Physical constants

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
radiator.

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
wavelength window

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!