Length Scales in Physics Chemistry Biology Length scales

Length Scales in Physics, Chemistry, Biology, … Length scales are useful to get a quick idea what will happen when making objects smaller and smaller. For example, quantum physics kicks in when structures become smaller than the wavelength of an electron in a solid. In that case, the electrons get squeezed into a “quantum box” and have to adapt to the shape of the solid by changing their wave function. Their wavelength gets shorter, and that increases their energy. Since the wave function of the outer electrons determines the chemical behavior, one is able to come close to realizing the medieval alchemist’s dream of turning one chemical element into another.

Fundamental Length Scales in Physics Quantum Well: Quantum Well Laser Electric Capacitor: Single Electron Transistor Magnetic Particle: Data Storage Media E 1 E 0 l d Energy Levels Charging Energy 3 h 2/8 m l 2 2 e 2/ d l < 7 nm d < 9 nm a = V 1/3 Spin Flip Barrier ½ M 2 a 3 a > 3 nm

Quantum Corral 48 iron atoms are assembled into a circular ring. The ripples inside the ring are electron waves.

Building a Quantum Corral for Manipulating Electron Wave Functions 1 2 3 4 Crommie and Eigler

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Kanji character for atom (lit. original child) Carbon monoxide man

1 nm ≈ 5 atoms Between an atom and a solid A chain of N atoms (each carrying one electron) creates N energy levels. With increasing chain length these become so dense that they form a band. As the bands become wider, the energy gap between them shrinks.

Quantum Length Scale Quantum Well, Corral: Quantum Well Laser E 1 E 0 l Energy Level Spacing: E 1 E 0 = 3 h 2/8 m l 2 E 1 E 0 > k. BT l < 7 nm Consider the two lowest energy levels of an electron in a box (in one dimension): The energy E of an electron is determined by its momentum p in classical physics: E = p 2/2 m (m = electron mass) Quantum physics relates the momentum p to the wavelength of the electron: p = h/ (De Broglie) (h=Planck’s constant) That produces an inversely quadratic relation between E and : E = h 2/2 m 2 The quantum box restricts : 1 = l 0 = 2 l

Electric Length Scale Capacitor, Quantum Dot: Single Electron Transistor Consider a metallic sphere with a single electron spread out over its surface. It is embedded into an substrate with dielectric constant , forming a capacitor with a positive countercharge at infinity. The electrostatic energy stored in this capacitor is given by Coulomb’s law : d Charging Energy EC = 2 e 2/ d E C > k. B T d < 9 nm EC = 2 e 2/ d (e = electron charge) (d = sphere diameter) ( =12 used, i. e. silicon)

Magnetic Length Scale Consider a needle-shaped magnetic particle with two possible magnetization directions: Magnetic Particle: Data Storage Media The magnetic energy barrier is proportional to the volume of the particle, i. e. the third power of its average dimension a : EM = ½ M 2 a 3 a = V 1/3 Spin Flip Barrier EM = ½ M 2 a 3 E M > k. B T a > 3 nm (e = electron charge) (a = average diameter) (cgs unit system) The magnetization M is estimated from the magnetic moment 2 B = eh/2 mc of an iron atom in a magnet and the iron atom density.

Scattering Lengths Elastic E = 0 Scattering Potential Diffraction, Phase Shift Inelastic E > 0 Electron e- h+ e- e- Semicond: long Metal: long 1000 nm Electron. Phonon phonon Trapping at an Impurity e- e- 10 nm 100 nm e- (Room temperature, longer at low temp. ) Consequences: • Ballistic electrons at small distances (extra speed gain in small transistors) • Recombination of electron-hole pairs at defects (energy loss in a solar cell) • Loss of spin information (optimum thickness of a magnetic hard disk sensor)

Screening Lengths l ~ 1 / n Metals: Electrons at EFermi Thomas-Fermi: 0. 1 nm (n = Density of screening charges) Semiconductors: Electrolytes: Electrons, Holes Debye: 1 -1000 nm Ions Debye-Hückel: 0. 1 -100 nm V -r/l e V(r) q r l r Exponential cutoff of the Coulomb potential (dotted) at the screening length l.

Length Scales in Electrochemistry Screening Debye-Hückel Length Electrolyte Electric: ECoulomb = k. BT Bjerrum Length, Gouy-Chapman Length Dielectric l. GC l. B ni, qi=ezi -e e l. DH = ( k. BT / 4 niqi 2 ) ½ l. B = e 2 / k. BT , = 1 / (4 l. B nizi 2 ) ½ = r. Coulomb 0. 1 Molar Na+Cl- Pure H 2 O l. DH = 1. 0 nm l. B = 0. 7 nm - e l. GC = 2 / l. Be

Length Scales in Polymers (including Biopolymers, such as DNA and Proteins) Random Walk, Entropy Stiffness vs. k. BT Radius of Gyration (overall size, N straight segments) Persistence Length (straight segment) l. P RG RG l. P N a cos = 1/e l. P = / k. BT Copolymers DNA (double) Polystyrene RG 20 -50 nm l. P 1 nm

Self-Organization via two Competing Length Scales Short Range Attraction versus Long Range Repulsion Ferromagnetic Exchange: Magnetic Dipole Interaction: Diblock Copolymer Hydrophilic versus Hydrophobic Depends on the relative block size