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

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

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

> Cluster size

Number

Cluster consisting of 2 basic clusters

> Cluster number

of atom 573K223K Number

Cluster consisting of 3 basic clusters

> Cluster number

of atom 573K223K Number

10 2 0 17 3 0 23 0 5 28 0 7 32 0 1

11 43 4 18 9 8 24 2 28 29 0 21 33 0 6

12 297 149 19 46 227 25 15 108 30 2 54 34 0 17

17 59 151 24 46 507 30 19 503 35 4 322 39 0 156

18 6 0 25 9 134 31 12 305 36 5 284 40 1 162

26 2 22 32 4 138 37 3 239 **41 0 179** 

27 0 3 33 2 47 38 1 153 42 0 164

34 0 11 39 1 84 43 0 154

40 0 36 44 1 105

41 0 17 45 1 68

42 0 1 46 0 51

43 0 1 47 0 24

48 0 10

49 0 4

50 0 1

20 133 788 26 18 369 31 4 131 35 2 26

**21 213** 1736 27 26 620 32 1 193 36 0 66

**22** 182 **1883 28 36 708** 33 4 271 37 1 92

23 102 1167 29 25 638 **34 6 334** 38 1 120

Cluster size

Cluster consisting of 1 basic clusters

> Cluster number

> > 573K 223K

1180 2771

**1650 5962** 

1119 4992

412 1404

Cluster size

Number of atom

13

**14** 

15

16

(continued)

Liquid Metal Atoms) 191

Cluster number

of atom 573K223K Number

Cluster consisting of 5 basic clusters

of atom 573K223K

Cluster number

Cluster size

Cluster consisting of 4 basic clusters

Cluster size

Fig. 10. Relationship of the numbers of basic clusters in a group with the size of cluster (number of atoms contained in a cluster) at 350K in system of Al. (a) for 1 ~ 3 group; (b) for 4 ~ 6 group; (c) for 7 ~ 10 group.

223 K, as shown in Table 6. From Table 6, 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. It is also found that the positions of the peak value points of the numbers of clusters are corresponded to the magic number points. In liquid state, the magic numbers are in the order of 14, 21, 28, 34…, and it is not clear for the clusters contained more than five basic clusters. In solid state, the magic numbers are in the order of 14, 22, 28, 34, 41(43), 46(48), 52(54), 57(59), 61(66), 70(74), which are corresponding to the first, second, third, …. and tenth group levels, respectively, the numbers in the brackets are the second magic numbers corresponding to the same group level. The first four magic numbers are almost the same as in liquid state; thereafter, it is also not clear for the clusters contained more than ten basic clusters.

On the other hand, for further understanding the magic number characteristics of the group level of clusters, the relations of the number of clusters in each group level with the number of atoms contained in each cluster for twelve groups, and the total number of clusters in all the group levels enclosed at 223K are shown in Fig 11 (a), (b), (c) and (d).

Number of cluster in a group

Fig. 10. Relationship of the numbers of basic clusters in a group with the size of cluster (number of atoms contained in a cluster) at 350K in system of Al. (a) for 1 ~ 3 group;

51

48

for 1 basic cluster for 2 basic clusters for 3 basic clusters 350K

> 55 59

223 K, as shown in Table 6. From Table 6, 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. It is also found that the positions of the peak value points of the numbers of clusters are corresponded to the magic number points. In liquid state, the magic numbers are in the order of 14, 21, 28, 34…, and it is not clear for the clusters contained more than five basic clusters. In solid state, the magic numbers are in the order of 14, 22, 28, 34, 41(43), 46(48), 52(54), 57(59), 61(66), 70(74), which are corresponding to the first, second, third, …. and tenth group levels, respectively, the numbers in the brackets are the second magic numbers corresponding to the same group level. The first four magic numbers are almost the same as in liquid state; thereafter, it is also not clear for the clusters contained

Number of atom in cluster 30 40 50 60 70 80 90

61 65 67 for 7 basic clusters for 8 basic clusters for 9 basic clusters for 10 basic clusters

31

33

Number of atom in cluster 20 25 30 35 40 45 50 55

38

40

42 45

for 4 basic cluster for 5 basic clusters for 6 basic clusters 350K

350K

On the other hand, for further understanding the magic number characteristics of the group level of clusters, the relations of the number of clusters in each group level with the number of atoms contained in each cluster for twelve groups, and the total number of clusters in all the group levels enclosed at 223K are shown in Fig 11 (a), (b), (c)

(b) for 4 ~ 6 group; (c) for 7 ~ 10 group.

