Lecture 9, March 29

• PET Physics Review

• Image of the Week

Velocity

P2 – P1

T2 – T1

In Words:

Change in position vs. time

Acceleration

• Change in Velocity over time• Increase or decrease in speed of a

particle• A linear accelerator “accelerates” a

particle to an extremely high velocity.• The particle then slams into a target

atom.

Conservation of Energy

• Total inflow of energy into a system must equal the total outflow of energy from the system.

• Protons: one positive unit of charge

• Mass = 1.67 x 10-24grams, 1.00759 amu

• Defines the Element

• Neutrons: neutral charge

• 1.00898 amu

• Electrons: one negative unit of charge

• Mass = 9.1 x 10-28 grams = 1/1835 that of a proton.

E = MC2

• Einstein’s Equation

• Mass can be converted into energy

Mass Defect

• Definition: There exists a difference in mass between constituent particles and the total mass of the atom.

• Example: C-12 is 12 amu

• Individually, the particles add up to 12.10266 amu

• This difference is the binding energy of the atom.

Binding Energies

• The mass defect distributes through various levels of binding energy in both the nucleus and electron shells of the atom

• To simplify, electron shells are denotedBy letter values for shells they occupy, innermost being the K shell.

• Binding energy keeps the electron in the shell, and is highest for the innermost electrons.

Binding Energy Continued

At the nuclear level, binding energy is derived from the strong nuclear force and is the energy required to disassemble a nucleus into neutrons and protons. At the atomic level, binding energy is derived from electromagnetic interaction and is the energy required to disassemble electrons from the atom.

Units of Energy

• Force = mass x acceleration• Unit of force: dyne = grams x cm/sec2 • Erg: A unit of energy equal to the work done when

a force of one dyne acts through a distance of one centimeter.

• Electron Volt: The energy acquired by an electron falling through a potential difference of one volt, approximately 1.6 × 10-12 ergs

• Mass Defect of C-12 = 95.62779 MeV

Some Quantum Theory

• E = mc2

• Energy =

Mass in grams x speed of light squared (c2)

However, this is the expression for erg.

Therefore, in Einstein’s Equation, energy is given in ergs.

Quantum States

An electron does not stay in an excited state for very long - it soon returns to the ground states. When it does so, a photon is emitted that has the same energy as the difference in the energy level between the excited state and the ground state

Line Spectra From Hydrogen

Electromagnetic Radiation

Model of Shape of Electromagnetic

Radiation “Wave Function”

How Does Energy Enter Into The Equation?

We need a constant that gives us the energy per cycle during one second.

Discovered by Max Planck:

Planck’s constant: h

h = 6.626 x 10-34 J . sec

= 6.626 x 10-27 erg . sec

= 4.136 x 10-15 eV.sec

Energy of a Photon

• = h x f = 4.136 x 10-15 eV. sec x f

• In words, Planck’s Constant times the frequency

• But f = c/lambda

• Therefore, E can also be expressed as

(h x c)/ lambda = 4.136 x 10-15 eV . sec x

3.0 x 1010 cm/sec = 12.4/lambda

Calculations

• Given the frequency, or wavelength, of a photon, the Energy can be calculated.

• Likewise, given the energy, the frequency and wavelength can be calculated.

Example #1

• Find the energy of a photon with a wavelength of: 4.136 x 10-5 cm

• E = (h x c)/ lambda = 12.408/ lambda

• = 12.408 eV . cm/4.136 x 10-5cm

• = 300,000 eV = 300KeV = 3 Mev

Electromagnetic Photon Emission

Types: Gamma Radiation

X-radiation

Beta Plus Decay

• Also called “Positron Decay.”

• Occurs most often in lighter nuclei.

Positron Decay

Electron Capture

• Electron capture is one form of radioactivity. A parent nucleus may capture one of its orbital

electrons and emit a neutrino. This is a process which competes with positron emission and has

the same effect on the atomic number. Most commonly, it is a K-shell electron which is

captured, and this is referred to as K-capture. A typical example is

•

EC

Electron Capture Continued

• In the middle range of the periodic table, those isotopes which are lighter than the most stable isotopes tend to decay by electron capture, and those heavier decay by negative beta decay. An example of this pattern is seen with silver isotopes, with two stable isotopes plus one of lower mass which decays by electron capture and one of heavier mass which decays by beta emission.

Decay Schemes

Graphical Illustration of Decay Process Depicts Energy Levels from Excitation

Energy to Ground StateDirection of Arrows Identifies Type of

Radiation

Example of Decay Scheme

Production of Positron Emitters

Radioactive Materials • Our world is radioactive and has been since it was created.

Over 60 radionuclides (radioactive elements) can be found in nature, and they can be placed in three general categories:

– Primordial - from before the creation of the Earth

– Cosmogenic - formed as a result of cosmic ray interactions

– Human produced - enhanced or formed due to human actions (minor amounts compared to natural)

Requirements for Nuclear Medicine

Unfortunately, the radioactive materials found in nature are of really no utility in nuclear medicine imaging

For imaging, we need:

short half life, optimal gamma energy, good statistics (# of available photons)

Radionuclide Production

Methods by which radionuclides are produced. Radionuclides can be produced in a nuclear reactor, in a cyclotron or in a radionuclide generator.

Some Terms

• Flux: # of neutrons, photons, etc, passing through one cm2/instant of time

• Fluence: # of neutrons, photons, etc, that passed through one cm2 over a period of time.

• Cross Section: a probability of interaction, and thus transmutation after target bombardment.

Particle Accelerators (Cyclotrons)

Particle AcceleratorDid you know that you have a type of particle accelerator in your house right now? In fact, you are reading this slide with one! The cathode ray tube (CRT) of any TV or computer monitor is really a particle accelerator.

CRT Example

The CRT takes particles (electrons) from the cathode, speeds them up and changes their direction using electromagnets in a vacuum and then smashes them into phosphor molecules on the screen. The collision results in a lighted spot, or pixel, on your TV or computer monitor.

CRT Diagram

A cyclotron consists of a pair of hollow, semicircular metal electrodes (called "dees" because of their shape), positioned between the poles of a large electromagnet (not shown). The dees are

separated from one another by a narrow gap. Near the center of the dees is an ion source (typically an electrical arc device in a gas) that is used to generate charged particles.

Cyclotron

Principle of Operation

Period, T, of revolution is constantVelocity increases with each revolution.Radius of revolution increases with each

new period.This enable the application of alternating

electric fields, thus accelerating the particles.

Cyclotron Produced Radionuclides

• Cyclotron produced radionuclides include all PET nuclides in common use such as fluorine F -18, oxygen O -15, nitrogen N -13 and carbon C -11, which are activated by proton irradiation.

Reaction Equations

• X (n, p) Y

• Means neutron, proton reaction

• X, the parent nucleus is bombarded with a neutron.

• The product nucleus decays with the emission of a proton.

PET/CT Virtual Bronchoscopy