# 251 31.3 Substructure of the Nucleus

Substructure of the Nucleus
• Define and discuss the nucleus in an atom.
• Define atomic number.
• Define and discuss isotopes.
• Calculate the density of the nucleus.
• Explain nuclear force.

What is inside the nucleus? Why are some nuclei stable while others decay? (See [link].) Why are there different types of decay (αα size 12{α} {}, ββ size 12{β} {} and γγ size 12{γ} {})? Why are nuclear decay energies so large? Pursuing natural questions like these has led to far more fundamental discoveries than you might imagine. Why is most of the carbon in this coal stable (a), while the uranium in the disk (b) slowly decays over billions of years? Why is cesium in this ampule (c) even less stable than the uranium, decaying in far less than 1/1,000,000 the time? What is the reason uranium and cesium undergo different types of decay (αα size 12{α} {} and ββ size 12{β} {}, respectively)? (credits: (a) Bresson Thomas, Wikimedia Commons; (b) U.S. Department of Energy; (c) Tomihahndorf, Wikimedia Commons)

We have already identified protons as the particles that carry positive charge in the nuclei. However, there are actually two types of particles in the nuclei—the proton and the neutron, referred to collectively as nucleons, the constituents of nuclei. As its name implies, the neutron is a neutral particle (q=0q=0 size 12{q=0} {}) that has nearly the same mass and intrinsic spin as the proton. [link] compares the masses of protons, neutrons, and electrons. Note how close the proton and neutron masses are, but the neutron is slightly more massive once you look past the third digit. Both nucleons are much more massive than an electron. In fact, mp=1836memp=1836me size 12{m rSub { size 8{p} } =”1836″ m rSub { size 8{e} } } {} (as noted in Medical Applications of Nuclear Physics and mn=1839memn=1839me size 12{m rSub { size 8{n} } =”1839″ m rSub { size 8{e} } } {}.

[link] also gives masses in terms of mass units that are more convenient than kilograms on the atomic and nuclear scale. The first of these is the unified atomic mass unit (u), defined as

1 u=1.6605×10−27kg.1 u=1.6605×10−27kg. size 12{“1 u”=1 “.” “6605”´”10″ rSup { size 8{-“27″} } ” kg”} {}

This unit is defined so that a neutral carbon 12C12C atom has a mass of exactly 12 u. Masses are also expressed in units of MeV/c2MeV/c2. These units are very convenient when considering the conversion of mass into energy (and vice versa), as is so prominent in nuclear processes. Using E=mc2E=mc2 size 12{E= ital “mc” rSup { size 8{2} } } {} and units of mm size 12{m} {} in MeV/c2MeV/c2 size 12{“MeV/”c rSup { size 8{2} } } {}, we find that c2c2 size 12{c rSup { size 8{2} } } {} cancels and EE size 12{E} {} comes out conveniently in MeV. For example, if the rest mass of a proton is converted entirely into energy, then

E=mc2=(938.27 MeV/c2)c2=938.27 MeV.E=mc2=(938.27 MeV/c2)c2=938.27 MeV. size 12{E= ital “mc” rSup { size 8{2} } = ( “938” “.” “27” “MeV/”c rSup { size 8{2} } ) c rSup { size 8{2} } =”938” “.” “27”” MeV”} {}

It is useful to note that 1 u of mass converted to energy produces 931.5 MeV, or

1 u=931.5 MeV/c2.1 u=931.5 MeV/c2. size 12{“1 u”=”931” “.” 5″ MeV/”c rSup { size 8{2} } } {}

All properties of a nucleus are determined by the number of protons and neutrons it has. A specific combination of protons and neutrons is called a nuclide and is a unique nucleus. The following notation is used to represent a particular nuclide:

