Phedotikov / 1 / FreeEnergy_27.01.08 / !Информация / Nikolo Tesla / Modern Physics For Engineers  Nikola Tesla
.pdfMODERN PHYSICS FOR ENGINEERS PHY355
INDEX
3D infinite potential box .13 

allowed transitions 

1electron atoms.......... 
16 
manyelectron 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 
BoseEinstein 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 
DuaneHunt 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 
zeropoint ................... 

12 
energy distribution .......... 
18 

expectation value ............ 

11 
radial .......................... 

15 
Fermi energy................... 

19 
Fermi speed .................... 

19 
Fermi temperature........... 
19 

FermiDirac 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 

MaxwellBoltzmann factor18 

mean speed ..................... 

18 
MichelsonMorley 


experiment................... 

3 
minimum angle ............... 

17 
molecular speeds............. 

18 
momentum....................... 

4 
relativistic.................... 

4 
momentum operator ........ 

11 
momentumenergy relation 5 

momentumtemperature 


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 
headon........................ 

7 
xray............................ 

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 

spinorbit splitting........... 

17 
splitting due to spin......... 

17 
spring harmonics ............. 

12 
statistical physics ............ 

18 
StefanBoltzman 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 
xray 


Lalpha waves.............. 

9 
scattering ..................... 

9 
Young's double slit 


experiment................... 

5 
Zeeman splitting ....... 
16, 18 

zeropoint 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
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)
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 1 of 22
MAXWELL’S EQUATIONS
Gauss’s law for electricity 
Ñò 
EgdA = 

q 












ε0 





Gauss’s law for 
Ñò BgdA = 0 





magnetism 





Faraday’s law 
Ñò 
Egds = − 

dΦB 




dt 


Ñò 
Bgds = μ0ε0 
dΦ 

Generalized Ampere’s law 
E 
+ μ0 I 

dt 
LORENTZ FORCE LAW
Lorentz force law: F = qE + qv × B
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.
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.
FUNDAMENTAL FORCES
FORCE 
RELATIVE 
RANGE 

STRENGTH 




Strong 
1 
Short, ~1015m 

Electroweak 



Electromagnetic 
102 
Long, 1/r2 

Weak 
109 
Short, ~1015m 

Gravitational 
1039 
Long, 1/r2 
ATOMIC MASS
The mass of an atom is it's 
atomic number 

atomic number divided by the 



product of 1000 times 
1000 × Na 

Avogadro's number. 

KINETIC ENERGY
The kinetic energy of a particle (ideal 
K = 
3kT 

gas) in equilibrium with its 



surroundings is: 
2 
PHASE SPACE
A sixdimensional pseudospace populated by particles described by six position and velocity parameters:
position: (x, y, z) velocity: (vx, vy, vz)
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 2 of 22
RELATIVITY
WAVELENGTH λ





c = speed of light 2.998 × 108 m/s 

c = 
1 
= ln 

λ = wavelength [m] 

m0e0 

ν = (nu) radiation frequency [Hz] 











Å = (angstrom) unit of wavelength 

1Å = 1010m 



equal to 1010 m 






m = (meters) 






MichelsonMorley Experiment indicated that light was not influenced by the “flow of ether”.
LORENTZ TRANSFORMATION
Compares position and time in two coordinate systems moving with respect to each other along axis x.


x  vt 













x¢ = 






t − vx / c 
2 


























t′ = 






1 v2 / c2 









1− v2 / c2 












v = velocity of (x’,y’,z’) system along the xaxis. [m/s] 

t = time [s] 















c = speed of light 2.998 × 108 m/s 







or with b = 
v 

and 

g = 

1 





c 


1 v2 / c2 











so that x¢ = g( x  vt ) 
and 

t¢ = g(t bx / c) 
LIGHT WAVEFRONT
Position of the wavefront of a light source located at the origin, also called the spacetime distance.
x2 + y2 + z 2 = c2t2
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.
TIME DILATION
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 = 
T0¢ 

or 
T ¢ = 
T0 

where: 
1 v2 / c2 

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 xaxis. [m/s] c = speed of light 2.998 × 108 m/s
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¢ 

where: 

1 v2 / c2 



L0 = the proper length (longer). [m] L' = contracted length [m]
v = velocity of (x’,y’,z’) system along the xaxis. [m/s] c = speed of light 2.998 × 108 m/s
RELATIVISTIC VELOCITY ADDITION
Where frame K' moves along the xaxis of K with velocity v, and an object moves along the xaxis with velocity ux' with respect of K', the velocity of the object with respect to K is ux.






