Most recent edit on 2008-08-13 03:52:10 by CharlesFrancis

Additions:

← Scattering ↑ →

Deletions:

← Scattering →

Edited on 2008-05-26 10:28:26 by CharlesFrancis

Additions:

← Scattering →

Consider a target, A, consisting of a number of particles at rest, number density ρ_A particles per unit volume, and depth l_A. A beam, B, of particles with momentum q and number density ρ_B particles per unit volume is aimed at the target. In time t a beam length l_B = vt passes through the target. In real beams the number density of particles is generally larger at the centre than at the edges. The total number of scattering events per unit time, N_process, for a particular process is expected to be proportional to l_A, v, ρ_A, ρ_B, integrated over the area in which the beam intersects the target,
Scattering ↑ Regularisation and Renormalisation →

Deletions:

← Scattering →

Consider a target, A, consisting of a number of particles at rest, number density ρ_A particles per unit volume, and depth l_A. A beam, B, of particles with momentum q and number density ρ_B particles per unit volume is aimed at the target. In time t a beam length l_B =vt passes through the target. In real beams the number density of particles is generally larger at the centre than at the edges. The total number of scattering events per unit time, N_process, for a particular process is expected to be proportional to l_A, v, ρ_A, ρ_B, integrated over the area in which the beam intersects the target,
Scattering ↑ Relational Quantum Gravity →

Edited on 2008-05-05 01:33:38 by CharlesFrancis

Additions:
Scattering-50N

The leading order M-matrix element for scattering of two distinguishable particles with initial momenta q and q' and final momenta p and p' is

Deletions:
Scattering-50N

The leading order M-matrix element for scattering of two distinguishable particles with initial momenta q and q' and final momenta p and p' is

Edited on 2008-05-05 01:32:32 by CharlesFrancis

Additions:
Feynman-1

Consider a target, A, consisting of a number of particles at rest, number density ρ_A particles per unit volume, and depth l_A. A beam, B, of particles with momentum q and number density ρ_B particles per unit volume is aimed at the target. In time t a beam length l_B =vt passes through the target. In real beams the number density of particles is generally larger at the centre than at the edges. The total number of scattering events per unit time, N_process, for a particular process is expected to be proportional to l_A, v, ρ_A, ρ_B, integrated over the area in which the beam intersects the target,
Scattering-7

For scattering in the centre of mass frame for two particles with masses m and m', let the 4-momenta in the final state be p = (E, p) and p' = (E', p'), where the final energies are Scattering-43

and

. Integrate the differential cross section over p'. This enforces conservation of 3-momentum and sets p' = −p, giving
Scattering-50N

The leading order M-matrix element for scattering of two distinguishable particles with initial momenta q and q' and final momenta p and p' is

Deletions:
Feynman-1

Consider a target, A, consisting of a number of particles at rest, number density ρ_A particles per unit volume, and depth l_A. A beam, B, of particles with momentum q and number density ρ_B particles per unit volume is aimed at the target. In time t a beam length l_B =vt passes through the target. In real beams the number density of particles is generally larger at the centre than at the edges. The total number of scattering events per unit time, N_process, for a particular process is expected to be proportional to l_A, v, ρ_A, ρ_B, integrated over the area in which the beam intersects the target,
Scattering-7

For scattering in the centre of mass frame for two particles with masses m and m', let the 4-momenta in the final state be p = (E, p) and p' = (E', p'), where the final energies are Scattering-43

and

. Integrate the differential cross section over p'. This enforces conservation of 3-momentum and sets p' = −p, giving
Scattering-50

The leading order M-matrix element for scattering of two distinguishable particles with initial momenta q and q' and final momenta p and p' is

Edited on 2008-03-11 11:10:47 by CharlesFrancis

Additions:

← Scattering →

A principle application of quantum theory, and of quantum electrodynamics in particular, is in the interpretation of the results of scattering experiments. I describe the scattering cross section, show how it may be calculated from Feynman rules. The prototypical scattering experiment, on which much research into the properties of elementary particles has been based, was the Geiger-Marsden gold foil experiment» carried out under the supervision of Ernest Rutherford». I obtain the Rutherford formula for non-relativistic Coulomb scattering, verified by the experiment, and from which we deduce the structure of the atom containing a dense charged nucleus.

