbers of basic clusters contained in each group for two cases of liquid state at 943K and solid state at 350K, as shown in Table 5. From Table 5, it can be clearly seen that there is a peak value (maximum) of the numbers of clusters for each group and this is shown with a short underline in the table. As we compare this peak value with the abundance usually used in the research of cluster configurations, it is found that the two concepts are

Formation and Evolution Characteristics of Nano-Clusters (For Large-Scale Systems of 106

elements, especially in different states.

13

Liquid Metal Atoms) 189

750K

<sup>45</sup> <sup>48</sup> <sup>51</sup> <sup>59</sup> <sup>65</sup>

943K

350K

phase given by Harris et al is 13, 19, 23, 26, 29, 32, 34, 43, 46, 49, 55, 61, 64, 66... (see Fig.1 in Ref. ( Harris, Kidwell & Northby, 1984), and the sequence obtained from the mass spectra of Xe clusters formed in a supersaturated vapor phase given by Echt et al (Echt, Sattler & Recknagle, 1981) is 13, 19, 23, 25, 29, 55, 71... (see Fig.1 in this Ref.), these results are also in good agreement with our sequence in the same error range. That is to say that the this simulation result from metal Al is similar to those from inert gases Ar and Xe, and this similarity should reflect in certain degree some essential relations between different

Fig. 9. Variation of the number of clusters in system of Al with sizes of clusters (i. e. the

25-27

<sup>25</sup> <sup>27</sup>

31 33

<sup>38</sup> <sup>40</sup> 42

**4.3.2 Magic number sequence of nana-clusters for liquid metal Na** 

It is highly interesting that this magic number sequence is also in good agreement with the results, obtained by using MD simulation and other model potentials from Solov'yov's and Doye's works, such as 13, 19, 23, 26, 29, 32, 34, 43, 46, 49, 55, 61, 64, 71, … (see Fig.1 and 2 in Ref. (Solov'yov I A, Solov'yov A V & Greiner, 2003)), and 13, 19, 23, 26, 29, 34, 45, 51, 55,… (see Fig.1 and 2 in Ref. (Doye & Meyer, 2005)) , respectively. From these, it can be explained that as long as the methods used to solve the similar problem are reasonable, the results

Number of atom in cluster 10 20 30 40 50 60

Number of atom in cluster 20 30 40 50 60 70

For liquid metal Na, for deep understanding the size distribution of the clusters mentioned above, we also only analyze ten group levels in the system in turn by the numbers of basic clusters contained in each group level for two cases of liquid state at 573 K and solid state at

number of atoms contained in the cluster) at 350K.

19

Number of cluster in total groups

should also be similar.

0

2000

4000

6000

8000

10000

12000

Number of

cluster in

system

completely consistent with each other. As we display the relations of the numbers of clusters formed in system with the size (the number of atoms contained in them) of these clusters, it is further found that the positions of the peak value points of the numbers of clusters also correspond to the magic number points. It is also clearly seen that the numbers of clusters at 943K are much less than those for the same group level at 350K for the former five group levels and there are few or almost none for the latter five group levels; and the front five peak value positions of clusters at 943K are not all consistent with those at 350K, for convenience of discussion for magic numbers, we only show the simulation results at 350K in figure 9.

It is clear from figure 9 that the quantity of various clusters is sensitive to the size of a cluster, and the magic numbers do exist. In the solid state at 350K, the total magic number sequence of all groups are in turn as 13, 19, 25, 27, 31, 33, 38, 40, 42, 45, 48, 51, 59, 65, 67…. However, when the number of atoms contained in a cluster is more than 70, the position of its magic number would be ambiguous.

In order to further reveal the magic number characteristics of the above-mentioned groups, we show the variation of the numbers of clusters in the system with the numbers of atoms contained in the clusters for ten groups in Figure 10, respectively.

It is observed in Fig.10 that although the ranges of neighboring groups have overlapped each other, one or two partial magic numbers still can be obviously distinguished for each group, and all the partial magic numbers for the ten groups rather correspond to the total magic number sequence for the whole system as shown in Figure 10. Going further, the total magic number sequence can be classified again according to the order of the ten groups of clusters in the following sequence: 13 (first magic number), 19 (second), 25-27(third), 31- 33(fourth), 38-40(fifth), 42-45(sixth), 48-51(seventh), 55-59(eighth), 61-65(ninth) and 67(tenth). The ninth and tenth magic numbers are not so obvious in figure 10 because the numbers of clusters containing 9 and 10 basic clusters are insufficient, however, they stand out in figure 10 (c). For simplicity, the magic number sequence corresponding to the order of the ten groups of clusters can be listed again as 13, 19, 25(27), 31(33), 38(40), 42(45), 48(51), 55(59), 61(65) and 67, where the numbers in bracket are the secondary magic numbers of the corresponding groups of clusters. We think the above-mentioned analysis is very important for searching the origin of the magic number of clusters formed in the system.