K' 














K 




u' 



= 

u¢ 
+ v 










v 

ux 

x 













1+ (v 
/ c 
2 
)ux¢ 










































If there is uy' or uz' within the K' frame then 










u¢ 






¢ 





uy = 



y 
and uz = 




uz 





g ëé1 (v / c2 )ux¢ ûù 

g ëé1 (v / c2 )ux¢ ûù 







ux = velocity of an object in the x direction [m/s]
v = velocity of (x’,y’,z’) system along the xaxis. [m/s] c = speed of light 2.998 × 108 m/s
γ = 1/ 1− v2 / c2
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 tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 3 of 22
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 




c 
1 



slope = v = 
β 
= 4 





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.
SPACETIME INTERVAL s
The quantity s2 is invariant between two frames of reference with relative movement along the xaxis.
s2 = x2 − (ct )2 = x′2 − (ct′)2
Two events occurring at different times and locations in the Kframe may be characterized by their s2 quantity.
s2 = x2 − (c t )2
lightlike  
s2 = 0: In this case, 
x2 = c2 t2, and the two 
events can only be connected by a light signal. 

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.
MOMENTUM p
p = mv for a photon: p = hν c
p = momentum [kgm/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 [kgm/s], convertible to [eV/c] by multiplying by c/q.
γ = 1/ 1− u2 / c2 m = mass [kg]
u = velocity of object [m/s]
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 4 of 22
DOPPLER EFFECT
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.
ν = 
1+ β 
ν0 
For two systems receeding from 
1− β 
each other, reverse the signs. 









ν = (nu) the frequency of emitted light as perceived in the other system [Hz]
ν0 = (nu) the proper frequency of the emitted light (lower 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 speed of light 2.998 × 108 m/s
v = velocity of (x’,y’,z’) system along the xaxis. [m/s]
RELATIVISTIC KINETIC ENERGY K
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 = γmc2 − mc2 where:
K = relativistic kinetic energy [J], convertible to [eV] by dividing by q.
γ = 1/ 1− v2 / c2 m = mass [kg]
c = speed of light 2.998 × 108 m/s
REST ENERGY E0
Rest energy is the energy an object has due to its mass.
E0 = mc2
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 = γmc2 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− v2 / c2 m = mass [kg]
c = speed of light 2.998 × 108 m/s
MOMENTUMENERGY RELATION
(energy)2 = (kinetic energy)2 + (rest energy)2
E2 = p2c2 + m2c4 where:
E = total energy (Kinetic + Rest energies) [J] p = momentum [kgm/s]
m = mass [kg]
c = speed of light 2.998 × 108 m/s
BINDING ENERGY
∙the potential energy associated with holding a system together, such as the coulomb force between a hydrogen proton and its electron
∙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.
EB = åmi c2 − M bound system c2 i
for hydrogen and singleelectron ions, the binding energy of the electron in the ground state is
mZ 2e4
EB = 2h2 (4πε0 )2
EB = binding energy (can be negative or positive) [J] m = mass [kg]
Z = atomic number of the element e = q = electron charge [c]
h = Planck's constant divided by 2π [Js] ε0 = permittivity of free space 8.85 × 1012 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.
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 offcenter 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]
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 5 of 22
WIEN'S CONSTANT
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.
lmaxT = 2.898´10−3 m ×K
STEFANBOLTZMANN LAW
May be applied to a blackbody or any material for which the emissivity is known.
R(T ) = esT 4 where:
R(T) = power per unit area radiated at temperature T
[W/m2]
ε = emissivity (ε = 1 for ideal blackbody)
σ = constant 5.6705 × 108 W/(m2· K4)
T = temperature (K)
PLANCK'S RADIATION LAW
I (l, T ) = 
2pc2 h 

1 

where: 
l5 

ehc / λkT 1 




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 6.6260755×1034 Js
k = Boltzmann's constant 1.380658×1023 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.
PHOTON
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 = hn = pc [Joules]
Momentum: p = hn [kgm/s], convertible to [eV/c] by
c
multiplying by c/q.
Wavelength: l = c [meters]
n
h = Planck's constant 6.6260755×1034 Js
ν = (nu) frequency of the electromagnetic wave associated with the light given off by the photon [Hz]
c = speed of light 2.998 × 108 m/s
PHOTOELECTRIC EFFECT
This is the way the book shows the formula, but it is a units nightmare.

