Atom, Nucleus, and Radi-ation

Lec 2 of Intro Rad Sci March 11 2014

Electromagnetic RadiationWave viewpoint

• Changing B induces E• Changing E induces B• The inextricable exchange causes E and B fields to

propagate outward at the speed of light• c = 3 × 108 m/s in vacuum

(courtesy Dr. Naqvi)

Electromagnetic Spectrum

keV- MeV

~ eV range

Electromagnetic RadiationQuantum viewpoint

• A photon is a “packet” or “quantum” of EM radiation

• The photon energy, hn, is proportional to the frequency, and hence inversely proportional to the wavelength, l

• A photon has zero rest mass (m0c2=0), and can therefore travel, and hence according to relativity, can travel only at the speed of light, c

E = hnh = Planck’s constant = 6.63 × 10-34 J/Hz

(courtesy Dr. Naqvi)

Plum Pudding Model• In mid-nineteenth century, optical spectroscopy

– Balmer’s empirical formula Eq. (2.1) for visible spectra of H was derived theoretically by Bohr in 1913

– Eq. (2.1) was visible, and by replacing 22 by 12 or 32 (and 42, …) was ultraviolet or infrared, respectively

• J. J. Thomson in 1897– Charge-to-mass ratio of cathode rays (only ~1/1700 of H)– Atom ~ plum pudding model– Ionization by radiation

Rutherford Nuclear Atom• In 1909, large-angle deflection of α-ptls (as probes) was

evidence for the existence of a very small & massive nu-cleus of + charge– Planetary model with mostly empty space– Light e- move rapidly about the nucleus

• Nuclear force vs. Atomic force– Saturate within ~ 10-15 m vs. not saturate – i.e., a given nucleon interacts with only a few others vs. all pairs

of charges interact with one another– Radius of nucleus ≈ 1.3·A1/3 ×10-15 m vs. Atomic size of all ele-

ments is more or less the same (~ 10-10 m)

Bohr’s Theory of Hydrogen Atom• An accelerated charge emits EM radiation, but• Bohr’s theory

– w/o radiating only in certain discrete orbits about the nucleus (2.3)

– transition of e- from one orbit to another → emission or ab-sorption of a photon of orbital energy lost or gained by e- (2.4)

• Some definitions– Bohr radius– Fine-structure constant (1/137)– Ionization potential– Rydberg constant, RM & R∞

Energy Levels of Hydrogen Atom

Ionization Continuum

n = 1

n = 2

n = 3n = 4

-13.6 eV

0 eV

Lyman Series (ultraviolet)

Balmer Series (visible)

The normal condition of the atom, or ground state, is the state with n = 1 The atom is in it’s lowest possible energy state and it’s most stable condition

Problem with Bohr’s model & classical mechanics

• Only correct for the energy levels of H & He+

• Semi-classical mechanics, i.e., mixing classical me-chanic w/ quantizing certain variables + relativistic models

• de Broglie’s wave/particle dualism– X-ray diffraction vs. Compton scattering– e- diffraction in Ni-crystal– Optical microscopy vs. electron microscopy (SEM, TEM)

Quantum Mechanics• Heisenberg’s uncertainty principle (matrix mechanics)

Δp·Δx ≥ ħ & ΔE·Δt ≥ ħ in 1925– Uncertainties in momentum of e- in atomic orbit (10-10 m) and nu-

cleus (10-15 m) vs. position: a few hundred MeV vs. a few eV – But, betas from nuclei ~ a few MeV → neutron in 1932

• Schroedinger’s wave mechanics, 2πr = nλ, n = 1, 2, 3 ... – Linear differential equ (2nd order in space & 1st order in time) >

superposition > wave packets > ptl– Boundary condition > eigenvalue > discrete energy– Dirac’s 1st order in space & time for relativistic motion

Atomic StructureBohr vs. Modern Quantum Models

Find the energy gained by an electron (in eV) when acceler-ated through a potential difference of 50 kV in an x-ray tube

3

. .

