Impact parameter: Difference between revisions
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The impact parameter is related to the [[scattering]] angle <math>\theta</math> by<ref>Landau L. D. and Lifshitz E. M. (1976) ''Mechanics'', 3rd. ed., Pergamon Press. {{ISBN|0-08-021022-8}} (hardcover) and {{ISBN|0-08-029141-4}} (softcover).</ref> |
The impact parameter is related to the [[scattering]] angle <math>\theta</math> by<ref>Landau L. D. and Lifshitz E. M. (1976) ''Mechanics'', 3rd. ed., Pergamon Press. {{ISBN|0-08-021022-8}} (hardcover) and {{ISBN|0-08-029141-4}} (softcover).</ref> |
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: <math>\theta = \pi - 2b\int_{r_\text{min}}^\infty \frac{dr}{r^2\sqrt{1 - (b/r)^2 - 2U/(mv_\infty^2)}},</math> |
: <math>\theta = \pi - 2b\int_{r_\text{min}}^\infty \frac{dr}{r^2\sqrt{1 - (b/r)^2 - 2U/(mv_\infty^2)}},</math> |
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where <math>v_\infty</math> is the velocity of the projectile when it is far from the center, and <math>r_\text{min}</math> is its closest distance from the center. |
where <math>v_\infty</math> is the velocity of the projectile when it is far from the center, and <math>r_\text{min}</math> is its closest distance from the center.<ref name=":0">{{Cite web|title=What is the impact parameter in a scattering exper class 12 physics CBSE|url=https://www.vedantu.com/question-answer/impact-parameter-in-a-scattering-exper-class-12-physics-cbse-5f4514a7ce20ca61ed671f01|access-date=2021-09-03|website=www.vedantu.com}}</ref><ref>{{Cite web|last=mitopercourseware|first=MIT|date=03/09/2021|title=Notes|url=https://ocw.mit.edu/courses/nuclear-engineering/22-105-electromagnetic-interactions-fall-2005/readings/chap6.pdf|url-status=live}}</ref> |
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==Scattering from a hard sphere== |
==Scattering from a hard sphere== |
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The simplest example illustrating the use of the impact parameter is in the case of scattering from a sphere. Here, the object that the projectile is approaching is a hard sphere with radius <math>R</math>. In the case of a hard sphere, <math>U(r) = 0</math> when <math>r > R</math>, and <math>U(r) = \infty</math> for <math> r \leq R </math>. When <math> b > R </math>, the projectile misses the hard sphere. We immediately see that <math>\theta = 0</math>. When <math>b \leq R</math>, we find that <math>b = R \cos\left(\frac{\theta}{2}\right)</math>. |
The simplest example illustrating the use of the impact parameter is in the case of scattering from a sphere. Here, the object that the projectile is approaching is a hard sphere with radius <math>R</math>. In the case of a hard sphere, <math>U(r) = 0</math> when <math>r > R</math>, and <math>U(r) = \infty</math> for <math> r \leq R </math>. When <math> b > R </math>, the projectile misses the hard sphere. We immediately see that <math>\theta = 0</math>. When <math>b \leq R</math>, we find that <math>b = R \cos\left(\frac{\theta}{2}\right)</math>.<ref>{{Cite web|title=Impact Parameter for Nuclear Scattering|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/impar.html|access-date=2021-09-03|website=hyperphysics.phy-astr.gsu.edu}}</ref> |
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== Collision centrality == |
== Collision centrality == |
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In [[Particle physics|high-energy nuclear physics]] — specifically, in [[Collider|colliding-beam experiments]] — collisions may be classified according to their impact parameter. Central collisions have <math>b \approx 0</math>, peripheral collisions have <math>0 < b < 2R</math>, and ultraperipheral collisions (UPCs) have <math>b > 2R</math>, where the colliding [[Atomic nucleus|nuclei]] are viewed as hard spheres with radius <math>R</math>. |
In [[Particle physics|high-energy nuclear physics]] — specifically, in [[Collider|colliding-beam experiments]] — collisions may be classified according to their impact parameter. Central collisions have <math>b \approx 0</math>, peripheral collisions have <math>0 < b < 2R</math>, and ultraperipheral collisions (UPCs) have <math>b > 2R</math>, where the colliding [[Atomic nucleus|nuclei]] are viewed as hard spheres with radius <math>R</math>.<ref name=":0" /> |
