MODERN PHYSICS FOR ENGINEERS PHY355 INDEX 3D infinite potential box .13 allowed transitions 1-electron atoms..........16 many-electron atoms ...17 Angstrom ........................21 angular frequency............10 appendix .........................21 atomic mass ..................... 2 average momentum .........11 Avogadro's number....18, 21 binding energy ................. 5 binomial expansion .........21 blackbody......................... 6 Bohr magneton................21 Bohr model ...................... 8 Bohr radius ...................... 7 Boltzmann constant .........21 Bose-Einstein distribution19 boson ..............................19 Bragg's law ...................... 9 bremsstrahlung................. 6 classical physics ............... 1 classical wave equation ...10 Compton effect................. 7 conservation laws ............. 1 constants .........................21 coordinate systems ..........22 coordinate transformations22 de Broglie wavelength.....10 degenerate energy levels..13 density of energy states ...19 density of occupied states 20 doppler effect ................... 5 Duane-Hunt rule .............. 6 electron acceleration.................. 8 angular momentum ...... 7 filling..........................16 orbit radius .................. 8 scattering ..................... 9 velocity........................ 8 energy binding ........................ 5 density of states ..........19 Fermi..........................19 kinetic ......................... 5 relation to momentum .. 5 relativistic kinetic ........ 5 rest .............................. 5
splitting .......... 16, 17, 18 states ..........................19 total ............................. 5 zero-point ...................12 energy distribution ..........18 expectation value ............11 radial ..........................15 Fermi energy...................19 Fermi speed ....................19 Fermi temperature...........19 Fermi-Dirac distribution..19 fermion ...........................19 frequency angular .......................10 fundamental forces ........... 2 geometry.........................22 Greek alphabet................21 group velocity .................10 harmonic motion .............12 Heisenberg limit .............12 Heisenberg uncertainty principle .....................12 Hermite functions............12 impact parameter ............. 7 infinite square well .........12 intensity of light ............... 6 inverse photoelectric effect6 kinetic energy 2, 5, 9, 12, 13 Landé factor ....................17 lattice planes.................... 9 laws of thermodynamics ... 2 length contraction............. 3 light wavefront................. 3 lightlike ........................... 4 line spectra ...................... 5 Lorentz force law ............. 2 Lorentz transformation ..... 3 magnetic moment............16 Maxwell speed distribution ...................................18 Maxwell’s equations ........ 2 Maxwell-Boltzmann factor18 mean speed .....................18 Michelson-Morley experiment................... 3 minimum angle ...............17 molecular speeds.............18 momentum....................... 4 relativistic.................... 4
momentum operator ........11 momentum-energy relation 5 momentum-temperature relation ........................ 9 Moseley's equation ........... 9 most probable speed........18 Newton’s laws ................. 2 normalization ..................11 normalization constant ....14 normalizing functions......14 orbital angular momentum15 order of electron filling....16 particle in a box ........12, 13 phase constant.................10 phase space .................2, 19 phase velocity .................10 photoelectric effect........... 6 photon.............................. 6 momentum................... 4 Planck's constant .............21 Planck's radiation law....... 6 positron............................ 6 potential barrier ..............13 probability ......................11 radial ..........................15 probability density radial ..........................15 probability of location .....11 proper length.................... 3 proper time ...................... 3 quantum numbers............15 radial acceleration ............ 8 radial probability.............15 radial probability density.15 radial wave functions ......14 radiation power ................ 6 relativity .......................... 3 rest energy ....................... 5 root mean square speed ...18 Rutherform scattering....... 8 Rydberg constant.........9, 21 scattering ......................7, 8 electron........................ 9 head-on........................ 7 x-ray............................ 9 Schrödinger wave equation .............................11, 12 3D rectangular coord...13 3D spherical coord. .....14
simple harmonic motion ..12 spacelike.......................... 4 spacetime diagram ........... 4 spacetime distance ........... 3 spacetime interval ............ 4 spectral lines.................... 9 spectroscopic symbols .....16 speed of light ................... 3 spherical coordinates.......22 spin angular momentum ..16 spin-orbit splitting...........17 splitting due to spin.........17 spring harmonics .............12 statistical physics ............18 Stefan-Boltzman law ........ 6 temperature Fermi..........................19 temperature and momentum9 thermodynamics ............... 2 time dilation..................... 3 timelike ........................... 4 total angular momentum..16 total energy ...................... 5 trig identities...................22 tunneling.........................13 uncertainty of waves........10 uncertainty principle .......12 units................................21 velocity addition............... 3 wave functions ................10 wave number.............10, 11 wave uncertainties...........10 wavelength..................3, 10 spectrum.....................21 waves envelope .....................10 sum.............................10 Wien's constant ................ 6 work function................... 6 x-ray L-alpha waves.............. 9 scattering ..................... 9 Young's double slit experiment................... 5 Zeeman splitting .......16, 18 zero-point energy.............12
CLASSICAL PHYSICS CLASSICAL CONSERVATION LAWS Conservation of Energy: The total sum of energy (in all its forms) is conserved in all interactions. Conservation of Linear Momentum: In the absence of external force, linear momentum is conserved in all interactions (vector relation). naustalgic Tom Penick
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Conservation of Angular Momentum: In the absence of external torque, angular momentum is conserved in all interactions (vector relation). Conservation of Charge: Electric charge is conserved in all interactions. Conservation of Mass: (not valid) www.teicontrols.com/notes
12/12/1999 Page 1 of 22
FUNDAMENTAL FORCES MAXWELL’S EQUATIONS Gauss’s law for electricity
q Ñ∫ EgdA = ε0
Gauss’s law for magnetism
Ñ∫ BgdA = 0
dΦB dt dΦ Ñ∫ Bgds = µ0ε0 dt E + µ0 I
Strong Electroweak Electromagnetic Weak Gravitational
Ñ∫ Egds = −
Faraday’s law Generalized Ampere’s law
RELATIVE STRENGTH
FORCE
LORENTZ FORCE LAW Lorentz force law: F = qE + qv × B
RANGE
1
Short, ~10-15m
10-2 10-9 10-39
Long, 1/r2 Short, ~10-15m Long, 1/r2
ATOMIC MASS The mass of an atom is it's atomic number divided by the product of 1000 times Avogadro's number.
atomic number 1000 × N a
KINETIC ENERGY NEWTON’S LAWS Newton’s first law: Law of Inertia An object in motion with a constant velocity will continue in motion unless acted upon by some net external force. Newton’s second law: The acceleration a of a body is proportional to the net external force F and inversely proportional to the mass m of the body. F = ma Newton’s third law: law of action and reaction The force exerted by body 1 on body 2 is equal and opposite to the force that body 2 exerts on body 1.
The kinetic energy of a particle (ideal gas) in equilibrium with its surroundings is:
K=
3kT 2
PHASE SPACE A
six-dimensional pseudospace populated by particles described by six position and velocity parameters: position: (x, y, z) velocity: (vx, vy, vz)
LAWS OF THERMODYNAMICS First law of thermodynamics: The change in the internal energy ∆U of a system is equal to the heat Q added to the system minus the work W done by the system. Second law of thermodynamics: It is not possible to convert heat completely into work without some other change taking place. Third law of thermodynamics: It is not possible to achieve an absolute zero temperature. Zeroth law of thermodynamics: If two thermal systems are in thermodynamic equilibrium with a third system, they are in equilibrium with each other.
