- Email: [email protected]
Vibrational properties of nickel and gold clusters
NANoSTRUCTURED MATERIALS VOL. 3, PP. 385-390, 1993 COPYRIGHT(~)1993 PERGAMONPRESSLTD. ALL R~HTSRESERVED.
0965-9773/93 $6.00 + .00 PRINTEDIN THE USA
VIBRATIONAL PROPERTIES OF NICKEL AND GOLD CLUSTERS S. Carnalla 1, A. Posada 2,3 and I.L. Garz6n 3
lFacultad de Ciencias, Universidad Aut6noma de Baja California, 22800 Ensenada, Baja California, M6xico 2Centro de Investigaci6n en Fisica, Universidad de Sonora, 83190 Hermosillo, Sonora, M6xico 3Instituto de Fisica, Universidad Nacional Aut6noma de M6xico, Apartado Postal 2681, 22800 Ensenada, Baja California, M6xico
Abstract - - A normal mode analysis was done on Nin and Aun, n=12,13,14,19,20,55, clusters to characterize their vibrational properties at low temperatures. A numerical diagonalization of the dynamical matrix was used to obtain the normal frequencies and eigenvectors, corresponding to the cluster minimum energy structure as well as isomers with higher potential energy. The dynamical matrix was constructed using a n-body Gupta-like potential. Different features werefound in thefrequency distribution of the Nin and Au, clusters according to their size and structure. The cluster maximum frequency in nickel is higher than in gold for all the sizes studied.
1. INTRODUCTION One of the most important physical properties of atomic clusters and small particles is their structure (1). At the present time only indirect measurements through electron diffraction experiments have allowed to obtain some information on the geometry of isolated clusters (2). Electron microscopy techniques have been tools of extraordinary importance to get some insight into the structural properties of supported small particles (3). The chemical reactivity of metal clusters with molecular adsorbates also has been utilized to determine the structure of the clusteradsorbate system (4). However, a direct experimental determination of the structure of isolated and supported atomic clusters is not yet available. From the theoretical point of view, the calculation of the equilibrium structure of atomic clusters or small particles is more reliable but, of course, depends on the type of interatomic interactions used to describe the bonding in the system. Various studies have been done in this direction using empirical potentials, and Ab Initio calculations (1). The results on the cluster structures obtained from these theoretical methods are difficult to test due to the lack of clean experimental techniques, especially for cluster sizes in the range of 1-I0 nm. 385
386
S CARNALLA,A PO,SADAANDIL GAP.ZbN
An experimental alternative to probe the cluster structure could be the development of vibrational spectroscopies able to distinguish between different possible geometries. The existence of point symmetry groups in the cluster configuration could be efficiently used to detect the active spectral modes. The theoretical study of vibrational properties of clusters and small particles has not been very intense, due in part, to the scarce data from experimental work and the lack of adequate models to describe the atomic interactions in the cluster. In this work we present results for the vibrational normal frequencies of nickel and gold clusters of various sizes. A normal mode analysis through the diagonalization of the dynamical matrix was used to calculate the cluster normal frequencies. The n-body Gupta-like potential was used to model the n-body interactions present in the metal cluster. The present calculation makes a comparison between the frequency distribution of nickel and gold clusters and how it evolves with the cluster size. A brief description of the calculations is presented in section 2. The results and their discussion are presented in section 3. A summary is given in section 4.
