# KRANE NUCLEAR PHYSICS PDF

|NTRODUCTORY. NUCLEAR. PHYSICS. Kenneth S. Krane. Oregon State University. JOHN WILEY & SONS. New York • Chichester • Brisbane - Toronto •. Library of Congress Cataloging in Publication Data: Krane, Kenneth S. Introductory nuclear physics. Rev. ed. of Introductory nuclear physics/David Halliday. Krane - Introductory Nuclear sppn.info - Ebook download as PDF File .pdf) or read book online.

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sppn.info - Ebook download as PDF File .pdf) , Text File .txt) or view presentation slides online. download Introductory Nuclear Physics on sppn.info ✓ FREE SHIPPING on qualified orders. introductory and modern physics classes, and i have continued using Introductory Nuclear Physics Krane Solutions Manual Pdf Pdf introductory nuclear.

We therefore adopt in this text the phenomenological approach, discussing each type of measurement, the theoretical formulation used in its analysis, and the insight into nuclear structure gained from its interpretation. We begin with a summary of the basic aspects of nuclear theory, and then turn to the experiments that contribute to our knowledge of structure, first radioactive decay and then nuclear reactions.

Finally, we discuss special topics that contribute to micro- scopic nuclear structure, the relationship of nuclear physics to other disciplines, and applications to other areas of research and technology. This atom of material, invisible to the naked eye, was to Democritus the basic constituent particle of matter. For the next years, this idea remained only a speculation, until investigators in the early nineteenth century applied the methods of experimental science to this problem and from their studies obtained the evidence needed to raise the idea of atomism to the level of a full-fledged scientific theory.

Today, with our tendency toward the specialization and compartmentalization of science, we would probably classify these early scientists Dalton, Avogadro, Faraday as chemists. Once the chemists had elucidated the kinds of atoms, the rules governing their combinations in matter, and their systematic classification Mendeleevs periodic table , it was only natural that the next step would be a study of the fundamental properties of individual atoms of the various elements, an activity that we would today classify as atomic physics.

These studies led to the discovery in by Becquerel of the radioactivity of certain species of atoms and to the further identification of radioactive substances by the Curies in Rutherford next took up the study of these radiations and their properties; once he had achieved an understanding of the nature of the radiations, he turned them around and used them as probes of the atoms themselves.

In the process he proposed in the existence of the atomic nucleus, the confirmation of whch through the painstaking experiments of Geiger and Marsden provided a new branch of science, nuclear physics, dedicated to studying matter at its most fundamental level. Investigations into the properties of the nucleus have continued from Rutherfords time to the present.

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In the s and s, it was discovered that there was yet another level of structure even more elementary and fundamental than the nucleus.

Studies of the particles that contribute to the structure at this level are today carried out in the realm of elementary particle or high energy physics.

Thus nuclear physics can be regarded as the descendent of chemistry and atomic physics and in turn the progenitor of particle physics. Investigation of nuclear properties and the laws governing the structure of nuclei is an active and productive area of physical research in its own right, and practical applications, such as smoke detectors, cardiac pacemakers, and medical imaging devices, have become common.

Thus nuclear physics has in reality three aspects: probing the fundamental particles and their interactions, classifying and interpreting the properties of nuclei, and providing technological advances that benefit society. The fundamental positively charged particle in the nucleus is the proton, which is the nucleus of the simplest atom, hydrogen.

A nucleus of atomic number Z therefore contains Z protons, and an electrically neutral atom therefore must contain Z negatively charged electrons. The mass number of a nuclear species, indicated by the symbol A , is the integer nearest to the ratio between the nuclear mass and the fundamental mass unit, defined so that the proton has a mass of nearly one unit.

We will discuss mass units in more detail in Chapter 3. For nearly all nuclei, A is greater than 2, in most cases by a factor of two or more.

Thus there must be other massive components in the nucleus. Before , it was believed that the nucleus contained A protons, in order to provide the proper mass, along with A - 2 nudear electrons to give a net positive charge of Z e. However, the presence of electrons within the nucleus is unsatisfactory for several reasons: 1. The nuclear electrons would need to be bound to the protons by a very strong force, stronger even than the Coulomb force.