Number of cluster in a group

Number of atom in cluster 10 15 20 25 30 35 40 45

25-27

more than ten basic clusters.

and (d).

Number of cluster in a group

13

19


(continued)


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

magic numbers.

Liquid Metal Atoms) 193

levels at the same group level. For this point, as we consider the magic number of each group level as the corresponding partial magic number, the total magic number sequence of all the group levels can be considered as the superposition of all the partial

Therefore, the total magic number sequence can be analyzed according to the corresponding group level in the order of 14(the first magic number), 22(second), 28(third), 34(fourth), 41- 43(fifth), 46-48(sixth), 52-54 (seventh), 57-59(eighth), 61-66(ninth) and 70-74(tenth). However, the last three magic numbers also cannot be clearly distinguished in the Fig.11 (d), since the

Going further, it can be seen that not only have the experimental results reported by Schriver and Harris et al (Schriver et al., 1990; Harris, Kidwell & Northby, 1984) provided a vital experimental certification to our simulation results, but also our simulation results could provide a reasonable model explanation to those experimental results. As regards the magic numbers obtained from experimental researches, some of them can be explained as usual with the viewpoint of geometric shell structure of cluster configurations being closed regularly (for neutral clusters and charged clusters) (Knight et.al., 1984; Harris et al., 1984; Echt, Sattler & Recknagle, 1981; Schriver et al., 1990; Robles, Longo & Vega, 2002), and the others cannot be explained with the same viewpoint because they are corresponding to the geometric non-shell structure of cluster configurations. However, from our simulation, it can be clearly seen that during the forming process of larger clusters, only a few clusters accumulate and extend continuously with a basic cluster as the core according to a certain rule; most of them are formed with combining different numbers and different types of basic clusters. So, it is the normal case and can be explained to find more clusters with geometric non-shell

So far, the critical question is why the magic number sequence of clusters formed by solidification of liquid metal Al from our simulation is so similar to those magic number sequences of clusters formed by ionic spray and gaseous deposition of metal Al, inert gases Ar and Xe from experimental studies? We think the main reason is that the solidification process of liquid metal is essentially similar to the formation process of clusters in the above-mentioned experimental studies. We consider that in the solidification process of liquid metals, various cluster configurations could be formed by the rapid agglomerating of a large number of atoms as the system spreads over a large space for a short time, while in the formation process of clusters in the experiments, various cluster configurations could be formed by slow gathering of a few atoms as the system spreads over a small space for a long time, and both their final results could be similar each other on the whole (even though they are not be similar completely). On the other hand, at present, the essential differences between different elements, especially different states, are still not be distinguished in detail, it is necessary to analyze and compare in detail various similar and dissimilar magic

Therefore, it may be feasible to adopt magic numbers, especially the partial magic numbers of the group levels, obtained during the rapid solidification process of liquid metals to understand the magic number characteristics obtained with experimental

numbers of the larger clusters containing more basic clusters are not enough.

structure in their magic number sequence as above-noted.

numbers of these sequences in the future.

methods.

Table 6. Relations of the number of clusters consisting of 1-10 basic clusters with the cluster size (number of atoms included) for liquid metal Na.

