Download Some Elements of Nuclear Physics Physics 3070 Spring, 2003

Some Elements of Nuclear Physics
Physics 3070
Spring, 2003
A. Isotopes
Two \isotopes" of the same element have dierent numbers of neutrons, but the same number
of protons (because they're the same element). The chemical characteristics of a substance are
determined by the number of protons, so dierent isotopes of the same element typically behave
chemically very similarly. The number of neutrons, however, determines the radioactive stability
of the nucleus, as we discuss further below.
An isotope is denoted in one of two ways. First, let Z = number of protons, N = number of
neutrons, and A = Z + N = number of protons + neutrons = number of \nucleons". Isotopes of
the same element have the same Z but dierent N and A. Then for an element X , you can denote
the isotope as:
AX
Z N
235
For example, 238
92 U 146 and 92 U 143 are the two naturally occurring isotopes of Uranium. For simplicity, they're commonly denoted 238 U and 235 U . Other examples are
1H
1 0 proton = p
2H
1 1 deuterium = D
4 He helium, -particle = 2 2
Isotopes occur naturally if 1 Z 92 and 0 N 146.
The second way to represent an isotope is as follows:
(Z; N )
This notation is actually more commonly used to denote the nucleus of an isotope, rather than the
atom. This is a subtle dierence, but when we discuss nuclear decay below we'll use this notation
to mean that the nucleus transforms during nuclear decays and reactions.
B. The Nucleus
The nucleus is composed of Z protons and N neutrons and has a total charge Ze, where e is the
magnitude of the charge of an electron. Nuclear radii range from 2 - 7 fm, where 1 fm = 10,15 m.
Recall that an atom is about 10,10 m in radius, so the nucleus is 20,000 - 100,000 times smaller
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than the entire atom. Neutrons and protons are, in fact, packed tightly inside the nucleus, lling
about half the available volume. Electrons, on the other hand, are free to roam the huge connes
of the atom itself.
C. Forces in Nature
It's incredible that the protons can survive packed so tightly together in the nucleus, because they
repel each other via the \electrostatic" force. This force repels particles that have the same
electronic charge and attracts them if they have dierent charges; so protons repel one another,
but attract electrons. Whatever's holding the protons in the nucleus must be strong enough to
overcome the electrostatic repulsion of the protons, but should not operate on length-scales much
larger than the nucleus. Gravity is far too weak.
What holds the protons in the nucleus is another force called the \strong force" that is very
strong and attractive between nucleons that are separated by about 1 - 2 fm, but is very weak if
nucleons are separated by more than about 2 fm. So the strong force has the following properties:
1. Stronger than the electrostatic force,
2. Operates only at short range (< 2 fm),
3. Repulsive at very short range (< 1 fm) to keep nucleons from interpenetrating.
Because protons are aected by the repulsive electrostatic force and the attractive strong force
and neutrons only by the strong force, neutrons are more strongly bound inside the nucleus than
protons. For this reason, there are typically more neutrons than protons in the nucleus, particularly
for heavy nuclei. For heavy nuclei there are about 40% more neutrons than protons (N=Z 1:4).
The radioactive stability of nuclei depends on this ratio N=Z . If there are too many or too few
neutrons, the nucleus will be more unstable and have a greater tendency to decay. The type and
frequency of decay determine the danger of a particular isotope.
As discussed below, when nuclei decay they emit energy. The unit of energy commonly used
by physicists is the electron-volt, eV, where 1 eV = 1:6 10,19 J. Chemical reactions typically
liberate several eV per reaction, but nuclear decays liberate more energy, as we will see below.
D. Nuclear Decay (Radioactivity)
Henri Becquerel accidentally discovered that uranium compounds caused a photographic plate to
become fogged. (He was investigating the relationship between x-rays and uorescence using crystals of uranium potassium sulfate.) Radioactivity is the spontaneous disintegration of an unstable
atomic nucleus and the emission of particles or electromagnetic radiation. Pierre and Marie Curie
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investigated uranium ores using chemical separation. They discovered that pitchblende and chalcocite, naturally occurring ores, were highly radioactive due to the presence of radium.
All naturally occurring elements with atomic numbers greater than 83, as well as some isotopes
of lighter elements, are radioactive.
