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Calculating the mass of a pion

Taking the range of the strong nuclear force to be about 1 fermi ( 10 15 m size 12{"10" rSup { size 8{ - "15"} } m} {} ), calculate the approximate mass of the pion carrying the force, assuming it moves at nearly the speed of light.


The calculation is approximate because of the assumptions made about the range of the force and the speed of the pion, but also because a more accurate calculation would require the sophisticated mathematics of quantum mechanics. Here, we use the Heisenberg uncertainty principle in the simple form stated above, as developed in Probability: The Heisenberg Uncertainty Principle . First, we must calculate the time Δ t size 12{Δt} {} that the pion exists, given that the distance it travels at nearly the speed of light is about 1 fermi. Then, the Heisenberg uncertainty principle can be solved for the energy Δ E size 12{ΔE} {} , and from that the mass of the pion can be determined. We will use the units of MeV / c 2 size 12{"MeV"/c rSup { size 8{2} } } {} for mass, which are convenient since we are often considering converting mass to energy and vice versa.


The distance the pion travels is d c Δ t , and so the time during which it exists is approximately

Δ t d c = 10 15 m 3 . 0 × 10 8 m/s 3.3 × 10 24 s. alignl { stack { size 12{Δt approx { {d} over {c} } = { {"10" rSup { size 8{ - "15"} } `m} over {3 "." 0 times "10" rSup { size 8{8} } `"m/s"} } } {} #" " approx 3 "." 3 times "10" rSup { size 8{ - "24"} } `s "." {} } } {}

Now, solving the Heisenberg uncertainty principle for Δ E size 12{ΔE} {} gives

Δ E h 4 π Δ t 6 . 63 × 10 34 J s 3 . 3 × 10 24 s size 12{ΔE approx { {h} over {4πΔt} } approx { {6 "." "63" times "10" rSup { size 8{ - "34"} } `J cdot s} over {4π left (3 "." 3 times "10" rSup { size 8{ - "24"} } `s right )} } } {} .

Solving this and converting the energy to MeV gives

Δ E 1 . 6 × 10 11 J 1 MeV 1 . 6 × 10 13 J = 100 MeV size 12{ΔE approx left (1 "." 6 times "10" rSup { size 8{ - "11"} } `J right ) { {1`"MeV"} over {1 "." 6 times "10" rSup { size 8{ - "13"} } `J} } ="100"`"MeV"} {} .

Mass is related to energy by Δ E = mc 2 size 12{ΔE= ital "mc" rSup { size 8{2} } } {} , so that the mass of the pion is m = Δ E / c 2 size 12{m=ΔE/c rSup { size 8{2} } } {} , or

m 100 MeV/ c 2 size 12{m approx "100"`"MeV/"c rSup { size 8{2} } } {} .


This is about 200 times the mass of an electron and about one-tenth the mass of a nucleon. No such particles were known at the time Yukawa made his bold proposal.

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Yukawa’s proposal of particle exchange as the method of force transfer is intriguing. But how can we verify his proposal if we cannot observe the virtual pion directly? If sufficient energy is in a nucleus, it would be possible to free the pion—that is, to create its mass from external energy input. This can be accomplished by collisions of energetic particles with nuclei, but energies greater than 100 MeV are required to conserve both energy and momentum. In 1947, pions were observed in cosmic-ray experiments, which were designed to supply a small flux of high-energy protons that may collide with nuclei. Soon afterward, accelerators of sufficient energy were creating pions in the laboratory under controlled conditions. Three pions were discovered, two with charge and one neutral, and given the symbols π + , π , and π 0 size 12{π rSup { size 8{+{}} } ,`π rSup { size 8{ - {}} } ,`"and "`π rSup { size 8{0} } } {} , respectively. The masses of π + size 12{π rSup { size 8{+{}} } } {} and π size 12{π rSup { size 8{ - {}} } } {} are identical at 139 . 6 MeV/ c 2 size 12{"139" "." 6`"MeV/"c rSup { size 8{2} } } {} , whereas π 0 size 12{ π rSup { size 8{0} } } {} has a mass of 135 . 0 MeV/ c 2 size 12{"135" "." 0`"MeV/"c rSup { size 8{2} } } {} . These masses are close to the predicted value of 100 MeV/ c 2 size 12{"100"`"MeV/"c rSup { size 8{2} } } {} and, since they are intermediate between electron and nucleon masses, the particles are given the name meson    (now an entire class of particles, as we shall see in Particles, Patterns, and Conservation Laws ).

The pions, or π size 12{π} {} -mesons as they are also called, have masses close to those predicted and feel the strong nuclear force. Another previously unknown particle, now called the muon, was discovered during cosmic-ray experiments in 1936 (one of its discoverers, Seth Neddermeyer, also originated the idea of implosion for plutonium bombs). Since the mass of a muon is around 106 MeV/ c 2 size 12{"106"`"MeV/"c rSup { size 8{2} } } {} , at first it was thought to be the particle predicted by Yukawa. But it was soon realized that muons do not feel the strong nuclear force and could not be Yukawa’s particle. Their role was unknown, causing the respected physicist I. I. Rabi to comment, “Who ordered that?” This remains a valid question today. We have discovered hundreds of subatomic particles; the roles of some are only partially understood. But there are various patterns and relations to forces that have led to profound insights into nature’s secrets.


  • Yukawa’s idea of virtual particle exchange as the carrier of forces is crucial, with virtual particles being formed in temporary violation of the conservation of mass-energy as allowed by the Heisenberg uncertainty principle.


A virtual particle having an approximate mass of 10 14 GeV/ c 2 size 12{"10" rSup { size 8{"14"} } `"GeV/"c rSup { size 8{2} } } {} may be associated with the unification of the strong and electroweak forces. For what length of time could this virtual particle exist (in temporary violation of the conservation of mass-energy as allowed by the Heisenberg uncertainty principle)?

3 × 10 39 s size 12{3 times "10" rSup { size 8{ - "39"} } `s} {}

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Calculate the mass in GeV/ c 2 size 12{"GeV/"c rSup { size 8{2} } } {} of a virtual carrier particle that has a range limited to 10 30 size 12{"10" rSup { size 8{ - "30"} } } {} m by the Heisenberg uncertainty principle. Such a particle might be involved in the unification of the strong and electroweak forces.

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Another component of the strong nuclear force is transmitted by the exchange of virtual K -mesons. Taking K -mesons to have an average mass of 495 MeV/ c 2 size 12{"495"`"MeV/"c rSup { size 8{2} } } {} , what is the approximate range of this component of the strong force?

1 . 99 × 10 16 m ( 0 . 2 fm ) size 12{1 "." "99" times "10" rSup { size 8{ - "16"} } `m` \( 0 "." 2`"fm" \) } {}

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