Definition: The scattering cross section, σ, for a particular process is the number, N_process of events of the process per unit time, divided by the number, l_BAρ_B, of particles in the target, divided by the flux, vρ_A, of particles in the beam,
The beam is taken to be uniform across the region containing the target particle. So, the incident wave packets may be considered identical up to the displacement, or impact parameter, b = (b², b³), transverse to beam direction. The ket describing a particle in the beam is
Scattering ↑ Relational Quantum Gravity →

Deletions:

← Scattering →

Definition: The scattering cross section^», σ, for a particular process is the number of events of the process per unit time, divided by the number, l_BAρ_B of particles in the target and the flux, vρ_A of particles in the beam
The beam is taken to be uniform across the region containing the target particle, so the incident wave packets may be considered identical up to the displacement, or impact parameter, b = (b², b³), transverse to beam direction. The ket describing a particle in the beam is
Scattering ↑ The Coulomb Force →

Edited on 2008-02-23 03:43:55 by CharlesFrancis

Additions:
Scattering-71

Deletions:
Scattering-71

Edited on 2008-02-23 03:40:35 by CharlesFrancis

Additions:
where σ is a constant of proportionality, called the scattering cross section».

The leading order M-matrix element for scattering of two distinguishable particles with initial momenta q and q' and final momenta p and p' is
The fine structure constant is defined to be dimensionless. If e is measured in units of charge, the right hand side becomes dimensionless on division by Scattering-70

Definition: The fine structure constant is

Deletions:
where σ is a constant of proportionality, called the scattering section».

Definition: The scattering cross section^», σ, for a particular process is the number of events of the process per unit time, divided by the number, l_BAρ_{B of particles in the target, divided by the flux, vρ_{A of particles directed at the target}}
The beam is uniform across the region containing the target particle, so the incident wave packets may be considered identical up to the displacement, or impact parameter, b = (b², b³), transverse to beam direction. The ket describing a particle in the beam is
Scattering-50

The leading order M-matrix element for scattering of two distinguishable particles with intial momenta q and q' and final momenta p and p' is
The fine structure constant is defined to be dimensionless. If e is measured in units of charge, the right hand side becomes dimensionless on division by Scattering-70

Edited on 2008-02-23 03:16:05 by CharlesFrancis

Additions:
The range of the interaction is small compared to the beam width. Number density may be taken to be slowly varying across the beam. If beam density is treated as constant, or if the above formula is used to compute an effective beam area, A, then the number of scattering events per unit time is simply the product of the cross section with the total number, N_A = Al_Aρ_A, of particles in the target and the flux», vρ_B, of particles per unit time per unit area in the beam,

Deletions:
.
The range of the interaction is small compared to the beam width and number density may be taken to be slowly varying. If beam density is treated as constant, or if this formula is used to compute an effective beam area, A, then the number of scattering events per unit time is simply the product of the cross section with the total number, N_A = Al_Aρ_A, of particles in the target and the flux», vρ_B, of particles per unit time per unit area in the beam,

Definition: The scattering cross section^», σ, for a particular process is the number of events of the process per unit time, divided by the number, lBAρB of particles in the target, divided by the flux, vρA of particles directed at the target

Edited on 2008-02-23 03:11:34 by CharlesFrancis

Additions:
large double black diamond

Consider a target, A, consisting of a number of particles at rest, number density ρ_A particles per unit volume, and depth l_A. A beam, B, of particles with momentum q and number density ρ_B particles per unit volume is aimed at the target. In time t a beam length l_B =vt passes through the target. In real beams the number density of particles is generally larger at the centre than at the edges. The total number of scattering events per unit time, N_process, for a particular process is expected to be proportional to l_A, v, ρ_A, ρ_B, integrated over the area in which the beam intersects the target,

where σ is a constant of proportionality, called the scattering section».
.
The range of the interaction is small compared to the beam width and number density may be taken to be slowly varying. If beam density is treated as constant, or if this formula is used to compute an effective beam area, A, then the number of scattering events per unit time is simply the product of the cross section with the total number, N_A = Al_Aρ_A, of particles in the target and the flux», vρ_B, of particles per unit time per unit area in the beam,

Clearly the cross section has units of area. It corresponds to the cross sectional area of that part of a beam of classical particles of finite size which would collide with a single classical particle, also of finite size, in the target, on the assumption that scattering is due only to direct collision.