A nucleus (Z,N) can decay following three main modes:
-decay (Z,N) ! (Z-2, N-2) + -decay , : (Z,N) ! (Z+1, N-1) + e, + + : (Z,N) ! (Z-1, N+1) + e+ + -decay (Z,N) ! (Z,N) + Each particular isotope will tend to decay naturally in one principal decay mode.
All decays are subject to three conservation laws:
1. Conservation of charge.
2. Conservation of nucleon number.
3. Conservation of relativistic energy.
When a nucleus decays, the total charge and the total number of nucleons remains constant. The
number of protons or neutrons may change, but their sum remains xed. Relativistic energy,
which is the sum of kinetic energy and rest-mass energy, will be discussed later. Typically, mass
is not conserved in a nuclear reaction. The products formed have a slightly lower mass. The mass
dierence goes into energy, as discussed later for nuclear ssion.
Alpha-decay involves ejecting an -particle (helium nucleus, 2 protons and 2 neutrons) from
the nucleus. 235 U , 238 U , and 239 Pu are all alpha-emitters:
235 U143
92
238 U146
92
239 Pu145
94
!
!
!
231 Th141 + 4 He2
90
2
234 Th144 + 4 He2
90
2
235 U143 + 4 He2
92
2
Beta-decay is perhaps more interesting than alpha-radiation because it involves some exotic
particles. In , decay, a neutron changes into a proton, emitting an electron and another particle
called an anti-neutrino. In + decay, a proton changes into a neutron, emitting a positron (an
anti-electron that interacts with an electron almost immediately upon its formation to form an
energetic photon) and a neutrino. Gamma-decay does not involve one nucleus transmuting into
another but rather a nucleus decaying from an excited state to a state of lower excitation, like an
electron moving from a higher to a lower energy level. The nucleus then ejects a -ray, which is
just an energetic photon with an energy of about 10 MeV.
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The neutrino is an interesting particle that was conjectured to exist by Wolfgang Pauli in the
early 1930s, but was not \observed" until the 1950s. It has very low mass (perhaps it's massless
but few physicists believe this anymore), has no charge, and interacts very weakly with matter
through the \weak" force, about which we'll say nothing other than that it exists.
Radioactivity, therefore, involves the transmutation of atoms in which the predominant mode
of decay depends on the particular isotope. The products are what is usually meant by \radioactivity": -particles, electrons, and -rays. Health eects of this radioactivity will be discussed
briey below.
N(t)
N0
N0/2
t1/2
t
Figure 1: Radioactive decay.
The rate of radioactive decay is represented quantitatively by r = decay rate = number decays/unit time. If r is a constant, then the number N (t) of an substance will reduce from some
initial number N0 exponentially (see Figure 1):
N (t) = N0 e,rt :
This can also be represented in terms of the half-life, t1=2 , of a substance, which is the time it
takes for half of the substance to decay on average:
N0 = N0 ;
N (t) = t=t
2 1=2 2n
(1)
where n is the number of half-lives that have occurred from the initial state up to time t.
For example, the half-life of the isotopes most relevant to nuclear ssion are 704 My for 235 U ,
4.5 Gy for 238 U , and 24.1 ky for 239 Pu. Assume that you begin with N0 = 106 atoms of each of
these species. How many will be left after 1 My? To answer this we will use equation (1) and,
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therefore, will need n on the way to nding the number of atoms remaining, N (t):
n235 = 1My=704My = 0:0014 ! N (t) = 106 =20:0014 = 999; 016 ( 1000 decays)
n238 = 1My=4500My = 0:00022 ! N (t) = 106 =20:0022 = 999; 846 ( 150 decays)
n239 = 103 ky=24:1ky = 41:7 ! N (t) = 106 =241:7 0 ( 106 decays)
E. Radioactive Series
We can see then why 239 Pu is not found naturally, any natural Plutonium-239 that was around at
the formation of the earth would have decayed long ago. We also see why there is much more 238 U
than 235 U , 235 U decays much faster and primordial 235 U is largely gone. (99.28% 238 U , 0.72% 235 U ).
Because the half-life of 238 U is about the same as the Earth's age, about 50% of the primordial
238 U that existed during the Earth's formation remains inside the Earth today.
All radioactive substances undergo natural decay through a sequence of steps until they reach a
stable substance. This sequence of steps is called a \Radioactive Series", and these series govern
which substances remain in nature. For example, 238 U decays on average through the following
series:
238 U
!
