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| {{More footnotes|date=November 2011}}
| | == Nike Free 5.0 Uk Companies Must Engage Facebook Visitors == |
| In [[nuclear physics]], the '''semi-empirical mass formula''' ('''SEMF''') (sometimes also called '''Weizsäcker's formula''', or the '''Bethe-Weizsäcker formula''', or the '''Bethe-Weizsäcker mass formula''' to distinguish it from the [[Bethe–Weizsäcker process]]) is used to approximate the [[mass]] and various other properties of an [[atomic nucleus]]. As the name suggests, it is based partly on [[theory]] and partly on [[empirical]] measurements. The theory is based on the '''liquid drop model''' proposed by [[George Gamow]], which can account for most of the terms in the formula and gives rough estimates for the values of the coefficients. It was first formulated in 1935 by German physicist [[Carl Friedrich von Weizsäcker]], and although refinements have been made to the coefficients over the years, the structure of the formula remains the same today.<ref>
| |
| {{cite journal
| |
| |last=von Weizsäcker |first=C. F.
| |
| |year=1935
| |
| |title=Zur Theorie der Kernmassen
| |
| |language=German
| |
| |journal=[[Zeitschrift für Physik]]
| |
| |volume=96 |issue=7–8 |pages=431–458
| |
| |bibcode=1935ZPhy...96..431W
| |
| |doi=10.1007/BF01337700
| |
| }}</ref><ref name=Bailey>
| |
| {{cite web
| |
| |last=Bailey |first=D.
| |
| |title=Semi-empirical Nuclear Mass Formula
| |
| |url=http://www.upscale.utoronto.ca/GeneralInterest/DBailey/SubAtomic/Lectures/LectF25/Lect25.htm
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| |work=PHY357: Strings & Binding Energy
| |
| |publisher=[[University of Toronto]]
| |
| |accessdate=2011-03-31
| |
| }}</ref>
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|
| |
|
| The SEMF gives a good approximation for atomic masses and several other effects, but does not explain the appearance of [[magic number (physics)|magic numbers]] of protons and neutrons, and the extra binding-energy and measure of stability that are associated with these numbers of nucleons.
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| | <li>[http://swapsomething.net/forum.php?mod=viewthread&tid=196278 http://swapsomething.net/forum.php?mod=viewthread&tid=196278]</li> |
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| | </ul> |
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| [[File:Betheweizsaecker.jpg|right|thumb|500px|A graphical representation of the semi-empirical binding energy formula. The binding energy per nucleon in MeV (highest numbers in dark red, in excess of 8.5 MeV per nucleon) is plotted for various [[nuclide]]s as a function of ''Z'', the [[atomic number]] (on the y-axis), vs. '''N''', the [[neutron number]] (on the x-axis). The highest numbers are seen for ''Z'' = 26 (iron).]]
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| ==The liquid drop model and its analysis==
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| The liquid drop model in [[nuclear physics]] treats the [[atomic nucleus|nucleus]] as a drop of incompressible nuclear fluid. It was first proposed by [[George Gamow]] and then developed by [[Niels Bohr]] and [[John Archibald Wheeler]]. The fluid is made of [[nucleon]]s ([[proton]]s and [[neutron]]s), which are held together by the [[strong nuclear force]]. This is a crude model that does not explain all the properties of the nucleus, but does explain the spherical shape of most nuclei. It also helps to predict the binding energy of the nucleus.
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| | | <li>[http://www.thered.cn/bbs/forum.php?mod=viewthread&tid=43073&fromuid=6212 http://www.thered.cn/bbs/forum.php?mod=viewthread&tid=43073&fromuid=6212]</li> |
| Mathematical analysis of the theory delivers an equation which attempts to predict the binding energy of a nucleus in terms of the numbers of [[proton]]s and [[neutron]]s it contains. This equation has five terms on its right hand side. These correspond to the cohesive binding of all the [[nucleon]]s by the [[strong nuclear force]], the [[electrostatic]] mutual repulsion of the protons, a [[Surface tension|surface energy]] term, an asymmetry term (derivable from the protons and neutrons occupying independent [[quantum]] [[momentum]] states) and a pairing term (partly derivable from the protons and neutrons occupying independent [[quantum]] [[spin (physics)|spin]] states).
