Boltzmann's constant k= 1.38066E-23 J/K
k= 0.00008617 eV/K
Atomic mass unit u= 1.66E-27 kg
umev= 931.5 MeV/c^2
Wien constant 2.90E-03
Planck's constant h= 4.14E-15 eV s
h= 6.63E-34 J s
h/ 2 pi hbar 1.05E-34 J s
h/ 2 pi hbev 6.59E-16 eV s
Speed of light c= 3.00E+08 m/s
hc= 1.24E+03 eV nm
hbarc= 1.97E+02 eV nm
Electron charge e= 1.60E-19 C
electron volt eV= 1.60E-19 joule
electron mass mel 9.11E-31 kg
mel 5.11E-01 MeV/c^2
proton mass mp 1.67E-27 kg
mpev 9.38E+02 MeV/c^2
mpg 0.93827231 GeV/c^2
neutron mass mn 1.67E-27 kg
mnev 9.40E+02 MeV/c^2
mng 0.93956563 GeV/c^2
Stefan's constant= sigma= 5.67E-08 watt/m^2K^4
Nuclear distance unit fermi 1.00E-15 m (a femtometer)
cross section unit barn= 1.00E-28 m^2
barn= 100 fm^2
common combo for scattering ke^2 1.44 MeV fm
First Bohr orbit radius a0 a0 0.0529 nm
Bohr magnetion mub 5.79E-05 eV/T
muba 9.27E-24 A m^2 A=ampere
#18 Maximum length of laser cavity for containment time 1E-8 sec
Reflectivity = 0.99 = a
Length = c*tc*ln(1/a) = 3E8*1E-8*LN(1/.99) = 0.030151008 m
#19 Laser pumping power requirement
Laser wavelength 550 nm, bandwidth 1E-8 nm, radiation lifetime =1E-8 s, spontaneous decay lifetime=1E-8s
From pg 64, eq 3.13, dN/(dw*dV) = 8 pi/w^4 where w=wavelength
The number of modes within the bandwidth of the laser = Nm = 8 pi dw dV/w^4 =
Nm= 8*pi*(1E-17 m)*(2.7E-5 m^3)/(550E-9)^4 = 74157.12186 modes
This assumes cubical cavity of dimension .03m as in prob 18 above.
The population difference is given by N2-N1 = Nm*tau/tc = 74157*1E-8/1E-8= 74157.12186
The pumping power is given by P = (N2-N1)*hc/(tau*w) = 74157*1240 eV nm*1.6E-19J/eV/(1E-8 s * 550 nm)
P = 2.67505E-06 watts
#21 Compare laser photon flux with that of high temperature thermal
Laser w= 500 nm, delta w/w= 1E-11, power 1 watt, area of mirror 1E-4 m^2 .
a. Find photon flux of laser at distance of 1 meter
To find area of beam at one meter, presume that it is diffraction limited with the mirror as
the effective aperture.
In Example 13-5 on page 364 Rohlf uses d=L*theta= L*w/dm where dm = mirror diameter
But if you are going to use the angle to the first diffraction minimum as the size of the radiation,
then you ought to at least use the first minimum of a circular aperture I think.
The first diffraction minimum is given by d sin theta = 1.22*n*w
so that with distance L>>diameter d, you get theta = w/dm = d/L so that the spot diameter
is given by d = 1.22Lw/dm.
The area of the spot at 1 meter will be proportional to (d^2/dm^2)*Area of source
So the area of the spot will be (pi*1.22*L*w/4)^2/Area of source
Area of spot = (pi*1.22*1*500E-9/4)^2/1E-4 = 2.2953E-09 m^2
d=1.22*L*w/dm= 5.40598E-05 m which is less than the source diameter!
dm=sqrt(4*Am/pi)= 0.011283792 m = radius of source
So using diffraction limited approach is a swindle since this situation does not
meet the conditions for Fraunhofer diffraction.
Maybe the idea was that you took this size source and focused it so that it was
within the diffraction limit - that would indeed be the limit of focusing.
If you went blindly ahead and used diffraction limiting anyway, then
the expression for the
flux would be
flux = Power of laser/((hc/w)*(pi d^2/4)) = P*w*Am/(hc*(pi*1.22*L*w/4)^2)
flux = P*Am/(hc*w*(pi*1.22*L/4)^2)
flux = 1 J/s*1E-4 m^2*1E9 nm/m/(1240 eV nm*500 nm*(pi*1.22*1 m/4)^2*1.6E-19 J/eV)
flux = 1.09797E+27 1/m^2 s
But at 1 meter with that size source, the diffraction is in fact a negligible
amount, so the beam can
be considered to be equal to the source in size if diffraction is the only cause of divergence.
So a more realistic flux would be
flux = Pw/hcAm = 1J/s*500nm/(1240 eV nm*1E-4 m^2*1.6E-19 J/eV)= 2.52016E+22 /m^2 s
b. Compare the flux of a thermal source under the condition kT=hc/w,
which is pretty hot at w=500nm
Requires temperature = 19,200 K
From the Planck radiation formula, Eq 3.32 dR/dw = 2pihc^2/(w^5(exp(hc/wkT)-1))
This radiation goes in all directions, so it is spread over 4piL^2 in area, or 4pi m^2 since L=1m
The radiation in the visible would be (dR/dw)(1E-4 m^2/4pi m^2)(w/hc)delta w
For the whole visible range delta w = 300 nm (from 400 to 700 nm).
The radiation then shakes down to
flux=3E8m/s*1E-4 m^2*300 nm/(2*500^4 nm^4)(exp(1)-1)= 4.19023E+22 /m^2 s
c. The photon flux in the bandwidth of the laser = 4.19E22*1E-11*500nm/300nm
flux in bandwidth = 6.98333E+11 m /m^2 s
Besides being several orders of magnitude lower in flux in the laser
and having a temperature of 19,200 K, it would be pretty hot to handle.
The radiated power is given by the Stefan Boltzmann law as
P=5.67E-8*10E-4 m^2*19200^4 = 7705271.992 watts
= 7.705271992 megawatts!