Tom Penick
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RELATIVITY WAVELENGTH λ
TIME DILATION 8
c = speed of light 2.998 × 10 m/s λ = wavelength [m] ν = (nu) radiation frequency [Hz] Å = (angstrom) unit of wavelength
1 c= = λν µ 0ε 0 1Å = 10-10m
equal to 10-10 m m = (meters)
Given two systems moving at great speed relative to each other; the time interval between two events occurring at the same location as measured within the same system is the proper time and is shorter than the time interval as measured outside the system.
T=
Michelson-Morley Experiment indicated that light was not influenced by the “flow of ether”.
LORENTZ TRANSFORMATION
x′ =
1− v / c 2
2
t′ =
t − vx / c
2
1 − v2 / c2
v = velocity of (x’,y’,z’) system along the x-axis. [m/s] t = time [s] c = speed of light 2.998 × 108 m/s or with β = so that
v 1 and γ = c 1 − v2 / c2
x′ = γ ( x − vt )
and
t ′ = γ ( t − βx / c )
1− v / c 2
2
or T ′ =
T0
where:
1 − v2 / c2
T’0, T0 = the proper time (shorter). [s] T, T’ = time measured in the other system [m] v = velocity of (x’,y’,z’) system along the x-axis. [m/s] c = speed of light 2.998 × 108 m/s
Compares position and time in two coordinate systems moving with respect to each other along axis x.
x − vt
T0′
LENGTH CONTRACTION Given an object moving with great speed, the distance traveled as seen by a stationary observer is L0 and the distance seen by the object is L', which is contracted.
L0 =
L′ 1 − v2 / c2
where:
L0 = the proper length (longer). [m] L' = contracted length [m] v = velocity of (x’,y’,z’) system along the x-axis. [m/s] c = speed of light 2.998 × 108 m/s
RELATIVISTIC VELOCITY ADDITION
LIGHT WAVEFRONT Position of the wavefront of a light source located at the origin, also called the spacetime distance.
x 2 + y 2 + z 2 = c 2t 2 Proper time T0 The elapsed time between two events occurring at the same position in a system as recorded by a stationary clock in the system (shorter duration than other times). Objects moving at high speed age less. Proper length L0 a length that is not moving with respect to the observer. The proper length is longer than the length as observed outside the system. Objects moving at high speed become longer in the direction of motion.
Where frame K' moves along the x-axis of K with velocity v, and an object moves along the x-axis with velocity ux' with respect of K', the velocity of the object with respect to K is ux.
K' u'
K
ux =
v
u ′x + v 1 + ( v / c 2 ) u x′
If there is uy' or uz' within the K' frame then
uy =
u′y
γ 1 − ( v / c 2 ) u′x
and u z =
u ′z
γ 1 − ( v / c 2 ) u′x
ux = velocity of an object in the x direction [m/s] v = velocity of (x’,y’,z’) system along the x-axis. [m/s] c = speed of light 2.998 × 108 m/s γ = 1/ 1 − v 2 / c 2 For the situation where the velocity u with respect to the K frame is known, the relation may be rewritten exchanging the primes and changing the sign of v.
Tom Penick
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SPACETIME INTERVAL ∆ s
SPACETIME DIAGRAM The diagram is a means of representing events in two systems. The horizontal x axis represents distance in the K system and the vertical ct axis represents time multiplied by the speed of light so that it is in units of distance as well. A point on the diagram represents an event in terms of its location in the x direction and the time it takes place. So points that are equidistant from the x axis represent simultaneous events.
ct
ct'
v =c
Worldline v = 0.25c
The quantity ∆s2 is invariant between two frames of reference with relative movement along the x-axis.
s 2 = x 2 − ( ct ) = x′2 − ( ct ′ ) 2
2
Two events occurring at different times and locations in the K-frame may be characterized by their ∆s2 quantity.
∆s 2 = ∆x 2 − ( c∆t )
2
lightlike - ∆ s2 = 0: In this case, ∆x2 = c2∆t2, and the two events can only be connected by a light signal.
c 1 slope = v = =4 β
spacelike - ∆ s2 > 0: In this case, ∆x2 > c2∆t2, and there exists a K'-frame in which the two events occur simultaneously but at different locations.
timelike - ∆ s2 < 0: In this case, ∆x2 < c2∆t2, and there exists a K'-frame in which the two events occur at the same position but at different times. Events can be causally connected.
x' v slope = c = β = 0.25
x A system K’ traveling in the x direction at ¼the speed of light is represented by the line ct’ in this example, and is called a worldline. The line represents travel from one location to another over a period of time. The slope of the line is proportional to the velocity. A line with a slope of 1 (dashed line in illustration) indicates travel at the speed of light, so no worldline can have a slope less than 1. A straight line indicates zero acceleration. Simultaneous events occurring at t = t’ = 0 in the K’ system may be represented by points along the x’ axis. Other simultaneous events in the K’ system will be found on lines parallel to the x’ axis.
MOMENTUM p p = mv
for a photon: p =
hν c
p = momentum [kg-m/s], convertible to [eV/c] by multiplying by c/q. m = mass of the object in motion [kg] v = velocity of object [m/s] ν = (nu) the frequency of photon light [Hz] c = speed of light 2.998 × 108 m/s
RELATIVISTIC MOMENTUM p p = γmu where: p = relativistic momentum [kg-m/s], convertible to [eV/c] by multiplying by c/q.
γ = 1/ 1 − u 2 / c2 m = mass [kg] u = velocity of object [m/s]
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DOPPLER EFFECT
MOMENTUM-ENERGY RELATION
Given two systems approaching each other at velocity v, light emitted by one system at frequency ν0 (nu, proper) will be perceived at the higher frequency of ν (nu) in the other system.
(energy) 2 = (kinetic energy) 2 + (rest energy) 2
1+ β ν= ν0 1− β
For two systems receeding from each other, reverse the signs.
ν = (nu) the frequency of emitted light as perceived in the
E 2 = p 2c 2 + m 2c 4
where:
E = total energy (Kinetic + Rest energies) [J] p = momentum [kg-m/s] m = mass [kg] c = speed of light 2.998 × 108 m/s
other system [Hz]
ν0 = (nu) the proper frequency of the emitted light (lower
BINDING ENERGY
for approaching systems) [Hz]. Frequency is related to wavelength by c = λν. β = v/c where v is the closing velocity of the systems (Use a negative number for diverging systems.) and c is the 8 speed of light 2.998 × 10 m/s v = velocity of (x’,y’,z’) system along the x-axis. [m/s]
• the potential energy associated with holding a system together, such as the coulomb force between a hydrogen proton and its electron
RELATIVISTIC KINETIC ENERGY K
EB = ∑ mi c 2 − M bound system c 2
Relativistic kinetic energy is the total energy minus the rest energy. When the textbook speaks of a 50 Mev particle, it is talking about the particle's kinetic energy.
K = γmc 2 − mc 2
• the difference between the rest energies of the individual particles of a system and the rest energy of a the bound system • the work required to pull particles out of a bound system into free particles at rest.
i
for hydrogen and single-electron ions, the binding energy of the electron in the ground state is
EB =
where:
K = relativistic kinetic energy [J], convertible to [eV] by dividing by q.
2h 2 ( 4 πε0 )
2
EB = binding energy (can be negative or positive) [J] m = mass [kg] Z = atomic number of the element e = q = electron charge [c]
γ = 1/ 1 − v 2 / c 2 m = mass [kg] c = speed of light 2.998 × 108 m/s
h
REST ENERGY E0 Rest energy is the energy an object has due to its mass.
E0 = mc 2
= Planck's constant divided by 2π [J-s]
ε0 = permittivity of free space 8.85 × 10-12 F/m c = speed of light 2.998 × 108 m/s
LINE SPECTRA Light passing through a diffraction grating with thousands of ruling lines per centimeter is diffracted by an angle θ.