ZAXN,ZAXN, size 12{“” lSub { size 8{Z} } lSup { size 8{A} } X rSub { size 8{N} } } {}

where the symbols AA size 12{A} {}, XX size 12{X} {}, ZZ size 12{Z} {} , and NN size 12{N} {} are defined as follows: The number of protons in a nucleus is the atomic number ZZ size 12{Z} {}, as defined in Medical Applications of Nuclear Physics. X is the symbol for the element, such as Ca for calcium. However, once ZZ size 12{Z} {} is known, the element is known; hence, ZZ size 12{Z} {} and XX are redundant. For example, Z=20Z=20 size 12{Z=”20″} {} is always calcium, and calcium always has Z=20Z=20 size 12{Z=”20″} {}. NN size 12{N} {} is the number of neutrons in a nucleus. In the notation for a nuclide, the subscript NN size 12{N} {} is usually omitted. The symbol AA size 12{A} {} is defined as the number of nucleons or the total number of protons and neutrons,

A=N+Z,A=N+Z, size 12{A=N+Z} {}

where AA size 12{A} {} is also called the mass number. This name for AA size 12{A} {} is logical; the mass of an atom is nearly equal to the mass of its nucleus, since electrons have so little mass. The mass of the nucleus turns out to be nearly equal to the sum of the masses of the protons and neutrons in it, which is proportional to AA size 12{A} {}. In this context, it is particularly convenient to express masses in units of u. Both protons and neutrons have masses close to 1 u, and so the mass of an atom is close to AA size 12{A} {} u. For example, in an oxygen nucleus with eight protons and eight neutrons, A=16A=16 size 12{A=”16″} {}, and its mass is 16 u. As noticed, the unified atomic mass unit is defined so that a neutral carbon atom (actually a 12C12C size 12{“” lSup { size 8{“12”} } C} {} atom) has a mass of exactly 12 uu size 12{u} {}. Carbon was chosen as the standard, partly because of its importance in organic chemistry (see Appendix A).

Masses of the Proton, Neutron, and Electron
Particle Symbol kg u MeVc2
Proton p 1.67262×10−271.67262×10−27 size 12{1 “.” “67262” times “10” rSup { size 8{ – “27”} } } {} 1.007276 938.27
Neutron n 1.67493×10−271.67493×10−27 size 12{1 “.” “67493”´”10″ rSup { size 8{-“27”} } } {} 1.008665 939.57
Electron e 9.1094×10−319.1094×10−31 size 12{9 “.” “1094”´”10″ rSup { size 8{-“31”} } } {} 0.00054858 0.511

Let us look at a few examples of nuclides expressed in the ZAXNZAXN notation. The nucleus of the simplest atom, hydrogen, is a single proton, or
11H11H (the zero for no neutrons is often omitted). To check this symbol, refer to the periodic table—you see that the atomic number
ZZ of hydrogen is 1. Since you are given that there are no neutrons, the mass number
AA is also 1. Suppose you are told that the helium nucleus or
αα particle has two protons and two neutrons. You can then see that it is written
24He224He2. There is a scarce form of hydrogen found in nature called deuterium; its nucleus has one proton and one neutron and, hence, twice the mass of common hydrogen. The symbol for deuterium is, thus,
12H112H1 (sometimes
DD is used, as for deuterated water
D2OD2O). An even rarer—and radioactive—form of hydrogen is called tritium, since it has a single proton and two neutrons, and it is written
13H213H2. These three varieties of hydrogen have nearly identical chemistries, but the nuclei differ greatly in mass, stability, and other characteristics. Nuclei (such as those of hydrogen) having the same
ZZ size 12{Z} {} and different NN size 12{N} {} s are defined to be isotopes of the same element.

