Properties of the nucleus 8. Nuclear Physics Properties of nuclei Binding Energy Radioactive decay Natural radioactivity Consists of protons and neutrons Z = no. of protons (Atomic number) N = no. of neutrons (Neutron number) A = Z+N (Mass number) Notation :For element X with mass no. A and Atomic no. Z A X Z Example 4 He Isotopes Isotopes are nuclei that have the same no. of protons but different no. of neutrons. Plot of N vs Z for stable nuclei Stable Nuclei The chemical properties are the same but the nuclear properties are different. i.e. some isotopes may be unstable and are radioactive. Outline shows unstable nuclei eg. H H 3 H Hydrogen - stable Deuterium - stable Tritium - radioactive Some radioactive nuclei 4 6 C 3 5 P 3 53 I N 38 9 U Z Max Z=83 Size of the nucleus From scattering experiments with alpha particles He ++ Size of the nucleus Radius varies as the cube root of A r= r o A /3 Closest approach KE PE =kq q /r For gold nuclei closest r < 3x0-4 m This is smaller than the size of atoms r ~ 0-0 m where r o =.x0-5 m example For Uranium 38, A=38 38 9 U 5 / 3 5 r.x0 = (38 ) = 7.4x0 m
Forces in the nuclei Coulomb forces The protons repel each other with Coulomb forces. These are enormously large due to the small size. Nuclear forces +Z Equivalence of mass and energy A famous result from Einstein s Special Relativity Theory E= mc mass can be converted into energy Energy equivalent of an electron mass The nucleus is held together by the nuclear force. This force acts only at short range (~0-5 m) and is independent of charge (i.e. acts between proton-proton, proton-neutron and neutron-neutron). E=mc = (9.x0-3 kg)(3x0 8 m/s) = 8.x0-4 J =5.x0 5 ev= 0.5MeV An electron can be annihilated (converted completely to energy). A 0.5 MeV photon is produced. Mass changes when energy is lost or gained Gasoline + O -> CO +water + kinetic energy (heat) Σmc + ΣEnergy =constant The energy released is equal to the change in mass in the reaction. E= mc CO + water is lighter. Burning kg of gasoline releases 44x0 6 J of energy. The change in mass is 6 E 44x0 J 0 m= = = 5x0 kg small change 8 c (3x0 m/s) in mass Binding energy The binding energy of the nucleus can be determined by measuring the mass of the components and the final product. E= mc energy + p + n -> D distance m = mass (hydrogen atom) + mass (neutron) mass (deuterium atom) m =.00785u +.008665u -.040u=.39x0-3 u u=.660559 x0-7 kg (atomic mass unit) E= mc = (.39x0-3 u)(.66x0-7 kg/u)(3x0 8 m/s) =3.6x0-3 J E=.x0 6 ev =. MeV Binding energy of the deuteron E Binding energy per nucleon (E/A) N neutrons+ Z protons -> Atom + energy Binding energy per nucleon (E/A) fission fusion average no. of nuclear contacts increases favor higher A average no. of contacts constant. Electrostatic forces favor lower A Goes through a maxima at 56 Fe Goes through a maxima at 56 Fe Fusion (combine small nuclei) increases binding energy Fission (break large nuclei) increases binding energy
Radioactivity Unstable nuclei decay releasing energy and radiation. Three types of radiation 4 alpha (α) particles - He nuclei (+ charge) beta (β) particles - electrons (- charge) positrons (+ charge) Radiation Penetration depth alpha particles Stopped by a sheet of paper beta particles - Stopped by a mm of aluminum gamma particles - Stopped by a few cm of lead gamma (γ) particles - high frequency electromagnetic radiation. (uncharged) Increasing penetration Measuring radiation Geiger Counter Natural radioactivity Many elements found in nature are unstable and decay emitting radioactivity. radiation These include Uranium, 38 U, Radon 4 Ra and Potassium 40 K. Carbon 4 C, ionization of gas causes current flow. Natural radioactive decay Decay rate The rate of decay, R, for N nuclei is proportional to N Thorium decay gives a variety of unstable products. N R = =λn t λ = decay constant (units time - ) Activity (measure of the rate of radioactive decay) Units Curie, Ci = 3.7x0 0 Decays/s 3