TOTAL ENERGY E Total energy is the kinetic energy plus the rest energy. When the textbook speaks of a 50 Mev particle, it is talking about the particle's kinetic energy.
E = K + E0 or E = γmc 2
mZ 2e 4
where:
E = total energy [J], convertible to [eV] by dividing by q. K = kinetic energy [J], convertible to [eV] by dividing by q. E0 = rest energy [J], convertible to [eV] by dividing by q. γ = 1/ 1 − v 2 / c 2 m = mass [kg] c = speed of light 2.998 × 108 m/s
Tom Penick
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d sin θ = nλ The equation also applies to Young's double slit experiment, where for every integer n, there is a lighting maxima. The off-center distance of the maxima is y = l tan θ
d = distance between rulings [m] θ = angle of diffraction [degrees] n = the order number (integer) λ = wavelength [m] l = distance from slits to screen [m]
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WIEN'S CONSTANT
PHOTON
The product of the wavelength of peak intensity λ [m] and the temperature T [K] of a blackbody. A blackbody is an ideal device that absorbs all radiation falling on it.
λ maxT = 2.898 ×10
−3
A photon is a massless particle that travels at the speed of light. A photon is generated when an electron moves to a lower energy state (orbit). Photon energy: E = hν = pc [Joules]
m⋅K
Momentum: p =
[kg-m/s], convertible to [eV/c] by multiplying by c/q.
STEFAN-BOLTZMANN LAW Wavelength:
May be applied to a blackbody or any material for which the emissivity is known.
R(T ) = εσT 4
hν c
λ=
c ν
[meters]
h = Planck's constant 6.6260755×10-34 J-s ν = (nu) frequency of the electromagnetic wave associated
where:
R(T) = power per unit area radiated at temperature T [W/m2] ε = emissivity (ε = 1 for ideal blackbody) σ = constant 5.6705 × 10-8 W/(m2· K4) T = temperature (K)
c
with the light given off by the photon [Hz] 8 = speed of light 2.998 × 10 m/s
PHOTOELECTRIC EFFECT This is the way the book shows the formula, but it is a units nightmare.
PLANCK'S RADIATION LAW 2πc h 1 I (λ , T ) = hc / λkT 5 λ e −1
1 2 mvmax = eV0 = hυ − φ 2
2
where:
I(λ, T) = light intensity [W/(m2· λ)] λ = wavelength [m] T = temperature [K] c = speed of light 2.998 × 108 m/s h = Planck's constant 6.6260755×10-34 J-s k = Boltzmann's constant 1.380658×10-23 J/K positron – A particle having the same mass as an electron but with a positive charge bremsstrahlung – from the German word for braking radiation, the process of an electron slowing down and giving up energy in photons as it passes through matter.
where:
1 2 = energy in Joules, but convert to eV for the mvmax 2 formula by dividing by q. eV0 = potential required to stop electrons from leaving the metal [V] hν = Planck's constant [6.6260755×10-34 J-s] multiplied by the frequency of light [Hz]. This term will need to be divided by q to obtain eV. φ = work function, minimum energy required to get an electron to leave the metal [eV]
INVERSE PHOTOELECTRIC EFFECT hc eV0 = hυmax = where: λ min eV0 = the kinetic energy of an electron accelerated through a voltage V0 [eV]
hν = Planck's constant [6.6260755×10-34 J-s] multiplied by λmin
the frequency of light [Hz]. This term will need to be divided by q to obtain eV. = the minimum wavelength of light created when an electron gives up one photon of light energy [m]
DUANE-HUNT RULE λ min
Tom Penick
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1.2398 × 10−6 = V0
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ELECTRON ANGULAR MOMENTUM from the Bohr model:
L = mvr = nh
where:
2
L = angular momentum [kg-m /s?] m = mass [kg] v = velocity [m/s] r = radius [m] n = principle quantum number
h
= Planck's constant divided by 2π [J-s]
rmin = particle separation (measured center to center) at the
The Bohr radius is the radius of the orbit of the hydrogen electron in the ground state (n=1):
a0 =
4πε0h 2 mee 2
and for higher states (n>1):
time that the bombarding particle reverses direction [m] other variables are previously defined
rn = a0 n 2
a0, rn = Bohr radius 5.29177×10-11 m, quantized radius [m] ε0 = permittivity of free space 8.85 × 10-12 F/m me = electron mass 9.1093897×10-31 [kg] e = q = electron charge [c] n = principle quantum number
COMPTON EFFECT The scattering of a photon due to collision with a single electron results in a new wavelength λ' and a directional change of ∠θ and is described by the following relation:
∆λ = λ′ − λ =
= Planck's constant divided by 2π [J-s]
The impact parameter b is the distance that a bombarding particle deviates from the direct-hit approach path, and is related to the angle θ at which it will be deflected by the target particle.
Z1 Z2 e θ cot 8πε 0 K 2 2
b = direct path deviation [m] Z1 = atomic number of the incident particle Z2 = atomic number of the target particle e = q = electron charge [c] ε0 = permittivity of free space 8.85 × 10-12 F/m K = kinetic energy of the incident particle Z1 θ = angle of particle Z1 deflection or scattering
Tom Penick
h (1 − cos θ) mc
scattered photon
IMPACT PARAMETER b
b=
Z1 Z2 e2 4πε0 K
rmin =
a0 BOHR RADIUS [m]
h
HEAD-ON SCATTERING When a particle of kinetic energy K and atomic number Z1 is fired directly at the nucleus, it approaches to rmin before reversing direction. The entire kinetic energy is converted to Coulomb potential energy. Since rmin is measured to the center of the particles, they will just touch when rmin is the sum of their radii.
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E = hn' p= h l'
θ photon
φ
E = hn p= h l electron at rest Ei = mc 2
recoil electron
Ef = Ee
The φ relations come from the conservation of momentum:
px : py :
h h = cos θ + pe cos φ λ λ′ h sin θ = pe sin φ λ′
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RUTHERFORD SCATTERING A particle of kinetic energy K and atomic number Z1 when fired at a target film of thickness t and atomic number Z2, will be deflected by an angle θ.
ELECTRON VELOCITY This comes from the Bohr model and only applies to atoms and ions having a single electron.
vn =
2
N ( θ) =
2 2 N i nt e 2 Z1 Z 2 16 4πε0 r 2 K 2 sin 4 ( θ / 2 )
N(θ) = number of particles scattered per unit area [m-2] θ = angle of particle Z1 deflection or scattering Ni = total number of incident particles [kg] n = number of atoms per unit volume [m-3] n = ρN A N M Mg
where ρ is density [g/m3], NA is Avogadro's number, NM is the number of atoms per molecule, and MG is the gram-molecular weight [g/mole].
t = thickness of the target material [m] e = q = electron charge [c] ε0 = permittivity of free space 8.85 × 10-12 F/m Z1 = atomic number of the incident particle Z2 = atomic number of the target particle r = the radius at which the angle θ is measured [m] K = kinetic energy of the incident particle Z1
PROBABILITY OF A PARTICLE SCATTERING BY AN ANGLE GREATER THAN θ 2
Z Z e θ f = πnt 1 2 cot 2 2 8πε0 K 2
e Z 1 Ze 2 = n 4πε0 h 2 πε0 me r 1 424 3 14243 n -dependent
r -dependent
v = electron velocity [m/s] Z = atomic number or number of protons in the nucleus e = q = electron charge [c] n = the electron orbit or shell ε0 = permittivity of free space 8.85 × 10-12 F/m me = mass of an electron 9.1093897×10-31 kg
h
= Planck's constant divided by 2π [J-s]
r = the radius of the electron's orbit [m]
ELECTRON ORBIT RADIUS This comes from the Bohr model and only applies to atoms and ions having a single electron.