2. THEORETICAL BACKGROUND The structural and dynamical properties of an n-atom cluster are determined by the topology of the 3n-dimensional potential energy function. The local minima of these hypersurfaces correspond to stable cluster structures. To model the atomic interactions in the n-atom metal cluster we used the n-body size-dependent Gupta potential (5-8) which in reduced units of energy V* and interatomic distances r*ij is given by
1~ A~e-P(rq-l) ~e-2q(ro-l) n
n
*
n
*
2
[11
In Eq.[l] we used the values p = 9, q = 3, for nickel and p = 10.15, q = 4.13 for gold (5-8). The values A = 0.101036 for nickel and A = 0.118438 for gold are determined to yield the minim urn energy fcc structure of the bulk phase. The absolute potential energy V and distances rij are obtained from their corresponding reduced values V* and r*ij using material- and size-dependent units of energy Un and distance r0n: V = V* • Un, rij = r*ij r0n (5-8). Using constant-energy molecular dynamics and the technique of thermal quenching (i.e. repeatedly setting the momenta of the particles to zero), we obtained the minimum energy structures of the clusters as well as isomeric structures with higher potential energies. The thermal quenched procedure was stopped when the cluster temperature dropped below 10-6 K. The 3n X 3n dynamical matrix containing the second derivatives of the potential energy with respect to the Cartesian coordinates of the n atoms in the cluster was constructed using the potential given in eq. [1]. The second derivatives were evaluated using the equilibrium coordinates of the minimum energy cluster structure obtained with the thermal quenching method. The dynamical matrix was diagonalized using a numerical procedure to obtain the normal frequencies and eigenvectors. For the cluster structures with a high degree of symmetry, group theory was used to characterize the cluster normal modes.
VIBRATIONALPROPERTIESOF NICKELAND GOLD CLUSTERS
387
3. RESULTS AND DISCUSSION Nickel clusters have icosahedral symmetry in their lowest energy structure for all the sizes studied (5,7). For the n = 13 cluster the icosahedral geometry is the most stable with energy V* = - 12.463. The lowest energy configurations for n = 12 (V* = -11.292) and n = 14 (V* = - 13.398) clusters are obtained from the icosahedral n = 13 by removing one surface atom and adding one atom to a three fold surface site, respectively, and allowing the relaxation of the cluster. The double icosahedron is the most stable geometry for n = 19 (V* = -18.641). The most stable configuration for n = 20 (V* = -19.651) is the 19-atom double icosahedron with an additional atom placed over an edge of the central five-atom ring. The most stable structure ofNi55 (V* = -57.400) is a twoshell icosahedron. These cluster configurations were used to evaluate the dynamical matrix of second derivatives of the potential energy with respect to the atomic coordinates. A numerical diagonalization of this matrix was performed to produce the normal frequencies and eigenvectors. Figure 1 shows the distribution of normal frequencies for the six sizes studied. For Nil2 (Csv) the normal frequencies follow the symmetry structure F = 5A1 + IA2 + 6El +6E2 with twelve modes twofold- and six onefold-degenerate. In Nil3 (Ih) the normal modes are given by F = lAg + IGg + 2Hg + 2Ti u + IT2u + IGu + IHu which correspond to three modes with degeneracy 5, two modes with degeneracy 4, three modes with degeneracy 3 and one single mode non-degenerate. The normal modes for Nil4 (C3v) are giving by F = 9A1 + 3A2 + 12E which are equivalent to twelve modes onefold- and twofold-degenerate. The 51 vibrational modes of Nil9 (D5t0 are F = 5A'l + IA'2 + 6E'1 + 5E'2 + l A " l + 4A"2 + 5E"l + 4E"2. Twenty of these modes are twofold-degenerate and the remaining eleven are non-degenerate. Ni20 (C3v) has eighteen modes with degeneracy 1 and 2 given by F = 13A1 + 5A2 + 18E. Ni55 (Ih) has 159 normal modes with symmetry F = 2Ag + 4Tlg + 3T2g + 5Gg + 7Hg + IA1 + 6T|u + 4T2u + 5Gu + 6Hu. These correspond to thirteen fivefold-, ten fourfold-, seventeen threefold- and three non-degenerate normal frequencies. The