Quantum Physical Phenomena in
Life (and Medical) Sciences
IV. Röntgen (X-) rays
Activated
tube
Péter Maróti
Professor of Biophysics, University of Szeged, Hungary
Suggested texts:
S. Damjanovich, J. Fidy and J. Szőlősi: Medical Biophysics, Semmelweis, Budapest 2006
P. Maróti, L. Berkes and F. Tölgyesi: Biophysics Problems, A Textbook with Answers, Akadémiai Kiadó, Bp. 1998.
Schoolexample for
very fast and
successful
innovation
X-ray quantum
energy
in diagnostics:
30-200 keV
in therapy:
5-20 MeV
Hand mit Ringen (Hand with
Rings): print of Wilhelm
Röntgen’s first "medical" X-ray, of
his wife's hand, taken on 22
December 1895.
Properties of the X-rays
Röntgen demonstrated that X-rays:
- produced luminescence from the wall of the discharge tube,
- traveled in straight lines,
- were not deflected by a magnetic field,
- were absorbed more by denser metals,
- were scattered when passing through a body and
- could ionize gases.
For complete description of the X-ray, we need
- the anode voltage: it determines the energy of the X-ray quanta (photons),
- the composition (type) of material of the anode: it determines the wavelength
of the characteristic radiation and
- the quality and thickness of the filters: they determine the hardness (softness)
of the radiation.
This talk concentrates on the physical bases of the X-radiation and the very important
medical applications in diagnostics (e.g. CT) and therapy are left for other lectures.
Generation of X-rays
X-ray beams from small spot size
Coolidge (high vacuum) tube
Double focus x-ray tube
The smaller is
F the sharper
is the imaging
spot size, F
D
object
R = F·d/D
d
R
The smaller
is R the
larger is the
contrast
Short exposure
(long filement)
long exposure
(short filement)
X-ray tubes
Coolidge X-ray tube, from around 1917. The heated cathode is on the left, and the
anode is right. The X-rays are emitted downwards.
Different X-ray tubes
bulb diameter 7 cm
The larger tube has a length of about
25 centimeter. It has a regulator, a
small glass compartment on the tube
with a piece of charcoal which could
be heated to correct the internal gas
vacuum.
The small tube measures about 15
centimeter with 5 cm bulb.
Rotating anode tube
The anode can be rotated by
electromagnetic induction from
a series of stator windings
outside the evacuated tube.
A: Anode
C: cathode
T: Anode target
W: X-ray window
X-ray generation by accelerators
Linac
The linear accelerator, or linac, is
the electromagnetic catapult that
brings electrons from a standing
start to relativistic velocity - a
velocity near the speed of light.
The buncher accelerates the pulsing
electrons as they come out of the
electron gun and pack them into
bunches. To do this the buncher
receives powerful microwave
radiation which accelerates the
electrons in somewhat the same way
that ocean waves accelerate surfers on
surfboards.
When the
cyclotron
principle is
used to
accelerated
electrons, it
has been
historically
called a
betatron.
The betatron (Donald Kerst, 1940) is a
device for accelerating electrons to higher
speeds as they move in a circular orbit.
The main question is: how can one keep
the electrons in a circular orbit as they
move faster and faster?
The portable X-ray betatron is a compact
circular electron accelerator producing a
high energy directional X-ray beam.
etarget
α
The (angular) divergence of the X-ray:
Basic Technical Data
X-Ray output energy selector: 2 to 6 MeV
Dose rate at 3' (air): 3R (3cGY) per minute
Focal spot size: .01" x .039"
Radiation beam spread angle: 26 degrees
Radiographic sensitivity: down to 0.5%
AC power input: 110/240V 50/60 Hz
α = me·c2 /E
me is the mass of the electron,
c is the speed of the light and
E is the energy of the accelerated
electrons when they leave the betatron.
X-ray spectra
Characteristic radiation: the
energy levels of the electrons are
quantized, thus the produced
radiation is also quantized,
demonstrates spectral lines which
are characteristic on the target
material (element).
Bremsstrahlung: Incident electrons
are deflected by the negative charge
of electrons in the target. Any change
in velocity (speed or direction or both)
is an acceleration. Accelerating a
charge emits radiation. The deflected
electrons’ acceleration is NOT
QUANTIZED. Thus, the spectrum is
continuous.