!
!
230 Th !
226 Ra
234 Th !, 234 Pa !, 234 U !
218 Po !
214 Pb !, 214 Bi !, 214 Po
222 Rn !
206 Pb
210 Pb !, 210 Bi !, 210 Po !
where Th is Thorium, Pa is Protactinium, Ra is Radium, Rn is Radon, Po is Polonium, Pb is Lead,
Bi is Bismuth. This is a series of 14 decays, 6 of which are , - and 8 are -decays. Ultimately,
238 U will decay all the way to Lead-206 which is non-radioactive. The initial decay from 238 U to
234 Th has a half-life of 4.5 Gy. The other decays are much faster, the slowest being 230 Th which
has a half-life of 75 ky and 234 U with a half-life of about 245 ky. Most of the other decays have
half-lives measured in days or fractions of days. In particular, the decay of Radon to Polonium has
a half-life of only 3.8 days, which is one of the reasons why radon in your house is dangerous. The
other reasons are that Radon is a gas and can be inhaled easily (-radiation is most dangerous
when inhaled) and being a noble gas it is chemically inert, so it passes through walls and oors
readily without reacting.
It is important to understand that these decays are going on all the time right under our feet.
238 U is very common in most soils and we live in a sea of its radioactive o-spring. To see how
common Uranium is, note that on average Uranium appears in soils at a concentration of about
2 parts per million. This concentration is highly variable, and can be as high as 1 part in 103 in
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phosphate deposits. We can easily nd the mass of Uranium, mU , in the top 1 m beneath a 16 m
16 m house. The mass will be the product of the concentration of Uranium, cU , the volume of
soil, Vsoil, and the density of soil, soil:
Vsoil = (16 m)2 1 m 250 m3
mU = cU Vsoilsoil = (2 10,6 )(250 m3 )(2 103 kg/m3) 1 kg:
So, you can expect that there's about a kg of Uranium right beneath your feet in an
average sized house for average soil. Similarly, in the top 2 m beneath the main campus of
CU (A 1 km2 = 106 m2 , V 2 106 m3 ), mU is expected to be about 10 tonnes (metric tons).
Natural variations in the distribution of Uranium may mean that this estimate is much too small.
F. Nuclear Reactions { Fission
The term \nuclear reaction", in contrast with nuclear decay discussed above, refers to the
response of a nucleus to being bombarded by another particle, usually a neutron. The relevant
reaction that we will study is neutron-induced ssion, which was rst studied by Enrico Fermi in
the 1930s. It was discovered by Hahn and Strassman in 1938 and explained a year later by Meitner
and Frisch. Nuclear ssion does not (except perhaps in very rare circumstances) occur naturally
without human intervention.
\Fission" means to break into two parts of nearly the same size, which is quite dierent from
- and -decay which produce an ospring that is about the same size as the parent.
An example of neutron-induced ssion is the following reaction which is particularly important
in nuclear reactors and nuclear weapons:
90
236
143
n + 235
92 U143 ! 92 U144 ! 36 Kr54 + 56 Ba87 + 3n + energy;
(2)
in which a slow-neutron is absorbed by a 235 U nucleus and transformed into an excited state of
236 U which ssions to produce Krypton, Barium, some energy and three neutrons. We'll discuss
where the energy comes from later on. The asterisk means that the nucleus is in an excited state.
Note that although it's said that 235 U is ssionable, it's actually 236 U that ssions. In its
unexcited state, 236 U will -decay like most of the other isotopes of Uranium, with a half-life of
about 23 My. The nucleus of 236 U produced by the reaction in equation (2) is excited, however, and
will ssion almost immediately, long before it will -decay. 238 U will also absorb a slow neutron to
produce 239 U in an excited state, but the 239 U nucleus will not be suciently excited to ssion.
Rather, it will , -decay with a half-life of about 23 min.
Each ssion that follows the reaction in equation (2) emits about 200 MeV of energy: 170 MeV
of kinetic energy of the ssion fragments, 5 MeV of kinetic energy in the neutrons, 15 MeV of
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energy in electrons and gamma rays that result from rare ,-decays, and 10 MeV as energy of
antineutrinos. Contrast this with chemical reactions that produce only a few eV per reaction.
