length scales in physics, chemistry, biology,… length scales are useful to get a quick idea what...
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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 Electric Magnetic
Quantum Well:Quantum Well Laser
Capacitor:Single Electron Transistor
Magnetic Particle:Data Storage Media
a = V1/3
Charging Energy
2e2/ d
Spin Flip Barrier
½ M2a3
Energy Levels
3h2/8m l2
d
E1
E0
l
l < 7 nm d < 9 nm 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
Crommie and Eigler
21
3 4
http://www.almaden.ibm.com/vis/stm/gallery.html
Kanji character for atom (lit. original child)
Carbon monoxide man
1 nm ≈ 5 atomsBetween 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
Energy Level
Spacing:
E1E0 = 3h2/8m l2
E1
E0
l
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 = p2/2m (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 = h2/2m 2
The quantum box restricts :
1 = l
0 = 2 l
E1E0 > kBT
Electric Length Scale
Capacitor, Quantum Dot:Single Electron Transistor
Charging Energy
EC = 2e2/ d
d
d < 9 nm
EC > kBT
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 :
EC = 2e2/ d (e = electron charge)
(d = sphere diameter) ( =12 used, i.e. silicon)
Magnetic Length Scale
Magnetic Particle:Data Storage Media
a = V1/3
Spin Flip Barrier
EM = ½ M2a3
a > 3 nm
EM > kBT
Consider a needle-shaped magnetic particle with two possible magnetization directions:
The magnetic energy barrier is proportional to the volume of the particle, i.e. the third power of its average dimension a :
EM = ½ M2a3 (e = electron charge)
(a = average diameter) (cgs unit system) The magnetization M is estimated from the magnetic moment 2B = eh/2mc of
an iron atom in a magnet and the iron atom density.
Elastic Inelastic E = 0 E > 0
Scattering Potential Electron- Electron- Trapping atDiffraction, Phase Shift Electron Phonon an Impurity
Semicond: long long 10 nmMetal: long 1000 nm 100 nm
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)
e-
e-e-h+
e- e-
e-phonon
(Room temperature,longer at low temp.)
Scattering Lengths
Screening Lengths
l ~ 1 / n (n = Density of screening charges)
Metals: Semiconductors: Electrolytes:
Electrons at EFermi Electrons, Holes Ions
Thomas-Fermi: 0.1 nm Debye: 1-1000 nm Debye-Hückel: 0.1-
100 nm
Exponential cutoff of the Coulomb potential (dotted) at the screening length l .
V(r) qe-r/l
r
V
rl
Length Scales in Electrochemistry
Screening Electric: ECoulomb = kBT
Debye-Hückel Length Electrolyte
Bjerrum Length, Gouy-Chapman
Length Dielectric
Pure H2O
lB = 0.7 nm
lB = e2 / kBT , lGC = 2 / lBe
= rCoulomb
lDH = ( kBT / 4 niqi2
) ½
= 1 / (4 lB nizi2
) ½
0.1 Molar Na+Cl-
lDH = 1.0 nm
ni,qi=ezi
lGC
- e
lB
-e e
Length Scales in Polymers (including Biopolymers, such as DNA and
Proteins)Random Walk, Entropy Stiffness vs. kBT
Persistence Length (straight segment)
lP = / kBT
DNA (double) Polystyrene
lP 50 nm lP 1 nm
lPcos = 1/e
a
Radius of Gyration(overall size, N straight
segments)
RG lP N
Copolymers
RG 20-50 nm
RG
Self-Organization via two Competing Length Scales
Short Range Attraction versus Long Range Repulsion
Ferromagnetic Exchange: Magnetic Dipole Interaction:
Ferromagnet Diblock Copolymer
Hydrophilic versus Hydrophobic
Depends on the relative block size