. . (1 ) (50 10 )e V = 5 ke0 V

k e qV

k e

- +

Application: x-ray tube

50 kV

(x-ray tube pictureCourtesy ofHyperphysics)

CathodeAnode

Bremsstrahlung X-rays

Characteristic X-rays

Bremsstrahlung and Characteris-tic X-rays

Nucleus of Atom, XAz

• Assemblage of neutrons and protons clustered in a nucleus and surrounded by electrons whirling in a variety of orbits

• Atomic number, Z = No. of protons• Mass number, A = No. of nucleons (protons, Z

plus neutrons, N = A – Z)

Ir, IrIr, -192 (ii) IrIr, 192, -Ir Ir, (i) :Note

77192

77

115192

77192192

77

Isotopes, Isomers, Isobars• Isotopes = elements having the same Z

but different A, e.g., 131I, 125I, 127I• Isomers = identical elements, but different

nuclear energy states, e.g.,

• Isobars = elements having the same A but different Z

• Isotones = elements having the same N

TcTc, 9999m

Nuclear Structure and ForcesTUG-of-WAR between the

ATTRACTIVE STRONG NUCLEAR FORCEand the

REPULSIVE ELECTROMAGNETIC FORCE

Binding Energy

• The nucleons (protons & neutrons) are bound together by a net force which NUCLEAR ATTRACTION forces exceed the ELECTROSTATIC (COULOMB) REPUL-SION forces. Associated with the net force is a POTEN-TIAL ENERGY of BINDING

• In order to separate the nucleus into its component nu-cleons, energy must be supplied from the outside

• Binding Energy (BE) = total mass of separate particles - mass of the atom

Binding Energy

Natural Radioactive Series

Auger electron [1/3]

• Physical phenomenon in which the transition of an e- in an atom filling in an inner-shell vacancy causes the emission of another e-

• Releasing an energy equal to the dif-ference in binding energies, EK–ELI.

• As the alternative to photon emis-sion, this energy can be transferred to an LIII e-, ejecting it from the atom with a K.E. = EK – ELI – ELIII

• Emission of an Auger e- increases the number of vacancies in the atomic shells by one unit Vacancy by

P.E., internal conversion,PIXE, or orbital e- capture

Auger electron [2/3]

• K fluorescence yield = No. of K X-ray photons emitted per vacancy in K shell

• Auger cascades can occur in rela-tively heavy atoms, as inner-shell va-cancies are successively filled by the Auger process, with simultaneous ejection of more loosely bound atomic e-’s

• An original, singly charged ion with one inner-shell vacancy can thus be converted into a highly charged ion by an Auger cascade

Auger e

- Yield

Auger electron [3/3]

• 125I decays by electron capture. The ensuing cas-cade can release some 20 e-s, depositing a large amount of energy (~1 keV) within a few nanometers

• A highly charged 125Te ion is left behind; DNA strand breaks, chromatid aberra-tions, mutations, bacterio-phage inactivation, and cell killing

Gamma Emission vs. Internal Conversion

• Excited daughter nucleus decays to the stable nucleus via either g-emission or internal conversion– g-emission:

• isomeric (Z & A unchanged) • discrete in g-spectrum

– Internal Conversion (IC): • process in which the energy of an excited nuclear state is trans-

ferred to an atomic e-, most likely a K- or L-shell e-

• Ee = E* -EB

• atomic inner-shell vacancies and thus emits characteristic X-ray• isomeric (Z & A unchanged)• dominant in heavy nuclei with low-lying excited state (small E*)

Gamma Emission vs. Internal Conversion

Bam13756

10% IC

Eavg of emitting beta = 1/3 Emax

A long-lived excited nuclear state is termed metastable and is designated by the symbol m:e.g.,

Ru9944

Orbital Electron Capture

• Inverse beta decay• too many protons and insufficient energy to

emit a positron(>1.022MeV)• p+  +  e-  →  n0  +  νe

• usually from the K or L electron shell (K-electron capture, also K-capture, or L-elec-tron capture, L-capture)

• QEC=∆P-∆D-EB

• Characteristic X-rays & Auger e-

Radioactive Decay

teANA ll 0

• N, No of unstable nuclei left at time, t

• A, Activity (Bq or Ci) at time, t

dtNdN l teNN l

0

l= decay constant [s-1]N0 = initial No of unstable nuclei

Relation between half-life and decay constant

2-t/T1/2

e-lt

HALF-LIFE (T) REPRESENTATION

DECAY CONSTANT (l) REPRESENTATION

T1/2 is the time taken for 50% of the atoms to survive

1/l is the time taken for the fraction 1/e (37%) of the atoms to survive (i.e., mean-life time, t).