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| | | <li>[http://recit.cscapitale.qc.ca/~clj/prof/spip.php?article22 http://recit.cscapitale.qc.ca/~clj/prof/spip.php?article22]</li> |
| If we consider the sum of the following five types of energies, then the picture of a nucleus as a drop of incompressible liquid roughly accounts for the observed variation of binding energy of the nucleus:
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| | | <li>[http://www.haikuo78.com/news/html/?234216.html http://www.haikuo78.com/news/html/?234216.html]</li> |
| [[File:Liquid drop model.svg]]
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| | | <li>[http://www.film-video-dvd-production.com/spip.php?article6/ http://www.film-video-dvd-production.com/spip.php?article6/]</li> |
| '''Volume energy'''. When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. So, this nuclear energy is proportional to the volume.
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| | | <li>[http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/ http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/]</li> |
| '''Surface energy'''. A nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.
| | |
| | | </ul> |
| '''[[Coulomb's law|Coulomb]] Energy'''. The electric repulsion between each pair of protons in a nucleus contributes toward decreasing its binding energy.
| |
| | |
| '''Asymmetry energy''' (also called [[Wolfgang Pauli|Pauli]] Energy). An energy associated with the [[Pauli exclusion principle]]. Were it not for the Coulomb energy, the most stable form of nuclear matter would have the same number of neutrons as protons, since unequal numbers of neutrons and protons imply filling higher energy levels for one type of particle, while leaving lower energy levels vacant for the other type.
| |
| | |
| '''Pairing energy'''. An energy which is a correction term that arises from the tendency of proton pairs and neutron pairs to occur. An even number of particles is more stable than an odd number.
| |
| | |
| ==The formula==
| |
| In the following formulae, let ''A'' be the total number of [[nucleon]]s, ''Z'' the number of [[proton]]s, and ''N'' the number of [[neutron]]s.
| |
| | |
| The mass of an atomic nucleus is given by
| |
| :<math>m = Z m_{p} + N m_{n} - \frac{E_{B}}{c^{2}}</math>
| |
| where <math>m_{p}</math> and <math>m_{n}</math> are the rest mass of a proton and a neutron, respectively, and <math>E_{B}</math> is the [[binding energy]] of the nucleus. The semi-empirical mass formula states that the binding energy will take the following form:
| |
| :<math>E_{B} = a_{V} A - a_{S} A^{2/3} - a_{C} \frac{Z^2}{A^{1/3}} - a_{A} \frac{(A - 2Z)^{2}}{A} - \delta(A,Z)</math> | |
| | |
| Each of the terms in this formula has a theoretical basis, as will be explained below.
| |
| | |
| ==Terms==
| |
| | |
| ===Volume term===
| |
| | |
| The term <math>a_{V} A</math> is known as the ''volume term''. The volume of the nucleus is proportional to ''A'', so this term is proportional to the volume, hence the name.
| |
| | |
| The basis for this term is the [[Strong interaction|strong nuclear force]]. The strong force affects both protons and neutrons, and as expected, this term is independent of ''Z''. Because the number of pairs that can be taken from ''A'' particles is <math>\frac{A(A - 1)}{2}</math>, one might expect a term proportional to <math>A^{2}</math>. However, the strong force has a very limited range, and a given nucleon may only interact strongly with its nearest neighbors and next nearest neighbors. Therefore, the number of pairs of particles that actually interact is roughly proportional to ''A'', giving the volume term its form.
