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## Physics 321

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**Physics 321**Hour 37 Collisions in Three Dimensions**Assume we know the masses, the initial momentum of each body**(they’re the same), and the scattering angle. Collision in the CM Frame Θ Θ**Assume we know any energy loss as well.**Collision in the CM Frame • We can find the kinetic energies is we know the momenta and the masses.**First, we find P.**Collision in the CM Frame • Then we find T1 and T2. • Now we know everything!**If Eex=0,**Elastic Scattering • All the scattering can do is change the directions of the particles.**The cm frame is really easy…**• Theorists (almost) always work in the cm frame. • Experimentalists usually convert cross sections to the cm as well. • But, experiments are not usually conducted in the cm frame! The CM Frame**Now let’s assume we have the CM solution and we want to**get back to the lab frame. • We know CM to Lab**So we can find the lab quantities**CM to Lab**But usually we don’t know Θa priori.**CM to Lab This is a quadratic equation we can solve for p1. Then we can find Θand everything else.**CollisionKinematics.nb**Example**Physics 321**Hour 38 Scattering Cross Sections**In the cm frame, two particles approach each other and**scatter. Given the interaction, what is the distribution of scattered particles? The Scattering Problem Θ Θ**Given the distribution of scattered particles, what is the**interaction? • This is a very hard but very interesting problem. The Inverse Scattering Problem Θ Θ**The impact need not be head-on. – The distance between the**trajectories is the impact parameter, b. The Scattering Problem Θ b Θ**The first thing we need to do is find the relationship**between θand b for a given type of interaction. Relating θ to b Θ b Θ**First we note that the total angular momentum is**Coulomb Scattering – b(θ)**Next we use a result from the central force problem.**Coulomb Scattering – b(θ)**For hyperbolic orbits**Coulomb Scattering – b(θ) φ**Using**Coulomb Scattering – b(θ) φ**Consider a small beam striking a large target. The detector**has solid angle ΔΩand detects all particles scattered. Differential Cross Section - Definition detector θ beam target**Consider a large beam striking a small target. A detector**has solid angle ΔΩ. Differential Cross Section - Definition detector beam θ target**We usually take 1 target particle…**Differential Cross Section - Theoretical detector beam θ target**Differential Cross Section - Theoretical**Find the total number of beam particles per second between This is the number in a ring of radius . This is:**Differential Cross Section - Theoretical**2) Find the total number of scattered particles between This is:**3) Think of the detector as subtending angles and . The**solid angle is defined by: or more generally (If the area of the detector is small, this can be expressed in terms of the detector’s area: .) Differential Cross Section - Theoretical**3) Think of the detector as subtending angles and . The**solid angle is defined by: (If the area of the detector is small, this can be expressed in terms of the detector’s area: .) This is: Differential Cross Section - Theoretical**4) The number of particles per second scattering into is:**The number of those hitting the detector is Differential Cross Section - Theoretical**The number of particles detected per second is**Differential Cross Section - Theoretical Or since we want a positive value, we usually write**The total cross section is the differential cross section**integrated over all angles. Total Cross Section**The only thing that changes in lab vs cm is .**CM to Lab Conversion