where the fine structure constant» in natural units ( Scattering-68

) is
The fine structure constant is defined to be dimensionless. If e is measured in units of charge, the right hand side becomes dimensionless on division by Scattering-70

,
Scattering ↑ The Coulomb Force →

Deletions:
large black diamond

Consider a target, A, consisting of a number of particles at rest, number density ρ_A particles per unit volume, and depth l_A. A beam, B, of particles with momentum q and number density ρ_B particles per unit volume is aimed at the target. In time t a beam length l_B =vt passes through the target. The total number of scattering events per unit time, N_process, for a particular process is expected to be proportional to l_A, v, ρ_A, ρ_B, and to the area, A, in which the beam intersects the target.

Definition: The scattering cross section^», σ, for a particular process is

Clearly the scattering cross section has units of area. It corresponds to the cross sectional area of that part of a beam of classical particles of finite size which would collide with a single classical particle, also of finite size, in the target, on the assumption that scattering is due only to direct collision.

In real beams the number density of particles is generally larger at the centre than at the edges. To compute the event rate in a real accelerator, it is necessary to integrate over the beam area,

The range of the interaction is small compared to the beam width and number density may be taken to be slowly varying. If beam density is treated as constant, or if this formula is used to compute an effective beam area, A, then the number of scattering events per unit time is simply the product of the cross section with the total number, N_A = Al_Aρ_A, of particles in the target and the number per unit time, Avρ_B, in the beam, divided by the effective beam area, A,

where the fine structure constant» in natural units ( Scattering-68

) is
The fine structure constant is defined to be dimensionless. If e is measured in units of charge, the right hand side becomes dimensionless on division by Scattering-70

,
Scatterig ↑ The Coulomb Force →

Edited on 2008-02-22 09:57:01 by CharlesFrancis

Additions:
large black diamond

The number of events per unit time is the number of particles in the target multiplied by the flux, or rate of particles passing through the target, vρ_B, multiplied by the probability, P(b), of scattering, given impact parameter b, integrated over b,
Then the differential cross section for scattering to Scattering-18

may be written
where it is implicit that this quantity is integrated over the resolution, dp_f = d³p₁ … d³p_n, of the detectors. There is a possibility that there will be no interaction. So, the S-matrix has the form,
The initial states are are approximately states of pure momentum. So, we may put k_A = q_A and k_B = q_B in the M-matrix and in the delta function (since the momentum of the initial state is far more precisely determined in practice than that in the final state). Since the initial states are normalised to unity,
Scattering-42

For scattering in the centre of mass frame for two particles with masses m and m', let the 4-momenta in the final state be p = (E, p) and p' = (E', p'), where the final energies are Scattering-43

and

. Integrate the differential cross section over p'. This enforces conservation of 3-momentum and sets p' = −p, giving
where E_cm is the total initial energy in the centre of mass frame. Integrate over the magnitude of final momentum to find the standard form of the differential cross section in the chosen normalisation,
where the magnitude of final momentum, |p|, and the final energies, E and E', are determined from energy momentum conservation. Simplify, using E_cm = E + E',
For non-relativistic energies we have p ≈ (m, p), q ≈ (m, q) , so that (p − q)² ≈ −|p − q|², and u is well approximated by a two spinor,
In the Geiger-Marsden experiment» electrons were scattered by a stationary gold foil target, v' = 0. Velocities were non-relativistic so that, from conservation of energy, |p| = |q| ≈ mv. The mass of a gold nucleus is large compared to that of an electron, so that, E' ≈ E_cm, E ≈ m. For scattering angle θ ,

Deletions:
large black diamond

The number of events per unit time is the number of particles in the target, multiplied by the flux, or rate of particles passing through the target, vρ_B, multiplied by the probability, P(b) of scattering, given impact parameter b, integrated over b,
Then the differential cross section for scattering to Scattering-18

The initial states are are approximately states of pure momentum. So, we may put k_A = q_A and k_{B = q_B} in the M-matrix and in the delta function (since the momentum of the initial state is far more precisely determined in practice than that in the final state). Since the initial states are normalised to unity,
Scattering-42

For scattering in the centre of mass frame for two particles with masses m and m', let the 4-momenta in the final state be p = (E, p) and p' = (E', p'), where the final energies are Scattering-43