There is some redundancy in the symbols AA size 12{A} {}, XX size 12{X} {}, ZZ size 12{Z} {}, and NN size 12{N} {} . If the element XX size 12{X} {} is known, then ZZ size 12{Z} {} can be found in a periodic table and is always the same for a given element. If both AA size 12{A} {} and XX size 12{X} {} are known, then NN size 12{N} {} can also be determined (first find ZZ size 12{Z} {}; then, N=A−ZN=A−Z size 12{N=A-Z} {}). Thus the simpler notation for nuclides is

AX,AX, size 12{“” lSup { size 8{A} } X} {}

which is sufficient and is most commonly used. For example, in this simpler notation, the three isotopes of hydrogen are 1H, 2H,1H, 2H, and 3H,3H, while the αα particle is 4He4He. We read this backward, saying helium-4 for 4He4He, or uranium-238 for 238U238U. So for 238U238U, should we need to know, we can determine that Z=92Z=92 for uranium from the periodic table, and, thus, N=238−92=146N=238−92=146 size 12{N=”238″-“92″=”146”} {}.

A variety of experiments indicate that a nucleus behaves something like a tightly packed ball of nucleons, as illustrated in [link]. These nucleons have large kinetic energies and, thus, move rapidly in very close contact. Nucleons can be separated by a large force, such as in a collision with another nucleus, but resist strongly being pushed closer together. The most compelling evidence that nucleons are closely packed in a nucleus is that the radius of a nucleus, rr size 12{r} {}, is found to be given approximately by

r=r0A1/3,r=r0A1/3, size 12{r=r rSub { size 8{0} } A rSup { size 8{1/3} } } {}

where r0=1.2 fmr0=1.2 fm and AA is the mass number of the nucleus. Note that r3∝Ar3∝A size 12{r rSup { size 8{3} } µA} {}. Since many nuclei are spherical, and the volume of a sphere is V=(4/3)πr3V=(4/3)πr3 size 12{V= ( 4/3 ) pr rSup { size 8{3} } } {}, we see that V∝AV∝A size 12{V prop A} {} —that is, the volume of a nucleus is proportional to the number of nucleons in it. This is what would happen if you pack nucleons so closely that there is no empty space between them.

Nucleons are held together by nuclear forces and resist both being pulled apart and pushed inside one another. The volume of the nucleus is the sum of the volumes of the nucleons in it, here shown in different colors to represent protons and neutrons.

How Small and Dense Is a Nucleus?

(a) Find the radius of an iron-56 nucleus. (b) Find its approximate density in kg/m3kg/m3, approximating the mass of 56Fe56Fe to be 56 u.

Strategy and Concept

56Fe56Fe is a straightforward application of
r=r0A1/3,r=r0A1/3, given
A=56A=56. (b) To find the approximate density, we assume the nucleus is spherical (this one actually is), calculate its volume using the radius found in part (a), and then find its density from
ρ=m/Vρ=m/V. Finally, we will need to convert density from units of u/fm3u/fm3 to kg/m3kg/m3.

Solution

(a) The radius of a nucleus is given by

r=r0A1/3.r=r0A1/3.

Substituting the values for r0r0 and AA yields

r=(1.2 fm)(56)1/3=(1.2 fm)(3.83)=4.6 fm.r=(1.2 fm)(56)1/3=(1.2 fm)(3.83)=4.6 fm.

(b) Density is defined to be
ρ=m/Vρ=m/V, which for a sphere of radius
rr is

ρ=mV=m(4/3)πr3.ρ=mV=m(4/3)πr3.

Substituting known values gives

ρ=56 u(1.33)(3.14)(4.6 fm)3=0.138 u/fm3.ρ=56 u(1.33)(3.14)(4.6 fm)3=0.138 u/fm3.

Converting to units of kg/m3kg/m3, we find

ρ=(0.138 u/fm3)(1.66×10–27kg/u)(1 fm10–15m)=2.3×1017kg/m3.ρ=(0.138 u/fm3)(1.66×10–27kg/u)(1 fm10–15m)=2.3×1017kg/m3.