| |
| | |
| The coefficient <math>a_{V}</math> is smaller than the binding energy of the nucleons to their neighbours <math>E_b</math>, which is of order of 40 [[MeV]]. This is because the larger the number of [[nucleon]]s in the nucleus, the larger their kinetic energy is, due to the [[Pauli exclusion principle]]. If one treats the nucleus as a [[Fermi ball]] of <math>A</math> [[nucleon]]s, with equal numbers of protons and neutrons, then the total kinetic energy is <math>{3\over 5} A \epsilon_F</math>, with <math>\epsilon_F</math> the [[Fermi energy]] which is [[Fermi energy#Nucleus|estimated]] as 28 [[MeV]]. Thus the expected value of <math>a_{V}</math> in this model is <math>E_b - {3\over 5}\epsilon_F \sim 17\;\mathrm{MeV} </math>, not far from the measured value.
| |
| | |
| ===Surface term===
| |
| | |
| The term <math>a_{S} A^{2/3}</math> is known as the ''surface term''. This term, also based on the strong force, is a correction to the volume term.
| |
| | |
| The volume term suggests that each nucleon interacts with a constant number of nucleons, independent of ''A''. While this is very nearly true for nucleons deep within the nucleus, those nucleons on the surface of the nucleus have fewer nearest neighbors, justifying this correction. This can also be thought of as a [[surface tension]] term, and indeed a similar mechanism creates [[surface tension]] in liquids.
| |
| | |
| If the volume of the nucleus is proportional to ''A'', then the radius should be proportional to <math>A^{1/3}</math> and the surface area to <math>A^{2/3}</math>. This explains why the surface term is proportional to <math>A^{2/3}</math>. It can also be deduced that <math>a_S</math> should have a similar order of magnitude as <math>a_V</math>.
| |
| | |
| ===Coulomb term===
| |
| The term <math>a_{C} \frac{Z(Z-1)}{A^{1/3}}</math> or <math>a_{C}\frac{Z^2}{A^{1/3}}</math> is known as the ''Coulomb'' or ''electrostatic term''.
| |
| | |
| The basis for this term is the [[electromagnetic force|electrostatic repulsion]] between protons. To a very rough approximation, the nucleus can be considered a sphere of uniform [[electric charge|charge]] density. The [[potential energy]] of such a charge distribution can be shown to be | |
| :<math>E = \frac{3}{5} \left( \frac{1}{4 \pi \epsilon_{0}} \right) \frac{Q^{2}}{R}</math>
| |
| | |
| where ''Q'' is the total charge and ''R'' is the radius of the sphere. Identifying ''Q'' with <math>Z e</math>, and noting as above that the radius is proportional to <math>A^{1/3}</math>, we get close to the form of the Coulomb term. However, because electrostatic repulsion will only exist for more than one proton, <math>Z^2</math> becomes <math>Z(Z-1)</math>. The value of <math>a_{C}</math> can be approximately calculated using the equation above:
| |
| | |
| [[Nuclear size|Empirical nuclear radius]]: | |
| | |
| :<math>R = r_0 A^{\frac{1}{3}}.</math>
| |
| | |
| Quantum charge integers:
| |
| | |
| :<math>Q = Ze \ </math>
| |
| | |
| :<math>Z^2 = Z(Z - 1) \ .</math>
| |
| | |
| Integration by substitution:
| |
| :<math>E = \frac{3}{5} \left( \frac{1}{4 \pi \epsilon_{0}} \right) \frac{Q^{2}}{R} = \frac{3}{5} \left( \frac{1}{4 \pi \epsilon_{0}} \right) \frac{(Ze)^{2}}{(r_0 A^{\frac{1}{3}})} = \frac{3 e^2 Z^2}{20 \pi \epsilon_{0} r_0 A^{\frac{1}{3}}} \approx \frac{3 e^2 Z(Z - 1)}{20 \pi \epsilon_{0} r_0 A^{\frac{1}{3}}} = a_{C} \frac{Z(Z-1)}{A^{1/3}}</math>
| |
| | |