4πε0 n 2 h 2 rn = me Ze 2 rn = electron orbit radius in the n shell [m] other variables are previously defined
ar RADIAL ACCELERATION ar = the radial acceleration of an orbiting electron [m/s2]
f = the probability (a value between 0 and 1) n = number of atoms per unit volume [m-3] n = ρN A N M
v = tangential velocity of the electron [m/s] r = electron orbit radius [m]
ar =
v2 r
Mg
where ρ is density [g/m3], NA is Avogadro's number, NM is the number of atoms per molecule, and MG is the gram-molecular weight [g/mole].
t = thickness of the target material [m] Z1 = atomic number of the incident particle Z2 = atomic number of the target particle e = q = electron charge [c] ε0 = permittivity of free space 8.85 × 10-12 F/m K = kinetic energy of the incident particle Z1 θ = angle of particle Z1 deflection or scattering Alpha particle: Z=2 Proton: Z=1
Tom Penick
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12/12/1999 Page 8 of 22
BRAGG'S LAW
R∞ RYDBERG CONSTANT R∞ is used in the Bohr model and is a close approximation assuming an infinite nuclear mass. R is the adjusted value. These values are appropriate for hydrogen and single-electron ions.
R=
µe Z 2e 4
4πch3 ( 4πε 0 )
2
where µe =
me M me + M
R∞ = Rydberg constant 1.09678×107 m-1 (1.096776×107 m-1 for hydrogen) µe = adjusted electron mass Z = atomic number, or number of protons in the nucleus
ε0 = permittivity of free space 8.85 × 10-12 F/m c = speed of light 2.998 × 108 m/s
h
= Planck's constant divided by 2π [J-s] me = mass of an electron 9.1093897×10-31 kg M = mass of the nucleus (essentially the same as the mass of the atom ⇒ atomic number × 1.6605×10-27) [kg]
Lα MOSELEY'S EQUATION British physicist, Henry Moseley determined this equation experimentally for the frequency of Lα xrays. Lα waves are produced by an electron decaying from the n=3 orbit to the n=2 or L orbit.
ν Lα =
5 2 cR ( Z − 7.4) 36
ν = (nu) frequency [Hz] c = speed of light 2.998 × 108 m/s R = Adjusted Rydberg constant (see above) [m-1] Z = atomic number or number of protons in the nucleus
SPECTRAL LINES This formula gives the wavelength of light emitted when an electron in a single-electron atom or ion decays from orbit nu to nl.
1 1 1 = Z 2R 2 − 2 λ nl nu
X-ray Scattering - X-rays reflected from a crystal experience interference effects since rays reflecting from the interior of the material take a longer path than those reflecting from the surface. Compare to ELECTRON SCATTERING below. θ
nλ = 2d sin θ
d
2 d sin θ d sin θ
n = order of reflection (number of lattice planes in depth) λ = wavelength of the incident wave [m] d = distance between lattice planes (interatomic spacing in this case) [m] θ = angle of incidence; the angle between the incident wave and the surface of the material
ELECTRON SCATTERING Electrons directed into a crystalline material are scattered (reflected) at various angles depending on the arrangement of lattice planes. There is more than one set of lattice planes in a crystal. The technique can be used to explore the characteristics of a material. Compare to BRAGG'S LAW above.
θ
nλ = D sin φ
φ α α
D d
n = order of reflection (number of lattice planes in depth) λ = wavelength of the incident wave [m] D = interatomic spacing [m] d = distance between lattice planes [m] φ = angle between the incident and reflected waves
K CLASSICAL KINETIC ENERGY Two expressions for kinetic energy:
λ = wavelength [m] Z = atomic number or number of protons in the nucleus R = Rydberg constant (1.096776×107 m-1 for hydrogen) nl = the lower electron orbit number nu = the upper electron orbit number
p2 3 = K = kT 2m 2 lead to a momentum-temperature relation for particles:
p 2 = 3mkT p = momentum [kg-m/s] m = particle mass [kg] K = kinetic energy [J] k = Boltzmann's constant 1.380658×10-23 J/K T = temperature in Kelvin (273.15K = 0°C, ∆K = ∆C) (see page 5 for RELATIVISTIC KINETIC ENERGY)
Tom Penick
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12/12/1999 Page 9 of 22
WAVES
Ψ SUM OF TWO WAVES (see also WaveSummingExample.pdf)
Ψ WAVE FUNCTIONS ∂ Ψ 1 ∂ Ψ = ∂x 2 v 2 ∂t 2 2
Classical Wave Equation We did not use this equation:
2
This wave function fits the classical form, but is not a solution to the Schröedinger equation: wave
wave number
time
angular frequency
phase constant
amplitude
distance
time
The negative sign denotes wave motion in the positive x direction, assuming omega is positive.
More general wave functions which are solutions to the Schröedinger equation are: wave
wave number
time
angular frequency
Ψ (x,t) = Ae i(kx- ω t) = A[cos(kx - ωt) + i sin(kx - ωt)] distance
distance
amplitude
envelope
A = harmonic amplitude [various units?] ∆k = difference in wave numbers k1 - k2 [rad/m] kav = average wave number (k1 + k2)/2 [rad/m] ∆ω = difference in angular
frequencies ω1 - ω2 [rad/s] ωav = average angular frequency (ω1 + ω2)/2 [rad/s] x = distance [m] t = time [s]
Ψ (x,t) =A sin( kx - ωt + φ) distance
∆ω ∆k x− t cos ( k av x − ωavt ) Ψ1 + Ψ 2 = 2 A cos 1442443 2 2 144 42444 3 internal wave
Phase Velocity:
vph = ωav / kav
[m/s] velocity of a point on a wave
Group Velocity:
ugr = ∆ω / ∆k
[m/s] speed of the envelope
time The negative sign denotes wave motion in the positive x direction, assuming omega is positive.
λ de BROGLIE WAVELENGTH De Broglie extended the concept of waves to all matter.
k WAVE NUMBER A component of a wave function representing the wave density relative to distance, in units of radians per unit distance [rad/m].
k=
2π λ
2π ω= T
vph =
λ ω = T k
This has to do with the effects of combining different waves. In order to know precisely the position of the wave packet envelope (∆x small), we must have a large range of wave numbers (∆k large). In order to know precisely when the wave is at a given point (∆t small), we must have a large range of frequencies (∆ω large). Another result of this relationship, is that an electronic component must have a large bandwidth ∆ω in order for its signal to respond in a short time ∆t.
∆k ∆x = 2π
φ PHASE CONSTANT The angle by which the wave is offset from zero, i.e. the angle by which the wave's zero amplitude point is offset from t=0. [radians or degrees].
Tom Penick
λ = wavelength [m] h = Planck's constant 6.6260755×10-34 J-s p = momentum [kg-m/s], convertible to [eV/c] by multiplying
WAVE UNCERTAINTIES
vph PHASE VELOCITY The velocity of a point on a wave, e.g. the velocity of a wave peak [m/s].
h p
by c/q.
ω ANGULAR FREQUENCY A component of a wave function representing the wave density relative to time (better known as frequency), in units of radians per second [rad/s].
