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# Quantum Chromodynamics

## Practice

### practice problem 1

The mass of the top quark is 173 GeV/c2 — heavier than some atoms. Identify the heaviest element that is lighter than the top quark.

#### solution

The mass of subatomic particles is measured in electronvolts while chemical elements are weighed in atomic mass units. This is a question about unit conversion.

 mtop = 173 GeV/c2 103 MeV 1 u 1 GeV 931.494 MeV/c2

mtop = 185.723 u

Now we need a periodic table. I have a nice one in this book. Find the elements that are around 186 u.

 74 W 183.84
 75 Re 186.21
 76 Os 190.23

It looks like the top quark is heavier than an atom of every element up to element 75 — rhenium. This is the pretty good answer. However…

You may recall that the masses stated on a periodic table are averages. Atoms of every 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 with the top quark slipped in.

Isotopes with mass approximately equal to 186 u Source: LBL and KAERI
tungsten (z = 74)   rhenium (z = 75)   osmium (z = 76)
182W 181.948206 u 26.3%   182Re 181.951211 u     182Os 181.952186 u
183W 182.950225 u 14.3%   183Re 182.950821 u     183Os 182.953110 u
184W 183.950933 u 30.7%   184Re 183.952524 u     184Os 183.952491 u 0.02%
185W 184.953421 u     185Re 184.952956 u 37.4%   185Os 184.954043 u
mass of the top quark = 185.723 u
186W 185.954362 u 28.6%   186Re 185.954987 u     186Os 185.953838 u 1.58%
187W 186.957158 u     187Re 186.955751 u 62.6%   187Os 186.955748 u 1.60%
188W 187.958487 u     188Re 187.958112 u     188Os 187.955836 u 13.3%
189W 188.961912 u     189Re 188.959228 u     189Os 188.958145 u 16.1%
190W 189.963180 u     190Re 189.961816 u     190Os 189.958445 u 26.4%
191W n/a     191Re 190.963124 u     191Os 190.960928 u
192W n/a     192Re 191.965960 u     192Os 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 is the pedantic answer to this question.

### practice problem 2

What fraction of the mass of you and me, the air, earth, oceans, and everything else we deal with in our ordinary lives is due to the strong force?

#### solution

Protons have a rest mass of 938.272 MeV/c2. Neutrons have a rest mass of 939.565 MeV/c2. The rest masses of the up and down quarks are 2.3 MeV/c2 and 4.8 MeV/c2, respectively. This means that quarks make up…

 2mu +1md = 2(2.3 MeV/c2) + 1(4.8 MeV/c2) mp 938.272 MeV/c2 2mu +1md = 9.4 MeV/c2 mp 938.272 MeV/c2 2mu +1md = 1.00% of the mass of a proton mp 1mu +2md = 1(2.3 MeV/c2) + 2(4.8 MeV/c2) mn 939.565 MeV/c2 1mu +2md = 11.9 MeV/c2 mn 939.565 MeV/c2 1mu +2md = 1.27% of the mass of a neutron mn

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

The sigma baryons are a family of particles with two first generation quarks (u, d) and one higher generation quark (s, c, b). Top quarks probably cannot form sigma baryons since they decay before they can interact with other quarks. Sigma baryons can have a charge of −1e, 0e, +1e, or +2e.
1. What combination(s) of quarks will produce a sigma baryon with a charge of −1e?
2. What combination(s) of quarks will produce a sigma baryon with a charge of +0e?
3. What combination(s) of quarks will produce a sigma baryon with a charge of +1e?
4. What combination(s) of quarks will produce a sigma baryon with a charge of +2e?

#### solution

Don't try answering the parts of this question in the order given. Answer the whole question at once. First recognize that the are three possible first generation quark pairs that make up sigma baryons (uu, ud, dd). Then recognize that there are three possible remaining quarks (s, c, b). Set up a 3×3 table, pop in the charges, add them up, and pick out the combinations with the appropriate total charge.
The sigma baryons
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
1. There are two sigma baryons with a charge of −1 e

dds), ddbb)

2. There are three sigma baryons with a charge of 0 e

uds0), ddc0c), udb0b)

3. There are three sigma baryons with a charge of +1 e

uus+), udc+c), uub+b)

4. There is one sigma baryon with a charge of +2 e

uuc++c)

Write something.