The ssion reaction in equation (2) has two major limitations. First, 235 U will only absorb a
neutron if the neutron is suciently slow. The reason is that slow neutrons take longer to move
past the uranium nuclei and feel the strong force longer. They are, therefore, more likely to be
captured. Second, 235 U is rare in nature. 239 Pu will also ssion, but it is much more rare in nature
than 235 U , and for 239 Pu to be used in a nuclear reactor it must be manufactured.
G. Chain Reactions
In 1942, Enrico Fermi achieved the rst nuclear \chain reaction" beneath the football eld (Stagg
Field) at the University of Chicago, using about 50 tons of uranium. A chain reaction is based
on the fact that nuclear ssion, such as the reaction in equation (2), starts with the capture of a
neutron and the reaction generates more neutrons than it absorbs. So, the neutrons produced can
go forth and generate more ssions, hence more neutrons, ssions, and so forth. A chain reaction is
not guaranteed, however. Whether the reaction will zzle, continue at a controlled rate, or proceed
explosively can be determined if one knows the \reproduction factor":
k = number of neutrons from each reaction that produce another ssion.
If k < 1, the reaction will die out, if k is equal to or only slightly greater than 1 the reaction will
continue in at a moderate rate, or if k >> 1 the reaction will continue explosively.
The reproduction factor k depends on two main things. (1) k depends on the speed of the
neutrons needed to induce the ssion reaction. On average, the neutrons coming out of reaction
(2) move too fast to be absorbed eciently by 235 U . For a chain-reaction to continue, the neutrons
must either be slowed down or the nuclear fuel must be exceedingly pure. To slow the neutrons,
a \moderator" is used. The nuclear fuel is frequently embedded in the moderator, which has
light nuclei that carry away much of the kinetic energy on each collision between the neutrons and
the moderator. Regular water (H2 O), heavy water (D2 O), graphite (nearly pure Carbon), and
beryllium are frequently used as moderators. The rst three typically are used in nuclear reactors
and the latter two in atomic bombs. (2) k depends on the density of the nuclear material. High
density material increases the probability that a given neutron will intersect a Uranium nucleus
and be absorbed. In atomic bombs, the density of the nuclear material is increased by setting o
chemical explosives that compress it. At STP, there is a \critical mass" above which k increases
past 1. This is the mass at which the nuclear chain-reaction will proceed.
Because 238 U will absorb slow neutrons without ssioning, it acts to poison the ssion reaction.
This is most important if the desire is for a nuclear explosion, but is less severe for nuclear reac7
tors. The nuclear fuel in atomic weapons, in particular, has to be \enriched" in 235 U before the
chain-reaction can proceed as planned. Enrichment increases the percentage of 235 U from 0.3% of
naturally occurring Uranium, to about 3% for fuel used in nuclear reactors to very high percentages
in excess of 90% for high-yield ssion weapons.
H. Nuclear Reactors
In nuclear reactors, k has to be controlled to lie near 1. To do this, neutrons must be absorbed by
\control rods" made of a substance like Cadmium, which strongly absorbs neutrons. The control
rods are moved into and out of the nuclear fuel to keep k 1. The control rods heat up as they
absorb the neutrons, and are cooled by some coolant, typically water or gas.
Most nuclear reactors use 235 U as nuclear fuel, but 235 U is rare and becoming more rare with
time. 239 Pu is an alternative, but does not occur naturally, so it has to be made to use in nuclear
reactors in the same way that it is manufactured for use in atomic weapons. It can be made in the
following way from 238 U (which remains plentiful):
n + 238 U ! 239 U + 239 U !, 239 Np + e, + 239 Np !, 239 Pu + e, + 239 U and 239 Np have half-lives of 23 minutes and 2.4 days, respectively, so the reaction can proceed
fairly rapidly.
Note that when the 239 Pu ssions, it produces neutrons for the ssion chain-reaction. Some of
these neutrons can be used, however, to bombard 238 U to make more 239 Pu. This idea of creating
and burning Plutonium simultaneously is called a \breeder reactor". There are a few commercial
breeder reactors operating in the world, but the technology remains in the experimental stage of
development.