| Potential energy of charge distribution:
| |
| :<math>E = \frac{3 e^2 Z(Z - 1)}{20 \pi \epsilon_{0} r_0 A^{\frac{1}{3}}}</math>
| |
| | |
| Electrostatic Coulomb constant:
| |
| :<math>a_{C} = \frac{3 e^2}{20 \pi \epsilon_{0} r_0}</math>
| |
| | |
| The value of <math>a_C</math> using the [[fine structure constant]]:
| |
| | |
| :<math>a_{C} = \frac{3}{5} \left( \frac{\hbar c \alpha}{r_0} \right)=\frac{3}{5} \left( \frac{R_P}{r_0} \right)\alpha m_pc^2</math>
| |
| | |
| where <math>\alpha</math> is the [[fine structure constant]] and <math>r_0 A^{1/3}</math> is the [[Nuclear size|radius of a nucleus]], giving <math>r_0</math> to be approximately 1.25 [[femtometer]]s. <math>R_P</math> is the proton Compton radius and <math>m_p</math> the proton mass. This gives <math>a_C</math> an approximate theoretical value of 0.691 [[MeV]], not far from the measured value.
| |
| | |
| :<math>a_{C} = 0.691 \; \text{MeV}</math>
| |
| | |
| ===Asymmetry term===
| |
| [[Image:Semf asymmetric term.svg|right|Illustration of basis for asymmetric term]] | |
| | |
| The term <math>a_{A} \frac{(A - 2Z)^{2}}{A}</math> or <math>{4}a_{A}\frac{((A/2)-Z)^{2}}{A}</math> is known as the ''asymmetry term''. Note that as <math>A = N + Z</math>, the parenthesized expression can be rewritten as <math>(N - Z)</math>. The form <math>(A - 2Z)</math> is used to keep the dependence on ''A'' explicit, as will be important for a number of uses of the formula.
| |
| | |
| The theoretical justification for this term is more complex. The [[Pauli exclusion principle]] states that no two [[fermion]]s can occupy exactly the same [[quantum state]] in an atom. At a given energy level, there are only finitely many quantum states available for particles. What this means in the nucleus is that as more particles are "added", these particles must occupy higher energy levels, increasing the total energy of the nucleus (and decreasing the binding energy). Note that this effect is not based on any of the fundamental forces ([[Gravity|gravitational]], electromagnetic, etc.), only the Pauli exclusion principle.
| |
| | |
| Protons and neutrons, being distinct types of particles, occupy different quantum states. One can think of two different "pools" of states, one for protons and one for neutrons. Now, for example, if there are significantly more neutrons than protons in a nucleus, some of the neutrons will be higher in energy than the available states in the proton pool. If we could move some particles from the neutron pool to the proton pool, in other words change some neutrons into protons, we would significantly decrease the energy. The imbalance between the number of protons and neutrons causes the energy to be higher than it needs to be, ''for a given number of nucleons''. This is the basis for the asymmetry term.
| |
| | |
| The actual form of the asymmetry term can again be derived by modelling the nucleus as a Fermi ball of protons and neutrons. Its total kinetic energy is
| |
| :<math>E_k = {3 \over 5} (N_p {\epsilon_F}_p + N_n {\epsilon_F}_n)</math>
| |
| where <math>N_p</math>, <math>N_n</math> are the numbers of protons and neutrons and <math>{\epsilon_F}_p</math>, <math>{\epsilon_F}_n</math> are their [[fermi energy|Fermi energies]]. Since the latter are proportional to <math>{N_p}^{2/3}</math> and <math>{N_n}^{2/3}</math>, respectively, one gets
| |
| :<math>E_k = C (N_p^{5/3} + N_n^{5/3})</math> for some constant ''C''.