Discussion

(a) The radius of this medium-sized nucleus is found to be approximately 4.6 fm, and so its diameter is about 10 fm, or 10–14m10–14m. In our discussion of Rutherford’s discovery of the nucleus, we noticed that it is about 10–15m10–15m in diameter (which is for lighter nuclei), consistent with this result to an order of magnitude. The nucleus is much smaller in diameter than the typical atom, which has a diameter of the order of 10–10m10–10m.

(b) The density found here is so large as to cause disbelief. It is consistent with earlier discussions we have had about the nucleus being very small and containing nearly all of the mass of the atom. Nuclear densities, such as found here, are about 2×10142×1014 times greater than that of water, which has a density of “only”
103kg/m3103kg/m3. One cubic meter of nuclear matter, such as found in a neutron star, has the same mass as a cube of water 61 km on a side.

# Nuclear Forces and Stability

What forces hold a nucleus together? The nucleus is very small and its protons, being positive, exert tremendous repulsive forces on one another. (The Coulomb force increases as charges get closer, since it is proportional to 1/r21/r2 size 12{1/r rSup { size 8{2} } } {}, even at the tiny distances found in nuclei.) The answer is that two previously unknown forces hold the nucleus together and make it into a tightly packed ball of nucleons. These forces are called the weak and strong nuclear forces. Nuclear forces are so short ranged that they fall to zero strength when nucleons are separated by only a few fm. However, like glue, they are strongly attracted when the nucleons get close to one another. The strong nuclear force is about 100 times more attractive than the repulsive EM force, easily holding the nucleons together. Nuclear forces become extremely repulsive if the nucleons get too close, making nucleons strongly resist being pushed inside one another, something like ball bearings.

The fact that nuclear forces are very strong is responsible for the very large energies emitted in nuclear decay. During decay, the forces do work, and since work is force times the distance (W=FdcosθW=Fdcosθ size 12{W= ital “Fd””cos”θ} {}), a large force can result in a large emitted energy. In fact, we know that there are two distinct nuclear forces because of the different types of nuclear decay—the strong nuclear force is responsible for αα size 12{α} {} decay, while the weak nuclear force is responsible for ββ size 12{β} {} decay.

The many stable and unstable nuclei we have explored, and the hundreds we have not discussed, can be arranged in a table called the chart of the nuclides, a simplified version of which is shown in [link]. Nuclides are located on a plot of NN size 12{N} {} versus ZZ size 12{Z} {}. Examination of a detailed chart of the nuclides reveals patterns in the characteristics of nuclei, such as stability, abundance, and types of decay, analogous to but more complex than the systematics in the periodic table of the elements. Simplified chart of the nuclides, a graph of NN size 12{N} {} versus ZZ size 12{Z} {} for known nuclides. The patterns of stable and unstable nuclides reveal characteristics of the nuclear forces. The dashed line is for N=ZN=Z size 12{N=Z} {}. Numbers along diagonals are mass numbers AA size 12{A} {}.

In principle, a nucleus can have any combination of protons and neutrons, but [link] shows a definite pattern for those that are stable. For low-mass nuclei, there is a strong tendency for NN size 12{N} {} and ZZ size 12{Z} {} to be nearly equal. This means that the nuclear force is more attractive when N=ZN=Z size 12{N=Z} {}. More detailed examination reveals greater stability when NN size 12{N} {} and ZZ size 12{Z} {} are even numbers—nuclear forces are more attractive when neutrons and protons are in pairs. For increasingly higher masses, there are progressively more neutrons than protons in stable nuclei. This is due to the ever-growing repulsion between protons. Since nuclear forces are short ranged, and the Coulomb force is long ranged, an excess of neutrons keeps the protons a little farther apart, reducing Coulomb repulsion. Decay modes of nuclides out of the region of stability consistently produce nuclides closer to the region of stability. There are more stable nuclei having certain numbers of protons and neutrons, called magic numbers. Magic numbers indicate a shell structure for the nucleus in which closed shells are more stable. Nuclear shell theory has been very successful in explaining nuclear energy levels, nuclear decay, and the greater stability of nuclei with closed shells. We have been producing ever-heavier transuranic elements since the early 1940s, and we have now produced the element with Z=118Z=118 size 12{Z=”118″} {}. There are theoretical predictions of an island of relative stability for nuclei with such high ZZ size 12{Z} {} s. The German-born American physicist Maria Goeppert Mayer (1906–1972) shared the 1963 Nobel Prize in physics with J. Jensen for the creation of the nuclear shell model. This successful nuclear model has nucleons filling shells analogous to electron shells in atoms. It was inspired by patterns observed in nuclear properties. (credit: Nobel Foundation via Wikimedia Commons)