I. Why Energy is Liberated During Fission
Energy is released during a nuclear reaction because mass is a form of energy, and mass is reduced
during nuclear decay. More mass is lost during nuclear ssion than due to other decay processes.
The \rest-mass energy" of a mass m is
E = mc2 :
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(3)
Consider a nucleus with kinetic energy K1 and mass m that decays into two progeny with a total
kinetic energy K2 and total mass M . Conservation of relativistic energy tells us that:
K1 + mc2 = K2 + Mc2 ) K2 = (m , M )c2 + K1 > K1
because the initial mass m is greater than the nal mass M . Some of the mass of the original
nucleus was converted to kinetic energy. This kinetic energy is available to increase the temperature
of surrounding materials, and so heat it to form steam. If the kinetic energy is great enough, it can
do great damage.
J. Nuclear Fusion
Nuclear \fusion" is the opposite of ssion, where two light nuclei merge into a heavier nucleus
and release energy. An example of a fusion reaction is the following:
2 H + 2 H ! 3 H + 1 H + 4 MeV;
(4)
which occurs in \Hydrogen" bombs. In this reaction, two deuterons (deuterium nuclei) merge
to yield tritium (Hydrogen nucleus with 2 neutrons), a proton, and 4 MeV of energy. To cause
the deuterons to merge, the electrostatic repulsion between the protons in the nuclei has to be
overcome. Because the repulsion is so strong, the deuterons have to be moving at a very large speed
to overcome it. This means that they have to have a large kinetic energy, which only happens at
very high temperatures. Thus, nuclear fusion occurs at high temperatures, and is therefore called a
\thermo-nuclear" reaction. Note that equation (4) will only occur if the temperature is larger
than 107 deg C .
Equation (4) is only half of the reaction that occurs in a typical Hydrogen bomb. The tritium
can be used in another fusion reaction that produces an even larger amount of energy:
1 H + 3 H ! 4 He + 17.6 MeV:
(5)
This is the reason that the nuclear material in thermonuclear weapons is doped or enriched with
tritium. Equations (4) and (5) together imply that 3 deuterons and a tritium nucleus can merge to
produce 21.6 MeV at very high temperatures. A ssion bomb is exploded rst in order to achieve
these high temperatures needed in a Hydrogen bomb. Although a ssion reaction produces about
10 times more energy than this pair of fusion reactions, the energy per nucleon is higher for the
fusion weapon. Thus, pound-for-pound fusion based thermo-nuclear weapons have more destructive
capabilities than ssion weapons.
More importantly, the temperatures needed to produce fusion also occur naturally in the interior
of stars, and nuclear fusion is the principal source of stellar energy. There are several fusion
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reactions that can occur in stars depending on their chemical composition, which changes with
age. A relatively young star like the Sun is mostly composed of Hydrogen, and the main source of
energy is a sequence of three reactions called the \proton-proton cycle":
I. 1 H + 1 H ! 2 H + e+ + + 0.4 MeV
II. 2 H + 1 H ! 3 He + + 5.5 MeV
III. 3 He + 3 He ! 4 He + 1 H + 1 H + 12.9 Mev
This process begins with the merger of two protons during which one undergoes + -decay to a
neutron, which yields a deuteron. The deuteron combines with another proton to produce 3 He.
Two of these nuclei combine to produce an -particle and two protons to begin the process again.
Note that you need two 3 He nuclei in Step III, so Steps I and II each have to be run twice before
Step III can occur. The net eect is for 4 protons to merge to form an -particle and energy:
41 H ! 4 He + 2e+ + 2 + 24.6 MeV:
The two positrons almost immediately annihilate with two electrons to produce a further 2.0 MeV,
so 27 MeV results from each proton-proton cycle. Nuclear fusion in the Sun generates about
4 1020 MW of power and converts about 50 trillion tons of Hydrogen into Helium every day.
When a star has \burned" most of its Hydrogen to Helium, its source of energy is temporarily
exhausted. The star contracts and its central temperature rises until new fusion processes begin
involving the formation of Carbon from Helium. In large stars, when the Helium is exhausted, there
is a second transition process in which Carbon is fused to form heavy elements all the way up to
Iron. Iron is the heaviest element that can be formed by fusion with the release of energy. Heavier
elements, from Iron to Uranium and beyond, are formed by a succession of neutron captures followed
by -decays. This process can occur slowly in an old, but stable star or rapidly in a supernova
explosion. This explosion ejects an enormous outer shell of matter and leaves behind a tiny but
massive neutron star or black hole. The matter that is ejected, including some of the recently
formed heavier elements, is dispersed through interstellar space and, ultimately, forms new stars
and planets like our Sun and Earth. The presence of heavy elements in the Sun and planets is
evidence that they, and we, are descendents of many now-dead stars.