A section from the previous 3D spectrum at constant tube voltage (60 kV)
Kα
characteristic
K lines
Kβ
Bremsstrahlung
λmin
Spectrum of the X-rays emitted by an X-ray tube with a rhodium target, operated at 60
kV. The smooth, continuous curve is due to Bremsstrahlung, and the spikes are
characteristic K lines for rhodium atoms.
The X-rays are extra-nuclear and have two sources: (1). A characteristic (e.g. K- or L-)
X-ray photon is created and emitted when an electron drops down orbits to fill a vacancy
in an innermost shell. The wavelength of the photon is unique to the atom and affords an
unambigious method of identifying the element by use of an X-ray spectrometer .
(2). A bremsstrahlung photon is produced when a high speed electron decelerates when
passing through the electric field in the close vicinity of an atomic nucleus. The spectrum
in this case is continuous up to the maximum energy of the bombarding electrons.
Braking radiation („Bremsstrahlung”):
Duane–Hunt displacement law
It gives the maximum frequency of X-rays that can be emitted by Bremsstrahlung in an
X-ray tube by accelerating electrons of electric charge e through an excitation voltage V
into a metal target.
The maximum frequency νmax is given by
νmax = eV/h
which corresponds to a minimum wavelength
λmin = (hc)/(eV)
where h is the Planck’s constant and c is the speed of light.
The law is in effect a statement of conservation of energy. The maximum frequency
(minimum wavelength) is that at which all of the electrical energy E = eV given to the
electron is successfully converted into radiation energy E = hν. This process is also
known as the inverse photoelectric effect.
Bremsstrahlung: emitted power, efficiency
The total emitted energy (or power, if the radiation is stationary) is the area under
¥
the curve:
Wtotal =
ò
0
DN
× dE
DE
Let’s replace the function by straight line:
DN
» const × Z × (Emax - E )
DE
and calculate the integral (the area of the triangle):
Emax
Wtotal » const × Z ×
ò (E
0
After expressing Emax from the anode voltage of the tube:
max
2
Emax
- E )dE = const × Z ×
2
2
Wtotal = const/2 × Z × e 2 Vanode
The total emitted power increases proportionally with the number of electrons impacting the
anode within unit of time, i.e. with the anode current:
2
Ptotal = cRtg × Z I anode Vanode
Here Z is the atomic number of the material of the anode, Vanode is the accelerating voltage, Ianode
is the anode current and cRtg ≈ 1.1·10-9 V-1. The total emitted power is proportional to the square
of the anode (tube) voltage. The radiation production efficiency in an X-ray tube:
2
cRtg × Z I anode Vanode
Ptotal
=
= cRtg × Z Vanode
h=
Pinvested
I anode Vanode
For Tungsten anode at Vanode = 100 kV, η ≈ 0.008 < 1%. The energy is largely transforms to heat.
Characteristic radiation:
the Moseley's Law
the heavier
elements
have up to
three lines
The frequency of characteristic K-series X-rays
produced from tubes with different anode materials
(Ca to Zn) was measured. The ordering of the
wavelengths of the X-ray emissions of the elements
coincided with the ordering of the elements by
atomic number.
the lighter elements
have two distinctive
characteristic X-ray There is linear relationship between the atomic
lines
number of the element the anode consists of and the
square root of the frequency.
Moseley's law as an empirical law
n = k1 × (Z - k2 )
where ν is the frequency of the
main or K x-ray emission line,
k1 and k2 are constants that
depend on the type of the line.
For example, the values for k1
and k2 are the same for all Kα
lines, so the formula can be
rewritten thus
ν = 2.47·1015 ·(Z - 1)2 Hz
Moseley’s law: derived from Bohr’s atomic
structure and spectroscopic term system
The energy of the X-ray lines (as difference of two energy terms) can be derived in a very similar
way as done with the optical spectral lines of the H-atom in the Bohr model. The wavenumber of
the transition is
æ 1
ö
1
= n = R (Z - 1) × ç 2 - 2 ÷ × s
çn
÷
l
è f ni ø
1
2
where R is the Rydberg constant (1.097·107 m-1), Z is the atomic number, n is the main quantum
number in the final (f) and initial (i) states and σ is a constant (≈ 1).
initial
final
As the transitions
responsible for the X-rays
occur in the inner shell, the
atomic number (thus the
electric charge) of the
nucleus (Z) should have
strong influence on the
spectroscopic terms. This
is why Z appears in the
expression of the Röntgen
lines (Moseley’s law) but
not in those of the optical
spectral lines (Balmer’s
expression).
