Nuclear Physics ---------------------------------- Joseph F. Alward, PhD Department of Physics University of the Pacific

The Shroud of Turin

  Important Terms

atomic number: number of protons neutron number:  number of neutrons nucleon:  proton or neutron in nucleus atomic mass:  number of nucleons fermi:  10-15 m strong force:  extremely short-range force between nucleons binding energy:  the energy needed to break up a nucleus into its various nucleons atomic mass unit, u:  mass equivalent of 931.5 MeV of energy carbon-14: commonly used radioactive isotope of carbon used to date organic matter alpha particle:  a helium nucleus, commonly emitted in many radioactive disintegrations beta particle:  an electron, emitted in some radioactive disintegrations gamma ray:  a high-enery photon, extremely harmful to living organisms geiger counter:  device for measuring radioactivity scintillation counter: device for measuring radioactivity MRI, magnetic resonance imaging:  based on energy levels of hydrogen nucleus half-life, T1/2:  the amount of time it takes for half a sample of radioactive material to decay N(t):  population at time t A(t):  activity, the number of disintegrations per second decay constant:  used in the equations N(t) = N0 e-lt   and Activity = lN becquerel:  one disintegration per second curie:  another unit of activity, the number of disintegrations per second per gram of radium

   Nuclear Structure

Z = Atomic number N = Number of neutrons A = Atomic mass A = Z + N

 Relative Size of Nuclei

Enrico Fermi           (1901-1954) ------------------------------- One fermi (f) = 10-15 m

  Radius versus Atomic Mass

r = 1.2 A1/3     (in f) ------------------------- Helium:  A = 4     r = 1.2 (4)1/3        = 1.9 f ------------------------- Uranium:  A = 238     r = 1.2 (238)1/3        = 7.4 f

  The Strong Force

Protons and neutrons in the nucleus  are collectively referred to as  nucleons.

Protons which would otherwise strongly repel at close distances are held in place by an extremely strong, but extremely short range force called the strong force. Other names for the strong force are strong nuclear force, or nuclear force. The strong force between two protons is about the same as the strong force between two neutrons, or a proton and a neutron.

Beyond about one fermi the strong force declines extremely rapidly. As more protons are added to the nucleus, more neutrons are needed to bind the protons together, but the larger the nucleus becomes, the farther apart are the protons and the less effective is the strong force.

  Neutron Number versus Proton Number

Electric force is longer range than the strong force. Eventually separation becomes too great for the strong force to compensate for the repulsive forces. Nucleii spontaneously disintegrate for proton numbers larger than 83. The release of light and or particles which accompanies the disintegration is called radiation, first discovered by Henri Becquerel in 1896.

  The Binding Energy of a Nucleus 

The larger the binding energy of a nucleus, the more stable it is. The binding energy is the difference between the rest energies.

  Atomic Mass Unit

One atomic mass unit (amu) = 1.6605 x 10-27 kg --------------------------------------------------------------------  E = (1.6605 x 10-27 kg) (3 x 108 m/s)2     = 1.49 x 10-10 J 1.49 x 10-10 J / 1.6 x 10-19 J /eV = 9.31 x 108 eV                                                       931  x 106 eV                                                    = 931 MeV   one amu = 931 MeV     An amu is often abbreviated u.

  The Binding Energy of Helium

Dm = 4.0330 - 4.0026        = 0.0304 u E = (931 MeV /u) 0.0304 u    = 28.3 MeV There are four nucleons, so the binding energy per nucleon is about 28/4, or about 7 MeV per nucleon.

  Binding Energy per Nucleon

Nuclei with the largest binding energy per nucleon are the most stable. ----------------------------------------------------- The largest binding energy per nucleon is 8.7 MeV, for mass number A = 60. Beyond bismuth, A = 209, nuclei are unstable.

  Alpha Particle Emission

The alpha particle is a helium nucleus.

 Balancing Nuclear Decay Equations

92U238 --------> 90Th234  +  2He4 ------------------------------------------------- Subscripts are "proton numbers" Superscripts are "nucleon numbers"

Proton and nucleon counts must be the same: 92 = 90 + 2 238 = 234 + 4

   Distribution of Energy in Alpha Emission

Dm = 0.0046 u E = 0.0046 x 931    = 4.3 MeV ----------------------- Which particle has the greater kinetic energy?