Yet no evidence for this strong force exists between protons and atomic electrons. Electrons that are emitted from the nucleus in radioactive p decay have energies generally less than 1 MeV; never do we see decay electrons with 20 MeV energies. Thus the existence of 20 MeV electrons in the nucleus is not confirmed by observation.

The total intrinsic angular momentum spin of nuclei for which A - 2 is odd would disagree with observed values if A protons and A - 2 electrons were present in the nucleus. In region 2. In Chapter 4 we show how these concepts can be extended to three dimensions and applied to nucleon- nucleon scattering problems. The D term in 4. Note that for constant potentials. Step Potential. The resulting solution is illustrated in Figure 2. The de Broglie wavelength changes from X.

Although the mathematical forms may be different for nonconstant potentials V x. This is a simple example of a scattering problem. Equation 2. All classical particles are reflected at the boundary. The wave function decreases exponentially in the classically forbidden region. The classical particle is never directly observed in that region. The solution is illustrated in Figure 2. Barrier Potential. The wavelength is the same on both sides of the barrier.

The particle can never be ob- served. The wave undergoes reflec- tions at both boundaries. This phenomenon of barrier penetration or quantum mechanical tunneling has important applications in nuclear physics. The quantum wave can penetrate the barrier and give a nonzero probability to find the particle beyond the barrier.

The particle is incident from the left. A bead sliding without friction on a wire and bouncing elastically from the wails is a simple physical example. Inside the well..

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The energy E. The corresponding wave functions are 2. The energy spectrum is illustrated in Figure 2. These states are bound states. The wave function for each level is shown by the solid curve. The right side of these equations defines a circle of radius P. Contrast this with the infinite well. They can be solved numerically on a computer. To keep the wave function finite in region 1 when x 4 -GO.

The solutions are 4. The graphical solutions are easiest if we rewrite Equations 2. As we will discuss in Chapter 4. The solutions are determined by the points where the circle intersects the tangent function. The Simple Harmonic Oscillator Any reasonably well-behaved potential can be expanded in a Taylor series about the point xo: Compare with the infinite well shown in Figure 2.

SO If xo is a potential minimum. Thus to a first approximation. Summary By studying these one-dimensional problems. A wave packet can penetrate into the classically forbidden region and appear beyond a potential barrier that it does not have enough energy to overcome.

The degree of the polynomial the highest power of x that appears is determined by the quantum number n that labels the energy states. The study of the simple harmonic oscillator therefore is important for under- standing a variety of systems.

Notice that the probabilities resemble those of Figure 2. Some of the resulting wave functions are listed in Table 2. This solution also shows penetration of the probability density into the classically forbidden region.

Quantum waves can undergo reflection and transmission when they encoun- ter a potential barrier. For our system. A noteworthy feature of this solution is that the energy levels are equally spaced.

The function h x turns out to be a simple polynomial in x. When a potential confines a particle to a region of space. Inside the well. The particle is permitted only a set of discrete energy values. We will skip the mathematical details and give only the result of the calculation: This situation is known as degeneracy.

The energy is given in units of E. The lowest state. The first excited state has three possible sets of quantum numbers: Figure 2. Each of these distinct and independent states has a different wave function. Notice that the spacing and ordering do not have the regularity of the one-dimen- sional problem. We will soon discuss its similar role in the nuclear shell model. Degeneracy is extremely important for atomic structure since it tells us how many electrons can be put in each atomic subshell.

The solution is 1 2. When we search for separable solutions.. The solution. The differential equation for Q. These functions give the angular part of the solution to the Schrodinger equation for any central potential V r. For each potential V r. Theequation for O 8 is 1 d 2.

Abramowitz and I. To find the energy eigenvalues. From the tables we find j. As in the case of the Cartesian well.

This gives j. As an example. Handbook of Mathematical Functions New York: The quantum number n does not arise directly in the solution in this case. To find the radial probability density. Such three-dimensional distribu- tions are difficult to represent graphically. The angular dependence of the probability density for any central potential is given by I Ye.. This situation is very similar to the case of electronic orbits in atoms..

The spherical harmonics Ye. In fact. Note that all j r vanish at the origin except j o. The right side shows the corre- sponding-normalized radial probability density. Powell and B.