minimum energy structures for gold clusters are different than those found for nickel for all the sizes studied, with the exception of Aul3, which has an icosahedral symmetry with energy V* = -11.774 (6). The most stable structures of Aul2 and Aul4 were obtained recently (8). For n = 12 the cluster shows a low symmetry structure with energy V* = -10.804, whereas the n = 14 cluster structure with symmetry Cry is formed by two parallel 6-atom rings (rotated 30 degrees) with one atom on the top of each ring. It has an energy of V* = -12.745. AuÂ9, Au20 and Au55 have unsymmetrical structures as the most stable configurations with energies of V* = -17.562, V* = -18.509 and V* = -52.603, respectively (8). The distribution of normal frequencies for these clusters are given in Figure 2. The different structural patterns found in gold clusters with respect to nickel produce a distinct behavior in their normal frequency distribution. Aun, n = 12,19,20,55, clusters show a broad band of frequencies with no degeneracy. Aut3 (Ih) shows the same distribution as Nil3 and Aul4 (C3v) has twelve twofold- and onefold-degenerate normal modes. Figures 1 and 2 show that the cluster maximum frequency Okaax has a small variation with the cluster size in the range n = 12-55. Moreover, the values for tOmaxin Nin are higher than in Aun clusters. This result is consistent with the behavior of the cutoff frequency in the bulk phase. 4. SUMMARY The vibrational frequencies of Nin and Aun, n = 12,13,14,19,20,55 clusters were obtained through a normal mode analysis using an n-body Gupta-like potential to model the interatomic
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Figure 1. Normal frequency distribution for Nin clusters. The frequency co is in units of 1.282 x 1013 s -1.
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380
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interactions in the cluster. Nickel clusters have a frequency dislribution with a high degree of degeneracy due to the high symmetry of their lowest energy configurations. A broad band of frequencies with no degeneracy was obtained for gold clusters (except Aul3 and Aul4) due to the absence of symmetry in their minimum energy structure. For n = 13,14 the vibrational frequencies in gold are similar to nickel clusters. The width of the frequency distribution shows a small variation with the cluster size. The maximum frequency in nickel clusters is higher than in gold for all the sizes studied, consistent with the bulk values. The broadening of the cluster vibrational frequencies with increasing temperature can be studied via molecular dynamics sire ulations. Such calculation is now in progress for Nin and Aun clusters. The present study, as well as the one at finite temperature, is waiting for experimental verification to gain insight into the study of the vibrational properties of metal clusters. ACKNOWLEDGEMENT This work was supported by DGAPA-UNAM under Project IN- 103189 and by CONACYT, M6xico under grant D111-904360.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
8.
See, for example, in: Physics and Chemistry of Finite Systems:From Clusters to Crystals. P. Jena, S.N. Khanna, and B.K. Rao (eds.), Vol. 1-2. Kluwer Academic Publishers, Dordrecht, (1992) and references therein. A. Yokozeki and G.D. Stein, J. Appl. Phys. 49, 2224 (1978); J. Farges, M.E deFeraudy, B. Raoult and G. Torchet, J. Chem. Phys. 78, 5067 (1983). See, for example, in: Ultramicroscopy 20, No. 1/2 (1986) and references therein; M.J. Yacam~, R. Herrera, A. G6mez, S. Tehuacanero and P. Schabes-Retchkiman, Surface Sci. 237,248 (1990). K. Lin, E.K. Parks, S.C. Richtsmeir, L.G. Pobo, and S.J. Riley, J. Chem. Phys. 83, 2282 (1985); W.E Hoffman, E.K. Parks, G.C. Nieman, L.G. Pobo and S.J. Riley, Z. Phys. D7, 83 (1987). J. Jellinek and I.L. Gar~n, Z.Phys. D20, 239 (1991). I.L. Garz6n and J. Jellinek, Z.Phys. D20, 235 (1991). I.L. Garz6n and J. Jellinek, Physics and Chemistry of Finite Systems:From Clusters to Crystals. P. Jena, S.N. Khanna, and B.K. Rao (eds.), Vol. I, p. 405, Kluwer Academic Publishers, Dordrecht, (1992). I.L. Garz6n and J. Jellinek, to be published.
