The Linear Stability of the Relative Equilibria in the Coulomb (N + 1) Body Problem

Introduction: The (n + 1)-body problem The Periodic Table of the Elements Linear Stability Analysis Properties of the Relative Equilibrium

The linear stability of the relative equilibria in the Coulomb (n + 1) body problem

John E. Martin, III and Charles Jaﬀ´e West Virginia University

Conference on Hamiltonian Systems and Celestial Mechanics 2014

Charles Jaﬀ´e Relative equilibria in the Coulomb (n + 1) body problem Introduction: The (n + 1)-body problem The Periodic Table of the Elements Linear Stability Analysis Properties of the Relative Equilibrium Table of contents

1 Introduction: The (n + 1)-body problem The Kepler Problem The Coulomb Problem The Hamiltonians 2 The Periodic Table of the Elements Periodic Law Periodic Properties Experiment vs Theory 3 Linear Stability Analysis Linear Stability Symmetry Analysis Implications 4 Properties of the Relative Equilibrium The energy, the frequency, the radius and the eigenvalues What’s next? Charles Jaﬀ´e Relative equilibria in the Coulomb (n + 1) body problem Introduction: The (n + 1)-body problem The Kepler Problem The Periodic Table of the Elements The Coulomb Problem Linear Stability Analysis The Hamiltonians Properties of the Relative Equilibrium The Kepler (n + 1)-body problem

The (n + 1)-body problem was introduced by J. Clerk Maxwell in his study of the rings of Saturn in 1858. The system consists of n bodies of mass 1 and one body have mass M interacting via gravity where M >> 1.

Charles Jaﬀ´e Relative equilibria in the Coulomb (n + 1) body problem Introduction: The (n + 1)-body problem The Kepler Problem The Periodic Table of the Elements The Coulomb Problem Linear Stability Analysis The Hamiltonians Properties of the Relative Equilibrium Meyer’s summary

The study of relative equilibria (r.e.) of the N-body problem has had a long history starting with the famous collinear configuration of the 3-body problem found by Euler (1767). Over the intervening years many different technologies have been applied to the study of r.e. In the older papers of Euler (1767), Lagrange (1772), Hoppe (1879), Lehmann-Filhes (1891), and Moulton (1910), special coordinates, symmetries, and analytic techniques were used. In their investigations, Dziobek (1900) used the theory of determinants; Smale (1970) used Morse theory; Palmore (1975) used homology theory; Simo (1977) used a computer; and Moeckel (1985) used real algebraic geometry. Thus, the study of r.e. has been a testing ground for many different methodologies of mathematics.

The Coulomb (n + 1)-body problem was introduced by H. Nagaoka in his study of atomic structure in 1904. The system consists of n electrons of mass 1 and charge −1 and the nucleus having mass M with charge Z interacting via the electrostatic forces. (The gravitational forces are suﬃciently weak that they can be neglected.)

1.8 × 103 ≤ M ≤ 5.4 × 105 1 ≤ Z ≤ 118

Atomic Number vs. Atomic Mass

350

300

250

200

150

100 Atomic Mass (M) 50

0 0 20 40 60 80 100 120 140 Atomic Number (Z)

Charles Jaﬀ´e Relative equilibria in the Coulomb (n + 1) body problem

Figure : Experimental data exist for at least 352 diﬀerent combinations of M and Z Introduction: The (n + 1)-body problem The Kepler Problem The Periodic Table of the Elements The Coulomb Problem Linear Stability Analysis The Hamiltonians Properties of the Relative Equilibrium The Hamiltonians

n 2 ! n 1 X pθ X λ H = p2 + i + p2 − ρi 2 zi q 2 ρi 2 2 i=1 i=1 ρi + zi n−1 n X X 1 ± q 2 2 2 i=1 j=i (ρi + ρj − 2ρi ρj cos(θj − θi )) + (zj − zi )

where for the Kepler problem λ = M and the sign is taken to be negative. For the Coulomb problem λ = Z and the sign is taken to be positive.

Charles Jaﬀ´e Relative equilibria in the Coulomb (n + 1) body problem Introduction: The (n + 1)-body problem Periodic Law The Periodic Table of the Elements Periodic Properties Linear Stability Analysis Experiment vs Theory Properties of the Relative Equilibrium Linus Pauling: General Chemistry

“One of the most valuable parts of chemical theory is the periodic law. In its modern form this law states simply that the properties of the chemical elements are not arbitrary, but depend upon the electronic structure of the atom and vary with the atomic number in a systematic way. The important point is that this dependence involves a crude periodicity that show itself in the periodic recurrence of characteristic properties.”