T1/2 = ln(2) / l = 0.693 / l

Point Source in Vacuum

N [photons/s]

(1/12 ) ∙ I0 = N/A [photons/s/cm2]

(1/3)2 ∙ I0 = N/(32A )[photons/s/cm2]

(1/22 ) ∙ I0 = N/(22A) [photons/s/cm2]

020 )4

1()Area

1( Ir

II

Photon Beam Attenuation

drINdI

rNIdIr

0

r

rN

eI

eII

0

0

= σ •N = linear attenuation coefficient [cm-1]

Source

r [cm] dr

I(r)

I + dII0 [photons/cm2]

N [atoms or electrons/cm3]

Collimator Collimator

Detector

0

Photon Fluence in Matter• Point source in matter (collimated)

• Point source in matter (Broad)

2'02

0

4 reI

reII

rr

BFr

eIBFr

eIIrr

2'02

0

4

Relation between half-value layer and attn coefficient

2-x/HVL

e-x

HALF-VALUE LAYER (HVL) REPRESENTATION

ATTENUATION COEFFICIENT () REPRESENTATION

HVL is the thickness taken for 50% of the photons to survive

1/ is the thickness taken for the fraction 1/e (37%) of the photons to survive (i.e., mean-free path, xm).

HVL = ln(2) / 0.693 /

Linear vs. Semi-log Plotting of e-

µx or e-lt

Mono-energetic photons, µ = constant and, thus HVL = constant

Linear Semi-log

Beam Hardening: Selective Absorption of Low-Energy Photons

10

100

0 1 2 3 4 5 6Absorber Thickness (mm AL)

Tran

smitt

ance

(%)

1st HVL = 0.99mm

2nd HVL = 1.99 mm

3rd HVL = 2 mm

1st HVL < 2nd HVL < 3rd HVL

Ē1st < Ē2nd <Ē3rd

Parallels between nuclear decay and photon attenuation

Parallels of Exponential Behavior

Note: (i) The exponent of exp, e.g. D/D0, µx, lt, should be dimensionless

e.g., λ, T1/2, μ, and HVL• Which of the following

expressions is most ap-propriate?

a) A =A0·e-λ/t, I =I0·e-μ·x

b) A =A0·2-λ·t, I =I0·2-x/HVL

c) A =A0·e-t/T1/2, I =I0·2-μ·x

d) A =A0·e-ln2·t/T1/2, I =I0·2-ln2·x/HVL

e) A =A0·2-t/T1/2, I =I0·e-ln2·x/HVL

e.g., Mass and Atomic No. An atom of Zn-65 has a mass number of 65.

A. Number of protons in the zinc atom 1) 30 2) 35 3) 65

B. Number of neutrons in the zinc atom 1) 30 2) 35 3) 65

C. What is the mass number of a zinc isotope with 37 neutrons?

1) 37 2) 65 3) 67

H.W. #1• Calculate Q-value for 10B(n,α)7Li reaction• Calculate the atomic density of sodium, 0.97 g/cm3

• A sample contains 1.0 GBq of 90Sr and 0.62 GBq of 90Y.– What will be the total activity of the sample 10 days later?– What will be the total activity of the sample 29.12 years later?

• A high-energy e- strikes a lead atom and eject one of K-e-’s from the atom. – What wavelength radiation is emitted when an outer e- drops into the va-

cancy?– What is the probability for Auger e- emitted?

• Calculate the recoil energy of the technetium atom as a result of photon emission in the isomeric transition

• Find the binding energy of the nuclide 24Na • Turner Chap. 2: 11, 12, 13, 14, 15, 18, 19, 36, 37, 43, 53, 56• Turner Chap. 3: 3, 4, 8, 11, 17, 29