| |
| The leading expansion in the difference <math>N_n - N_p</math> is then
| |
| :<math>E_k = {C\over 2^{2/3}} \left((N_p+N_n)^{5/3} + {5\over 9}{(N_n-N_p)^2 \over (N_p+N_n)^{1/3}}\right) + O((N_n-N_p)^2).</math>
| |
| At the zeroth order expansion the kinetic energy is just the [[Fermi energy#Nucleus|Fermi energy]] <math>\epsilon_F\equiv {\epsilon_F}_p = {\epsilon_F}_n</math> multiplied by <math>{3\over 5}(N_p+N_n)^{2/3}</math>. Thus we get
| |
| :<math>E_k = {3\over 5}\epsilon_F (N_p+N_n)+ {1\over 3}\epsilon_F {(N_n-N_p)^2 \over (N_p+N_n)} + O((N_n-N_p)^4) | |
| = {3\over 5}\epsilon_F A+ {1\over 3}\epsilon_F {(A-2Z)^2 \over A} + O((A-2Z)^4).</math>
| |
| | |
| The first term contributes to the volume term in the semi-empirical mass formula, and the second term is minus the asymmetry term (remember the kinetic energy contributes to the total binding energy with a ''negative'' sign).
| |
| | |
| <math>\epsilon_F</math> is 38 [[MeV]], so calculating <math>a_A</math> from the equation above, we get only half the measured value. The discrepancy is explained by our model not being accurate: nucleons in fact interact with each other, and are not spread evenly across the nucleus. For example, in the [[Nuclear shell model|shell model]], a proton and a neutron with overlapping [[wavefunction]]s will have a greater [[strong interaction]] between them and stronger binding energy. This makes it energetically favourable (i.e. having lower energy) for protons and neutrons to have the same quantum numbers (other than [[isospin]]), and thus increase the energy cost of asymmetry between them.
| |
| | |
| One can also understand the asymmetry term intuitively, as follows. It should be dependent on the [[absolute difference]] <math>|N - Z|</math>, and the form <math>(A - 2Z)^{2}</math> is simple and [[Derivative|differentiable]], which is important for certain applications of the formula. In addition, small differences between ''Z'' and ''N'' do not have a high energy cost. The ''A'' in the denominator reflects the fact that a given difference <math>|N - Z|</math> is less significant for larger values of ''A''.
| |
| | |
| ===Pairing term===
| |
| The term <math>\delta(A,Z)</math> is known as the ''pairing term'' (possibly also known as the pairwise interaction). This term captures the effect of [[Spin (physics)|spin]]-coupling. It is given by:<ref>
| |
| {{cite book
| |
| |last=Krane |first=K.
| |
| |year=1988
| |
| |title=Introductory Nuclear Physics
| |
| |pages=68
| |
| |publisher=[[John Wiley & Sons]]
| |
| |isbn=0-471-85914-1
| |
| }}</ref>
| |
| | |
| :<math>\delta(A,Z) = \begin{cases} -\delta_{0} & Z,N \mbox{ even } (A \mbox{ even}) \\ 0 & A \mbox{ odd} \\ +\delta_{0} & Z,N \mbox{ odd } (A \mbox{ even})\end{cases}</math> | |
| where
| |
| :<math>\delta_{0} = \frac{a_{P}}{A^{1/2}}.</math>
| |
| | |
| Due to [[Pauli exclusion principle]] the nucleus would have a lower energy if the number of protons with spin up were equal to the number of protons with spin down. This is also true for neutrons. Only if both ''Z'' and ''N'' are even can both protons and neutrons have equal numbers of spin up and spin down particles. This is a similar effect to the asymmetry term.
