Sector Construcción
4.8 SECTOR TRANSPORTE, ALMACENAMIENTO, CORREO Y MENSAJERÍA
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Interactions
“When the intervals, passages, connections, weights, impulses, collisions, movement, order and position of the atoms interchange, so also must the things formed from them change.”
Lucretius (ca. 100 bc) Much of what is known about atoms and radiation from them was learned by aim-ing subatomic particles at various target materials. The discovery of cathode rays provided a source of projectiles that led to the discovery of x-rays and subsequently radioactivity. Scattering of alpha particles emitted from radioactive substances gave the first clues and insight that the positive charges of atoms are coalesced into a very small nucleus. Interactions of alpha particles, protons, deuterons, neu-trons, and light nuclei with various target nuclei have produced many new products including transmutation and/or fission of heavy elements and fusion of light atoms.
4.1
Production of X-rays
Roentgen was able to describe most of the known characteristics of x-rays after his monumental discovery by conducting several experiments; however, it was not possible to explain how x-rays were produced until the concepts of atoms, parti-cles, and quanta were understood. It is now known that x-ray production occurs, as shown in Figure 4-1, when a negatively charged electron of kinetic energy eV enters the force field of the positively charged nucleus of a target atom. This force field, which is strongest for high-Z materials like tungsten, deflects and acceler-ates the electron, which causes the emission of electromagnetic radiation as it is bent near the nucleus. This is consistent with classical electromagnetic theory because the electron is not bound. Because radiation is emitted and energy is lost in the process, the electron must slow down, so that when it escapes the force field of the nucleus it has less energy. Overall, the electron experiences a net deceleration, and its energy after being decelerated is eV – hm where hm appears as electromagnetic radiation. Roentgen named these radiations x-rays to characterize their unknown sta-tus. This process of radiation being produced by an overall net deceleration of the electrons is called Bremsstrahlung, a German word meaning braking radiation.
Physics for Radiation Protection: A Handbook. James E. Martin Copyright 3 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40611-5
Fig. 4-1 Production of x-rays in which accelerated electrons emit bremsstrahlung.
Fig. 4-2 X-ray spectra of intensityI(v) versus electron energy E for tungsten (W) and molybdenum (Mo) targets, each of which is operated at 35 kV.
X-ray production is a probablistic process because any given electron may take any path past a target nucleus including one in which all of its energy is lost.
Bremsstrahlung photons are thus emitted at all energies up to the accelerating energy eV and in all directions, including absorbtion in the target. As shown in Figure 4-2 for tungsten (W) and molybdenum (Mo), x-ray spectra have a continu-ous distribution of energies up to the maximum energy Emaxof the incoming elec-4 Interactions
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tron. The value of Emaxdoes not depend on the target material, but is directly pro-portional to the maximum voltage. About 98% of the kinetic energy of the acceler-ated electrons is lost as heat because most of the impinging electrons expend their energy in ionizing target atoms. Other aspects of the production and uses of x-rays are described in Chapter 15.
4.2
Characteristic X-rays
Figure 4-2 shows discrete lines superimposed on the continuous x-ray spectrum for a molybdenum target because the 35 keV electrons can overcome the 20 keV binding energy of inner shell electrons in the molybdenum target. However, this does not occur for the tungsten target spectrum because the inner shell electrons of tungsten are tightly bound at 69.5 keV. The vacancy created by a dislodged or-bital electron can be filled by an outer shell (or free) electron changing its energy state, or, as Bohr described it, jumping to a lower potential energy state with the emission of electromagnetic radiation; the emitted energy is just the difference between the binding energy of the shell being filled and that of the shell from whence it came. And since the electrons in each element have unique energy states, these emissions of electromagnetic radiation are “characteristic” of the ele-ment, hence the term “characteristic x-rays” (see Figure 4-3). They uniquely iden-tify each element.
Fig. 4-3 Emission of a characteristic x-ray due to a higher energy electron giving up energy to fill a particular shell vacancy.
If an electron vacancy exists in the K shell, the characteristic x-rays that are emitted in the process of filling this vacancy are known as K-shell x-rays, or simply K x-rays. Although the filling electrons can come from the L, M, N, etc., shells, characteristic x-rays are known by the shell that is filled. Further, the K x-ray that originates from the L shell is known as the Kax-ray; if the transition is from the LIIIsubshell it is a Ka1x-ray, and if from the LIIsubshell a Ka2x-ray. A transition from the LIsubshell is forbidden by the laws of quantum mechanics. In a similar fashion those from M, N, and O shells and subshells are known as Kb, Kc, Kdand so forth with appropriate numerical designations for the originating subshells.
Figure 4-4 illustrates these transitions in molybdenum, including the energies of the L subshells.