K. Health Eects of Radiation
This is a complicated subject, about which we can only provide a little information. Radioactivity
comes in three forms: -particles, electrons, and photons (usually - or X -rays). These three
forms of radioactivity have very dierent capabilities to penetrate skin. On average, electrons and
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-rays are more penetrative than -particles, so -emitters are more of a concern than -emitters,
everything else being equal. Radioactive species with long half-lives are much less dangerous than
those with short half-lives.
Damage to cells is most severe during cellular reproduction. Therefore, rapidly reproducing
cells like bone-marrow or tumors are most aected by radiation, and children are more aected
than adults.
-particles cannot penetrate the skin outside your body, in fact it does not penetrate the
surface dead skin layer to intersect live tissue for the most part. Uranium, a common -emitter, is
so common in soils (about 2 parts in 106 ) that skin has evolved to protect you from alpha-radiation.
Uranium and other -emitters are considered harmful mostly if they're inhaled, because the tissue
inside your body is more delicate. Because Uranium, in particular, is heavy it does not suspend
readily in air. Other alpha-emitters, particularly Radon, that appear in the natural radioactive
decay series of Uranium, occur normally as gases, and they're a health concern in houses that are
not well ventilated. Uranium, if swallowed, tends to be removed mostly by feces but also by your
kidneys. A kidney is a robust organ and is not strongly aected by -particles unless the dose is
very strong. Uranium may be absorbed by bone, but usually in the outer layers well away from the
bone marrow that could be damaged if irradiated with alpha-radiation. Uranium is also chemically
toxic, but we will not discuss this as it does not relate to the radioactive characteristics of the
substance.
There's much more to be said and science is still evolving on this subject. You might want to
keep the following in mind:
1. Danger depends on the type of radioactivity and the nature of the exposure: skin contact,
breathing, ingestion.
2. -emitters are only dangerous when inhaled or swallowed.
3. The danger depends on the activity level of the radioactive substance, the longer the half-life
the less dangerous the substance.
4. Background radioactivity at the Earth's surface is already pretty high because Uranium is very
common in soils and, of course, the Sun is bombarding us with photons.
5. Frequently, it is the progeny of the main radioactive agent that is a greater health risk than the
parent. For example, radon is a bigger concern than Uranium in homes because, although it is
an -emitter, it has a much shorter half-life than Uranium and is a gas so it is readily inhaled.
Also, Strontium-90, a decay product of ssion, is chemically indistinguishable from Calcium,
so it is absorbed readily by bones, including bone-marrow. It poses a risk to bone-marrow.
6. In naturally occurring radiation, progeny only occur when the parent has been in place a
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long time { because the progeny occur through a natural radioactive series that may take
thousands or even millions of years to develop.
There are measures of exposure to radioactivity. \Dosimetry" is the measurement of radiation
and the study of its eects on living organisms. There are several dierent units used to measure
radiation. The absorbed dose describes the amount of energy deposited per kilogram of exposure
time, measured in the gray (Gy).
1 Gy = 1 J/kg = 100 Rads
(Rads are non-SI, but are in general use.) The biological damage produced on a given organism is
called the dose equivalent, measured in sieverts (Sv).
1 Sv = 100 rem = 105 mrem
(rem { rad equivalent man) The dose equivalent(Sv) = absorbed dose(Gy) a quality factor(Q).
The quality factor is a number assigned to each type of radiation to describe its biological eects.
The eect that absorbed radiation has on dierent types of tissues varies. There remains disagreement among scientists about the cumulative eects of low dosage exposure to radiation. Much
research is still needed into the long-term biological eects of radiation. The becquerel (Bq) is the
activity of a source produced when one disintegration per second occurs from a radioactive source.
1 Bq = 1 disintegration per second
This unit does not distinguish between the eects of dierent types of radiation.
1 curie (Ci) = 3.7 1010 Bq.
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