λ=
[email protected]
∆ω∆t = 2π
∆k = the range of wave numbers, see WAVE NUMBER ∆x = the width of the wave envelope ∆ω = the range of wave frequencies ∆t = a time interval
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12/12/1999 Page 10 of 22
SCHRÖDINGER'S WAVE EQUATION +
K
U
=
E
2 ∂Ψ ( x, t ) h ∂ Ψ ( x, t ) − + V Ψ ( x, t ) = i h 2 2m ∂x ∂t 2
time-independent form:
−
h d ψ ( x) + V ( x ) ψ ( x ) = Eψ ( x ) 2m dx 2
or
h
2
2
−
d ψ ( x) h = E −V ( x) 2mψ ( x ) dx 2 2
2
= Planck's constant divided by 2π [J-s] Ψ(x,t) = wave function V = voltage; can be a function of space and time m = mass [kg]
or
where
A sin ( kx ) + B cos ( kx )
find the probability that a particle is located between x1 and x2. Normalize the wave function: 2
∫
∞
0
A2 ψ 2 dx = 1
with A known, find the probability: P =
∫
x2
x1
A2 ψ 2 dx
〈 x〉, 〉, 〈x 〈 2〉 EXPECTATION VALUES average value: ∞
x = ∫ ψ * ( x ) x ψ ( x ) dx −∞
average x2 value:
(x,t)
Two separate solutions to the time-independent equation have the form:
Aeikx + Be −ikx
PROBABILITY OF LOCATION Given the wave function: ψ ( x, t )
time-dependent form:
k = 2m ( E − V ) / h
Note that the wave number k is consistent in both solutions, but that the constants A and B are not consistent from one solution to the other. The values of constants A and B will be determined from boundary conditions and will also depend on which solution is chosen.
∞
x 2 = ∫ ψ * ( x ) x 2 ψ ( x ) dx −∞
pˆ
MOMENTUM OPERATOR
An operator transforms one function into another function. The momentum operator is:
pˆ = −ih
d dx
For example, to find the average momentum of a particle described by wave function ψ: ∞ ∞ d p = ∫ ψ * pˆ ψ dx = ∫ ψ * −ih ψ dx −∞ −∞ dx
PROBABILITY A probability is a value from zero to one. The probability may be found by the following steps: Multiply the function by its complex conjugate and take the integral from negative infinity to positive infinity with respect to the variable in question, multiply all this by the square of a constant c and set equal to one. ∞
c 2 ∫ F * F dx = 1 −∞
Solve for the probability constant c. The probability from x1 to x1 is: P = c
2
Tom Penick
∫
x2
x1
F * F dx
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SIMPLE HARMONIC MOTION
INFINITE SQUARE-WELL POTENTIAL
Examples of simple harmonic motion include a mass on a spring and a pendulum. The average potential energy equals the average kinetic energy equals half of the total energy. In simple harmonic motion, k is the spring constant, not the wave number. spring constant k:
ω=
k m
potential energy V:
V=
1 2 kx 2
Schrödinger Wave Equation for simple harmonic motion: where α 2 =
mk h2
The wave equation solutions are:
force: F = kx
d 2ψ = ( α2 x2 − β) ψ dx 2 2mE and β = h2 ψ n = H n ( x ) e −αx
2
/2
where Hn(x) are polynomials of order n, where n = 0,1,2,· · · and x is the variable taken to the power of n. The functions Hn(x) are related by a constant to the Hermite polynomial functions.
α ψ0 = π
α ψ1 = π
1/4
1/4
e −αx
2
/2
2α xe −αx
α ψ2 = π
2 1 2αx 2 − 1) e −αx / 2 ( 2
α ψ3 = π
2 1 x α ( 2αx 2 − 3) e−αx / 2 3
1/4
1/4
(
2
/2
or "Particle in a Box" This is a concept that applies to many physical situations. Consider a two-dimensional box in which a particle may be trapped by an infinite voltage potential on either side. The problem is an application of the Schrödinger Wave Equation.
V ( x)
0
L
x
The particle may have various energies represented by waves that must have an amplitude of zero at each boundary 0 and L. Thus, the energies are quantized. The probability of the particle's location is also expressed by a wave function with zero values at the boundaries. Wave function:
nπx ψ n ( x ) = A sin L
Energy levels:
π2 h 2 En = n 2mL2 2
Probability of a particle being found between x1 and x2:
A=
2 L
P=∫
x2
x = x1
Ψ *Ψ dx
normalization constant
2 a useful identity: sin θ =
1 (1 − cos 2θ ) 2
)
…and they call this simple!
quantized energy levels:
1 En = n + hω 2
The zero-point energy, or Heisenberg limit is the minimum energy allowed by the uncertainty principle; the energy at n=0:
E0 =
1 hω 2
HEISENBERG UNCERTAINTY PRINCIPLE These relations apply to Gaussian wave packets. They describe the limits in determining the factors below.
∆px ∆x ≥ h / 2
∆E ∆t ≥ h / 2
∆px = the uncertainty in the momentum along the x-axis ∆x = the uncertainty of location along the x-axis ∆E = the uncertainty of the energy ∆t = the uncertainty of time. This also happens to be the particle lifetime. Particles you can measure the mass of (E=mc2) have a long lifetime.
Tom Penick
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12/12/1999 Page 12 of 22
POTENTIAL BARRIER
3D INFINITE POTENTIAL BOX
V (x )
When a particle of energy E encounters a barrier of potential V0, there is a possibility of either a reflected wave or a transmitted wave.
particle
V0 0
x
L
Region I Region II Region III
for E > V0: kinetic energy: K = E − V0 wave number:
k II = 2m ( E − V0 ) / h reflected wave: ϕI = Be
V sin ( k II L ) T = 1 + 0 4 E ( E − V0 ) reflection probability: R = 1 − T
0
L1
x
h 2 ∂ 2ψ ∂ 2ψ ∂ 2 ψ + + + V ψ = Eψ 2m ∂x 2 ∂y2 ∂z 2
kI x 2
L2
n πx n πy n πz ψ n1 n2 n3 = A sin 1 sin 2 sin 3 L1 L2 L3
− ik I x
2
L3
Wave equation for the 3D infinite potential box:
ik I x
transmitted wave: ϕIII = Fe
y
z
Time-independent Schrödinger Wave Equation in three dimensions:
−
k I = k III = 2mE / h
incident wave: ϕI = Ae
Consider a three-dimensional box with zero voltage potential inside the box and infinite voltage outside. A particle trapped in the box is described by a wave function and has quantized energy levels.
−1
Energy levels:
En1 n2 n3 =
trans. probability:
π 2h 2 n12 n22 n32 + + 2m L12 L22 L32
Degenerate energy levels may exist—that is, different combinations of n-values may produce equal energy values.
for E < V0: Classically, it is not possible for a particle of energy E to cross a greater potential V0, but there is a quantum mechanical possibility for this to happen called tunneling. kinetic energy: K = V0 − E wave #, region II: κ =
2m (V0 − E ) / h
V0 2 sinh 2 ( κL ) trans. probability: T = 1 + 4 E (V0 − E ) E E −2 κL when κL ? 1 : T = 16 1 − e V0 V0
Tom Penick
−1
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12/12/1999 Page 13 of 22
SCHRÖDINGER'S EQUATION – 3D SPHERICAL spherical coordinate form:
1 ∂ 2 ∂ψ 1 ∂ ∂ψ 1 ∂2ψ 2m r + sin θ + + ( E −V) ψ= 0 r2 ∂r ∂r r2 sin θ∂θ ∂θ r2 sin2 θ ∂θ2 h2 separation of variables using: Ψ ( r, θ, φ) = R ( r ) f ( θ) g ( φ) We can obtain a form with terms of g on one side and terms of R and f on the other. These are set equal to the constant ml2. ml turns out to be an integer. Another seperation is performed for R and f and the constant is l(l+1), where l is an integer. The three equations are:
NORMALIZING WAVE FUNCTIONS To normalize a function, multiply the function by its complex conjugate and by the square of the normalization constant A. Integrate the result from -∞ to ∞ and set equal to 1 to find the value of A. The normalized function is the original function multiplied by A. To normalize the wave function Ψ(x):
∫
2
∞
−∞
AΨ dx
∞
A2 ∫ Ψ 2 dx
→
−∞
Where Ψ is an even function, we can simplify to: ∞
2A2 ∫ Ψ 2dx 0
∞
and find A:
2 A2 ∫ Ψ 2 dx = 1 0
Some relations for definite integrals will be useful in solving this equation; see CalculusSummary.pdf page 3.