, and

. Integrate the differential cross section over p'. This enforces conservation of 3-momentum and sets p' = −p, giving
where E_cm is the total initial energy in the centre of mass frame. Integrate over the magnitude of final momentum to find the standard form of the differential cross section in the chosen normalisation
where the magnitude of final momentum, |p|, and the final energies, E and E', are determined from energy momentum conservation. Simplify, using E_{cm = E + E'},
For non-relativistic energies we have p ≈ (m, p, q ≈ (m, q , so that (p − q)² ≈ −|p − q|², and u is well approximated by a two spinor,
In the Geiger-Marsden experiment» electrons were scattered by a stationary gold foil target, v' = 0. Velocities were non-relativistic so that, from conservation of energy, |p| = |q| ≈ mv. The mass of a gold nucleus is large compared to that of an electron, so that, E' ≈ E_cm , E ≈ m. For scattering angle θ

Edited on 2008-02-22 09:35:48 by CharlesFrancis

Additions:
The range of the interaction is small compared to the beam width and number density may be taken to be slowly varying. If beam density is treated as constant, or if this formula is used to compute an effective beam area, A, then the number of scattering events per unit time is simply the product of the cross section with the total number, N_A = Al_Aρ_A, of particles in the target and the number per unit time, Avρ_B, in the beam, divided by the effective beam area, A,
Since the number density, ρ_B, of particles in the beam is taken as constant, we conclude that the cross section is
where it is implicit that this quantity is integrated over the resolution, dp_f = d³p₁ … d³p_n, of the detectors. There is a possibility that there will be no interaction, so the S-matrix has the form,

Deletions:
The range of the interaction is small compared to the beam width and number density may be taken to be slowly varying. If beam densities are treated as constant, or if this formula is used to compute an effective beam area, A, then the number of scattering events per unit time is simply the product of the cross section with the total number, N_A = Al_Aρ_A, of particles in the target and the number per unit time, Avρ_B, in the beam, divided by the effective beam area A,
Since the number density,ρ_B is taken as constant, we conclude that the cross section is
where it is implicit that this quantity is integrated over the resolution, dp_f = d³p₁ … d³p_n of the detectors. There is a possibility that there will be no interaction, so the "S""-matrix has the form,

Edited on 2008-02-22 09:29:41 by CharlesFrancis

Additions:
The number of events per unit time is the number of particles in the target, multiplied by the flux, or rate of particles passing through the target, vρ_B, multiplied by the probability, P(b) of scattering, given impact parameter b, integrated over b,
Assuming we are not interested in the trivial case when no interaction takes place, we may ignore the 1 in this expression and consider only the T-matrix. T (and S) contain a factor of an energy-momentum conserving delta function. Extracting this factor gives the M-matrix, defined by
Scattering-50

The leading order M-matrix element for scattering of two distinguishable particles with intial momenta q and q' and final momenta p and p' is

Deletions:
The number of events per unit time is the number of particles in the target, multiplied by the flux, or rate of particles passing through the target, vρ_B, multiplied by the probability, P(b) of scattering, given impact parameter b, integrated over b, Assuming we are not interested in the trivial case when no interaction takes place, we may ignore the 1 in this expression and consider only the T-matrix. T (and S) contain a factor of an energy-momentum conserving delta function. Extracting this factor gives the M-matrix, defined by
Scattering-50

The leading order M-matrix element for scattering of two distinguishable particles with intial momenta q and q' and final momenta p and p'"" is

Edited on 2008-02-22 09:25:19 by CharlesFrancis

Additions:

Definition: The scattering cross section^», σ, for a particular process is
The range of the interaction is small compared to the beam width and number density may be taken to be slowly varying. If beam densities are treated as constant, or if this formula is used to compute an effective beam area, A, then the number of scattering events per unit time is simply the product of the cross section with the total number, N_A = Al_Aρ_A, of particles in the target and the number per unit time, Avρ_B, in the beam, divided by the effective beam area A,

Cross Section and Probability

For definitenes I will designate the beam direction 1, and the two transverse directions 2 and 3. Particles in the beam and the target are assumed to described by wave packets with approximate momenta in the 1-direction q_B and q_A, respectively (the target may be, but is not necessarily, stationary. For example, it may be a second beam with opposite momentum). The beam may be considered as a collection of particles, described by wave packets with momentum centred on beam momentum. The initial state of a particle in the target is described by a ket,
The beam is uniform across the region containing the target particle, so the incident wave packets may be considered identical up to the displacement, or impact parameter, b = (b², b³), transverse to beam direction. The ket describing a particle in the beam is
Without loss of generality, the initial states of beam and target particles are normalised to unity,

