Introduction to Cluster Dynamics Paul-Gerhard Reinhard, Eric Suraud 2004 WILEY-VCH Verlag GmbH & Co. B Gross properties of atoms and solids Some basic properties and characteristics of atoms and bulk have been discussed in a general manner in Chapter 1. The aim of this appendix is to provide quantitative information complementing the more qualitative discussions of Chapter 1. B.1 The periodic table of elements The Mendeleev table of elements provides a classification of elements on the basis of their properties, in particular at the chemical level. It is shown on the next page. The layout follows a standard pattern. The atomic number Z (top left) is the number of protons in the nucleus. The atomic mass (bottom) is weighted by isotopic abundances in the Earth s surface, relative to the mass of the 12 C isotope, defined to be exactly 12 unified atomic mass units (amu). Relative isotopic abundances often vary considerably, both in natural and commercial samples. A number in parentheses is the mass of the longest lived isotope of that element when no stable isotope exists. For elements 110 to 112, the atomic numbers of known isotopes are given. These data are from [AW93, oawa96]. The names given below for elements 104 to 109 are those recommended by the International Union of Pure and Applied Chemistry in late 1997. Elements belonging to a vertical column form a group, while elements along a horizontal line constitute a period. The elements of each group have consistently high or low values of certain physical and chemical properties. As is well known the small energies involved in chemistry imply that only the least bound electrons do play a role in the interactions between atoms, and hence in molecules. A key factor is thus the occupation of the last (valence) electronic shell. As a consequence shell closure as observed in rare gases (group 18, He, Ne, Ar,Kr,Xe...) makestheseatomsratherinert. Inturn,groups 1 (alkalines) and 17 (halogens) are chemically particularly active, because of their closeness to rare gases. Atoms of group 1 have a single weakly bound s electron, they are strongly metallic, usually soft and very reactive with oxygen or water, in particular. Halogens (group 17) possess a p subshell lacking one electron for shell closure and they thus develop a strong tendency to gain an electron from outside. They can in particular form especially stable arrangements with alkalines. In between groups 1 and 17 tendencies interpolate between the ones observed in alkalines or halogens and they are globally less marked. Groups 3 to 12 correspond to the so called transition metals in which the energetic position of the d and f orbitals relative to s and p shells is fluctuating from element to element. This inhibits a prediction of the properties of these elements by simple systematics. Introduction to cluster dynamics. Paul-Gerhard Reinhard, Eric Suraud Copyright c 2004 Wiley-VCH Verlag GmbH & Co. KGaA ISBN: 3-527-40345-0
260 B Gross properties of atoms and solids
B.2 Atomic trends 261 B.2 Atomic trends The evolution of gross atomic properties with atomic number helps one to understand qualitatively the behavior of an element inside a compound. Particularly important is how atomic properties vary within a given group or period. A basic aspect is electron binding as a function of the atomic number Z and thus the way atomic radii and ionization potentials (IP) evolve. Intuitively, when Z increases (alonga periodordownagroup) onemightexpectthatelectrons become more strongly bound. But the larger the Z, the larger the number of bound electrons and the larger the effect of the Pauli principle to make the least bound electrons farther and farther away from the nucleus, and thus the electrons are less and less strongly bound. Both effects of course act against each other, but altogether it is the shielding effect which dominates. The distance factor thus wins and electrons are globally less and less strongly bound when Z increases. This means an increase of atomic radius and a decrease of ionization potential. These trends are represented in Figure B.1 showing schematically the evolution of atomic gross properties along groups (vertical) and periods (horizontal). It is also interesting to consider, in this respect of size and IP, the anion and cation corresponding to a given atom. Starting from the neutral species X with radius R and ionization potential ε 0, the radius and IP of X are smaller than R and ε 0 respectively, while the radius and IP of X + are larger. The trends shown in Figure B.1 are global. There are of course fine details, in particular due to quantum effects, which cannot be seen from this figure. As a complement we thus display in Figure B.2 the explicit (experimental) values of two key atomic observables, radii and ionization potentials. We recover, of course, the general trends outlined in Figure B.1, along periods or groups, but now with fine quantum details, which nevertheless do not alter the general scheme outlined above. Figure B.1: Trends for evolution of radius, metalicity, ionization potential and electron affinity with changing element. The directions have the same meaning as in the periodic table: horizontal direction stands for changes within one period and vertical direction within one group.
