S- and P-Wave Cross Section Explained

In summary: So, in summary, the s-wave cross section refers to the scattering amplitude of an incoming plane wave with a linear combination of spherical waves, represented as a "Partial Wave expansion". The l-quantum number represents the classical angular momenta of the incoming particle with respect to the centre of the scattering potential and is quantized. The cross section is proportional to the scattering amplitude modulus square, with s-cross section having only l=0 and p-cross section having only l=1. This is analogous to how one particle appears to another, with a ball-like target particle having a classic 3D cross-section but an excited particle having a different appearance and vice versa.
  • #1
touqra
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What does it mean when they say s wave cross section, p wave cross section .. ?
 
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  • #2
touqra said:
What does it mean when they say s wave cross section, p wave cross section .. ?

the [itex]l[/itex]-quantum number of the incoming wave. You often have a plane wave coming into the target. The plane wave you write as a linear combination of spherical waves, and you call this "Partial Wave expansion".

http://farside.ph.utexas.edu/teaching/qm/lectures/node70.html

The [itex]l[/itex] that you see in eq 957 is then the "[itex]l[/itex] - QM number".
And [itex]l[/itex] is denoted by, 0 = s, 1 = p, 2 = d, etc, same as in atomic physics notation.

Semiclassicaly, you can see the [itex]l[/itex] as the classical angular momenta of the incoming particle with respect to the centre of the scattering potential. And also the [itex]l[/itex] is quantisized, so only some values of [itex]l[/itex] are allowed.

Now since the sum goes to infinity in eq 957, we cut of where we expect no partial waves to contribute. And that is often assigned by [tex] l_{max} \approx R\cdot k [/tex]
Where R is the range of the potential and k is the momenta of the incoming particle (wave number).

Now the cross section is proportional to the scattering amplitude modulus square, i.e the modulus square of eq. 965 times a constant with a lot of pi's hbar's etc.

So the s-cross section, you only have [itex]l[/itex] = 0 in you sum, and p-cross section only [itex]l[/itex] = 1. etc.

I hope you got the idea =)
 
  • #3
Hello,
To munch the QM into an analogy:
Another way to look at is - how does one particle look to another. If you assume the target particle to be ball-like, then in its basic form (ground state), you'll get the classic 3D cross-section - this is how it will appear to the incoming particle and such it will be scattered from the target particle. But if the particle is excited to a higher state, then it will no longer appear as a ball but something else entirely. And vice versa.
Smoochie
 

1. What are S- and P-waves?

S-waves (secondary waves) and P-waves (primary waves) are two types of seismic waves that are generated during an earthquake. S-waves are transverse waves that travel slower than P-waves and can only travel through solid materials. P-waves are longitudinal waves that can travel through both solid and liquid materials.

2. How do S- and P-waves differ in terms of their motion?

S-waves move perpendicular to the direction of wave propagation, while P-waves move parallel to the direction of wave propagation. This means that S-waves cause particles in the ground to move up and down or side to side, while P-waves cause particles to move back and forth.

3. How are S- and P-waves detected?

S- and P-waves are detected using seismometers, which are instruments that measure ground motion. Seismometers can detect the arrival time and amplitude of both S- and P-waves, and this data can be used to determine the location and magnitude of an earthquake.

4. What is the difference between S- and P-wave cross sections?

S-wave cross sections show the amplitude of S-waves as they travel through the Earth's interior, while P-wave cross sections show the amplitude of P-waves. These cross sections can help scientists understand the structure and composition of the Earth's interior.

5. How do S- and P-waves help us understand earthquakes?

By studying S- and P-waves, scientists can learn more about the properties of the Earth's interior, such as the density and elasticity of different layers. This information can then be used to better understand how earthquakes occur and how they affect different areas. S- and P-waves can also help with earthquake prediction and hazard assessment.

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