0965-9773/93 $6.00 + .00 PRINTEDIN THE USA
VIBRATIONAL PROPERTIES OF NICKEL AND GOLD CLUSTERS S. Carnalla 1, A. Posada 2,3 and I.L. Garz6n 3
lFacultad de Ciencias, Universidad Aut6noma de Baja California, 22800 Ensenada, Baja California, M6xico 2Centro de Investigaci6n en Fisica, Universidad de Sonora, 83190 Hermosillo, Sonora, M6xico 3Instituto de Fisica, Universidad Nacional Aut6noma de M6xico, Apartado Postal 2681, 22800 Ensenada, Baja California, M6xico
Abstract - - A normal mode analysis was done on Nin and Aun, n=12,13,14,19,20,55, clusters to characterize their vibrational properties at low temperatures. A numerical diagonalization of the dynamical matrix was used to obtain the normal frequencies and eigenvectors, corresponding to the cluster minimum energy structure as well as isomers with higher potential energy. The dynamical matrix was constructed using a n-body Gupta-like potential. Different features werefound in thefrequency distribution of the Nin and Au, clusters according to their size and structure. The cluster maximum frequency in nickel is higher than in gold for all the sizes studied.
1. INTRODUCTION One of the most important physical properties of atomic clusters and small particles is their structure (1). At the present time only indirect measurements through electron diffraction experiments have allowed to obtain some information on the geometry of isolated clusters (2). Electron microscopy techniques have been tools of extraordinary importance to get some insight into the structural properties of supported small particles (3). The chemical reactivity of metal clusters with molecular adsorbates also has been utilized to determine the structure of the clusteradsorbate system (4). However, a direct experimental determination of the structure of isolated and supported atomic clusters is not yet available. From the theoretical point of view, the calculation of the equilibrium structure of atomic clusters or small particles is more reliable but, of course, depends on the type of interatomic interactions used to describe the bonding in the system. Various studies have been done in this direction using empirical potentials, and Ab Initio calculations (1). The results on the cluster structures obtained from these theoretical methods are difficult to test due to the lack of clean experimental techniques, especially for cluster sizes in the range of 1-I0 nm. 385
386
S CARNALLA,A PO,SADAANDIL GAP.ZbN
An experimental alternative to probe the cluster structure could be the development of vibrational spectroscopies able to distinguish between different possible geometries. The existence of point symmetry groups in the cluster configuration could be efficiently used to detect the active spectral modes. The theoretical study of vibrational properties of clusters and small particles has not been very intense, due in part, to the scarce data from experimental work and the lack of adequate models to describe the atomic interactions in the cluster. In this work we present results for the vibrational normal frequencies of nickel and gold clusters of various sizes. A normal mode analysis through the diagonalization of the dynamical matrix was used to calculate the cluster normal frequencies. The n-body Gupta-like potential was used to model the n-body interactions present in the metal cluster. The present calculation makes a comparison between the frequency distribution of nickel and gold clusters and how it evolves with the cluster size. A brief description of the calculations is presented in section 2. The results and their discussion are presented in section 3. A summary is given in section 4.
2. THEORETICAL BACKGROUND The structural and dynamical properties of an n-atom cluster are determined by the topology of the 3n-dimensional potential energy function. The local minima of these hypersurfaces correspond to stable cluster structures. To model the atomic interactions in the n-atom metal cluster we used the n-body size-dependent Gupta potential (5-8) which in reduced units of energy V* and interatomic distances r*ij is given by
1~ A~e-P(rq-l) ~e-2q(ro-l) n
n
*
n
*
2
[11
In Eq.[l] we used the values p = 9, q = 3, for nickel and p = 10.15, q = 4.13 for gold (5-8). The values A = 0.101036 for nickel and A = 0.118438 for gold are determined to yield the minim urn energy fcc structure of the bulk phase. The absolute potential energy V and distances rij are obtained from their corresponding reduced values V* and r*ij using material- and size-dependent units of energy Un and distance r0n: V = V* • Un, rij = r*ij r0n (5-8). Using constant-energy molecular dynamics and the technique of thermal quenching (i.e. repeatedly setting the momenta of the particles to zero), we obtained the minimum energy structures of the clusters as well as isomeric structures with higher potential energies. The thermal quenched procedure was stopped when the cluster temperature dropped below 10-6 K. The 3n X 3n dynamical matrix containing the second derivatives of the potential energy with respect to the Cartesian coordinates of the n atoms in the cluster was constructed using the potential given in eq. [1]. The second derivatives were evaluated using the equilibrium coordinates of the minimum energy cluster structure obtained with the thermal quenching method. The dynamical matrix was diagonalized using a numerical procedure to obtain the normal frequencies and eigenvectors. For the cluster structures with a high degree of symmetry, group theory was used to characterize the cluster normal modes.