Highly interesting is that though the ranges of neighboring group levels are overlapped each other as shown in Fig 11, the magic number of each group level is still clearly corresponded to the magic number of the total magic number sequence for all the group

Cluster number

of atom 573K223K Number

Cluster consisting of 9 basic clusters

> Cluster number

of atom 573K223K Number

Cluster size

Cluster consisting of 10 basicclusters

of atom 573K223K

Cluster number

Cluster size

Cluster consisting of 8 basic clusters

Cluster size

Cluster consisting of 6 basic clusters

> Cluster number

> > 573K 223K

Cluster size

Number of atom

Cluster consisting of 7 basic clusters

size (number of atoms included) for liquid metal Na.

Highly interesting is that though the ranges of neighboring group levels are overlapped each other as shown in Fig 11, the magic number of each group level is still clearly corresponded to the magic number of the total magic number sequence for all the group

Cluster number

of atom 573K223K Number

Cluster size

Number

levels at the same group level. For this point, as we consider the magic number of each group level as the corresponding partial magic number, the total magic number sequence of all the group levels can be considered as the superposition of all the partial magic numbers.

Therefore, the total magic number sequence can be analyzed according to the corresponding group level in the order of 14(the first magic number), 22(second), 28(third), 34(fourth), 41- 43(fifth), 46-48(sixth), 52-54 (seventh), 57-59(eighth), 61-66(ninth) and 70-74(tenth). However, the last three magic numbers also cannot be clearly distinguished in the Fig.11 (d), since the numbers of the larger clusters containing more basic clusters are not enough.

Going further, it can be seen that not only have the experimental results reported by Schriver and Harris et al (Schriver et al., 1990; Harris, Kidwell & Northby, 1984) provided a vital experimental certification to our simulation results, but also our simulation results could provide a reasonable model explanation to those experimental results. As regards the magic numbers obtained from experimental researches, some of them can be explained as usual with the viewpoint of geometric shell structure of cluster configurations being closed regularly (for neutral clusters and charged clusters) (Knight et.al., 1984; Harris et al., 1984; Echt, Sattler & Recknagle, 1981; Schriver et al., 1990; Robles, Longo & Vega, 2002), and the others cannot be explained with the same viewpoint because they are corresponding to the geometric non-shell structure of cluster configurations. However, from our simulation, it can be clearly seen that during the forming process of larger clusters, only a few clusters accumulate and extend continuously with a basic cluster as the core according to a certain rule; most of them are formed with combining different numbers and different types of basic clusters. So, it is the normal case and can be explained to find more clusters with geometric non-shell structure in their magic number sequence as above-noted.

So far, the critical question is why the magic number sequence of clusters formed by solidification of liquid metal Al from our simulation is so similar to those magic number sequences of clusters formed by ionic spray and gaseous deposition of metal Al, inert gases Ar and Xe from experimental studies? We think the main reason is that the solidification process of liquid metal is essentially similar to the formation process of clusters in the above-mentioned experimental studies. We consider that in the solidification process of liquid metals, various cluster configurations could be formed by the rapid agglomerating of a large number of atoms as the system spreads over a large space for a short time, while in the formation process of clusters in the experiments, various cluster configurations could be formed by slow gathering of a few atoms as the system spreads over a small space for a long time, and both their final results could be similar each other on the whole (even though they are not be similar completely). On the other hand, at present, the essential differences between different elements, especially different states, are still not be distinguished in detail, it is necessary to analyze and compare in detail various similar and dissimilar magic numbers of these sequences in the future.

Therefore, it may be feasible to adopt magic numbers, especially the partial magic numbers of the group levels, obtained during the rapid solidification process of liquid metals to understand the magic number characteristics obtained with experimental methods.

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

stability and higher heredity, and so on.

atoms of basic clusters).

Liquid Metal Atoms) 195

minority of atoms in which the central atoms of basic clusters are connected tightly each other with multi-bonded, would be more stable than others and they would possess better

Fig. 12. Schematic diagram of three larger clusters consisting of 43, 53 and 69 atoms within 7 basic clusters with connecting bonds, respectively, at 223K (the gray spheres are the center

These features can be shown in Fig.12. It is the schematic diagram of three larger clusters consisting of 43, 53 and 69 atoms within the same group level of 7 basic clusters, with

Fig. 11. Relations of the number of cluster in a group with the size of cluster (i. e. atoms included in cluster) at 223K .(a) for 1 ~ 3 groups; (b) for 4 ~ 6 groups; (c) for 7 ~ 9 groups; (d) for 10 ~ 12 group levels of clusters .