As in the one-dimensional case. The solutions are discussed in J. In this case the polynomials are called Laguerre polynomials. From the mathematical solution of the radial equation. The Simple Harmonic Oscillator We consider a central oscillator potential. The energy levels are given by E. Chapter 7. Some representative solutions are listed in Table 2. The energy does not depend on t'. The general properties of the one-dimensional solutions are also present in this case: The angular.

The Coulomb Potential The attractive Coulomb potential energy also has a simple central form. Quantum Mechanics Reading. Since the energies do not depend on m. The vertical arrows mark the classical turning points.

As in Figure 2. The total degeneracy of each.. Pauling and E. There are two important new features in the three-dimensional calculations that do not arise in the one-dimensional calcula- tions: The degeneracies will have the same effect in the nuclear shell model that the 't and me degeneracies of the energy levels of the Coulomb potential have in the atomic shell model-they tell us how many particles can occupy each energy level.

Chapter 5. Table 2. For a discussion of these solutions. These new features will have important consequences when we discuss. Introduction to Quantum Mechanics New York: Summary The mathematical techniques of finding and using solutions to the Schrodinger equation in three dimensions are similar to the techniques we illustrated previ- ously for one-dimensional problems.

The Bohr radius a. This angular momentum quantum number has the same function in all three-dimensional problems involving central potentials. The states are labeled with n. In classical physics. We must first find a quantum mechanical operator for The behavior of angular momentum in quantum theory is discussed in the next section.

This can be done simply by replacing the components of p with their operator equivalents: In atomic physics. We first consider the magnitude of the angular momentum. The atomic substates with a given t j value are labeled using spectroscopic notation. When we now try to find the direction of 4 we run into a barrier imposed by the uncertainty principle: These are summarized in Table 2.

Why doesn't the "centrifugal repulsion" appear to occur in this case? P Figure 2. It is the very act of measuring one component that makes the other two indeterminate. The spin can be treated as an angular momentum although it cannot be represented in terms of classical variables. When we measure lA. By convention.. Once we determine the value of one component. The conventional vector representation of this indeterminacy is shown in Figure 2.

For the electron. The complete description of an electronic state in an atom requires the introduction of a new quantum number. This is a fundamental limitation. It is not at all obvious that a similar picture is useful for nucleons inside the nucleus. We discuss this topic in detail when we consider the nuclear shell model in Chapter 5.

The combined t and s vectors rotate or precess about the direction of j. In interpreting both figures. If a system is left unchanged by the parity operation. The vectors t and s have definite lengths. We would indicate these states as p? The vector j precesses about the z direction so that j. The vector coupling of Equation 2. When there is an additional quantum number. From Equations 2. If the potential V r is left unchanged by the parity operation.

For some of the solutions. This is usually indicated along with the total angular momentum for that state. Mixed-parity wave functions are not permitted. In it was discovered that certain nuclear processes p decays gave observable quantities whose measured values did not respect the parity symmety. The parity of the combined wave function will be even if the combined wave function repre- sents any number of even-parity particles or an even number of odd-parity particles.

The second consequence of the parity rule is based on its converse. A description of these experiments is given in Section 9. Recall our solutions for the one-dimensional harmonic oscillator.

Polynomials mixing odd and even powers do not occur. Thus nuclear states can be assigned a definite parity. On the other hand. That is. The wave functions listed in Table 2.

The establish- ment of parity violation in p decay was one of the most dramatic discoveries in nuclear physics and has had profound influences on the development of theories of fundamental interactions between particles.

In Chapter 10 we will discuss how the parity of a state can be determined experimentally. The wave function for a system of many particles is formed from the product of the wave functions for the individual particles. Notice the solutions illustrated in Figure 2.

When A is the same as B. This is of course. The probability to Jind two identical particles of half-integral spin in the same quantum state must always vanish. All combined wave functions representing identical particles must be either completely symmetric or completely antisymmetric We can regard A and B as representing a set of quantum numbers.

A special case arises when we have identical quantum states A and B. We therefore have two cases. If we choose the minus sign then the result is an antisymmetric wave function.

If the ex- change changes the sign. Let us consider the case of two particles.. If we choose the plus sign. When we turn to our laboratory experiments to verify these assertions. Probability densities must be invariant with respect to exchange of identical particles. If the electrons are truly indistinguishable.