# Section Summary

• Two particles, both called nucleons, are found inside nuclei. The two types of nucleons are protons and neutrons; they are very similar, except that the proton is positively charged while the neutron is neutral. Some of their characteristics are given in [link] and compared with those of the electron. A mass unit convenient to atomic and nuclear processes is the unified atomic mass unit (u), defined to be
1 u=1.6605×10−27 kg=931.46 MeV/c2.1 u=1.6605×10−27 kg=931.46 MeV/c2.
• A nuclide is a specific combination of protons and neutrons, denoted by
ZAXNor simplyAX,ZAXNor simplyAX, size 12{“” lSup { size 8{A} } X} {}

ZZ size 12{Z} {} is the number of protons or atomic number, X is the symbol for the element, NN size 12{N} {} is the number of neutrons, and AA size 12{A} {} is the mass number or the total number of protons and neutrons,

A=N+Z.A=N+Z. size 12{A=N+Z} {}
• Nuclides having the same ZZ size 12{Z} {} but different NN size 12{N} {} are isotopes of the same element.
• The radius of a nucleus, rr size 12{r} {}, is approximately
r=r0A1/3,r=r0A1/3,

where r0=1.2 fmr0=1.2 fm. Nuclear volumes are proportional to AA size 12{A} {}. There are two nuclear forces, the weak and the strong. Systematics in nuclear stability seen on the chart of the nuclides indicate that there are shell closures in nuclei for values of ZZ size 12{Z} {} and NN size 12{N} {} equal to the magic numbers, which correspond to highly stable nuclei.

# Conceptual Questions

The weak and strong nuclear forces are basic to the structure of matter. Why we do not experience them directly?

Define and make clear distinctions between the terms neutron, nucleon, nucleus, nuclide, and neutrino.

What are isotopes? Why do different isotopes of the same element have similar chemistries?

# Problems & Exercises

Verify that a 2.3×1017kg2.3×1017kg size 12{2 “.” 3 times “10” rSup { size 8{“17”} } “kg”} {} mass of water at normal density would make a cube 60 km on a side, as claimed in [link]. (This mass at nuclear density would make a cube 1.0 m on a side.)

m=ρV=ρd3⇒a=mρ1/3=2.3×1017kg1000 kg/m313=61×103 m=61 kmm=ρV=ρd3⇒a=mρ1/3=2.3×1017kg1000 kg/m313=61×103 m=61 km

Find the length of a side of a cube having a mass of 1.0 kg and the density of nuclear matter, taking this to be 2.3×1017 kg/m32.3×1017 kg/m3 size 12{2 “.” 3´”10″ rSup { size 8{“17″} } ” kg/m” rSup { size 8{3} } } {}.

What is the radius of an αα size 12{α} {} particle?

1.9 fm1.9 fm size 12{1 “.” 9″ fm”} {}

Find the radius of a 238Pu238Pu size 12{“” lSup { size 8{“238”} } “Pu”} {} nucleus. 238Pu238Pu size 12{“” lSup { size 8{“238”} } “Pu”} {} is a manufactured nuclide that is used as a power source on some space probes.

(a) Calculate the radius of 58Ni58Ni size 12{“” lSup { size 8{“58”} } “Ni”} {}, one of the most tightly bound stable nuclei.

