College Physics for AP® Courses

# Section Summary

• Some nuclei are radioactive—they spontaneously decay destroying some part of their mass and emitting energetic rays, a process called nuclear radioactivity.
• Nuclear radiation, like x rays, is ionizing radiation, because energy sufficient to ionize matter is emitted in each decay.
• The range (or distance traveled in a material) of ionizing radiation is directly related to the charge of the emitted particle and its energy, with greater-charge and lower-energy particles having the shortest ranges.
• Radiation detectors are based directly or indirectly upon the ionization created by radiation, as are the effects of radiation on living and inert materials.

• Radiation detectors are based directly or indirectly upon the ionization created by radiation, as are the effects of radiation on living and inert materials.

### 31.3Substructure of the Nucleus

• 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 Table 31.2 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
$1 u = 1.6605 × 10 − 27 kg = 931.46 MeV / c 2 . 1 u = 1.6605 × 10 − 27 kg = 931.46 MeV / c 2 .$
• A nuclide is a specific combination of protons and neutrons, denoted by
$ZAXN or simplyAX,ZAXN or simplyAX, size 12{"" lSup { size 8{A} } X} {}$
$ZZ size 12{Z} {}$ 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.

### 31.4Nuclear Decay and Conservation Laws

• When a parent nucleus decays, it produces a daughter nucleus following rules and conservation laws. There are three major types of nuclear decay, called alpha $α,α, size 12{ left (α right ),} {}$ beta $β,β, size 12{ left (β right ),} {}$ and gamma $γγ size 12{ left (γ right )} {}$. The $αα size 12{α} {}$ decay equation is
$ZAXN→Z−2A−4YN−2+24He2.ZAXN→Z−2A−4YN−2+24He2. size 12{"" lSub { size 8{Z} } lSup { size 8{A} } X rSub { size 8{N} } rightarrow "" lSub { size 8{Z - 2} } lSup { size 8{A - 4} } Y rSub { size 8{N - 2} } +"" lSub { size 8{2} } lSup { size 8{4} } "He" rSub { size 8{2} } } {}$
• Nuclear decay releases an amount of energy $EE size 12{E} {}$ related to the mass destroyed $ΔmΔm$ by
$E=(Δm)c2.E=(Δm)c2. size 12{E= $$Δm$$ c rSup { size 8{2} } } {}$
• There are three forms of beta decay. The $β−β− size 12{β rSup { size 8{ - {}} } } {}$decay equation is
$ZAXN→Z+1AYN−1+β−+ν¯e.ZAXN→Z+1AYN−1+β−+ν¯e.$
• The $β+β+$ decay equation is
$ZAXN→Z−1AYN+1+β++νe.ZAXN→Z−1AYN+1+β++νe.$
• The electron capture equation is
$ZAXN+e−→Z−1AYN+1+νe.ZAXN+e−→Z−1AYN+1+νe.$
• $β−β−$ is an electron, $β+β+ size 12{β rSup { size 8{+{}} } } {}$ is an antielectron or positron, $νeνe size 12{v rSub { size 8{e} } } {}$ represents an electron’s neutrino, and $ν¯eν¯e size 12{ {overline {ν rSub { size 8{e} } }} } {}$ is an electron’s antineutrino. In addition to all previously known conservation laws, two new ones arise— conservation of electron family number and conservation of the total number of nucleons. The $γγ$ decay equation is
$Z A X N * → Z A X N + γ 1 + γ 2 + ⋯ Z A X N * → Z A X N + γ 1 + γ 2 + ⋯ size 12{"" lSub { size 8{Z} } lSup { size 8{A} } X rSub { size 8{N} } rSup { size 8{*} } rightarrow "" lSub { size 8{Z} } lSup { size 8{A} } X rSub { size 8{N} } +γ rSub { size 8{1} } +γ rSub { size 8{2} } + dotsaxis } {}$
$γγ size 12{γ} {}$ is a high-energy photon originating in a nucleus.

### 31.5Half-Life and Activity

• Half-life $t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}$ is the time in which there is a 50% chance that a nucleus will decay. The number of nuclei $NN size 12{N} {}$ as a function of time is
$N=N0e−λt,N=N0e−λt, size 12{N=N rSub { size 8{0} } e rSup { size 8{ - λt} } } {}$
where $N0N0 size 12{N rSub { size 8{0} } } {}$ is the number present at $t=0t=0 size 12{t=0} {}$, and $λλ size 12{λ} {}$ is the decay constant, related to the half-life by
$λ=0.693t1/2.λ=0.693t1/2. size 12{λ= { {0 "." "693"} over {t rSub { size 8{1/2} } } } } {}$
• One of the applications of radioactive decay is radioactive dating, in which the age of a material is determined by the amount of radioactive decay that occurs. The rate of decay is called the activity $RR size 12{R} {}$:
$R= Δ N Δ t .R= Δ N Δ t . size 12{R= { {ΔN} over {Δt} } } {}$
• The SI unit for $RR size 12{R} {}$ is the becquerel (Bq), defined by
$1 Bq=1 decay/s. 1 Bq=1 decay/s. size 12{1" Bq"="1 decay/s"} {}$
• $RR size 12{R} {}$ is also expressed in terms of curies (Ci), where
$1Ci=3.70×1010 Bq.1Ci=3.70×1010 Bq. size 12{1" Ci"=3 "." "70" times "10" rSup { size 8{"10"} } " Bq"} {}$
• The activity $RR size 12{R} {}$ of a source is related to $NN size 12{N} {}$ and $t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}$ by
$R=0.693Nt1/2.R=0.693Nt1/2. size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {}$
• Since $NN size 12{N} {}$ has an exponential behavior as in the equation $N=N0e−λtN=N0e−λt size 12{N=N rSub { size 8{0} } e rSup { size 8{ - λt} } } {}$, the activity also has an exponential behavior, given by
$R=R0e−λt,R=R0e−λt, size 12{R=R rSub { size 8{0} } e rSup { size 8{ - λt} } } {}$
where $R0R0 size 12{R rSub { size 8{0} } } {}$ is the activity at $t=0t=0 size 12{t=0} {}$.

### 31.6Binding Energy

• The binding energy (BE) of a nucleus is the energy needed to separate it into individual protons and neutrons. In terms of atomic masses,
$BE={[Zm(1H)+Nmn]−m(AX)}c2,BE={[Zm(1H)+Nmn]−m(AX)}c2,$
where $m1Hm1H size 12{m left ("" lSup { size 8{1} } H right )} {}$ is the mass of a hydrogen atom, $mAXmAX$ is the atomic mass of the nuclide, and $mnmn$ is the mass of a neutron. Patterns in the binding energy per nucleon, $BE/ABE/A$, reveal details of the nuclear force. The larger the $BE/ABE/A size 12{"BE"/A} {}$, the more stable the nucleus.

### 31.7Tunneling

• Tunneling is a quantum mechanical process of potential energy barrier penetration. The concept was first applied to explain $αα size 12{α} {}$ decay, but tunneling is found to occur in other quantum mechanical systems.