density, flux & luminosity

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density is a consideration factor for scattering experiment. In low density, both for target and the beam, then the probability of collision will be small and experiment will be time consuming and uncertainly increase. Remember that the size of nuclear is 1000 times less then the atom. the cross section area of it will be 1000 x 1000 times lesser. the chance for a nucleus-nucleus collision is very small. for example, if there is only 1 particle in the area on 1 atom, the chance for hitting the nucleus is 1 / ( \pi 10^6) = 0.000003, 3 in 1 million. it is just more than nothing. thus, in order to have a hit, we have to send more then 3 million particles for 1 atom. in some case, the beam density is small, say, 0.3 million particles per second on an area of 1 atom. then we have to wait 10 second for 1 hit.

density is measured in particle per area for target .

for beam, since particle is moving in it, time is included in the unit. there are 2 units – flux and luminosity. flux is particle per second, and luminosity is energy per second per area. since energy of the beam is solely by the number of particle. so, density of beam is particle per second per area. but in particle physics, the energy of particle was stated. thus, the luminosity is equally understood as density of beam, and their units are the same as particle per second per area.

In daily life, density is measured by mass per volume. although the unit are different, they are the same thing – ” how dense is it? ”

in solid, the density is highest compare to other state of matter. from wiki, we can check the density. and coveted it in to the unit we want. for example, copper has density 8.94 g / cm^3 . its molar mass is 29 g / mole . thus, it has 0.31 N_A copper atom in 1 cm^3 . and N_A = 6.022 \times 10^{23} , which is a huge number, so, the number of atom on 1 cm^2 is 3.25 \times 10^{15} .

how about gas? the density depends on temperature and pressure, at 0^o C and 1 atm pressure, helium has density  1.79 \times 10^{-4} g / cm^3 and the molar mass is 2. thus, the number of Helium atom in 1 cm^2 is 1.42 \times 10^{13} . when the temperature go to -100 degree, the density will increase.

beside of the number of atom per area. we have to consider the thickness of the target. think about a target is a layer structure, each layer has certain number of atom per area. if the particle from the beams miss the 1st layer, there will be another layer and other chance for it to hit. thus. more the thickness, more chance to hit.

For a light beam, the power P and the wavelength \lambda determine the flux of photon. power is energy per second. and energy of single photon is inversely proportional to its wavelength. the density of a light beam is given by :

L = n/area = P \frac { \lambda} { h c} = P \lambda \times 5 \times 10^{15} [W^{-1}][nm^{-1}][s^{-1}][m^{-2}]

where, L is the luminosity and n is the flux. for typical green class 4 laser, which has power more then 0.5 W and wavelength is about 500nm. the flux is about  n=1.3 \times 10^18 [s^{-1}] photons per second per unit area. if the laser spot light is about 5 mm in diameter. thus, the density of the beam is L=1.7^{18} [s^{-1}][cm^2].

for laser pointer in office, which is class 1 laser. the power is less then0.4 mW, say, 0.1 mW. for same spot size, the luminosity  is still as high as 6.6^{14} [s^{1}][cm^2] .

on LHC, the beam flux can be 10^{34} [s^{-1}][cm^2] . by compare the the density of solid copper. it is much denser. thus, a collision in LHC is just like smashing 2 solid head to head and see what is going on.


Differential Cross Section

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In nuclear physics, cross section is a raw data from experiment. Or more precisely differential cross section, which is some angle of the cross section, coz we cannot measure every scatter angle and the differential cross section gives us more detail on how the scattering going on.

The differential cross section (d.s.c.) is the square of the scattering amplitude of the scatter spherical wave, which is the Fourier transform of the density.

d.s.c = |f(\theta)|^2 = Fourier ( \rho (r), \Delta p , r )

Where the angle θ come from the momentum change. So, sometime we will see the graph is plotted against momentum change instead of angle.

By measuring the yield of different angle. Yield is the intensity of scattered particle. We can plot a graph of the Form factor, and then find out the density of the nuclear or particle.

However, the density is not in usual meaning, it depends on what kind of particle we are using as detector. For example, if we use electron, which is carry elected charge, than it can feel the coulomb potential by the proton and it reflected on the “density”, so we can think it is kind of charge density.

Another cross section is the total cross section, which is sum over the d.s.c. in all angle. Thus, the plot always is against energy. This plot give us the spectrum of the particle, like excitation energy, different energy levels.