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Because the [[Strong interaction|color force]] has an extremely short range, it cannot couple quarks that are separated by much more than one nucleon's radius; hence, strong interactions are suppressed in peripheral and ultraperipheral collisions. This means that final-state particle multiplicity{{clarify|reason="multiplicity" has several different meanings, need to define what it means here|date=January 2021}} is typically greatest in the most central collisions, due to the [[Parton (particle physics)|partons]] involved having the greatest probability of interacting in some way. This has led to [[charged particle]] multiplicity being used as a common measure of collision centrality (charged particles are much easier to detect than uncharged particles). |
Because the [[Strong interaction|color force]] has an extremely short range, it cannot couple quarks that are separated by much more than one nucleon's radius; hence, strong interactions are suppressed in peripheral and ultraperipheral collisions. This means that final-state particle multiplicity{{clarify|reason="multiplicity" has several different meanings, need to define what it means here|date=January 2021}} is typically greatest in the most central collisions, due to the [[Parton (particle physics)|partons]] involved having the greatest probability of interacting in some way. This has led to [[charged particle]] multiplicity being used as a common measure of collision centrality (charged particles are much easier to detect than uncharged particles). |
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Revision as of 16:46, 3 September 2021
The impact parameter is defined as the perpendicular distance between the path of a projectile and the center of a potential field created by an object that the projectile is approaching (see diagram). It is often referred to in nuclear physics (see Rutherford scattering) and in classical mechanics.
The impact parameter is related to the scattering angle by[1]
where is the velocity of the projectile when it is far from the center, and is its closest distance from the center.[2][3]
Scattering from a hard sphere
The simplest example illustrating the use of the impact parameter is in the case of scattering from a sphere. Here, the object that the projectile is approaching is a hard sphere with radius . In the case of a hard sphere, when , and for . When , the projectile misses the hard sphere. We immediately see that . When , we find that .[4]
Collision centrality
 |
In high-energy nuclear physics — specifically, in colliding-beam experiments — collisions may be classified according to their impact parameter. Central collisions have , peripheral collisions have , and ultraperipheral collisions (UPCs) have , where the colliding nuclei are viewed as hard spheres with radius .[2]
Because the color force has an extremely short range, it cannot couple quarks that are separated by much more than one nucleon's radius; hence, strong interactions are suppressed in peripheral and ultraperipheral collisions. This means that final-state particle multiplicity[clarification needed] is typically greatest in the most central collisions, due to the partons involved having the greatest probability of interacting in some way. This has led to charged particle multiplicity being used as a common measure of collision centrality (charged particles are much easier to detect than uncharged particles).
Because strong interactions are effectively impossible in ultraperipheral collisions, they may be used to study electromagnetic interactions — i.e. photon–photon, photon–nucleon, or photon–nucleus interactions — with low background contamination. Because UPCs typically produce only two to four final-state particles, they are also relatively "clean" when compared to central collisions, which may produce hundreds of particles per event.
See also
References
- ^ Landau L. D. and Lifshitz E. M. (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).
- ^ a b "What is the impact parameter in a scattering exper class 12 physics CBSE". www.vedantu.com. Retrieved 2021-09-03.
- ^ mitopercourseware, MIT (03/09/2021). "Notes" (PDF).
{{cite web}}: Check date values in:|date=(help)CS1 maint: url-status (link) - ^ "Impact Parameter for Nuclear Scattering". hyperphysics.phy-astr.gsu.edu. Retrieved 2021-09-03.
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