λmin
Problem
You are operating an X-ray tube with a chromium (Cr) target by applying an acceleration
potential of 60 kV. Draw a schematic of the x-ray spectrum emitted by this tube and label
on it the characteristic wavelengths of the Kα line of the characteristic radiation and
minimum wavelength of the Bremsstrahlung (λmin).
Solution
The characteristic X-ray spectrum of Cr will show the characteristic Kβ and Kα lines on
top of the continuous spectrum or Bremsstrahlung. We may quantify the wavelengths of
Kα and (λmin).
24Cr has Z = 24 atomic number: The wavelength of the K line can be calculated from
α
the Moseley’s law: λKα = c/ν = (3·108 m/s)/[2.47· 1015 (Z-1)2 ·1/s] = 230 pm.
The shortest wavelength limit of the Bremsstrahlung can be calculated from the DuaneHunt displacement law: λmin = hc/(eV) = 20.7 pm.
X-ray „fluorescence” (XRF)
Georg von Hevesy applied X-rays to excite characteristic X-rays from a sample and to
use diffraction to identify its constituent elements. This is termed X-ray fluorescence.
Later he introduced a method of activation analysis based on neutron bombardment
(neutron activation); this method yields better detection limits.
He received the 1943 Nobel Prize in
Chemistry for his work on the use of isotopes
as tracers in studying chemical processes.
Hevesy was the first to apply the radioactive
tracer technique to biology, and he later used
it in medical research.
Typical wavelength dispersive XRF
spectrum; note the systematic increase
in atomic number toward the left.
Diffraction of the X-rays
WAVE + CRYSTAL ↔ DIFFRACTION
Passage of waves of light through crystalline arrangement of atoms (crystals)
should cause diffraction or interference phenomena.
It can be reversed: if the radiation shows diffraction pattern in crystal, then the
radiation should consist of wave and not of particle.
Von Laue took diffraction as positive proof that X-rays are electromagnetic
radiation (waves), not particles.
Each dot, called a
reflection, in this
diffraction pattern
forms from the
constructive
interference of
scattered X-rays
passing through a
crystal. The data
can be used to
determine the
crystalline structure.
Bragg's law of diffraction and the structure of NaCl
The pattern of spots in the Laue diffractogram could be explained by reflection of
waves from crystal planes. L. W. Bragg studied the diffraction of Pt-Lα X-rays by NaCl
crystal. He concluded that salt consisted of a three dimensional lattice of Na+ and Clions (At that time chemists refused believe that NaCl contains no NaCl molecules, just
alternating array of Na+ and Cl- ions!)
For a family of parallel crystal planes with spacing d, a monochromatic incident beam
of wavelength λ will appear to be reflected by the planes if the incident angle Θ obeys
the Bragg’s law:
2 d sin Q = n × l
n = 0,1,2,...
The extra path length traveled by the ray scattered by
the top layer relative to the next layer down is 2d·sinΘ
Applicaton of the Bragg’s law to crystals
Reflection planes (shaded area) in
simple crystals
Diatomic crystal: simple cubic latice with
two scattering elements at each point
Bragg’s reflection at a set of planes from
the cubic lattice in a well defined direction
X-ray crystallography
The phase problem: the intensities of the spots can be measured (and the amplitudes
can be calculated) but the phase differences are lost. Methods to get back partly this
information:
Electron density map
1) Fourier-transfer; Fourier-refinement, 2) Multiple
isomorphous replacement (heavy atom replacement at specific
loci), 3) Comparison with known structures of similar proteins
(biomolecules)
Goniometer
3D structure
Diffraction patter
reaction center
protein from
photosynthetic
bacteria; ~100 kDa
acidic,
basic,
histidine.
X-ray crystallography: what the DNA looks like at
the atomic level?
In the early 1950s, James Watson and Francis Crick
(Cambridge University) proposed the double-helix structure
of (B-)DNA, which has been called the most important
biological work of the past century.
The unusual DNA structures (called a Holliday junctions)
play a key role in DNA's ability to repair itself – a vital
biological function that can be applied in biomedicine and
other related fields.