262 B Gross properties of atoms and solids Figure B.2: Upper: Atomic radii (in Å) as a function of atomic number Z up to Z =92. The overall evolution is a soft increase of radii as a function of Z. This evolution is strongly affected by shell effects which lead to a marked oscillatory pattern. Halogens have consistently small radii while alkalines with their weakly bound s valence electron have consistently large radii. Lower: Atomic IP s (in ev) as a function of atomic number Z. The general trend is a soft decrease of IPs as a function of Z. As radii, IPs are strongly affected by shell effects, which leads to marked oscillations on top of the average evolution. Rare gas atoms consistently exhibit the largest IP reflecting their stability. In turn alkalines exhibit consistently small IPs reflecting the fact that their valence s electron is weakly bound.
B.3 Electronic structure of atoms 263 B.3 Electronic structure of atoms The next step is to consider the explicit electronic structure of atoms. As discussed in Section 1.1.1, the explicit electronic structure of atoms is involved. Nevertheless, and particularly in light atoms, Hund s rules provide a reliable scheme for predicting the electronic structures (see Section 1.1.1.1). These electronic structures are reported in Table B.1. The tables provide the atomic number, name and electronic structure for most elements in standard notation. Table B.1: Table of electronic structures. The electronic configurations are taken from [MW95]. Standard notation is used. This means, e.g., for the element Si: starting point is aneelectroniccore(seeelectronicstructureofne),two3s and two 3p electrons are added. Z Element Configuration 1 H Hydrogen 1s 2 He Helium 1s 2 3 Li Lithium (He)2s 4 Be Beryllium (He)2s 2 5 B Boron (He)2s 2 2p 6 C Carbon (He)2s 2 2p 2 7 N Nitrogen (He)2s 2 2p 3 8 O Oxygen (He)2s 2 2p 4 9 F Fluorine (He)2s 2 2p 5 10 Ne Neon (He)2s 2 2p 6 11 Na Sodium (Ne)3s 12 Mg Magnesium (Ne)3s 2 13 Al Aluminum (Ne)3s 2 3p 14 Si Silicon (Ne)3s 2 3p 2 15 P Phosphorus (Ne)3s 2 3p 3 16 S Sulfur (Ne)3s 2 3p 4 17 Cl Chlorine (Ne)3s 2 3p 5 18 Ar Argon (Ne)3s 2 3p 6 19 K Potassium (Ar)4s 20 Ca Calcium (Ar)4s 2 21 Sc Scandium (Ar)3d4s 2 22 Ti Titanium (Ar)3d 2 4s 2 23 V Vanadium (Ar)3d 3 4s 2 24 Cr Chromium (Ar)3d 5 4s 25 Mn Manganese (Ar)3d 5 4s 2 26 Fe Iron (Ar)3d 6 4s 2 27 Co Cobalt (Ar)3d 7 4s 2 28 Ni Nickel (Ar)3d 8 4s 2 29 Cu Copper (Ar)3d 10 4s 30 Zn Zinc (Ar)3d 10 4s 2
264 B Gross properties of atoms and solids Z Element Configuration 31 Ga Gallium (Ar)3d 10 4s 2 4p 32 Ge Germanium (Ar)3d 10 4s 2 4p 2 33 As Arsenic (Ar)3d 10 4s 2 4p 3 34 Se Selenium (Ar)3d 10 4s 2 4p 4 35 Br Bromine (Ar)3d 10 4s 2 4p 5 36 Kr Krypton (Ar)3d 10 4s 2 4p 6 37 Rb Rubidium (Kr)5s 38 Sr Strontium (Kr)5s 2 39 Y Yttrium (Kr)4d5s 2 40 Zr Zirconium (Kr)4d 2 5s 2 41 Nb Niobium (Kr)4d 4 5s 42 Mo Molybdenum (Kr)4d 5 5s 43 Tc Technetium (Kr)4d 5 5s 2 44 Ru Ruthenium (Kr)4d 7 5s 45 Rh Rhodium (Kr)4d 8 5s 46 Pd Palladium (Kr)4d 10 47 Ag Silver (Kr)4d 10 5s 2 48 Cd Cadmium (Kr)4d 10 5s 2 49 In Indium (Kr)4d 10 5s 2 5p 50 Sn Tin (Kr)4d 10 5s 2 5p 2 51 Sb Antimony (Kr)4d 10 5s 2 5p 3 52 Te Tellurium (Kr)4d 10 5s 2 5p 4 53 I Iodine (Kr)4d 10 5s 2 5p 5 54 Xe Xenon (Kr)4d 10 5s 2 5p 6 55 Cs Cesium (Xe)6s 56 Ba Barium (Xe)6s 2 57 La Lanthanum (Xe)5d6s 2 58 Ce Cerium (Xe)4f5d6s 2 59 Pr Praseodymium (Xe)4f 3 6s 2 60 Nd Neodymium (Xe)4f 4 6s 2 61 Pm Promethium (Xe)4f 5 6s 2 62 Sm Samarium (Xe)4f 6 6s 2 63 Eu Europium (Xe)4f 7 6s 2 64 Gd Gadolinium (Xe)4f 7 5d6s 2 65 Tb Terbium (Xe)4f 9 6s 2 66 Dy Dysprosium (Xe)4f 10 6s 2 67 Ho Holmium (Xe)4f 11 6s 2 68 Er Erbium (Xe)4f 12 6s 2 69 Tm Thulium (Xe)4f 13 6s 2 70 Yb Ytterbium (Xe)4f 14 6s 2 71 Lu Lutetium (Xe)4f 14 5d6s 2 72 Hf Hafnium (Xe)4f 14 5d 2 6s 2