VIBRATIONALPROPERTIESOF NICKELAND GOLD CLUSTERS
387
3. RESULTS AND DISCUSSION Nickel clusters have icosahedral symmetry in their lowest energy structure for all the sizes studied (5,7). For the n = 13 cluster the icosahedral geometry is the most stable with energy V* = - 12.463. The lowest energy configurations for n = 12 (V* = -11.292) and n = 14 (V* = - 13.398) clusters are obtained from the icosahedral n = 13 by removing one surface atom and adding one atom to a three fold surface site, respectively, and allowing the relaxation of the cluster. The double icosahedron is the most stable geometry for n = 19 (V* = -18.641). The most stable configuration for n = 20 (V* = -19.651) is the 19-atom double icosahedron with an additional atom placed over an edge of the central five-atom ring. The most stable structure ofNi55 (V* = -57.400) is a twoshell icosahedron. These cluster configurations were used to evaluate the dynamical matrix of second derivatives of the potential energy with respect to the atomic coordinates. A numerical diagonalization of this matrix was performed to produce the normal frequencies and eigenvectors. Figure 1 shows the distribution of normal frequencies for the six sizes studied. For Nil2 (Csv) the normal frequencies follow the symmetry structure F = 5A1 + IA2 + 6El +6E2 with twelve modes twofold- and six onefold-degenerate. In Nil3 (Ih) the normal modes are given by F = lAg + IGg + 2Hg + 2Ti u + IT2u + IGu + IHu which correspond to three modes with degeneracy 5, two modes with degeneracy 4, three modes with degeneracy 3 and one single mode non-degenerate. The normal modes for Nil4 (C3v) are giving by F = 9A1 + 3A2 + 12E which are equivalent to twelve modes onefold- and twofold-degenerate. The 51 vibrational modes of Nil9 (D5t0 are F = 5A'l + IA'2 + 6E'1 + 5E'2 + l A " l + 4A"2 + 5E"l + 4E"2. Twenty of these modes are twofold-degenerate and the remaining eleven are non-degenerate. Ni20 (C3v) has eighteen modes with degeneracy 1 and 2 given by F = 13A1 + 5A2 + 18E. Ni55 (Ih) has 159 normal modes with symmetry F = 2Ag + 4Tlg + 3T2g + 5Gg + 7Hg + IA1 + 6T|u + 4T2u + 5Gu + 6Hu. These correspond to thirteen fivefold-, ten fourfold-, seventeen threefold- and three non-degenerate normal frequencies. The minimum energy structures for gold clusters are different than those found for nickel for all the sizes studied, with the exception of Aul3, which has an icosahedral symmetry with energy V* = -11.774 (6). The most stable structures of Aul2 and Aul4 were obtained recently (8). For n = 12 the cluster shows a low symmetry structure with energy V* = -10.804, whereas the n = 14 cluster structure with symmetry Cry is formed by two parallel 6-atom rings (rotated 30 degrees) with one atom on the top of each ring. It has an energy of V* = -12.745. AuÂ9, Au20 and Au55 have unsymmetrical structures as the most stable configurations with energies of V* = -17.562, V* = -18.509 and V* = -52.603, respectively (8). The distribution of normal frequencies for these clusters are given in Figure 2. The different structural patterns found in gold clusters with respect to nickel produce a distinct behavior in their normal frequency distribution. Aun, n = 12,19,20,55, clusters show a broad band of frequencies with no degeneracy. Aut3 (Ih) shows the same distribution as Nil3 and Aul4 (C3v) has twelve twofold- and onefold-degenerate normal modes. Figures 1 and 2 show that the cluster maximum frequency Okaax has a small variation with the cluster size in the range n = 12-55. Moreover, the values for tOmaxin Nin are higher than in Aun clusters. This result is consistent with the behavior of the cutoff frequency in the bulk phase. 4. SUMMARY The vibrational frequencies of Nin and Aun, n = 12,13,14,19,20,55 clusters were obtained through a normal mode analysis using an n-body Gupta-like potential to model the interatomic
388
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I
D
I
~
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=1 m I
i
2..0
U
A POSADA AND IL GARZ0N
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i
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Figure 1. Normal frequency distribution for Nin clusters. The frequency co is in units of 1.282 x 1013 s -1.