Azimuthal equation:
1 d2g 2 + ml = 0 ⇒ g = Aeiml φ 2 g dφ
To normalize the wave function Ψ(r), where r is the radius in spherical coordinates:
∫
Radial equation:
l ( l + 1) 1 d 2 dR 2m r + E − V R − R=0 ( ) r 2 dr dr h2 r2
Angular Equation: 2 df ml 1 d sin θ + l l + 1 − ) f =0 ( sin θ d θ dθ sin 2 θ
ml = magnetic quantum number; integers ranging from –l
∞
0
2
∞
A2 ∫ r 2 Ψ 2 dr = 1
r 2 AΨ dr →
0
Note that we integrate from 0 to ∞ since r has no negative values. To normalize the wave function Ψ(r,θ θ ,φ φ): ∞
A2 ∫ dr r 2 AΨ
2
0
∫
π
0
2π
d θ sin θ ∫ d φ = 1 0
Note that dr, dθ, and dφ are moved to the front of their respective integrals for clarity.
to +l
l = orbital angular momentum quantum number h = Planck's constant divided by 2π [J-s] E = energy V = voltage; can be a function of space and time (x,t) m = mass [kg]
Rnl(r) RADIAL WAVE FUNCTIONS for the hydrogen atom n 1
l 0
Rnl(r)
2 a0
3/2
e − r / a0
0
r e − r / 2 a0 w− a0 ( 2a0 )3 / 2
2
1
r a0
3
0
3
1
2
3
2
e − r / 2 a0 3 ( 2a0 ) 1
( a0 )3 / 2 1
( a0 )
3/2
1
( a0 )
3/2
3/2
2 r r 2 − r /3 a0 27 − 18 + 2 2 e a0 a0 81 3 4 r r − r /3 a0 6 − e a0 a0 81 6 4 r 2 − r /3 a0 e 2 81 30 a0
a0 = Bohr radius 5.29177×10-11 m
Tom Penick
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12/12/1999 Page 14 of 22
ATOMS
P(r)dr RADIAL PROBABILITY The radial probability is a value from 0 to 1 indicating the probability of a particle occupying a certain area radially distant from the center of orbit. The value is found by integrating the right-hand side of the expression over the interval in question:
P(r ) dr = r 2 R ( r ) dr 2
r = orbit radius R(r) = radial wave function, normalized to unity
P(r) RADIAL PROBABILITY DENSITY The radial probability density depends only on n and l.
P(r ) = r 2 R ( r )
2
r = orbit radius R(r) = radial wave function, normalized to unity
〈 r〉〉 RADIAL EXPECTATION VALUE average radius (radial wave function):
r =∫
∞
r =0
r P ( r ) dr = ∫
∞
r =0
P(r) = probability distribution function
r R ( r ) dr 3
QUANTUM NUMBERS n = principal quantum number, shell number, may have values of 1, 2, 3, …
l = orbital angular momentum quantum number, subshell number, may have values of 0 to n-1. These values are sometimes expressed as letters: s=0, p=1, d=2, f=3, g=4, h=5, … ml = magnetic quantum number, may have integer values from -l to +l for each l. (p251) ms = magnetic spin quantum number, may have values of +½ or -½ Then we introduce these new ones: s = intrinsic quantum number, s =1/2 (p238) j = total angular momentum quantum number, j = l ± s, but j is not less than 0. (p257) mj = magnetic angular momentum quantum number, may have values from -j to +j (p257) Example, for n = 3:
l=
0
j=
1/2
1/2
3/2
3/2
5/2
-1/2 +1/2
-1/2 +1/2
-3/2 -1/2 +1/2 +3/2
-3/2 -1/2 +1/2 +3/2
-5/2 -3/2 -1/2 +1/2 +3/2 +5/2
mj =
P ( r ) = r 2 R ( r ) dr
1
2
2
R(r) = radial wave function, normalized to unity
ml =
0
-1
0
+1
-2
-1
0
+1
+2
ms =
-1/2 +1/2
-1/2 +1/2
-1/2 +1/2
-1/2 +1/2
-1/2 +1/2
-1/2 +1/2
-1/2 +1/2
-1/2 +1/2
-1/2 +1/2
L ORBITAL ANGULAR MOMENTUM Classically, orbital angular momentum is ρr or mvr. The orbital angular momentum L is a vector quantity. It components are as follows: Magnitude: Z-axis value:
L = h l ( l + 1)
Lz = ml h
The values of Lx and Ly cannot be determined exactly but obey the following relation:
L2 = L2x + L2y + L2z h = Planck's constant divided by 2π [J-s] l = orbital angular momentum quantum number ml = magnetic quantum number; integers ranging from –l to +l The orbital angular momentum quantum number was originally given letter values resulting from early visual observations: sharp, principal, diffuse, fundamental
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l =0 1 2 3 4 5 s p df gh
12/12/1999 Page 15 of 22
MANY-ELECTRON ATOMS
S SPIN ANGULAR MOMENTUM The spin angular momentum is (insert some illuminating explanation here). Magnitude:
S = h s ( s + 1) = h 3 / 4
z 1 +2h
z component:
S z = ms h = ± h / 2
SPECTROSCOPIC SYMBOLS The energy state of an atom having 1 or 2 electrons in its outer shell can be represented in the form
S=h
3/4
n 2S +1 L j n = shell number S = intrinsic spin angular momentum quantum number; ½
- 1h 2
J TOTAL ANGULAR MOMENTUM The vector sum of the orbital angular momentum and the spin angular momentum. This applies to 1electron and many-electron atoms.
J = L+S
for a single-electron shell, 0 or 1 (S1 + S2) for the 2electron shell L = angular momentum quantum number; l for singleelectron shell, L1 + L2 for a 2-electron shell, expressed as a capital letter: S=0, P=1, D=2, F=3, G=4, H=5, I=6. j = total angular momentum quantum number j = l ± s. I'm not sure how to tell whether it's plus or minus, but I think it has to be the lower value of j to be in the ground state. j is positive only.
J (the magnitude?) is an integer value from |L-S| to L+S.
ORDER OF ELECTRON FILLING ALLOWED TRANSITIONS The allowed energy level transitions for 1-electron atoms are ∆n: any
∆l: ±1
∆mj: 0, ±1
∆j: 0, ±1
ZEEMAN SPLITTING ("ZAY·
mahn")
When a single-electron atom is under the influence of an external magnetic field (taken to be in the z-axis direction), each energy level (n=1,2,3,…) is split into multiple levels, one for each quantum number ml. The difference in energy is:
∆E = µ B Bml ∆E = difference in energy between two energy levels [J] µB = Bohr magneton 9.274078×10-24 J/T B = magnetic field [T] ml = magnetic quantum number; integers ranging from –l to +l
Here's a way to remember the order in which the outer shells of atoms are filled by electrons: Form groups of l-numbers like this. The first group is just the lowest value for l: s. The next value of l is p; form a new group of p with s. The third value of l is d; form the third group with d, p, and s. You get a list of groups like this: Now, in a column, write each group twice beginning with the single s that is the first group. Next number each s beginning with 1, placing the number in front of the s. This is as far as I have gone with the list at right. The next step is to number each p beginning with the number 2. Then number each d beginning with the number 3. Number each f beginning with 4, and so on.