If the sign does not change upon exchange of the particles. The energy of the state is precisely determined. We also construct some simple antisymmetric wave functions for the quarks that make up nucleons and other similar particles. In transitions between atomic or nuclear excited states. The Heisenberg relationship. Thus the total decay energy must be constant.

This vanishing of the antisymmetric wave function is the mathemati- cal basis of the Pauli principle. Thus a state with an exact energy lives forever. A nonstationary state has a nonzero energy uncertainty A E. The expectation values of physical observ- ables. The decay probability or transition probability X the number of decays per uir-IMJme is inversely related to the mean lifetime 7: In particular.

The lifetime T of this state the mean or average time it lives before making a transition to a lower state can be estimated from the uncertainty principle by associating 7 with the time At during which we are permitted to carry out a measurement of the energy of the state. We can do this if we have knowledge of 1 the initial. No such vanishing occurs for the symmetric combination. Even though a system may make a transition from an initial energy state Ei to a final state E.

If the final state E. Later in this text. We still describe the various levels as if they were eigenstates of the system. Elementary Modern Physics. We will merely state the result. Principles of Modern Physics New York: Weidner and R. If there is a large density of states near E. The quantity p E. New York: Quantum Physics of Atoms. The density of final states must be computed based on the type of decay that occurs. Modern Physics New York: Quantum Mechanics. French and E..

Elementary Quantum Mechanics San Francisco: Be sure to take special notice that the decay probability depends on the square of the transition matrix element. Concepts of Modern Physics. This terminol- ogy comes from an alternative formulation of quantum mechanics based on matrices instead of differential equations. Quantum mechanics references at about the same level as the present text are listed below: The integral Vf[ is sometimes called the matrix element of the transition operator V I.

The calculation of X is too detailed for this text. Eisberg and R. It is the number of states per unit energy interval at E. Allyn and Bacon. Quantum Mechanics New York: Advanced quantum texts. Introduction to the Quantum Theory. Continue Figure 2. Evaluate all undetermined coefficients in terms of a single common coefficient.

Solve the Schrodinger equation for the following potential: For the ground state and first two excited states of the one-dimensional simple harmonic oscillator. Comment on the behavior of T. Find p. Derive Equation 2. Show that the first four radial wave functions listed in Table 2.

Carry out the separation procedure necessary to obtain the solution to Equation 2. First couple two of the spins. Is the potential invariant with respect to parity? Are the wave functions? Discuss the assignment of odd and even parity to the solutions. Some values of S may occur more than once. For instance. Both the Coulomb potential that binds the atom and the resulting electronic charge distribution extend to infinity.

Such a property would be exceedingly difficult to measure. Are there a few physical properties that can be listed to give an adequate description of any nucleus? To a considerable extent. This also leads to some difficulties since we obtain different radii for an atom when it is in different compounds or in different valence states.

These are the static properties of nuclei that we consider in this chapter. In later chapters we discuss the dynamic properties of nuclei. We must therefore select a different approach and try to specify the overall characteristics of the entire nucleus. To understand these static and dynamic properties in terms of the interaction between individual nucleons is the formidable challenge that confronts the nuclear physicist.

These experiments would then determine the distribution of nuclear charge primarily the distribution of protons but also involving somewhat the distribution of neutrons. In some experiments. Figure 3. Beams of electrons with energies MeV to 1 GeV can be produced with high-energy accelerators. Let us try to make this problem more quantitative.

The problems we face result from the difficulty in determining just what it is that the distribution is describing. In other experiments. These minima do not fall to zero like diffraction minima seen with light incident on an opaque disk. Several minima in the diffractionlike pattern can be seen. In Section 5 of this chapter. The first minimum in the diffractionlike pattern can clearly be seen.

To see the object and its details. For nuclei. It is therefore relatively natural to characterize the nuclear shape with two parameters: The Distribution of Nuclear Charge Our usual means for determining the size and shape of an object is to examine the radiation scattered from it which is. These are. As we will soon discuss. The shape of the cross section is somewhat similar to that of diffraction patterns obtained with light waves..

Ehrenberg et al. The interaction V Y depends on the nuclear charge density Zep. Measuring the scattering probability as a function of the angle Y then gives us the dependence. This also shows diffractionlike behavior. Since we originally assumed that the scattering was elastic. Heisenberg et al. Note the different vertical and horizontal scales for the two energies.

