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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
Quantum Well:
Quantum Well Laser
Electric
Capacitor:
Single Electron Transistor
Magnetic
Magnetic Particle:
Data Storage Media
E1
E0
d
a = V1/3
Energy Levels
Charging Energy
Spin Flip Barrier
3h2/8m l2
2e2/ d
½ M2a3
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
1
2
3
4
Crommie
and Eigler
http://www.almaden.ibm.com/vis/stm/gallery.html
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
E1
E0
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)
l
Energy Level Spacing:
E1E0 = 3h2/8m l2
E1E0 > kBT

l < 7 nm
That produces an inversely quadratic
relation between E and  :
E = h2/2m 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 = 2e2/ d
EC > kBT

d < 9 nm
EC = 2e2/ 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 M2a3
a = V1/3
Spin Flip Barrier
EM = ½ M2a3
EM > kBT

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
ElectronElectron
h+
Trapping at
an Impurity
phonon
e-
e-
e-
Semicond: long
Metal:
long
ElectronPhonon
long
 1000 nm
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 = kBT
Bjerrum Length, Gouy-Chapman Length
Dielectric
lGC
lB
ni,qi=ezi
-e
e
lDH = (  kBT / 4 niqi2 ) ½
lB = e2 /  kBT ,
= 1 / (4 lB nizi2 ) ½
= rCoulomb
0.1 Molar Na+Cl-
Pure H2O
lDH = 1.0 nm
lB = 0.7 nm
-
e
lGC = 2 / lBe 
Length Scales in Polymers
(including Biopolymers, such as DNA and Proteins)
Random Walk, Entropy
Stiffness  vs. kBT
Radius of Gyration
(overall size, N straight segments)
Persistence Length
(straight segment)
lP
RG
RG  lP N
cos = 1/e
a
lP =  / kBT
Copolymers
DNA (double)
Polystyrene
RG  20-50 nm
lP  50 nm
lP  1 nm
Self-Organization via two Competing Length Scales
Short Range Attraction versus Long Range Repulsion
Ferromagnet
Ferromagnetic Exchange:

Magnetic Dipole Interaction: 
Diblock Copolymer
Hydrophilic versus Hydrophobic
Depends on the relative block size