Cross Section and the Matrix Element

Deletions:

Definition: The scattering cross section, σ, for a particular process is The range of the interaction is small compared to the beam width and number density may be taken to be slowly varying. If beam densities are treated as constant, or if this formula is used to compute an effective beam area, A, then the number of scattering events per unit time is simply the product of the cross section with the total number, N_A = Al_Ar_A, of particles in the target and the number per unit time, Avρ_B, in the beam, divided by the effective beam area A,
Cross-Section and Probability
Scattering-7

For definitenes I will designate the beam direction 1, and the two transverse directions 2 and 3. Particles in the beam and the target are assumed to described by wave packets with approximate momenta in the 1-direction q_B and qA, respectively (the target may be, but is not necessarily, stationary. For example, it may be a second beam with opposite momentum). The initial state of a particle in the target is described by a ket,
The beam may be considered as a collection of particles, described by wave packets with momentum centred on beam momentum. The beam is uniform across the region containing the targe particle, so the incident wave packets may be considered identical up to the displacement, or impact parameter, ""b = (b2, b3), transverse to beam direction. The ket describing a particle in the beam is
The kets for the target and the beam are normalised to unity.
Cross Section and the Matrix Element

Edited on 2008-02-22 09:07:36 by CharlesFrancis

Additions:

Definition: The scattering cross section, σ, for a particular process is
is the cross section for scattering at angle θ per unit of solid angle.

Deletions:

Definition: The scattering cross section σ for a particular process is
is the cross section for scattering at angle θ per unit of solid angle.""

Edited on 2008-02-22 09:05:08 by CharlesFrancis

Additions:
Feynman-1

Consider a target, A, consisting of a number of particles at rest, number density ρ_A particles per unit volume, and depth l_A. A beam, B, of particles with momentum q and number density ρ_B particles per unit volume is aimed at the target. In time t a beam length l_B =vt passes through the target. The total number of scattering events per unit time, N_process, for a particular process is expected to be proportional to l_A, v, ρ_A, ρ_B, and to the area, A, in which the beam intersects the target.

Definition: The scattering cross section σ for a particular process is

Deletions:
Feynman-1

Definition: The scattering cross section» σ for a particular process is

Edited on 2008-02-22 08:58:36 by CharlesFrancis

Additions:
A principle application of quantum theory, and of quantum electrodynamics in particular, is in the interpretation of the results of scattering experiments. I describe the scattering cross section, show how it may be calculated from Feynman rules. The prototypical scattering experiment, on which much research into the properties of elementary particles has been based, was the Geiger-Marsden gold foil experiment» carried out under the supervision of Ernest Rutherford». I obtain the Rutherford formula for non-relativistic Coulomb scattering, verified by the experiment, and from which we deduce the structure of the atom containing a dense charged nucleus.
Feynman-1

Consider a target, A, consisting of a number of particles at rest, number density ρ_A particles per unit volume, and depth l_A. A beam, B, of particles with momentum q and number density ρB particles per unit volume is aimed at the target. In time t a beam length lB =vt passes through the target. The total number of scattering events per unit time, Nprocess, for a particular process is expected to be proportional to lA, v, ρA, ρB, and to the area, A"", in which the beam intersects the target.
Scatterig ↑ The Coulomb Force →

Deletions:
A principle application of quantum theory, and of quantum electrodynamics in particular, is in the interpretation of the results of scattering experiments. I describe the scattering cross section, show how it may be calculated from Feynman rules. The prototypical scattering experiment on which much research into the properties of elementary particles has been based was the Geiger-Marsden gold foil experiment» carried out under the supervision of Ernest Rutherford». I obtain the Rutherford formula for non-relativistic Coulomb scattering, verified by the experiment, and from which we deduce the structure of the atom containing a dense charged nucleus.
Feynman-1

Oldest known version of this page was edited on 2008-02-22 08:54:24 by CharlesFrancis []

Page view:

← Scattering →

A principle application of quantum theory, and of quantum electrodynamics in particular, is in the interpretation of the results of scattering experiments. I describe the scattering cross section, show how it may be calculated from Feynman rules. The prototypical scattering experiment on which much research into the properties of elementary particles has been based was the Geiger-Marsden gold foil experiment» carried out under the supervision of Ernest Rutherford». I obtain the Rutherford formula for non-relativistic Coulomb scattering, verified by the experiment, and from which we deduce the structure of the atom containing a dense charged nucleus.