Ze2 p. The distance over which this drop occurs is nearly independent of the size of the nucleus. The origin of coordinates is located arbitrarily. Thus R cc All3. The results of this procedure for several different nuclei are shown in Figure 3.

## Introductory-Nuclear-Physics-new-Krane.pdf

Thus the number of nucleons per unit volume is roughly constant: One remarkable conclusion is obvious. The conclusion from measurements of the nuclear matter distribution is the same. The charge density is roughly constant out to a certain point and then drops relatively slowly to zero.

Nucleons do not seem to congregate near the center of the nucleus. We define the skin. These measurements give the most detailed descrip- tions of the complete nuclear charge distribution.

The quantity P q is known as a form factor. The skin thickness t is shown for 0. Nuclear Sizes and Structure Oxford: These distributions were adapted from R. Since real nuclei are not points. From the slope of the line. Barrett and D. The root mean square rms radius. The value of t is approximately 2. The nuclear charge density can also be examined by a careful study of atomic transitions. As a rough approximation. The central density changes very little from the lightest nuclei to the heaviest.

V r will not tend toward infinity for a nucleus with a nonzero radius. In solving the Schrodinger equation for the case of the atom with a single electron. More complete listings of data and references can be found in the review of C. Using the 1s hydrogenic wave function from Table 2. The line is not a true fit to the data points.

The effect of the spherical nucleus is thus to change the energy of the electronic states. The slope of the straight line gives R.

Atomic Data and Nuclear Data Tables Since no such nuclei exist. The latter is a fair approximation to real nuclei as suggested by Figure 3. Detailed calculations that treat the 1s electron relativistically and take into account the effect of other electrons give a more realistic relationshp between the slope of Figure 3.

Thus EK A. Because these electronic orbits lie much further from the nucleus than the 1s orbit. The agreement with the A 2 I 3 dependence is excellent. Thus a single measurement of the energy of a K X ray cannot be used to deduce the nuclear radius. The slope. The resulting values are in the range of 1. It is also possible to measure isotope shifts for the optical radiations in atoms those transitions among the outer electronic shells that produce visible light.

The calculated K X-ray energies. Lee et al. Since the Bohr radius depends inversely on the mass. The reason these effects are so small has to do with the difference in scale of l o 4 between the Bohr radius a. The energy of the K X ray in Hg is about keV. Both groups. If the optical transitions involve s states.

Measurements across a large range of nuclei are consistent with R. For integrals of the form of Equation 3. We can improve on this situation by using a muonic atom. These effects of the nuclear size on X-ray and optical transitions are very small. The muon is a particle identical to the electron in all characteristics except its mass. The data are taken from P. Beams of the resulting muons are then focused onto a suitably chosen target. A Data taken from J. Compare these results with Figure 3.

The optical transition used for these measure- ments has a wavelength of Muons are not present in ordinary matter. Bonn et al. Since ordinary K X rays are in the energy range of tens of keV.. These data were obtained through laser spectroscopy. In contrast to the case with electronic K X rays. The energy levels of atomic hydrogen depend directly on the electronic mass. Initially the muon is in a state of very high principal quantum number n.

As we discuss in Chapter 4. The energy difference between 3He and 3H is thus a measure of the Coulomb energy of the second proton.

The isotope shift can clearly be seen as the change in energy of the transitions. From E. The data are roughly consistent with R. Changing a proton into a neutron should therefore not affect the nuclear energy of the three-nucleon system. Yet another way to determine the nuclear charge radius is from direct measurement of the Coulomb energy differences of nuclei. To get from 3He to 3H. The effect is about 0. The two peaks show the 2p. Shera et al.

As in Figure 3. The difference in Coulomb energy. Consider now a more complex nucleus.

The Coulomb energy of a uniformly charged sphere of radius R is 3. Because neutrons and protons each must obey the Pauli principle. The situation is resolved if we choose a case as with 3He-3H in which no change of orbital is involved. The slope of the line gives R. Engfer et al. The Z of the first nucleus must equal the N of the second and thus the N of the first equals the 2 of the second. Examples of such pairs of mirror nuclei are l. If we try to change a proton to a neutron we now have a very different circumstance.