4.2 Characteristic X-rays 151
4 Interactions
Fig. 4-4 Emission of characteristic x-rays from a molybdenum target due to electron vacancies in various shells followed by a higher energy electron giving up energy to fill a particular shell vacancy.
L x-rays are produced when the bombarding electrons knock loose an electron from the L shell and electrons from higher levels drop down to fill these L-shell vacancies. The lowest-energy x-ray of the L series is known as La, and the other L x-rays are labeled in order of increasing energy, which corresponds to being filled by electrons from higher energy orbits as shown in Figure 4-4. Characteristic x-rays for M, N, etc., shells are designated by this same pattern. Interactions in the higher-energy shells occurs with lower probability than for K-shell x-rays because target-atom electrons in outer shells are spread out over a larger volume, thus pre-senting a smaller target to incoming electrons.
A wide array of characteristic x-rays can be observed for a given element due to the subshells within the major shells. For example, the Kax-ray could originate from any one of the subshells of the n = 2 level, as shown on the left-hand side of Figure 4-4. The energies of these different transitions will be slightly different, and emissions associated with each are possible. The energy differences are very small for most shells, but worth noting for the K and L shells especially for the higher Z elements. Listings of emitted radiations associated with radioactive transformations denote these as Ka1, Ka2, La1, La2, etc. (see Chapter 3 and Appen-dix D).
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4.2 Characteristic X-rays Example 4-1. From the data in Figure 4-4, compute the energy of Ka1, Ka2, Kb, Kc, and Lacharacteristic x-rays.
Solution. The emitted x-rays occur because electrons change energy states to fill shell vacancies. Each characteristic x-ray energy is the difference in potential ener-gy the electron has before and after the transition. For K x-rays:
Ka1= –2.52 – (–20.0) = 17.48 keV Ka2= –2.63 – (–20.0) = 17.37 keV Kb= –0.50 – (–20.0) = 19.50 keV Kc= –0.06 – (–20.0) @ 20.00 keV
For L-shell vacancies, it is presumed that the lowest energy subshell is filled from the lowest energy M shell, or
La= –0.51 – (–2.87) = 2.36 keV
Other permutations are also possible, which would produce a large array of discrete characteristic x-rays, all of which can be resolved with modern x-ray spectrometers.
4.2.1
X-rays and Atomic Structure
Study of characteristic x-rays led to fundamental information on atomic structure.
Moseley’s analyses of K x-rays proved that the periodic table should be ordered by increasing Z instead of by mass and that certain elements were out of order and others were not yet discovered. The relative positions of the Kaand Kb lines in Moseley’s original photographic images are shown in Figure 4-5 for elements from calcium to copper; the images clearly show that the wavelengths decrease in a regular way as the atomic number increases. The gap between the calcium and titanium lines represents the positions of the lines of scandium, which occurs be-tween those two elements in the periodic system.
Before Moseley’s work the periodic table was ordered according to increasing mass, which placed certain elements in the wrong group (e.g., cobalt and nickel or iodine and tellurium). Moseley found that when the elements were ordered according to Z the chemical properties corresponded to the proper group. He also found gaps corresponding to yet undiscovered elements. For example, the radio-active element technetium (Z = 43) does not exist in nature and was not known at the time, but was later discovered based on his work.
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Fig. 4-5 Moseley’s photographs of Kaand Kbx-ray lines for various light elements.
4.2.2
Auger Electrons
A characteristic x-ray photon is produced any time an electron vacancy is filled by a higher-energy electron. Many of these characteristic x-rays are emitted from the atom, but many do not exit the atom because of a phenomenon called the Auger effect. When this occurs the x-ray interacts with an electron from a shell farther out, thus ejecting it from the atom. These are known as Auger electrons.
The Auger effect is similar to an “inner photoelectric effect” as shown schemati-cally in Figure 4-6. First a K-shell vacancy occurs, which can be filled by any high-er-energy electron but most likely from the L shell with the energy released as a characteristic x-ray. If, instead of leaving the atom, the x-ray interacts with a sec-ond L-shell electron, it is ejected from the atom and the L shell loses two electrons.
These are then replaced by electrons from the M shell or farther out, sometimes producing a cascade of Auger electrons. The probability of such nonradiative pro-cesses, which compete with x-ray emission, has been found to decrease with increasing nuclear charge, as shown in Figure 4-7 where the fraction of total emis-sions that occur by Auger electrons is denoted by 1 – g. In light atoms with low Z, the ejection of Auger electrons far outweighs characteristic x-ray emissions (Fig-ure 4-7), but is the reverse for high-Z atoms.
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4.2 Characteristic X-rays
Fig. 4-6 Auger electron emission (right) competes with emis-sion by characteristic x-rays (left).
