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Opus in profectus

# Quantum Chromodynamics

## Practice

### practice problem 1

The mass of the top quark is 172.69 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 = 172.69 GeV/c2 103 MeV 1 u 1 GeV 931.494 MeV/c2

mtop = 185.390 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 185 u Source: LBL and KAERI
tungsten (z = 74) rhenium (z = 75) osmium (z = 76)
182W 181.948 206 u 26.3% 182Re 181.951 211 u 182Os 181.952 186 u
183W 182.950 225 u 14.3% 183Re 182.950 821 u 183Os 182.953 110 u
184W 183.950 933 u 30.7% 184Re 183.952 524 u 184Os 183.952 491 u 0.02%
185W 184.953 421 u 185Re 184.952 956u 37.4% 185Os 184.954 043 u
mass of the top quark = 185.390 u
186W 185.954 362 u 28.6% 186Re 185.954 987 u 186Os 185.953 838 u 1.58%
187W 186.957 158 u 187Re 186.955 751 u 62.6% 187Os 186.955 748 u 1.60%
188W 187.958 487 u 188Re 187.958 112 u 188Os 187.955 836 u 13.3%
189W 188.961 912 u 189Re 188.959 228 u 189Os 188.958 145 u 16.1%
190W 189.963 180 u 190Re 189.961 816 u 190Os 189.958 445 u 26.4%
191W n/a 191Re 190.963 124 u 191Os 190.960 928 u
192W n/a 192Re 191.965 960 u 192Os 191.961 479 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 mass of 938.272 MeV/c2. Neutrons have a mass of 939.565 MeV/c2. The 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)

### practice problem 4

Two protons in a helium nucleus are sparated by a typical distance of 1.2 fm. Determine…
1. the electrostatic force between them. (About how big is this?)
2. the gravitational force between them. (About how big is this?)