#### **4.3.3 Stability of nana-clusters**

From the above mentioned, it can be clearly seen that the larger clusters within a same group level should have not the same number of atoms because they contained different basic clusters containing different number of atoms. Therefore, these larger clusters would have different number of atoms. It can be clearly seen that those larger clusters containing

Number of clusters in system

0

200

400

600

800

for 1 basic clusters for 2 basic clusters for 3 basic clusters for 4 basic clusters for 5 basic clusters for 6 basic clusters for 7 basic clusters for 8 basic clusters for 9 basic clusters for 10 basic clusters for 11 basic clusters for 12 basic clusters

(a) (b)

Number of atoms in a cluster 10 15 20 25 30 35

22

6000 for the total cluster

223K for Na <sup>14</sup>

for the total cluster for 1 basic clusters for 2 basic clusters for 3 basic clusters for 4 basic clusters for 5 basic clusters for 6 basic clusters for 7 basic clusters for 8 basic clusters for 9 basic clusters for 10 basic clusters for 11 basic clusters for 12 basic clusters

28

(c) (d)

57 59

57 59

61

61

Number of atoms in a cluster 45 50 55 60 65

52 54

(d) for 10 ~ 12 group levels of clusters .

223K for Na

52 54

**4.3.3 Stability of nana-clusters** 

Fig. 11. Relations of the number of cluster in a group with the size of cluster (i. e. atoms included in cluster) at 223K .(a) for 1 ~ 3 groups; (b) for 4 ~ 6 groups; (c) for 7 ~ 9 groups;

Number of clusters in system

0

50

61

66

66

70

223K for Na

34 43

34

100

150

200

61

Number of atoms in a cluster 60 65 70 75 80 85 90

<sup>74</sup> <sup>70</sup>

for 12 basic clusters 223K for Na

Number of atoms in a cluster 30 35 40 45 50 55

46 48

<sup>46</sup> <sup>48</sup>

41

<sup>74</sup> <sup>79</sup> <sup>83</sup>

79 82

84

for the total cluster for 1 basic clusters for 2 basic clusters for 3 basic clusters for 4 basic clusters for 5 basic clusters for 6 basic clusters for 7 basic clusters for 8 basic clusters for 9 basic clusters for 10 basic clusters for 11 basic clusters

for the total cluster for 1 basic clusters for 2 basic clusters for 3 basic clusters for 4 basic clusters for 5 basic clusters for 6 basic clusters for 7 basic clusters for 8 basic clusters for 9 basic clusters for 10 basic clusters for 11 basic clusters for 12 basic clusters

87

From the above mentioned, it can be clearly seen that the larger clusters within a same group level should have not the same number of atoms because they contained different basic clusters containing different number of atoms. Therefore, these larger clusters would have different number of atoms. It can be clearly seen that those larger clusters containing

0

100

48

200

300

Number of clusters in system

> 46 48

46

Number of clusters in system

minority of atoms in which the central atoms of basic clusters are connected tightly each other with multi-bonded, would be more stable than others and they would possess better stability and higher heredity, and so on.

Fig. 12. Schematic diagram of three larger clusters consisting of 43, 53 and 69 atoms within 7 basic clusters with connecting bonds, respectively, at 223K (the gray spheres are the center atoms of basic clusters).

These features can be shown in Fig.12. It is the schematic diagram of three larger clusters consisting of 43, 53 and 69 atoms within the same group level of 7 basic clusters, with

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

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connecting bonds, respectively, at 223K (the gray spheres are the center atoms of basic clusters). From the diagrams of their center atoms with multi-bonded or single-bonded each other, it can be clearly seen that the cluster consisting of 43 atoms has a dense connecting of all atoms and would possess better stability and higher heredity than other two clusters consisting of 53 and 69 atoms, respectively, in turn.