Radioactive decay Radioactive decay is a random process the amount of material remaining varies exponentially with time. t N= N e λ o Decay constant λ ( units /time) This can also be expressed as Half Life = N No t/t ln 0.693 T = = λ λ Example 9.3 The half life of radium Ra is.6x0 3 yr. If the sample contains 3.00x0 6 nuclei. Find the activity in curies. (Ci=3.7x0 0 decays/s) ln 0.693 T = = λ λ 0.693 0.693 λ= = 3 T/.6x0 yr(365day / yr)(4hr / day)(3600s /hr) =.37x0 s R=λN R=.37x0 - s - (3.00x0 6 nuclei) =4.x0 5 decays/s R=.x0-5 Ci or µcuries 3.7x0 0 decays/sci Example 9.3 The half life of radium Ra is.6x0 3 yr. If the sample contains 3.00x0 6 nuclei. Find the number of nuclei after 4.8x0 3 yr. Example 9.3 The half life of radium Ra is.6x0 3 yr. If the sample contains 3.00x0 6 nuclei. Find the activity after 4.8x0 3 yr. = N No t/t / N= 3.00x0 3 3 4.8x0 /.6x0 6 N=3.00x0 6 (/8)=3.75x0 5 nuclei R=λN After this time since the no. of nuclei is reduced by a factor of 8 the decay rate will also be reduced by a factor of 8. R = µci/8 =.4 µci Radioactive dating Smoke Detector 4 C is continually formed by cosmic rays in the upper atmosphere. n + 4 N -> 4 C + p so that the concentration of 4 C is relatively constant over long times ( longer than the half life). 4 C 4 C decays T // = 5730 years Ionization of air by a radioactive source produces a current. Smoke traps the electrons and reduces the current. setting off the alarm. The fraction of 4 C decays exponentially with time. 4
Medical Applications. Radiation Damage. Nuclear particles have much higher energies than chemical bonds. Radiation breaks chemical bonds forming reactive chemical species radicals. Reactive chemicals cause radiation damage to biological systems often reaction with DNA Radiation is often used in treating cancer. Radiation Therapy external radiation Radioimmunotherapy Iodine 3 Properties of 3 I New methods for can deliver radiation more specifically to target cells Half-life 8.07 days Beta particle maximum energy- 807 kev average energy - 8 kev Range in tissue -.4 mm Treatment of non-hodgkins lymphoma with radioimmunotherapy Common clinical applications Radioimmunotherapy, thyroid ablation for benign and malignant disease Medical Imaging X-ray Computer axial tomography (CAT) Positron emission tomography (PET) Magnetic resonance imaging (MRI) Contrast x-ray absorption may use heavy elements to increase contrast i.e I, Ba CAT scan Contrast Resolution. A three- dimensional image is reconstructed from many two dimensional pictures. 5
Computer tomography CAT scan of a chest incident x-ray a b detector c d detector detector 3 detector 4 at each detector absorbance is due to the sum of the absorptions from each segment. A (detector) = A i +A j For 4 detectors -> 4 linear equations Solve for and 4 segment - > 4 unknowns each absorbance Positron Emission Tomography Emission of gamma ray photons traveling in opposite direction by Positron-Electron annihilation. (conservation of momentum) Positrons are produced by decay of short lived radioactive nuclei such as 8 F (T / =0 min) F O+ e + +ν 8 8 9 8 PET imaging system e Annihilation produces 0.5 MeV photons + e γ+γ + hf hf Two coincident detectors are used to detect the gamma rays. The source is in a line directly between the two detectors. PET scan of the human brain 5 O ( T / = min) marks the consumption of O due to brain activity. 6