VIBRATIONALPROPERTIESOFNICKELANDGOLDCLUSTERS
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Figure 2. Normal frequency distribution for Aun clusters. The frequency o~ is in units of 0.699 x 1013 s q.
380
S CARNALLA,A POSADAANOIL GARZ6N
interactions in the cluster. Nickel clusters have a frequency dislribution with a high degree of degeneracy due to the high symmetry of their lowest energy configurations. A broad band of frequencies with no degeneracy was obtained for gold clusters (except Aul3 and Aul4) due to the absence of symmetry in their minimum energy structure. For n = 13,14 the vibrational frequencies in gold are similar to nickel clusters. The width of the frequency distribution shows a small variation with the cluster size. The maximum frequency in nickel clusters is higher than in gold for all the sizes studied, consistent with the bulk values. The broadening of the cluster vibrational frequencies with increasing temperature can be studied via molecular dynamics sire ulations. Such calculation is now in progress for Nin and Aun clusters. The present study, as well as the one at finite temperature, is waiting for experimental verification to gain insight into the study of the vibrational properties of metal clusters. ACKNOWLEDGEMENT This work was supported by DGAPA-UNAM under Project IN- 103189 and by CONACYT, M6xico under grant D111-904360.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
8.
See, for example, in: Physics and Chemistry of Finite Systems:From Clusters to Crystals. P. Jena, S.N. Khanna, and B.K. Rao (eds.), Vol. 1-2. Kluwer Academic Publishers, Dordrecht, (1992) and references therein. A. Yokozeki and G.D. Stein, J. Appl. Phys. 49, 2224 (1978); J. Farges, M.E deFeraudy, B. Raoult and G. Torchet, J. Chem. Phys. 78, 5067 (1983). See, for example, in: Ultramicroscopy 20, No. 1/2 (1986) and references therein; M.J. Yacam~, R. Herrera, A. G6mez, S. Tehuacanero and P. Schabes-Retchkiman, Surface Sci. 237,248 (1990). K. Lin, E.K. Parks, S.C. Richtsmeir, L.G. Pobo, and S.J. Riley, J. Chem. Phys. 83, 2282 (1985); W.E Hoffman, E.K. Parks, G.C. Nieman, L.G. Pobo and S.J. Riley, Z. Phys. D7, 83 (1987). J. Jellinek and I.L. Gar~n, Z.Phys. D20, 239 (1991). I.L. Garz6n and J. Jellinek, Z.Phys. D20, 235 (1991). I.L. Garz6n and J. Jellinek, Physics and Chemistry of Finite Systems:From Clusters to Crystals. P. Jena, S.N. Khanna, and B.K. Rao (eds.), Vol. I, p. 405, Kluwer Academic Publishers, Dordrecht, (1992). I.L. Garz6n and J. Jellinek, to be published.