The Scattering Cross Section

Consider a target, A, consisting of a number of particles at rest, number density ρ_A particles per unit volume, and depth l_{A. A beam,}B, of particles with momentum q and number density ρ_B particles per unit volume is aimed at the target. In time t a beam length lB =vt passes through the target. The total number of scattering events per unit time, Nprocess, for a particular process is expected to be proportional to lA, v, ρA, ρB, and to the area, A, in which the beam intersects the target. </en.wikipedia.org/wiki/Cross_section_%28physics%29 scattering cross section]] σ for a particular process is <img title="The scattering cross section" alt="Scattering-3" src="images/scattering/Scattering-3g.gif" align="texttop" vspace="3"> << Clearly the scattering cross section has units of area. It corresponds to the cross sectional area of that part of a beam of classical particles of finite size which would collide with a single classical particle, also of finite size, in the target, on the assumption that scattering is due only to direct collision. We are particularly interested in the dependency on the cross section on the scattering angle. <Definition: The differential cross section, <img title="The differential cross section" alt="Scattering-4" src="images/scattering/Scattering-4g.gif" align="texttop" vspace="3"> is the cross section for scattering at angle θ per unit of solid angle.

In real beams the number density of particles is generally larger at the centre than at the edges. To compute the event rate in a real accelerator, it is necessary to integrate over the beam area,

The range of the interaction is small compared to the beam width and number density may be taken to be slowly varying. If beam densities are treated as constant, or if this formula is used to compute an effective beam area, A, then the number of scattering events per unit time is simply the product of the cross section with the total number, N_A = Al_Ar_A, of particles in the target and the number per unit time, Avρ_B, in the beam, divided by the effective beam area A,

Cross-Section and Probability
Scattering-7

The beam may be considered as a collection of particles, described by wave packets with momentum centred on beam momentum. The beam is uniform across the region containing the targe particle, so the incident wave packets may be considered identical up to the displacement, or impact parameter, b = (b², b³), transverse to beam direction. The ket describing a particle in the beam is <img title="The initial state of a beam particle" alt="Scattering-10" src="images/scattering/Scattering-10.gif" align="texttop" vspace="3"> The kets for the target and the beam are normalised to unity. <img title="Normalisation of initial states" alt="Scattering-11" src="images/scattering/Scattering-11.gif" align="texttop" vspace="3"> The number of events per unit time is the number of particles in the target, multiplied by the flux, or rate of particles passing through the target, vρB, multiplied by the probability, P(b) of scattering, given impact parameter b, integrated over b,

Since the number density,ρ_B is taken as constant, we conclude that the cross section is

Cross Section and the Matrix Element
The detectors» for final-state particles typically measure momentum (position is not resolved at the level of the de Broglie wavelength»). It will be sufficient to treat final states as eigenstates of momentum,

The total probability for scattering is the integral of the probability density with respect to the momenta of particles in the final state,

where the probability density in momentum space is

Then the differential cross section for scattering to Scattering-18

may be written

where it is implicit that this quantity is integrated over the resolution, dp_f = d³p₁ … d³p_n of the detectors. There is a possibility that there will be no interaction, so the "S-matrix has the form, <img title="The T-matrix defined from the S-matrix" alt="Scattering-20" src="images/scattering/Scattering-20.gif" align="texttop" vspace="3"> Assuming we are not interested in the trivial case when no interaction takes place, we may ignore the 1 in this expression and consider only the T-matrix. T (and S) contain a factor of an energy-momentum conserving delta function. Extracting this factor gives the M-matrix, defined by

expanding

and

in the integrand for the differential cross section,

The integration over b gives Scattering-27

. Together with the four delta functions in the factor

we can carry out all six integrations over Scattering-29

. Since

, only the integration in the 1-direction is non-trivial,

where the generalised scaling property of the delta function has been used, and v_A and vB are the velocities of target and beam in the laboratory frame. In the last line, we have Scattering-34

and

. In addition we have, for i = 2, 3, Scattering-36

and

, and two equations from the mass shell condition. So Scattering-38

and

. With these results, the differential cross section is

Two Particle Scattering

For scattering in the centre of mass frame for two particles with masses m and m', let the 4-momenta in the final state be p = (E, p) and p' = (E', p'), where the final energies are Scattering-43