  Energy Distribution in Radioactive Decay

Ratio of kinetic energies:  KEm / KEM:   (1/2 mV2) / (1/2 Mv2) = (m/M)(V2/v2)                                      = (m/M)(V/v)2        (1)

Conservation of momentum: Mv = mV                             (2) Rearranging, we get V/v = M/m                           (3)     Substitute (3) into (1): Ratio = (m/M)(M/m)2         (4)          = M/m Smaller mass gets more energy

   Radioactivity in Radium

In "balancing" a nuclear disintegration equation, note that the   subscripts and superscripts add up.

  Marie Curie

Marie Sklodowska Curie                 (1867-1934)

Lithograph entitled "Radium" appeared in the December 22, 1904 issue of Vanity Fair.

Marie and Pierre Curie isolated 1/30 ounce of radium from one ton of uranium ore.   She alsodiscovered polonium, which she named after her native country. Marie contracted leukemia from extended contact with the radium; the pages of her lab notebook were later found to be contaminated with radioactive fingerprints.

  Plutonium Powered Light Sphere

94Pu239 -----> 92U235 + a

Six ounces (170 grams) of plutonium dioxide inside graphite-iridium container. Alpha particles colliding with graphic casing heats it to 1000 degrees centigrade. Sphere radiates about 100 watts of light energy, and will continue to do so for decades.

  Smoke Detector

Alpha particles emitted from source ionize the air and provide the charge necessary to conduct current through the air. Charges stick to the heavy smoke particles and the current drops, causing the alarm to buzz.

   Beta Particle (Electron) Emission

90Th234   ------>  91Pa234  + -1e0

The neutron number of an electron is zero, and the proton number is negative one.   Negative beta particles are emitted when a neutron is transformed into a proton and an electron.

   Beta Particle (Electron) Emission by Carbon-14

6C14 -----> 7N14 + -1e0 The subscripts represent the "proton" number (electrons have a negative) proton number. Superscripts represent the nucleon number; electrons are not nucleons, so their nucleon number is zero.

  Beta Particle (Positron) Emission by Oxygen-15

A positron has the same mass as the electron, but  opposite charge.

8O15 -----> 7N15 + 1e0 The subscripts represent the "proton" number (a positron has a positive proton number) Superscripts represent the nucleon number; positrons are not nucleons, so their nucleon number is zero.

   Reaching Stability Through Gamma Ray Emission

Nucleii with excess energy emit gamma-rays, which are extremely short- wavelength electro- magnetic waves, i.e., very high energy photons. 1 MeV is typical.

  Wavelength of a Gamma Ray

What is the wavelength of a 1 MeV gamma ray? Using the 1234 rule: l = 1234 eV-nm / E    = 1234 eV-nm / 1 x 106 eV    = 1.23 x 10-6 nm    = 1.23 x 10-15 m    = 1.23 fermi This gamma radiation is extraordinarily harmful to humans and other living things since its wavelength is comparable to the diameter of a nucleon; transmutations are likely when such radiation reaches nuclei.

 Brain Surgery with the Gamma Knife

  The Geiger Counter

Hans Geiger invented the "Geiger counter". ---------------------------------- It was Hans Geiger who, while working in Ernest Rutherford's lab, was the first to see the alpha particles reverse direction in the alpha particle experiment, but it was Rutherford's calculation which proved the existence of the nucleus.

  The Scintillation Counter

Scintillator is material which will emit photons when struck by high energy charged particles or high energy photons. Photon strikes metal plate, ejecting electrons which are pulled toward 100 V anode. The anode is coated with a material which is easily ionizable and releases two or more electrons for each one that strikes it.

 Hydrogen Energy Levels in Magnetic Field

  Magnetic Resonance Imaging (MRI)

DC current coils create a non-uniform magnetic field B. Hydrogen nucleii have energy levels whose separation is proportional to B AC current coils emit electromagnetic which is absorbed and then reemitted by the hydrogen nuclei Radio-frequency waves are much safer than X-rays.