1 18 IA VIIIA 11A 8A 1 Periodic Table of the Elements 2 2 13 14 15 16 17 HydrogenH IIA IIIA IVA VA VIA VIIA HeHelium 1.008 2A 3A 4A 5A 6A 7A 4.003 3 4 5 6 7 8 9 10

LithiumLi BerylliumBe BoronB CarbonC NitrogenN OxygenO FluorineF NeNeon 6.941 9.012 10.811 12.011 14.007 15.999 18.998 20.180 11 12 13 14 15 16 17 18 3 4 5 6 7 8 9 10 11 12 NaSodium MagnesiumMg IIIB IVB VB VIB VIIB VIII IB IIB AluminumAl SiliconSi PhosphorusP SulfurS ChlorineCl ArArgon 22.990 24.305 3B 4B 5B 6B 7B 8 1B 2B 26.982 28.086 30.974 32.066 35.453 39.948 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

PotassiumK CaCalcium ScandiumSc TitaniumTi VanadiumV ChromiumCr ManganeseMn FeIron CoCobalt NickelNi CuCopper ZnZinc GaGallium GermaniumGe AsArsenic SeleniumSe BromineBr KryptonKr 39.098 40.078 44.956 47.88 50.942 51.996 54.938 55.933 58.933 58.693 63.546 65.39 69.732 72.61 74.922 78.09 79.904 84.80 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

RubidiumRb StrontiumSr YttriumY ZirconiumZr NiobiumNb MolybdenumMo TechnetiumTc RutheniumRu RhodiumRh PalladiumPd AgSilver CadmiumCd IndiumIn SnTin AntimonySb TelluriumTe IodineI XeXenon 84.468 87.62 88.906 91.224 92.906 95.94 98.907 101.07 102.906 106.42 107.868 112.411 114.818 118.71 121.760 127.6 126.904 131.29 55 56 57-71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86

CsCesium BaBarium HafniumHf TantalumTa TungstenW RheniumRe OsmiumOs IridiumIr PlatinumPt AuGold HgMercury ThalliumTl PbLead BismuthBi PoloniumPo AstatineAt RnRadon 132.905 137.327 178.49 180.948 183.85 186.207 190.23 192.22 195.08 196.967 200.59 204.383 207.2 208.980 [208.982] 209.987 222.018 87 88 89-103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118

FranciumFr RaRadium RutherfordiumRf DubniumDb SeaborgiumSg BohriumBh HassiumHs MeitneriumMt DarmstadtiumDs RoentgeniumRg CoperniciumCn UnuntriumUut FleroviumFl UnunpentiumUup LivermoriumLv UnunseptiumUus UnunoctiumUuo 223.020 226.025 [261] [262] [266] [264] [269] [268] [269] [272] [277] unknown [289] unknown [298] unknown unknown

57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 Lanthanide Series LanthanumLa CeCerium PraseodymiumPr NeodymiumNd PromethiumPm SamariumSm EuropiumEu GadoliniumGd TerbiumTb DysprosiumDy HolmiumHo ErbiumEr TmThulium YtterbiumYb LutetiumLu 138.906 140.115 140.908 144.24 144.913 150.36 151.966 157.25 158.925 162.50 164.930 167.26 168.934 173.04 174.967 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Actinide Series ActiniumAc ThoriumTh ProtactiniumPa UraniumU NeptuniumNp PlutoniumPu AmericiumAm CmCurium BerkeliumBk CaliforniumCf EinsteiniumEs FmFermium MendeleviumMd NobeliumNo LawrenciumLr 227.028 232.038 231.036 238.029 237.048 244.064 243.061 247.070 247.070 251.080 [254] 257.095 258.1 259.101 [262]

Alkali Alkaline Transition Basic Noble Semimetal Nonmetal Halogen Lanthanide Actinide Metal Earth Metal Metal Gas

Ionization Energy vs Atomic Number

Data from Clementi et al (1967)

Ionization Energy (eV) 0 0 20 40 60 80 100 120 Atomic Number (Z)

Electron Affinity vs Atomic Number

Electron Affinity (kJ/mol)

400

350

300

250

200

150

100 Electron Affinity 50

0 0 10 20 30 40 50 60 70 80 90 Atomic Number (Z)