(b) What is the ratio of the radius of 58Ni58Ni size 12{“” lSup { size 8{“58”} } “Ni”} {} to that of 258Ha258Ha size 12{“” lSup { size 8{“258”} } “Ha”} {}, one of the largest nuclei ever made? Note that the radius of the largest nucleus is still much smaller than the size of an atom.

(a) 4.6 fm4.6 fm size 12{4 “.” “6 fm”} {}

(b) 0.61 to 10.61 to 1 size 12{0 “.” “61 to 1”} {}

The unified atomic mass unit is defined to be 1 u=1.6605×10−27kg1 u=1.6605×10−27kg size 12{1″ u”=1 “.” “6605”×”10″ rSup { size 8{-“27”} } “kg”} {}. Verify that this amount of mass converted to energy yields 931.5 MeV. Note that you must use four-digit or better values for cc size 12{c} {} and ∣qe∣∣qe∣ size 12{ lline q rSub { size 8{e} } rline } {}.

What is the ratio of the velocity of a ββ size 12{β} {} particle to that of an αα size 12{α} {} particle, if they have the same nonrelativistic kinetic energy?

85.4 to 185.4 to 1 size 12{“85” “.” “4 to 1”} {}

If a 1.50-cm-thick piece of lead can absorb 90.0% of the γγ size 12{γ} {} rays from a radioactive source, how many centimeters of lead are needed to absorb all but 0.100% of the γγ size 12{γ} {} rays?

The detail observable using a probe is limited by its wavelength. Calculate the energy of a γγ size 12{γ} {}-ray photon that has a wavelength of 1×10−16m1×10−16m size 12{1 times “10” rSup { size 8{ – “16”} } m} {}, small enough to detect details about one-tenth the size of a nucleon. Note that a photon having this energy is difficult to produce and interacts poorly with the nucleus, limiting the practicability of this probe.

12.4 GeV12.4 GeV size 12{“12” “.” “4 GeV”} {}

(a) Show that if you assume the average nucleus is spherical with a radius r=r0A1/3r=r0A1/3 size 12{r=r rSub { size 8{0} } A rSup { size 8{1/3} } } {}, and with a mass of AA size 12{A} {} u, then its density is independent of AA size 12{A} {}.

(b) Calculate that density in u/fm3u/fm3 size 12{“u/fm” rSup { size 8{3} } } {} and kg/m3kg/m3 size 12{“kg/m” rSup { size 8{3} } } {}, and compare your results with those found in [link] for 56Fe56Fe size 12{“” lSup { size 8{“56”} } “Fe”} {}.

What is the ratio of the velocity of a 5.00-MeV ββ size 12{β} {} ray to that of an αα size 12{β} {} particle with the same kinetic energy? This should confirm that ββ size 12{β} {}s travel much faster than αα size 12{β} {}s even when relativity is taken into consideration. (See also [link].)

19.3 to 1

(a) What is the kinetic energy in MeV of a ββ size 12{β} {} ray that is traveling at 0.998c0.998c? This gives some idea of how energetic a ββ size 12{β} {} ray must be to travel at nearly the same speed as a γγ ray. (b) What is the velocity of the γγ ray relative to the ββ size 12{β} {} ray?

## Glossary

atomic mass
the total mass of the protons, neutrons, and electrons in a single atom
atomic number
number of protons in a nucleus
chart of the nuclides
a table comprising stable and unstable nuclei
isotopes
nuclei having the same ZZ size 12{Z} {} and different NN size 12{Ns} {}s
magic numbers
a number that indicates a shell structure for the nucleus in which closed shells are more stable
mass number
number of nucleons in a nucleus
neutron
a neutral particle that is found in a nucleus
nucleons
the particles found inside nuclei
nucleus
a region consisting of protons and neutrons at the center of an atom
nuclide
a type of atom whose nucleus has specific numbers of protons and neutrons
protons
the positively charged nucleons found in a nucleus 