1 
mv2 
= eV = hu  f 
where: 




2 
max 
0 





1 mvmax2 = energy in Joules, but convert to eV for the
2
formula by dividing by q.
eV0 = potential required to stop electrons from leaving the metal [V]
hν = Planck's constant [6.6260755×1034 Js] 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
eV = hu 

= 
hc 

where: 

lmin 

0 
max 



eV0 = the kinetic energy of an electron accelerated through a voltage V0 [eV]
hν = Planck's constant [6.6260755×1034 Js] multiplied by the frequency of light [Hz]. This term will need to be divided by q to obtain eV.
λmin = the minimum wavelength of light created when an electron gives up one photon of light energy [m]
DUANEHUNT RULE
lmin = 1.2398´10−6
V0
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 6 of 22
ELECTRON ANGULAR MOMENTUM
from the Bohr model:
L = mvr = nh where:
L = angular momentum [kgm2/s?]
m= mass [kg]
v= velocity [m/s]
r= radius [m]
n= principle quantum number
h = Planck's constant divided by 2p [Js]
a0 BOHR RADIUS [m]
The Bohr radius is the radius of the orbit of the hydrogen electron in the ground state (n=1):


4πε0h 
2 

and for higher 



a 
= 


r = a n2 

0 

mee 
2 


states (n>1): 
n 
0 








a0, rn = Bohr radius 5.29177×1011 m, quantized radius [m]
ε0 = permittivity of free space 8.85 × 1012 F/m me = electron mass 9.1093897×1031 [kg]
e = q = electron charge [c]
n = principle quantum number
h = Planck's constant divided by 2p [Js]
IMPACT PARAMETER b
The impact parameter b is the distance that a bombarding particle deviates from the directhit approach path, and is related to the angle q at which it will be deflected by the target particle.
b = Z1 Z2 e2 cot θ 8πε0 K 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 × 1012 F/m K = kinetic energy of the incident particle Z1
θ = angle of particle Z1 deflection or scattering
HEADON 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.
rmin = Z1 Z2e2
4πε0 K
rmin = particle separation (measured center to center) at the time that the bombarding particle reverses direction [m]
other variables are previously defined
COMPTON EFFECT
The scattering of a photon due to collision with a single electron results in a new wavelength l' and a directional change of Ðq and is described by the following relation:

′ 
h 
(1− cos θ) 

Δλ = λ − λ = mc 



scattered photon 






E = h n' 




p = 
h 



θ 
' 





l 

photon 

φ 



E = h n 





p = h 


recoil electron 

l 
electron at rest 
E 
f 
= E 



e 
Ei = mc 2
The f relations come from the conservation of momentum:
p 

: 
h 
= 
h 
cos θ + p cos φ 

x 





λ λ′ 
e 






p 

: 
h 
sin θ = p sin φ 

y 




λ′ 
e 





Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 7 of 22
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.
N (q) = 
Ni nt æ 
e2 ö2 



Z12Z22 



ç 

÷ 









16 

r 
2 
K 
2 
sin 
4 
(q/ 2) 



è 4pe0 ø 




N(θ) = number of particles scattered per unit area [m2]
θ = angle of particle Z1 deflection or scattering Ni = total number of incident particles [kg]
n = number of atoms per unit volume [m3] n =
t = thickness of the target material [m] e = q = electron charge [c]
ε0 = permittivity of free space 8.85 × 1012 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
æ Z Z e2 
ö2 

q 


f = pnt ç 
1 2 
÷ 
cot2 


8pe0 K 
2 


è 
ø 


f = the probability (a value between 0 and 1)
3 
ρN A N M 

n = number of atoms per unit volume [m 
] n = 


M g where ρ is density [g/m3], NA is Avogadro's number, NM is the number of atoms per molecule, and MG is the grammolecular 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 × 1012 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
ELECTRON VELOCITY
This comes from the Bohr model and only applies to atoms and ions having a single electron.
v = 
1 

Ze2 
= 
e Z 
n 
n 4pe0h 

2 pe0mer 





14243 

14243 


ndependent 

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 × 1012 F/m me = mass of an electron 9.1093897×1031 kg h = Planck's constant divided by 2π [Js]
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.

r 
= 
4pe0n2h2 









n 

m Ze2 











e 


















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] 




ar 
= 
v 
2 



v = tangential velocity of the electron [m/s] 




r 


r = electron orbit radius 
[m] 























Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 8 of 22
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 singleelectron ions.
R = 
m 
e 
Z 2e4 

where me = 
m M 



e 

4pch3 (4pe0 )2 

me + M 
R∞ = Rydberg constant 1.09678×107 m1 (1.096776×107 m1 for hydrogen)
μe = adjusted electron mass
Z = atomic number, or number of protons in the nucleus
ε0 = permittivity of free space 8.85 × 1012 F/m c = speed of light 2.998 × 108 m/s
h = Planck's constant divided by 2π [Js] me = mass of an electron 9.1093897×1031 kg
M = mass of the nucleus (essentially the same as the mass of the atom Þ atomic number × 1.6605×1027) [kg]
Lα MOSELEY'S EQUATION
British physicist, Henry Moseley determined this equation experimentally for the frequency of Lα x rays. Lα waves are produced by an electron decaying from the n=3 orbit to the n=2 or L orbit.
nL 
= 
5 
cR (Z  7.4)2 


α 
36 





ν = (nu) frequency [Hz]
c = speed of light 2.998 × 108 m/s
R = Adjusted Rydberg constant (see above) [m1]
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 singleelectron atom or ion decays from orbit nu to nl.
1 
æ 
1 

1 
ö 


= Z 2 R ç 

 


÷ 
l 
2 
nu 
2 

è nl 

ø 
λ = wavelength [m]
Z = atomic number or number of protons in the nucleus R = Rydberg constant (1.096776×107 m1 for hydrogen) nl = the lower electron orbit number
nu = the upper electron orbit number
BRAGG'S LAW
Xray Scattering  Xrays 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
nl = 2d sin q 
d 

2d 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.

α 
φ 

θ 
α 






nl = D sin f 



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:
p2 = K = 3 kT
2m 2
lead to a momentumtemperature relation for particles:
p2 = 3mkT
p = momentum [kgm/s] m = particle mass [kg] K = kinetic energy [J]
k = Boltzmann's constant 1.380658×1023 J/K
T = temperature in Kelvin (273.15K = 0°C, DK = DC) (see page 5 for RELATIVISTIC KINETIC ENERGY)
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 9 of 22
WAVES
Ψ WAVE FUNCTIONS
Classical Wave Equation 
¶2 Y 
= 
1 ¶2 Y 

We did not use this equation: 
¶x2 
v2 

¶t 2 
This wave function fits the classical form, but is not a solution to the Schröedinger equation:
wave 
time 
wave 
angular 
phase 

number 
frequency 
constant 





Ψ(x,t) = Asin(kx  ωt + 
φ) 

distance 
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 time
Ψ (x,t) =
distance
amplitude
wave 
angular 
number 
frequency 
Ae i( kx ωt) 
= A[cos(kx  ωt) + i sin(kx  ωt)] 
distance time
The negative sign denotes wave motion in the positive x direction, assuming omega is positive.
k WAVE NUMBER
A component of a wave function 




k = 
2p 


representing the wave density relative to 






l 


distance, in units of radians per unit 























distance [rad/m]. 


































ω ANGULAR FREQUENCY 







A component of a wave function 












representing the wave density relative to 


w = 2p 


time (better known as frequency), in units 







T 



of radians per second [rad/s]. 

















































vph PHASE VELOCITY 











The velocity of a point on a wave, 
v 

= l 
= w 


e.g. the velocity of a wave peak 

ph 



T 



k 


[m/s]. 




































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].
Ψ SUM OF TWO WAVES
(see also WaveSummingExample.pdf)

Y 

+ Y 

= 2A cos æ 
Dk 
x  
Dw 
t 
öcos (k 

x  w t ) 

1 
2 


av 




ç 

2 

÷ 

av 






è 2 

ø 
1442443 






14424443 
internal wave 







envelope 





A = harmonic amplitude [various units?] 





k = difference in wave numbers k1  k2 [rad/m] 


kav = average wave number (k1 + k2)/2 
[rad/m] 



Δω = difference in angular 







frequencies ω1  ω2
[rad/s]
ωav = average angular frequency (ω1 + ω2)/2
[rad/s]
x = distance [m] t = time [s]
Phase Velocity:
vph = wav / kav [m/s] velocity of a point on a wave
Group Velocity:
[m/s] speed of the envelope
ugr = Dw/ Dk
λ de BROGLIE WAVELENGTH
De Broglie extended the concept of 
l = 
h 

waves to all matter. 


p 










λ
h = Planck's constant 6.6260755×1034 Js
p= momentum [kgm/s], convertible to [eV/c] by multiplying by c/q.
WAVE UNCERTAINTIES
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 (Δω large). Another result of this relationship, is that an electronic component must have a large bandwidth Δω in order for its signal to respond in a short time t.
Dk Dx = 2p DwDt = 2p
k = the range of wave numbers, see WAVE NUMBER x = the width of the wave envelope
Δω = the range of wave frequencies t = a time interval
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 10 of 22