Fig. 4-7 Ratio g of characteristic x-ray emission to Auger elec-tron production as a function of the atomic numberZ.
The kinetic energy of an ejected Auger electron is equal to the energy hm of the characteristic x-ray (if it were to be emitted) minus the binding energy of the ejected electron in its respective shell, or
KEauger= hm – BEe
Example 4-2. Silver is bombarded with 59.1 keV Karadiation from tungsten. If the binding energies of the K- and L-shell electrons in Ag are 25.4 and 3.34 keV,
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respectively, what are the energies of (a) electrons ejected from the K and L shells by direct ionization and (b) Auger electrons ejected from the L shell by the Kaand Kbx-rays?
Solution. (a) The energy of photoelectrons from the L shell of Ag is 59.1 – 3.34 = 55.76 keV
and from the K shell, 59.1 – 25.4 = 33.7 keV
(b) The energy of Auger electrons ejected from the L shell by Kbx-rays is 24.9 – 3.34 = 21.56 keV
and by Kax-rays 22.1 – 3.34 = 18.76 keV
4.3
Nuclear Interactions
Nuclear interactions are those that involve a bombarding particle and a target nucleus (in x-ray production the electron is not absorbed by the nucleus). Projec-tiles used in nuclear interactions can be alpha particles (a), protons (p), deuterons (d), neutrons (n), or light nuclei such as tritium (3H) or helium (3He). Interaction of a projectile with a target atom (the reactants) yields first a compound nucleus which then breaks up to produce the final products:
X + x fi [compound nucleus] fi Y + y
where X is the target nucleus, x is the bombarding particle or projectile, Y is the product nucleus, and y is the emitted product (either a particle, a nucleus, or a photon). Such reactions are also shown in condensed form as
Target (projectile, emission) product
The total charge (total Z) and the total number of nucleons (total A) are the same before and after the reaction, and momentum and energy must be conserved.
Three categories of interactions can occur as illustrated in Figure 4-8: (a) scatter-ing, in which the projectile bounces off the target nucleus with a transfer of some of its energy; (b) pickup and stripping reactions in which a high-energy projectile either collects (picks up) or loses (strips) nucleons from a target atom, and 156
4.3 Nuclear Interactions (c) absorption of the projectile into the target nucleus to form a new (or excited) atom that then undergoes change. Each type of interaction will produce recoil of the nucleus and deceleration (or stopping) of the particle, which alters its momen-tum and energy.
Fig. 4-8 Nuclear interactions in which a bombarding particle is scattered by the target nucleus, absorbed to form a com-pound nucleus with excess energy, or picks up or strips nucleons.
Scattering reactions produce a decrease in the energy of the projectile by elastic or inelastic scattering. If the residual nucleus is left in its lowest or ground state, the scattering is elastic; if left in an excited state, the scattering is called inelastic.
Pickup and stripping reactions usually occur when the projectile has high energy, and in such reactions the nucleon enters or leaves a definite “shell” of the target nucleus without disturbing the other nucleons in the target. Rutherford’s (a, p) reaction with nitrogen may be thought of as a stripping reaction.
Absorption reactions occur when the incident projectile is fully absorbed into a target atom to form a new compound nucleus which lives for a very short time in an excited state and then breaks up. The new nucleus only exists for l0–16s or so;
thus, it cannot be observed directly, but this is much longer than the 10–21seconds required for a projectile to traverse the nucleus. It is therefore assumed that the compound nucleus does not “remember” how it was formed, and consequently it can break up any number of ways depending only on the excitation energy avail-able. The formation and breakup of64Zn is one example:
63Cu þ11Hfi fi 63Zn þ n
64Zn fi 62Cu þ n þ p
60Ni þ42Hefi fi 62Zn þ 2n
The relative probabilities of forming any of the three interaction products from the compound nucleus64Zn* are essentially identical, even though the initial
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ticles are different. Bombardment of lithium with protons is another example (see below) of multiple ways the compound nucleus can break up, yielding different products.
4.3.1 Cross-Section
The charge, mass, and energy of a bombarding particle (or projectile) determine its interaction probability with a target atom, which is assumed to be at rest. Their mass and charge also affect the likelihood of interaction. A particle(s) from the compound nucleus also has mass, charge, and kinetic energy which also influ-ence the probability of any given interaction. These effects are collectively described by the “cross-section” of the interaction, denoted as riwhere i refers to the product of the interaction. There is no guarantee that a particular bombarding projectile will interact with a target nucleus to bring about a given reaction; thus, ronly provides a measure of the probability that it will occur. The cross-section is dependent on the target material and features of the incident “particle,” which may in fact be a photon. These include the energy, charge, mass, and de Broglie wavelength of the projectile, as shown schematically in Figure 4-9, and the vibra-tional frequency, the spin, and the energy states of nucleons in the target atom.