It is fundamentally important in
explaining the actual biological
function of genes - in particular,
such issues as genetic
"expression," DNA mutation
and repair, and why some DNA
structures are inherently prone
to damage and mutation.
Understanding DNA structure is
just as necessary as knowing
gene sequence. The human
genome project (detailed
explanation of the genetic
sequence of the entire human
genome), is one side of the coin.
The other side is understanding
how the three-dimensional
structure of different types of
DNA are defined by those
sequences, and, ultimately, how
that defines biological function.
PIXE: Particle-induced X-ray
Emissions
(particle)
A bombardment from an alpha source can be used to stimulate atomic x-ray transitions in a
number of materials. The x-ray energies reveal the atomic level spacings with great clarity, and
can be used to identify the atoms. The characteristic x-ray lines of the elements are utilized to
determine the composition of unknown samples.
Attenuation (absorption) of X-radiation:
the Beer’s law
The absorption of X-ray radiation is described
by the general exponential law of radiation
attenuation (Beer’s law):
I = I 0 × e- m x
where I0 is the intensity of the rays that are
incident perpendicularly on the absorbent, which
is a homogeneous layer of thickness x, I is the
intensity of the rays after passing through the layer
and μ is the attenuation coefficient, which includes
all the significant properties of the absorbent (e.g.
tissues in medical applications) and the interaction
between the absorbent and the radiation.
I0
0
ln I/I0
μ
slope: -μ
x
x
Half-value thickness. This is that
thickness of the material which reduces
the intensity of a narrow homogeneous
beam to half its original value:
xH = (ln 2)/μ
Mass attenuation coefficient:
μm = μ/ρ
is the attenuation per unit mass of
material traversed (ρ is the density).
I
dI(x)=μ·I(x)dx
Mass attenuation coefficient μ/ρ
Energy
of X-ray
quantum
E
(MeV)
(cm2/g)
air
water
fat
muscle
bone
Z = 7,78
ρ = 0,0012
Z = 7,51
ρ = 0,9982
Z = 6,46
ρ = 0,92
Z = 7,64
ρ = 1,04
Z = 12,31
ρ = 1,65
0,01
5,12
5,329
3,268
5,356
28,51
0,1
0,1541
0,1707
0,1688
0,169
0,186
1
0,06358
0,07072 0,0708
0,0701
0,0657
10
0,02045
0,02219 0,0214
0,0219
0,0231
20
0,01705
0,01813 0,017
0,0179
0,0207
Energy-dependence of the attenuation
coefficient of X-ray in water
I (x) = I(0)·exp(-μ·x)
Classic scattering
sk
Linear attenuation coefficient
water
Photoeffect
t
Compton-effect
sC
Pair production
k
Energy of the X-ray quantum
The X-ray quanta are scattering
on the electrons elastically
(without energy loss).
External photoeffect: the
ionizing radiation liberates an
electron from the shell.
Compton-effect: scattering of
photons (X-ray quanta) on free
or loosley bound electrons.
The high energy (hν >1,02
MeV) photon (X-ray
quantum) produces an
electron and a positron with
opposite direction of speeds
in the close vicinity of the
electric (Coulomb) field of
the nucleus of the absorbing
material.
Classic scattering
Photoelectric absorption
(photoeffect)
Compton-scattering
Pair production
Mechanisms of
attenuation
Comparison of the different attenuation
mechanisms of radiation
μ: attenuation coefficient, E: energy of the quantum, Z: atomic number of the
chemical element of the material
Range of energy in soft tissues
mechanisms
Coherent scattering
Photoelectric absorption
independent
Compton-effect
Pair production
Ionization processes induced by
photons of radiation in matter
Matter
Survey of different posible interactions
when photon enters the matter.
Wavy lines = Paths of photons
: photoeffect
Pair production
Comptonscattering
Harringbones indicate the paths
of the ion pairs produced by the
radiation in the matter.
Gamma- or
X-rays
Triplet production
The density of the herringbones
offers the frequency (density) of
the ionization process.
: Pair annihilation
Straight lines = paths of electrons and positrons
Problem.
X-rays of the same initial intensity I0 pass through soft tissue x1 = 18 cm thick and
through the same soft tissue containing a bone x2 = 4 cm thick. The attenuation
coefficients are μ1 = 0.19 cm-1 for the soft tissue and μ2 = 0.42 cm-1 for the bone. What
is the intensity ratio of the emerging rays?