, and

. Integrate the differential cross section over p'. This enforces conservation of 3-momentum and sets p' = −p, giving

where E_cm is the total initial energy in the centre of mass frame. Integrate over the magnitude of final momentum to find the standard form of the differential cross section in the chosen normalisation

From the generalised scaling property of the delta function,

where the magnitude of final momentum, |p|, and the final energies, E and E', are determined from energy momentum conservation. Simplify, using E_{cm = E + E'},

Rutherford Scattering

The leading order M-matrix element for scattering of two distinguishable particles with intial momenta q and q' and final momenta p and p' is <img title="the matrix element for two particle scattering" alt="Scattering-51" src="images/scattering/Scattering-51.gif" align="texttop" vspace="3"> Applying <a href=http://www.teleconnection.info/rqg/FeynmanDiagrams#FeynmanRules>Feynman» rules</a>, <img title="applying Feynman rules" alt="Scattering-52" src="images/scattering/Scattering-52.gif" align="texttop" vspace="3"> For non-relativistic energies we have p ≈ (m, p, q ≈ (m, q , so that (p − q)2 ≈ −|p − q|2, and u is well approximated by a two spinor, <img title="non-relativistic spinor" alt="Scattering-56" src="images/scattering/Scattering-56.gif" align="texttop" vspace="3"> We need consider only the first two components. <img title="drop second two spin components" alt="Scattering-57" src="images/scattering/Scattering-57.gif" align="texttop" vspace="3"> and, for a ≠ 0, <img title="drop second two spin components" alt="Scattering-59" src="images/scattering/Scattering-59.gif" align="texttop" vspace="3"> In other words spin is conserved for non-relativistic energies. The matrix element reduces to <img title="simplify the matrix element" alt="Scattering-60" src="images/scattering/Scattering-60.gif" align="texttop" vspace="3"> Substituting this into the differential cross section, <img title="substitute in differential cross section" alt="Scattering-61" src="images/scattering/Scattering-61.gif" align="texttop" vspace="3"> In the [[http://en.wikipedia.org/wiki/Geiger-Marsden_experiment Geiger-Marsden experiment]] electrons were scattered by a stationary gold foil target, v' = 0. Velocities were non-relativistic so that, from conservation of energy, |p| = |q| ≈ mv. The mass of a gold nucleus is large compared to that of an electron, so that, E' ≈ Ecm , E ≈ m. For scattering angle θ <img title="the change in magnitude of momentum in terms of the scattering angle" alt="Scattering-66" src="images/scattering/Scattering-66.gif" align="texttop" vspace="3"> Thus, the cross section reduces to the [[http://en.wikipedia.org/wiki/Rutherford_scattering Rutherford formula]] for scattering in a Coulomb potential, <img title="the Rutherford scattering formula" alt="Scattering-67" src="images/scattering/Scattering-67.gif" align="texttop" vspace="3"> where the [[http://en.wikipedia.org/wiki/Fine_structure_constant fine structure constant]] in natural units (<img title="natural units" alt="Scattering-68" src="images/scattering/Scattering-68.gif" align="texttop" vspace="3">) is <img title="the fine structure constant in natural units" alt="Scattering-69" src="images/scattering/Scattering-69.gif" align="texttop" vspace="3"> The fine structure constant is defined to be dimensionless. If e is measured in units of charge, the right hand side becomes dimensionless on division by <img title="conversion factor to conventional natural units" alt="Scattering-70" src="images/scattering/Scattering-70.gif" align="texttop" vspace="3">, <img title="the fine structure constant in conventional units" alt="Scattering-71" src="images/scattering/Scattering-71.gif" align="texttop" vspace="3">""

Scattering ↑ The Coulomb Force →

RQG : Scattering

← Scattering ↑ →

← Scattering →

← Scattering →

← Scattering →

← Scattering →

← Scattering →

Cross Section and Probability

Cross Section and the Matrix Element

← Scattering →

The Scattering Cross Section

Two Particle Scattering

Rutherford Scattering