µ MAGNETIC MOMENT Both the magnetic moment µ and the orbital angular momentum L are vectors:
ì =−
The result will be the order of filling (there are a few exceptions) and will look like this: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d and so on.
e L 2m
s ps dps fdps gfdps hgfdps 1s
2s p 3s p 4s d p 5s f d p 6s f d p 7s and so on.
m = mass of the orbiting particle [kg]
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12/12/1999 Page 16 of 22
g LANDÉ g FACTOR
SPLITTING DUE TO SPIN
A dimensionless number that helps make physics complicated. Used in ANOMALOUS ZEEMAN SPLITTING
g = 1+
J ( J + 1) + S ( S + 1) − L ( L + 1) 2 J ( J + 1)
ALLOWED PHOTON TRANSITIONS The allowed photon energy level transitions for manyelectron atoms are ∆L: ±1
∆mj: 0, ±1, but can't transition from 0 to 0 when ∆J=0. Other transitions are possible—just not likely.
θ MINIMUM ANGLE BETWEEN J AND THE Z-AXIS There were exercises where we had to calculate this. I don't know what the significance is. This is done similarly for L and S as well.
j=
Example: j×h
h j ( j + 1)
cos 2 θ =
5 2
∆E =
+
→
+ 32
j2 → j cos 2 θ = j ( j + 1) ( j + 1)
cos θ =
j j +1
+ -
1 2 1 2 3 2 5 2
h h
θ
J = h j(j+1)
h h
S=h
3/4
- 1h 2
hc ∆λ λ2
∆E = difference in energy between two (split) energy levels ms = ±1/2 [J]
∆λ = difference in wavelengths for the transitions to the ground state for each energy level [m] λ = wavelength for the transitions to the ground state for the lower of the two energy levels (the greater of the two wavelengths) [m] h = Planck's constant 6.6260755×10-34 J-s c = speed of light 2.998 × 108 m/s
z 5 2
z 1 +2h
The lower of the two energy levels is aligned with the magnetic field.
∆J: 0, ±1, but J can't transition from 0 to 0.
∆S: 0
cos θ =
For each state described by quantum numbers n, l, ml, there are two states defined by the magnetic spin numbers ms = ±1/2. These two levels have the same energy except when the atom is influenced by an external magnetic field.
SPIN-ORBIT ENERGY SPLITTING Spin-orbit energy splitting is the splitting of energy levels caused by an internal magnetic field due to spin. This produces a greater ∆E than the spin splitting described above. p265 P.E. due to spin
h
V = −ì s ·B
z-component µ = − g eh J z z s
h
2me h
energy level difference
∆E = g s
eh B me
e = q = electron charge 1.6022×10-19 C
h
= Planck's constant divided by 2π [J-s]
jz = z-component of the total angular momentum ∆E = difference in energy between two (split) energy levels ms = ±1/2 [J]
gs = 2, the gyromagnetic ratio me = mass of an electron 9.1093897×10-31 kg B = internal magnetic field [T]
Tom Penick
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12/12/1999 Page 17 of 22
STATISTICAL PHYSICS
ANOMALOUS ZEEMAN SPLITTING ("ZAY· mahn")
In addition to the Zeeman splitting of the ml energy levels described previously, and the spin-orbit energy splitting described above, there is a splitting of the mj levels when an external magnetic field is present. The difference in energy between levels is:
V = µ B Bext gm j
v*, v , vrms MOLECULAR SPEEDS [m/s] F ( v ) dv = 4πCe
v* most probable
v* =
2 2kT = m βm
v=
4 2π
speed:
V = difference in energy between two energy levels [J] µB = Bohr magneton 9.274078×10-24 J/T Bext = external magnetic field [T] g = Landé factor [no units] mj = magnetic angular momentum quantum number; halfintegers ranging from –j to +j
1 − β mv2 2
Maxwell speed distribution:
v
mean speed:
vrms root mean square speed:
v 2 dv
kT m 1/2
vrms = v 2
3kT m
=
v = velocity [m/s] C = normalization constant k = Boltzmann's constant 1.380658×10-23 J/K T = temperature [K] m = mass of the molecule [kg] β = the parameter 1/kT [J-1]
ENERGY DISTRIBUTION Derived from Maxwell's speed distribution:
F (E) =
8πC −βE 1 / 2 e E 2m3 / 2
FMB MAXWELL-BOLTZMANN FACTOR The Maxwell-Boltzmann factor is a value between 0 and 1 representing the probability that an energy level E is occupied by an electron (at temperature T). This is for classical systems, such as ideal gases. One way to determine if Maxwell-Boltzmann statistics are valid is to compare the de Broglie wavelength λ = h/p of a typical particle with the average interparticle spacing d. If λ<
FMB = Ae −βE
1/3
V d = N
A = normalization constant β = the parameter 1/kT [J-1] d = space between atoms [m] N = number of particles in volume V. Note that Avogadro's number, 6.022×1023, is the number of gas molecules in 22.4 liters, or 22.4×10-3 m3, at 0°C and 1 atmosphere. Also, gas volume is proportional to temperature: V1/T1=V2/T2.
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12/12/1999 Page 18 of 22
FFD FERMI-DIRAC DISTRIBUTION A value between 0 and 1 indicating the probability than an energy state is occupied by an electron. The Fermi-Dirac distribution is valid for fermions, particles with half-integer spins that obey the Pauli principle. Atoms and molecules consisting of an even number of fermions must be considered bosons when taken as a whole because their total spin will be zero or an integer.
FFD =
TF FERMI TEMPERATURE The Fermi temperature may be quite high, 80,000 K for copper.
TF =
EF k
EF = Fermi Energy [eV] k = Boltzmann's constant 1.380658×10-23 J/K
1
uF FERMI SPEED
βE
B1e + 1
The Fermi speed, uF =
B1 = normalization constant β = the parameter 1/kT [J-1]
2 EF m
comes from the definition: EF =
FBE BOSE-EINSTEIN DISTRIBUTION The Bose-Einstein distribution is valid for bosons, particles with zero or integer spins that do no obey the Pauli principle. Photons, pions, and liquid 4He are bosons.
FBE =
1 βE
B2 e − 1
The Fermi energy depends on the density of electrons in the material. The Fermi-Dirac distribution is modified to include the Fermi energy:
1 β( E − EF )
+1
h2 3N 8m πL3
2/3
spacing
n3
where E1 = h 2 / 8mL2 (a constant equal to 1/3 of the ground state energy)
3 2 1
and r = n12 + n22 + n32
1
and E is whatever energy the problem is concerned with.