I1
I2
What is the contrast of the bone?
x2
Solution. The ratio of intensities I1 and I2 is
x1
soft tissue
I0
bone
I0
I1
I 0 × exp(- m1 x1 )
=
= exp(- ( m1 - m 2 ) x2 ) = 2.5
I 2 I 0 × exp(- m1 ( x1 - x2 ) - m 2 x2 )
In radiography, the relative X-ray intensity difference between the background tissue
and the anatomic structure of interest is referred to as the subject contrast (also known
as tissue contrast):
CS =
I bgd - IS
I bgd
where IS and Ibgd are the intensity of X-rays passing through the anatomic structure of
interest and the adjacent background tissue, respectively. If I2 is substituted for IS and
I1 for Ibgd, then the contrast of the bone in the present problem is CS = 0.6.
Problems for seminar
1. Determine the wavelength of Kα and λmin for molybdenum (Mo).
(Z = 42; for Kα line ni = 2 and nf =1; σ =1)
2. Identify the element giving rise to Kα with λKα = 251 pm.
(Solution: Z = 23, Vanadium)
3. The anode of an X–ray tube is made of Tungsten (atomic number 74). What are the
frequencies of the characteristic Kα (ni = 2) and Kβ (ni = 3) radiations?
4. For radiotherapic purposes, electrons are used which are accelerated by betatron to
25 -45 MeV energy. What will be the angular divergence of the X-ray produced by
target metal hit by these electrons at the outlet of the betatron?
5. What is the maximum frequency of the X-ray quantum in a 10 kV X-ray tube if the
kinetic energy of the electron hitting the anode would convert completely to radiation?
Problems for seminar
6. The anode voltage and current in an X-ray tube are set to 80 kV and 6 mA,
respectively. The anode material is tungsten.
a) What is the maximum energy of the X-ray quanta?
b) What is the short-wavelength limit of the X-ray spectrum?
c) What is the radiative power of the X-rays produced in this tube?
d) What is the efficiency of X-ray production?
e) How much heat is generated per minute?
f) What is the velocity of electrons at the anode?
g) How many electrons hit the anode per second?
7. The anode voltage of an X-ray tube is supplied by a charged capacitor of high
capacitance C = 1 μF. What is the percentage decrease of the initial accelerating voltage
of V = 100 kV after an exposure current and time of 5 mA and 2 s, respectively? Why is
it important to maintain constant anode voltage during exposure?
Problems for seminar
8. X-rays for a chest examination are produced by an X-ray tube at a voltage of V = 80
kV and a current of I = 5 mA with an efficiency of η = 0.65%.
a) What is the intensity I0 of X-rays at a distance r = 1 m from the focal point of the
tube, if they distribute uniformly in a solid angle 2π (i.e. in a hemisphere)?
b) What is the dose absorbed in the chest r = 1 m distant from the tube during
transillumination for t = 10 s? The chest is assumed to have a uniform thickness of
x = 10 cm. The average attenuation coefficient is μ = 0.18 cm-1, and the average
density is ρ = 1.05 g·cm-3.
9. What is the subject contrast given by the blood in a soft tissue environment? The
radius of the aorta is x = 1 cm, and the attenuation coefficients of the blood and the
surrounding soft tissue are 0.215 cm-1 and 0.211 cm-1, respectively. How much is the
contrast increased if iodine is injected into the blood circulation and the attenuation
coefficient of the blood is increased to 0.284 cm-1?
10. The lattice of NaCl is cubic with a unit-cell dimension of 5.64 Å. Each unit cell
contains 4 Na+ and 4 Cl- ions. The density of NaCl is 2.163 g·cm-3. Calculate the
Avogadro’s number using this information.
Problems for seminar
11. For λ = 154 pm X-rays, obtain the first order (n = 1) Bragg angles for interplanar
spacings of 500 pm, 1 nm, and 100 nm.
12. From the Bragg’s equation of diffraction it can be seen that for a fixed λ, the
interplanar spacing d is inversely proportional to sin Θ. In other words, scattering
corresponding to the smallest spacing occurs at the maximum value of sin Θ (when
Θ = 90o). If λ = 154 pm X-rays are used, what is the theoretical limit of resolution (i.e.
what is the minimum spacing observable)?