Quantum Chromodynamics
Practice
practice problem 1
solution
The mass of subatomic particles is measured in electron volts while chemical elements are weighed in atomic mass units. This is a question about unit conversion.
m_{top} = | 173.21 GeV/c^{2} | 10^{3} MeV | 1 u | ||
1 | GeV | 931.494 MeV/c^{2} |
m_{top} = 185.949 u
Now we need a periodic table. I have a nice one in this book. Find the greatest atomic mass less than 185.798 u. Rhenium satisfies this condition with a mass of 186.21 u. (The next lighter element is tungsten at 183.84 u. The next heavier is osmium at 190.23 u.) This must be the answer. The top quark is heavier than an atom of every element up to element 75 — rhenium.
Well, it isn't quite the answer. You may recall that the masses of the elements stated on a periodic table are averages. Atoms of any element can be found with a variety of masses. These variations are called isotopes. Here's a fragment of the table of isotopes for tungsten, rhenium, and osmium.
tungsten (z = 74) | rhenium (z = 75) | osmium (z = 76) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
^{182}W | 181.948206 u | 26.3% | ^{182}Re | 181.951211 u | ^{182}Os | 181.952186 u | |||||
^{183}W | 182.950225 u | 14.3% | ^{183}Re | 182.950821 u | ^{183}Os | 182.953110 u | |||||
^{184}W | 183.950933 u | 30.7% | ^{184}Re | 183.952524 u | ^{184}Os | 183.952491 u | 0.02% | ||||
^{185}W | 184.953421 u | ^{185}Re | 184.952956 u | 37.4% | ^{185}Os | 184.954043 u | |||||
mass of the top quark = 185.949 u | |||||||||||
^{186}W | 185.954362 u | 28.6% | ^{186}Re | 185.954987 u | ^{186}Os | 185.953838 u | 1.58% | ||||
^{187}W | 186.957158 u | ^{187}Re | 186.955751 u | 62.6% | ^{187}Os | 186.955748 u | 1.60% | ||||
^{188}W | 187.958487 u | ^{188}Re | 187.958112 u | ^{188}Os | 187.955836 u | 13.3% | |||||
^{189}W | 188.961912 u | ^{189}Re | 188.959228 u | ^{189}Os | 188.958145 u | 16.1% | |||||
^{190}W | 189.963180 u | ^{190}Re | 189.961816 u | ^{190}Os | 189.958445 u | 26.4% | |||||
^{191}W | n/a | ^{191}Re | 190.963124 u | ^{191}Os | 190.960928 u | ||||||
^{192}W | n/a | ^{192}Re | 191.965960 u | ^{192}Os | 191.961479 u | 41.0% |
The top quark is heavier than most naturally ocurring tungsten atoms, lighter than most naturally ocurring rhenium atoms, and lighter than nearly all naturally ocurring osmium atoms. This answer is accurate, but a bit overly detailed.
practice problem 2
solution
Protons have a rest mass of 938.272 MeV/c^{2}. Neutrons have a rest mass of 939.565 MeV/c^{2}. The rest masses of the up and down quarks are 2.3 MeV/c^{2} and 4.8 MeV/c^{2}, respectively. This means that quarks make up…
2m_{u} +1m_{d} | = | 2(2.3 MeV/c^{2}) + 1(4.8 MeV/c^{2}) |
m_{p} | 938.272 MeV/c^{2} | |
2m_{u} +1m_{d} | = | 9.4 MeV/c^{2} |
m_{p} | 938.272 MeV/c^{2} | |
2m_{u} +1m_{d} | = | 1.00% of the mass of a proton |
m_{p} | ||
1m_{u} +2m_{d} | = | 1(2.3 MeV/c^{2}) + 2(4.8 MeV/c^{2}) |
m_{n} | 939.565 MeV/c^{2} | |
1m_{u} +2m_{d} | = | 11.9 MeV/c^{2} |
m_{n} | 939.565 MeV/c^{2} | |
1m_{u} +2m_{d} | = | 1.27% of the mass of a neutron |
m_{n} |
Most of the universe is hydrogen, but the Earth is more than just hydrogen. There's oxygen, silicon, carbon, aluminum, and more. These elements are made of nuclei that are roughly half protons and half neutrons. I can't tell you anything better than that. Let's agree to be reasonable in our precision.
The strong force is responsible for nearly 99 per cent of the mass we deal with in our everyday lives.
practice problem 3
- What combination(s) of quarks will produce a sigma baryon with a charge of −1 e?
- What combination(s) of quarks will produce a sigma baryon with a charge of +0 e?
- What combination(s) of quarks will produce a sigma baryon with a charge of +1 e?
- What combination(s) of quarks will produce a sigma baryon with a charge of +2 e?
solution
uu | ud | dd | ||||
---|---|---|---|---|---|---|
s | uus (+⅔ e)(+⅔ e)(−⅓ e) +1 e |
uds (+⅔ e)(−⅓ e)(−⅓ e) +0 e |
dds (−⅓ e)(−⅓ e)(−⅓ e) −1 e |
|||
c | uuc (+⅔ e)(+⅔ e)(+⅔ e) +2 e |
udc (+⅔ e)(−⅓ e)(+⅔ e) +1 e |
ddc (−⅓ e)(−⅓ e)(+⅔ e) +0 e |
|||
b | uub (+⅔ e)(+⅔ e)(−⅓ e) +1 e |
udb (+⅔ e)(−⅓ e)(−⅓ e) +0 e |
ddb (−⅓ e)(−⅓ e)(−⅓ e) −1 e |
- There are two sigma baryons with a charge of −1 e
dds (Σ^{−}), ddb (Σ^{−}_{b})
- There are three sigma baryons with a charge of 0 e
uds (Σ^{0}), ddc (Σ^{0}_{c}), udb (Σ^{0}_{b})
- There are three sigma baryons with a charge of +1 e
uus (Σ^{+}), udc (Σ^{+}_{c}), uub (Σ^{+}_{b})
- There is one sigma baryon with a charge of +2 e
uuc (Σ^{++}_{c})
practice problem 4
solution
Answer it.