We compare the total magic number sequence mentioned above to the experimental results of the photo-ionization mass spectra of clusters, formed through supersonic deposition from supersaturated gaseous phase Al, obtained by Schriver et al as shown in Fig.3 of Ref. (Schriver et al., 1990), it can be clearly seen that the magic numbers reported (14, 17, 23, 29, 37, 43, 47, 55, 67…), and those not reported (19, 21, 25, 33, and 39) (they can be clearly seen in the same Fig.3, maybe the authors thought those numbers were not consistent with the magic number rule at that time), are almost all consistent with our magic number sequence (in the error range of ±1). Thus, it can be said that the magic number sequence from our simulation is supported by the experimental results, but their clusters are produced by both different formation processes even though they are of the same element, Al.

In particular, as we further compare the magic number sequence from our simulation to the experimental results of inert gas clusters, it can be also clearly seen that the magic number sequence obtained from the mass spectra of Ar clusters formed in a supersaturated ionic

completely consistent with each other. As we display the relations of the numbers of clusters formed in system with the size (the number of atoms contained in them) of these clusters, it is further found that the positions of the peak value points of the numbers of clusters also correspond to the magic number points. It is also clearly seen that the numbers of clusters at 943K are much less than those for the same group level at 350K for the former five group levels and there are few or almost none for the latter five group levels; and the front five peak value positions of clusters at 943K are not all consistent with those at 350K, for convenience of discussion for magic numbers, we only show the simulation results at 350K

It is clear from figure 9 that the quantity of various clusters is sensitive to the size of a cluster, and the magic numbers do exist. In the solid state at 350K, the total magic number sequence of all groups are in turn as 13, 19, 25, 27, 31, 33, 38, 40, 42, 45, 48, 51, 59, 65, 67…. However, when the number of atoms contained in a cluster is more than 70, the position of

In order to further reveal the magic number characteristics of the above-mentioned groups, we show the variation of the numbers of clusters in the system with the numbers of atoms

It is observed in Fig.10 that although the ranges of neighboring groups have overlapped each other, one or two partial magic numbers still can be obviously distinguished for each group, and all the partial magic numbers for the ten groups rather correspond to the total magic number sequence for the whole system as shown in Figure 10. Going further, the total magic number sequence can be classified again according to the order of the ten groups of clusters in the following sequence: 13 (first magic number), 19 (second), 25-27(third), 31- 33(fourth), 38-40(fifth), 42-45(sixth), 48-51(seventh), 55-59(eighth), 61-65(ninth) and 67(tenth). The ninth and tenth magic numbers are not so obvious in figure 10 because the numbers of clusters containing 9 and 10 basic clusters are insufficient, however, they stand out in figure 10 (c). For simplicity, the magic number sequence corresponding to the order of the ten groups of clusters can be listed again as 13, 19, 25(27), 31(33), 38(40), 42(45), 48(51), 55(59), 61(65) and 67, where the numbers in bracket are the secondary magic numbers of the corresponding groups of clusters. We think the above-mentioned analysis is very important

in figure 9.

its magic number would be ambiguous.

contained in the clusters for ten groups in Figure 10, respectively.

for searching the origin of the magic number of clusters formed in the system.

different formation processes even though they are of the same element, Al.

We compare the total magic number sequence mentioned above to the experimental results of the photo-ionization mass spectra of clusters, formed through supersonic deposition from supersaturated gaseous phase Al, obtained by Schriver et al as shown in Fig.3 of Ref. (Schriver et al., 1990), it can be clearly seen that the magic numbers reported (14, 17, 23, 29, 37, 43, 47, 55, 67…), and those not reported (19, 21, 25, 33, and 39) (they can be clearly seen in the same Fig.3, maybe the authors thought those numbers were not consistent with the magic number rule at that time), are almost all consistent with our magic number sequence (in the error range of ±1). Thus, it can be said that the magic number sequence from our simulation is supported by the experimental results, but their clusters are produced by both

In particular, as we further compare the magic number sequence from our simulation to the experimental results of inert gas clusters, it can be also clearly seen that the magic number sequence obtained from the mass spectra of Ar clusters formed in a supersaturated ionic phase given by Harris et al is 13, 19, 23, 26, 29, 32, 34, 43, 46, 49, 55, 61, 64, 66... (see Fig.1 in Ref. ( Harris, Kidwell & Northby, 1984), and the sequence obtained from the mass spectra of Xe clusters formed in a supersaturated vapor phase given by Echt et al (Echt, Sattler & Recknagle, 1981) is 13, 19, 23, 25, 29, 55, 71... (see Fig.1 in this Ref.), these results are also in good agreement with our sequence in the same error range. That is to say that the this simulation result from metal Al is similar to those from inert gases Ar and Xe, and this similarity should reflect in certain degree some essential relations between different elements, especially in different states.

Fig. 9. Variation of the number of clusters in system of Al with sizes of clusters (i. e. the number of atoms contained in the cluster) at 350K.

It is highly interesting that this magic number sequence is also in good agreement with the results, obtained by using MD simulation and other model potentials from Solov'yov's and Doye's works, such as 13, 19, 23, 26, 29, 32, 34, 43, 46, 49, 55, 61, 64, 71, … (see Fig.1 and 2 in Ref. (Solov'yov I A, Solov'yov A V & Greiner, 2003)), and 13, 19, 23, 26, 29, 34, 45, 51, 55,… (see Fig.1 and 2 in Ref. (Doye & Meyer, 2005)) , respectively. From these, it can be explained that as long as the methods used to solve the similar problem are reasonable, the results should also be similar.