The data are taken from a review of muonic X-ray determinations of nuclear charge distribu- tions by R.. The data show the expected dependence. The minimum proton energy necessary to cause this reaction is a measure of the energy difference between " B and l l C.

The maximum energy of the positron is a measure of the energy difference between the nuclei. As expected from Equation 3. A second method of measuring the energy difference is through nuclear reactions. R Since 2 represents the nucleus of higher atomic number. The probabil- ity for scattering at a certain angle depends on the energy of the incident particle exactly as predicted by the Rutherford formula.

The Distribution of Nuclear Matter An experiment that involves the nuclear force between two nuclei will often provide a measure of the nuclear radius. The difference between the two techniques results from differences between muons and rrr mesons: A third method for determining the nuclear matter radius is the measurement of the energy of rrr-mesic X rays.

This method is very similar to the muonic X-ray technique discussed above for measuring the charge radius. These calculated values depend on the nuclear matter radius R.

## Krane - Introductory Nuclear Physics.pdf

As the energy of the incident a particle is increased. For another example. As an example of a measurement that determines the size of the nuclear matter distribution. In this case the radius is characteristic of the nuclear. If the separation between the two nuclei is always greater than the sum of their radii.

When the rrr-meson wave functions begin to overlap with the nucleus. In addition. Like the muons. The deterinination of the spatial variation of the force between nuclei enables the calculation of the nuclear radii. The a decay probabilities can be calculated from a standard barrier-penetration approach using the Schrodinger equation. All of these effects could in principle be used as a basis for deducing the nuclear radius. The a particle must escape the nuclear potential and penetrate a Coulomb potential barrier.

This situation is known as Rutherford scattering and is discussed in Chapter In this case the Rutherford formula no longer holds. The point at which this breakdown occurs gives a measure of the size of the nucleus.

We should instead use a distribution. For these calculations it is therefore very wrong to use the "uniform sphere" model of assuming a constant density out to R and zero beyond R. We will not go into the details of the calculations. Both show the A l l 3 dependence with R. We merely give the result. When the incident a particle gets close enough to the target Pb nucleus so that they can interact through the nuclear force in addition to the Coulomb force that acts when they are far apart the Rutherford formula no longer holds.

Eisberg and C. Adapted from a review of Y particle scattering by R. The deuteron is relatively weakly bound and thus this number is rather low compared with typical nuclei. As we probe ever deeper into the constituents of matter.

In a hydrogen atom. The measured half-lives can thus be used to determine the radius R where the nuclear force ends and the Coulomb repulsion begins. At a yet deeper level. In a simple nucleus. If so. The masses of the quarks are not known no free quarks have yet been confirmed experimentally and quarks may not be permitted to exist in a free state. It may therefore be necessary to correct for the mass and binding energy of the electrons. It is therefore not possible to separate a discussion of nuclear mass from a discussion of nuclear binding energy.

Even though we must analyze the energy balance in nuclear reactions and decays using nuclear masses. The half-life for Y emission depends on the probability to penetrate the barrier. The separated masses may be focused to make an image on a photographic plate. The E field would exert a force qE that would tend to divert the ions upward in Figure 3. Equally as important. To determine the nuclear masses and relative abundances in a sample of ordinary matter.

All mass spectroscopes begin with an ion source. The next element is a velocity selector. Mass spectrometry was the first technique of high precision available to the experimenter. In this section. Often a vapor of the material under study is bombarded with electrons to produce the ions. A schematic diagram of a typical mass spectrograph is shown in Figure 3. Ions emerging from the source have a broad range of velocities.

In the next section we analyze the measured masses to determine the binding energy. Ions pass through undeflected if the forces cancel. The measurement of nuclear masses occupies an extremely important place in the development of nuclear physics. Not so with atomic physics -nineteenth-century measurements of average atomic weights led to discrepancies in the atomic periodic table.

To measure masses to precisions of order requires instruments of much greater sophistication. The fixed point on the atomic mass scale is 12C. An ion source produces a beam with a thermal distribution of velocities. To determine the mass of another atom. It would be preferable to measure the smaller difference between two nearly equal masses.