 MRI Laboratory

  MRI Image

   Half-Life 

T1/2 = time for half the sample             to disintegrate ---------------------------------------- T1/2   =  5 years ---------------------------------------- Number of nucleii present at time t = 0: N0     = 1000 --------------------------------------- When t = 5 yrs,    N = 500           t = 10 yrs,   N = 250           t = 20 yrs,   N = 125.

  The Uranium Decay Series

About half of the uranium created at the time of the creation of the universe billions of years ago is still around. The only radium that exists today is that which is created as a result of the decay of uranium.

  Measuring the Age of Organic Matter

A German tourist in the Italian Alps discovered the remains of the "Iceman" in the ice of a glacier in 1991.

  Calculating the Iceman's Age

The current activity per gram of carbon  is 0.23 Bq per gram. Iceman's carbon  showed 0.121, or about half what it  would be if the Iceman were alive. Since the half-life of carbon-14 is  about 5700 years, the Iceman's remains are about 5700 years old.

 The Shroud of Turin

Since the1354 AD, a yellowing piece of linen 14-ft long has been stored in Turin, Italy.   It bears the image of a person who seems to be wearing a crown of thorns. Could the Shroud of Turin have been the burial cloth of a person who died two thousand years ago?

  Dating of the Shroud of Turin

At the time of the public exhibition of the shroud in 1354, a bishop declared it to be fraud.  Most religious bodies take a neutral stance on the shroud's authenticity. In 1988, three laboratories were given four pieces of fabric; three were control pieces similar in appearance, and one was a piece from the shroud.  The labs all agreed that the shroud was 608-728 years old, which means that it came into existence sometime between 1260 and 1380 AD, a time span which includes the year the shroud was first shown to the public.

 Radioactivity Equations

N(t)  =  population at time t N(0) =  population at time zero N0   =   N(0) l     =   decay constant   N(t) = N0 e-lt

Example:    N0 = 1000 l = 2 x 10-3 years -1 When will N = 200? ------------------------------------ N   = N0 e-lt                     (1) e-lt = N /N0                      (2) ln (e-lt) = ln (N /N0)         (3)    -l t = ln (N /N0)             (4)

(continued) t = - [ln (N /N0)] / l                      (5)   = - [ln (200/1000)] /2 x10-3      (6)   =  805 years

  Radioactivity Units

A  =  number of disintegrations          per second, activity A  = lN       One becquerel (Bq) is one disintegration per second.   One curie is the number of disintegrations per second (the "activity") of one gram of radium, or about 3.7 x 10 10 Bq.

 The Activity of Carbon

One gram of carbon contains mainly carbon-12. To find the number of  carbon-14 atoms, divide the number of carbon atoms in the sample by 8.3 x 1011. One gram of carbon from a living organism has an activity of 0.23 Bq.

    Half-Life Problem

The half-life of a radioactive substance is 10 hours.   What is the decay constant, l? -------------------------------------------------------- N = N0 e-lt                                    (1) 0.50 N0 = N0 e-l10                       (2) e-l10 = 0.50                                  (3) ln(e-l10) = ln(0.50)                       (4)     -10 l  = -0.693                        (5)           l   = 0.0693 hrs-1              (6)

How much time will it take for the sample's activity to fall to only 20% of what it was originally? ---------------------------------------------- N = 0.20 N0                              (7) 0.20 N0 = N0 e-0.0693 t             (8) -0.0693 t = ln (0.20)                  (9)             t = 23 hours

  Decay Constant
  and Half-Life

N = N0 e-lt                    (1) 0.50 N0 =  N0 e-lT                  (2) (T = half-life, usually T1/2)       e-lT =  0.50                       (3)  ln(e-lT) =  ln(0.50)                   (4)        -lT =  -0.693                    (5)             T =  0.693/l                   (6)           l  = 0.693/T                   (7)

 Half-Life Example

38Sr90 (strontium-90) has a half-life of 28.5 years. How long will it take for 98% of a sample of strontium-90 to disappear? ------------------------------------------------------------------ l  = 0.693/T1/2     = 0.693 / 28.5     = 0.0243 years-1

(continued) 0.02 = e-0.0243 t t = - ln(0.02) /0.0243 years-1   =  161 years