B.3 Electronic structure of atoms 265 Z Element Configuration 73 Ta Tantalum (Xe)4f 14 5d 3 6s 2 74 W Tungsten (Xe)4f 14 5d 4 6s 2 75 Re Rhenium (Xe)4f 14 5d 5 6s 2 76 Os Osmium (Xe)4f 14 5d 6 6s 2 77 Ir Iridium (Xe)4f 14 5d 7 6s 2 78 Pt Platinum (Xe)4f 14 5d 9 6s 79 Au Gold (Xe)4f 14 5d 10 6s 80 Hg Mercury (Xe)4f 14 5d 10 6s 2 81 Tl Thallium (Xe)4f 14 5d 10 6s 2 6p 82 Pb Lead (Xe)4f 14 5d 10 6s 2 6p 2 83 Bi Bismuth (Xe)4f 14 5d 10 6s 2 6p 3 84 Po Polonium (Xe)4f 14 5d 10 6s 2 6p 4 85 At Astatine (Xe)4f 14 5d 10 6s 2 6p 5 86 Rn Radon (Xe)4f 14 5d 10 6s 2 6p 6 87 Fr Francium (Rn)7s 88 Ra Radium (Rn)7s 2 89 Ac Actinium (Rn)6d7s 2 90 Th Thorium (Rn)6d 2 7s 2 91 Pa Protactinium (Rn)5f 2 6d7s 2 92 U Uranium (Rn)5f 3 6d7s 2 93 Np Neptunium (Rn)5f 4 6d7s 2 94 Pu Plutonium (Rn)5f 6 7s 2 95 Am Americium (Rn)5f 7 7s 2 96 Cm Curium (Rn)5f 7 6d7s 2 97 Bk Berkelium (Rn)5f 9 7s 2 98 Cf Californium (Rn)5f 10 7s 2 99 Es Einsteinium (Rn)5f 11 7s 2 100 Fm Fermium (Rn)5f 12 7s 2 101 Md Mendelevium (Rn)5f 13 7s 2 102 No Nobelium (Rn)5f 14 7s 2
266 B Gross properties of atoms and solids B.4 Properties of bulk material We provide here some data on bulk materials. Most of the discussions in this book have been based on pictures stemming from atoms rather than from the bulk. Condensed matter techniques provide valuable tools for very large systems and they have also been often used in cluster physics. We thus give here in table form a few properties of solids. Solid state physics is a huge field. We restrict ourselves here to the few basic numbers used at some places in the book: the Wigner Seitz radius and the work function. The Wigner Seitz radius r s is defined as the radius occupied by one atom in a sample. In other words, if ρ is the material density, r s = (3/(4πρ)) 1/3. The Wigner Seitz radius thus provides information on the compactness of a material. The work function W is just the bulk analogue of the ionization potential in an atom or a cluster. It provides a measure of the global electronic binding of the material. Table B.2 shows r s and W for a selection of simple and noble metals, most of them being extensively used in examples throughout the book. As expected, one can see that the values of W are relatively small, even as compared to atomic IP. This reflects the particular softness of metals among other materials. Table B.2: Measured gross properties of simple and noble metals. We list the Wigner Seitz radius r s of the material (second column) and the work function W (third column). Element r s (a 0 ) W (ev) Li 3.25 2.38 Na 3.96 2.35 K 4.86 2.22 Rb 5.20 2.16 Cs 5.62 1.81 Cu 2.67 4.4 Ag 3.02 4.3 Au 3.01 4.3 Be 1.87 3.92 Mg 2.66 3.64 Ca 3.27 2.80 Sr 3.57 2.35 Ba 3.71 2.49 Nb 3.07 3.99 Element r s (a 0 ) W (ev) Fe 2.12 4.31 Mn 2.14 3.83 Zn 2.30 4.24 Cd 2.59 4.1 Hg 2.65 4.52 Al 2.07 4.25 Ga 2.19 3.96 In 2.41 3.8 Tl 2.48 3.7 Sn 2.22 4.38 Pb 2.30 4.0 Bi 2.25 4.4 Sb 2.14 4.08