The relationship between the Fermi energy and the number density of particles is:
EF =
ENERGY STATES IN "PHASE SPACE"
E = r 2 E1
EF FERMI ENERGY [eV]
e
EF = Fermi Energy [eV] m = mass (probably of the electron) [kg]
The points in the 1/8 sphere represent the energy states of a particle in a cube, see 3D INFINITE POTENTIAL BOX p13. integer Energy level at radius r
B2 = normalization constant β = the parameter 1/kT [J-1]
FFD =
1 2 muF 2
2 3
energy states energy levels
n2 ∆r
1 2 3
r
n1
Nr Number of energy states in a sphere of radius r, i.e. the number of energy states there are with energy less than E: 3/2
π E 1 4 N r = 2 πr 3 or N r = 3 E1 8 3
FFD = Fermi-Dirac distribution, a value from 0 to 1 indicating the probability that an energy state is occupied β = the parameter 1/kT [J-1] h = Planck's constant 6.6260755×10-34 J-s m = mass of the particle [kg] N/L3 = number density of the particles [m-3]
where the factor of 2 is due to spin degeneracy, and the factor of 1/8 is because the energy states only occupy 1/8 of the sphere where n1, n2, n3 are all positive.
g(E) DENSITY OF ENERGY STATES Number of states per unit energy
g (E) =
π −3 / 2 1/ 2 E1 E 2
g (E) =
∆N r dN r = ∆E dE
Nr = number of energy states in a sphere of radius r E1 = a constant equal to 1/3 of the ground state energy
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12/12/1999 Page 19 of 22
n(E) DENSITY OF OCCUPIED STATES Number of occupied states per unit energy
n ( E ) = FFD ⋅ g ( E ) g ( E ) , for E < EF for E > EF 0,
at T=0, n ( E ) =
FFD = Fermi-Dirac distribution, a value from 0 to 1 indicating the probability that an energy state is occupied
Tom Penick
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APPENDIX CONSTANTS
BINOMIAL EXPANSION For x < 1 : n( n − 1) 2 n(n − 1)(n − 2) 3 = 1 ± nx + x ± x +L 2! 3!
Avogadro’s number
N A = 6.0221367 ×1023
[molecules/mole]
µB =
Bohr magneton
qh = 9.27407836 × 10−24 J/T 2me
(1 ± x )
k = 1.380658 ×10−23 J/K K = 8.62 × 10 −5 eV/K
Boltzmann’s constant or Earth to Moon distance
≈ 384 ×106 m
Elementary charge
q = 1.60 × 10 −19 C kg
me = 0.51100 MeV/c
2
kg
mneutron = 939.57 MeV/c −27
m p = 1.6726231×10
TV, FM
kg
Microwave Infrared light
− 12
Visible light
F/m
h = 6.6260755 ×10−34 J-s
violet
= 4.14 × 10 −15 eV-s
blue
Speed of sound (air 0°C)
R = 1.097373 × 10 m kT = 0.0259 eV c = 2.998 × 108 m/s vs = 331.29 m/s
1 Å (angstrom) 1 µm (micron) 1 nm = 10Å = 10-7 cm 1 eV = 1.6 × 10-19 J 1 V = 1 J/C 1 N/C = 1 V/m
10-8 cm = 10-10 M 10-4 cm 273.15K = 0°C 1 W = 1 J/S = 1 VA 1 J = 1 N· m = 1 C· V
7
Rydberg constant kT @ room temperature Speed of light
Shortwave radio
2
ε0 = 8.8541878 ×10
Planck’s constant
Longwave radio
2
m p = 938.27 MeV/c Permittivity of free space
Joules ×
Kg ×
yellow orange Red Ultraviolet light X-rays Gamma rays Cosmic rays
Α Β Χ ∆ Ε Φ Γ Η
1 = eV q
c = eV/c2 q
c eV Momentum: kg ⋅ m × = q c s
Tom Penick
1013 - 1015 1012 - 1013 1011 - 1012 109 - 1011 107 - 109 8000 - 107
1 - 100 km 100 - 1000 m 10 - 100 m 0.1 - 10 m 1 - 100 mm
3600 - 6900 3600 4300 4900 5600 6000 6900 100 - 3900 0.05 - 100 0.001 - 0.05 < 0.001
GREEK ALPHABET
2
Mass:
ANGSTROMS
0.8 - 1000 µm 360 - 690 nm 360 nm 430 nm 490 nm 560 nm 600 nm 690 nm 10 - 390 nm 5 - 10,000 pm 100 - 5000 fm < 100 fm
green
-1
UNITS Energy:
METERS
Standard Broadcast
−27
mneutron = 1.6749 ×10
Proton mass
(1 ± x )n = 1 ± nx
WAVELENGTH SPECTRUM
me = 9.1093897 × 10
Neutron mass
When x is much less than 1:
BAND −31
Electron mass
n
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α β χ δ ε φ γ η
alpha beta chi delta epsilon phi gamma eta
Ι Κ Λ Μ Ν Ο Π Θ
ι κ λ µ ν ο π θ
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iota kappa lambda mu nu omicron pi theta
Ρ Σ Τ Υ Ω Ξ Ψ Ζ
ρ σ τ υ ω ξ ψ ζ
rho sigma tau upsilon omega xi psi zeta
12/12/1999 Page 21 of 22
TRIG IDENTITIES − ix
i 2sin x = e − e 2 cos x = eix + e− ix ix
COORDINATE TRANSFORMATIONS −x
2 sinh x = e − e 2 cosh x = e x + e − x x
eix = cos x + i sin x sin ( A ± B ) = sin A cos B ± cos A sin B
cos ( A ± B ) = cos A cos B m sin A sin B
A+ B A−B sin A + sin B = 2 sin cos 2 2 A+ B A−B cos A + cos B = 2 cos cos 2 2
GEOMETRY SPHERE Area A = 4πr 2
4 Volume V = πr 3 3
ELLIPSE
Area A = πAB
Circumference a2 + b2 2
L ≈ 2π
COORDINATE SYSTEMS Cartesian or Rectangular Coordinates:
r ( x, y, z ) = xxˆ + yyˆ + zzˆ
xˆ is a unit vector
r = x2 + y2 + z 2 r is distance from center θ is angle from vertical φ is the CCW angle from the x-axis
rˆ , èˆ , and φˆ are functions of position—their orientation depends on where they are located. Cylindrical Coordinates:
C(r , φ, z )
To obtain: A ( r , φ , z ) = rˆ A r + φˆ A φ + zˆ A z
Ar = x 2 + y 2 y φ = tan −1 x z=z
rˆ = xˆ cos φ + yˆ sin φ φˆ = − xˆ sin φ + yˆ cos φ zˆ = zˆ
Cylindrical to Rectangular:
ˆ + yyˆ + zzˆ To obtain: r ( x, y, z ) = xx x = r cos φ y = r sin φ z=z
xˆ = rˆ cos φ − φˆ cos φ φˆ = rˆ sin φ + yˆ cos φ zˆ = zˆ
Rectangular to Spherical:
ˆA To obtain: A ( r , θ, φ) = rˆ Ar + èˆ Aθ + φ φ Ar = x 2 + y 2 + z 2 rˆ = xˆ sin θ cos φ + yˆ sin θ sin φ + zˆ cos θ θ=
z cos −1 x2 + y2 + z2
èˆ = xˆ cos θ cos φ + yˆ cos θ sin φ − zˆ sin θ y φ = tan −1 φˆ = − xˆ sin φ + yˆ cos φ x Spherical to Rectangular:
Spherical Coordinates:
P(r , θ, φ)
Rectangular to Cylindrical:
r is distance from the vertical (z) axis φ is the CCW angle from the x-axis
To obtain: r ( x, y, z ) = xxˆ + yyˆ + zzˆ
x = r sin θ cos φ xˆ = rˆ sin θ cos φ − èˆ cos θ cos φ − φˆ sin φ y = r sin θ sin φ yˆ = rˆ sin θ sin φ + èˆ cos θ sin φ + φˆ cos φ z = r cos θ
zˆ = rˆ cos θ − èˆ sin θ
z is the vertical distance from origin
Tom Penick
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