These cannot be predicted directly by nuclear theory; thus, cross-sections for any given arrangement of projectile and target are usually determined by measuring the number of projectiles per unit area (i.e., the flux) before and after they impinge on a target containing N atoms. As shown in Figure 4-10, the reduction in the flux of projectile after passing through the target is a direct measure of the number of interactions that occur in the target; it is a function of f and the inter-action probability rN of the target atoms:
du dx¼ rNu
which can be integrated to yield f(x) = f0e–rNx
Fig. 4-9 Parameters related to interactions of particles with target nuclei.
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4.3 Nuclear Interactions
Fig. 4-10 Incident flux f0of projectiles on a target containing N atoms/cm3is reduced to f(x) after traversing a thickness x (cm).
The units of r will be in terms of apparent area (m2or cm2) per target atom.
Although the cross-section has the dimensions of area, it is not the physical area presented by the nucleus to the incoming particle but is better described as the sum total of those nuclear properties that determine whether a reaction is favor-able or not. When such measurements and calculations were first made for a series of elements, the researchers were surprised that the calculated values of r were as large as observed. Apparently, one of them exclaimed that it was “as big as a barn,” and the name stuck. One barn, or b, is
1 b = 10–24cm2or 10–28m2
which is of the order of the square of a nuclear radius. Cross-sections vary with the energy of the bombarding particle, the target nucleus, and the type of interac-tion.
Example 4-3. A flux of 106neutrons/s in a circular beam of 1 cm radius is incident on a foil of aluminum (density = 2.7 g/cm3) that was measured in several places with a micrometer to average 0.5 cm thickness. After passing through the foil the beam contained 9.93 R 105neutrons/s. Calculate the cross-section rc.
Solution. The number of atoms/cm3in the foil is ð2:7 g=cm3Þð6:022 · 1023atoms=molÞ
26:982 g=mol ¼ 6:026 · 1022atoms=cm3
lnuðxÞu
0 ¼ –rcNx ¼ rcR 6.026 R 1022atoms/cm2R 0.5 cm ln9:93 · 105
106 ¼ –3.013 R 1022rc
and rc¼ 0.233 R 10–24cm2or 0.233 b
Cross-sections are highly dependent on incident particle energy and target mate-rial as shown in Figure 4-11 for neutrons on cadmium and boron. Such plots show a general fall-off of r with increasing neutron energy, i.e., a 1/v dependence of r on neutron speed, and often contain resonance energies where the
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tion is very high. The National Nuclear Data Center at Brookhaven National Labo-ratory ([email protected]) is the authoritative source for similar information on cross-section data for a host of nuclear interactions, including protons, deuterons, alpha particles, and other projectiles, as well as neutrons.
Fig. 4-11 Neutron absorption cross-sections versus energy for cadmium and boron.
4.3.2
Q-values for Nuclear Reactions
The energy balance, or Q-value, of a reaction is a function of the mass change (since mass is proportional to energy) that occurs between a particle and a target and the resulting products:
Q = (MX+ mx)c2– (MY+ my)c2
where mx, MX, my, and MYrepresent the masses of the incident particle, target nucleus, product particle, and product nucleus, respectively. Thus, the Q-value is easily calculated by subtracting the masses of the products from the masses of the reactants. If the Q-value is positive, the kinetic energy of the products is greater than that of the reactants and the reaction is “exoergic,” i.e., energy is gained at the expense of the mass of the reactants. If the Q-value is negative, the reactions are “endoergic” and energy must be supplied for the reaction to occur.
The amount of energy actually needed to bring about an endoergic reaction is somewhat greater than the Q-value. When the incident particle collides with the target nucleus, conservation of momentum requires that the fraction 160
4.3 Nuclear Interactions mx/(mx+ MX) of the kinetic energy of the incident particle must be retained by the products as kinetic energy; thus, only the fraction MX/(mx+ MX) of the energy of the incident particle is available for the reaction. The threshold energy Eth, which is somewhat larger than the Q-value, is the kinetic energy the incident particle must have for the reaction to be energetically possible; it is given by
Eth¼ Q MXþ mx
MX
Example 4-4. Calculate the Q-value for the14N (a,p) O17reaction and the threshold energy.
Solution. The atomic masses from Appendix B are:
Mass of reactants
The mass difference is –0.001280 u, which has an energy equivalent of Q = –1.19 MeV; thus, the reaction is endoergic (i.e., energy must be supplied). The
The mass difference is –0.001280 u, which has an energy equivalent of Q = –1.19 MeV; thus, the reaction is endoergic (i.e., energy must be supplied). The