Atomic Radii vs Atomic Number

Data from Clementi et at (1967)

350

300

250 Period 1 Period 2 200 Period 3 Period 4 150 Period 5 Period 6 100

Atomic Radii (pm) 50

0 0 10 20 30 40 50 60 70 80 90 100 Atomic Number (Z)

Keep the number of electrons n ﬁxed while varying the nuclear charge Z. Example for n = 2: H−1, He, Li +1, Be+2, B+3, C +4, N+5, O+6, F +7

Ionic Radii for Isoelectronic Series

250 He (2e) Ne (10e) Ar (18e) 200 Ni (28e) Zn (30e) Kr (36e) 150 Pd (46e) Xe (54e) Er (68e) 100 Tm (69e) Yb (70e) Pt (78e) Hg (80e) 50

Ionic Radii (pm) Rn (86e) Ra (88e) 0 0 10 20 30 40 50 60 70 80 90 100 Atomic Number (Z)

Ionic Radii

The He (2e) Isoelectronic Series

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100

Ionic Radii (pm) 50

0 0 1 2 3 4 5 6 7 8 910 Atomic Number (Z)

Experimental Theory

Charles Jaﬀ´e Relative equilibria in the Coulomb (n + 1) body problem Introduction: The (n + 1)-body problem Linear Stability The Periodic Table of the Elements Symmetry Analysis Linear Stability Analysis Implications Properties of the Relative Equilibrium Linearized Equations of Motion

The stability of the relative equilibria is studied by ﬁrst linearizing the equations of motion

z˙ = J∇Hz

where ∇H is the Hessian of the Hamiltonian, J is

0 I J = −I 0

and I is the n × n unit matrix. The stability is then determined by examining the eigenvalues of the Jacobian J∇H.

1 Transform to rotating frame. 2 Transform to symmetry variables. 3 Decouple the coordinates from the momenta. 4 Diagonalize and balance the Hessian via a sequence of rotations and scalings. 5 Transform to complex coordinates. 6 Multiply by J. Criteria: The relative equilibrium is said to be linearly stable if none of the eigenvalues have positive real part.

The equilibrium conﬁguration in the rotating frame has belongs to the Cnh point group. As an example consider n = 3 or C3h.

C3h E 2C3 σh 2S3 A0 1 1 1 1 A00 1 1 -1 -1 E 0 2 -1 2 -1 E 00 2 -1 -2 1

0 00 0 00 ΓLi = 2A + A + 2E + E

For even n consider n = 6 or C6h. As the in-plane and out-of-plane motion decouples, one can work in C6.

C6h E 2C6 2C3 C2 σh 2S6 2S3 i 0 Ag 1 1 1 1 1 1 1 1 A 0 Bu 1 -1 1 -1 1 -1 1 -1 B 0 E1u 2 1 -1 -2 2 1 -1 -2 E1 0 E2g 2 -1 -1 2 2 -1 -1 2 E2 00 Au 1 1 1 1 -1 -1 -1 -1 A 00 Bg 1 -1 1 -1 -1 1 -1 1 B 00 E1g 2 1 -1 -2 -2 -1 1 2 E1 00 E2u 2 -1 -1 2 -2 1 1 -2 E2

The irreducible representations for odd n:

0 00 ΓH = 2A + A

0 00 0 00 ΓLi = 2A + A + 2E + E

0 00 0 00 0 00 ΓB = 2A + A + 2E1 + E1 + 2E2 + E2

0 00 0 00 0 00 0 00 ΓN = 2A + A + 2E1 + E1 + 2E2 + E2 + 2E3 + E3

The irreducible representations for even n:

0 00 0 00 ΓHe = 2A + A + 2B + B

0 00 0 00 0 00 ΓBe = 2A + A + 2B + B + 2E + E

0 00 0 00 0 00 0 00 ΓC = 2A + A + 2B + B + 2E1 + E1 + 2E2 + E2

0 00 0 00 0 00 0 00 0 00 ΓO = 2A + A + 2B + B + 2E1 + E1 + 2E2 + E2 + 2E3 + E3

Charles Jaffé Relative equilibria in the Coulomb (n + 1) body problem Irreducible representations with a single prime corresponds to in-plane motion. Those with double prime correspond to out-of-plane motion. The 2A0 modes leave the relative configuration of the system unchanged; one is the rotation about the z-axis and the other is the symmetric breathing mode.