Often the magnetic fields of the velocity and momentum selectors are common. A velocity selector passes only those ions with a particular velocity others being deflected as shown.. In practice we could calibrate for one particular mass. Neglecting corrections for the difference in the molecular binding energies of the two molecules which is of the order of l o p 9 u. Measuring the current passing through an exit slit which replaces the photographic plate of Figure 3.

From the relative areas of the peaks. Nuclide Abundances The mass spectrometer also permits us to measure the relative abundances of the various isotopes of an element. The measured Q value is Notice in particular how the 1 part in lo6 uncertainties in the measured A values give uncertainties in the deduced atomic masses of the order of 1 part in l o 8 or lo9.

It is also possible to determine mass differences by measuring the energies of particles in nuclear reactions. The nuclide 12N is unstable and decays with a half-life of only 0. The nuclear reaction method allows us to determine the masses of unstable nuclides whose masses cannot be measured directly. By measuring the kinetic energies of the reacting particles. Some mass spectrometers are designed to process large quantities of material often at the expense of another character- istic of the equipment.

Separated isotopes. If we add the measured masses of the six stable isotopes with the abundances as relative weighting factors. Separated Isotopes If we set the mass spectrometer on a single mass and collect for a very long time.

The ordinates for the peaks at mass positions 78 and 80 should be divided by 10 to show these peaks in their true relation to the others. A typical sample of natural krypton would consist of a mixture of the six stable isotopes with the above relative composition.

If we surround a plant with an atmosphere of CO. A schematic representation of the process is shown in Figure 3. The first laser is tuned to the transition corresponding to the resonant excitation of isotope A. The second laser has a broad energy profile. A second laser beam is set to a wavelength that corresponds to ionization of the excited atoms.. The beam of neutral atoms from the oven is a mixture of four isotopes A. After passing through the second laser. The A. A beam of neutral atoms passes through a laser beam..

Laser Isotope Separation A completely different technique for separating iso- topes takes advantage of the extremely sharp that is. As discussed in the last section. Laser beams are sufficiently sharp so that they can be tuned to excite electrons in one isotope of a mixture of isotopes but not in the others. The longest-lived radioactive isotope of nitrogen has a half-life of 10 min. The final energy states of the free electron are continuous rather than quantized Other useful and interesting properties that are often tabulated are the neutron and proton separation energies.

We occasionally find atomic mass tables in which. Grouping the 2 proton and electron masses into 2 neutral hydrogen atoms.

In a similar way we can define the proton separation energy Sp as the energy needed to remove a proton: Given the mass defect. The neutron and proton separation energies are analogous to the ionization energies in atomic physics-they tell us about the binding of the outermost or. Even this l o p 6 precision does not affect measurements in nuclear physics because we usually work with dzferences in mass energies.

Electronic binding energies are of order keV in heavy atoms. Several remarkable features are immediately apparent. Attempting to understand this curve of binding energy leads us to the semiempirical mass formula.

Since the binding energy increases more or less linearly with A. The average binding energy of most nuclei is. Since B varies linearly with A. If every nucleon attracted all of the others.

Table 3. First of all. Just like the atomic ionization energies. A where a. As with many other nuclear properties that we will discuss.

We therefore delay discussion of the systematics of separation energies until we discuss nuclear models in Chapter 5. From electron scattering we learned that the nuclear density is roughly constant. These important subjects are discussed in Chapters 13 and This linear dependence of B on A is in fact somewhat surprising. The constants evaluate to 0. These nucleons do not contribute to B quite as much as those in the center.

Thus the surface nucleons contribute to the binding energy a term of the form -a. If our binding energy formula is to be realistic in describing the stable nuclei that are. We also note. A overestimates B by giving full weight to the surface nucleons.

An exception to the above argument is a nucleon on the nuclear surface. Since each proton repels all of the others. We must therefore subtract from B a term proportional to the nuclear surface area. The explanation for this effect will come from our discussion of the shell model in Chapter 5. The surface area of the nucleus is proportional to R 2 or to A Our binding energy formula must also include the Coulomb repulsion of the protons. For heavy nuclei. The parabola will be centered about the point where Equation 3.David Halliday.

The nucleon-nucleon force includes a repulsive term. It is therefore not necessary to write 2. All physically meaningful wave functions must be properly normalized. The conventional vector representation of this indeterminacy is shown in Figure 2.

Since p.

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