Introduction: The (n + 1)-body problem Linear Stability The Periodic Table of the Elements Symmetry Analysis Linear Stability Analysis Implications Properties of the Relative Equilibrium Block Diagonalization

The introduction of symmetry variables block diagonalizes the Hessian matrix. Each block corresponding to one of the irreducible representations.

Charles Jaffé Relative equilibria in the Coulomb (n + 1) body problem The 2A0 modes leave the relative configuration of the system unchanged; one is the rotation about the z-axis and the other is the symmetric breathing mode.

The introduction of symmetry variables block diagonalizes the Hessian matrix. Each block corresponding to one of the irreducible representations. Irreducible representations with a single prime corresponds to in-plane motion. Those with double prime correspond to out-of-plane motion. The 2A0 modes leave the relative conﬁguration of the system unchanged; one is the rotation about the z-axis and the other is the symmetric breathing mode.

The A0 irreducible representation is totally symmetric. Its block is a 4 × 4 with determinant is equal to zero.

The eigenvalues are (0, 0, ±iω0) The two zeros are associated with the rotation. The complex pair is associated with the symmetric breathing mode.

The A00 representation corresponds to out-of-plane motion. The A00 representation corresponds to a 2 × 2 block with determinant not equal to zero.

The eigenvalues are (±iω0) for n = 1 and (±iω1 = ±ia/Z) 3 for n > 1 where a = r0.

The B0 representation corresponds to in-plane motion. The B0 representation corresponds to a 4 × 4 block with determinant not equal to zero.

The eigenvalues are (±λ, ±iω2).

The B00 representation corresponds to out-of-plane motion. The B00 representation corresponds to a 2 × 2 block with determinant not equal to zero.

The eigenvalues are (±iω0) if n = 2 and (±iω3) for n > 2.

0 The E1 representation corresponds to in-plane motion. 0 The E1 representation corresponds to an 8 × 8 block with determinant not equal to zero. It can be further reduced to two 4 × 4 blocks. The blocks being complex conjugates of each other.

The eigenvalues are (±iω4, ±iω4, a ± ib, −a ± ib)

00 The E1 representation corresponds to out-of-plane motion. 00 The E1 representation corresponds to a 4 × 4 block with determinant not equal to zero. It can be further reduced to two 2 × 2 blocks. The blocks being complex conjugates of each other.

The eigenvalues are (±iω0, ±iω0) if n = 3 and (±iω5, ±iω5) for n > 3

The principle relative equilibria for the Coulomb (n + 1)-body problem is not linearly stable. In fact, it is a highly degenerate (n − 1)-rank saddle.

n Element ω0 Saddle Single Double modes 1 H 2+1 0 0 0 3 2 He 2+1 1 2 0 6 3 Li 3+1 2 1 2 9 4 Be 3+1 3 3 2 12 5 B 3+1 4 1 6 15 6 C 3+1 5 3 6 18 7 N 3+1 6 1 10 21 8 O 3+1 7 3 10 24 9 F 3+1 8 1 14 27 10 Ne 3+1 9 3 14 30

Charles Jaﬀ´e Relative equilibria in the Coulomb (n + 1) body problem Introduction: The (n + 1)-body problem The Periodic Table of the Elements The energy, the frequency, the radius and the eigenvalues Linear Stability Analysis What’s next? Properties of the Relative Equilibrium The Screening Factors

Screening Factor

Units of Charge

s 15 µ 10

0 0 5 10 15 20 25 30 35 40 45 n

Energy vs. Atomic Number

Z=n

4 S P D 2 Log(|E|) 0

-2 0 2 4 6 8 10 12 14 16 18 20 Z

Radii

Z=n

1.4

1.2

1 S 0.8 P 0.6 D Radius 0.4

0.2

0 0 2 4 6 8 10 12 14 16 18 20 n

Frequency

Z=n

10 S ) P ω 5 D Ln( 0 0 2 4 6 8 10 12 14 16 18 20 -5

-10 n

Other Relative Equilibria? Choreographies? Use correct symmetry to include quantum eﬀects (permutation-inversion group). Unravel the dynamics of the complex quartic (a rotating saddle)! Semiclassical Quantization (already done for n = 1, submitted to Accounts of Theoretical Chemistry)!

John E. Martin, III Jes´usPalaci´an Patricia Yanguas Turgay Uzer

Charles Jaﬀ´e Relative equilibria in the Coulomb (n + 1) body problem