| |
| | |
| The factor <math>A^{-1/2}</math> is not easily explained theoretically. The Fermi ball calculation we have used above, based on the liquid drop model<!-- Was Self-redirect! --> but neglecting interactions, will give an <math>A^{-1}</math> dependence, as in the asymmetry term. This means that the actual effect for large nuclei will be larger than expected by that model. This should be explained by the interactions between nucleons; For example, in the [[nuclear shell model|shell model]], two protons with the same quantum numbers (other than [[Spin (physics)|spin]]) will have completely overlapping [[wavefunction]]s and will thus have greater [[strong interaction]] between them and stronger binding energy. This makes it energetically favourable (i.e. having lower energy) for protons to pair in pairs of opposite spin. The same is true for neutrons.
| |
| | |
| ==Calculating the coefficients== | |
| | |
| The coefficients are calculated by fitting to experimentally measured masses of nuclei. Their values can vary depending on how they are fitted to the data. Several examples are as shown below, with units of megaelectronvolts.
| |
| | |
| {|class="wikitable sortable"
| |
| !
| |
| ! Least-squares fit
| |
| ! Wapstra<ref>
| |
| {{cite book
| |
| |last=Wapstra |first=A. H.
| |
| |year=1958
| |
| |chapter=Atomic Masses of Nuclides
| |
| |title=External Properties of Atomic Nuclei
| |
| |pages=1–37
| |
| |publisher=[[Springer (publisher)|Springer]]
| |
| |doi=10.1007/978-3-642-45901-6_1
| |
| |isbn=978-3-642-45902-3
| |
| }}</ref>
| |
| ! Rohlf<ref>
| |
| {{cite book
| |
| |last=Rohlf |first=J. W.
| |
| |year=1994
| |
| |title=Modern Physics from α to Z<sup>0</sup>
| |
| |publisher=[[John Wiley & Sons]]
| |
| |isbn=978-0471572701
| |
| }}</ref>
| |
| |-
| |
| ! <math>a_V</math>
| |
| | 15.8
| |
| | 14.1
| |
| | 15.75
| |
| |-
| |
| ! <math>a_S</math>
| |
| | 18.3
| |
| | 13
| |
| | 17.8
| |
| |-
| |
| ! <math>a_C</math>
| |
| | 0.714
| |
| | 0.595
| |
| | 0.711
| |
| |-
| |
| ! <math>a_A</math>
| |
| | 23.2
| |
| | 19
| |
| | 23.7
| |
| |-
| |
| ! <math>a_P</math><!-- Is a_P or |δ| constant? -->
| |
| | 12
| |
| | ''n/a''
| |
| | ''n/a''
| |
| |-
| |
| ! <math>\delta</math> (even-even)<!-- Is a_P or |δ| constant? -->
| |
| | ''n/a''
| |
| | -33.5
| |
| | +11.18
| |
| |-
| |
| ! <math>\delta</math> (odd-odd)<!-- Is a_P or |δ| constant? -->
| |
| | ''n/a''
| |
| | +33.5
| |
| | -11.18
| |
| |-
| |
| ! <math>\delta</math> (even-odd)
| |
| | ''n/a''
| |
| | 0
| |
| | 0
| |
| |}
| |
| | |
| The semi-empirical mass formula provides a good fit to heavier nuclei, and a poor fit to very light nuclei, especially [[helium|<sup>4</sup>He]]. This is because the formula does not consider the internal [[Nuclear shell model|shell structure]] of the nucleus. For light nuclei, it is usually better to use a model that takes this structure into account.
| |
| | |
| ==Examples for consequences of the formula==
| |
| By maximizing ''B''(''A'',''Z'') with respect to ''Z'', we find the number of protons ''Z'' of the stable nucleus of atomic weight ''A''.{{citation-needed|date=January 2014}}
| |
| We get
| |
| :<math>Z \approx {1\over 2} {A\over 1 + A^{2/3} {a_C\over 4 a_A}}.</math>
| |
| This is roughly ''A''/2 for light nuclei, but for heavy nuclei there is an even better agreement with [[periodic table|nature]].
| |
| | |
| By substituting the above value of ''Z'' back into ''B'' one obtains the binding energy as a function of the atomic weight, ''B''(''A'').
| |
| Maximizing ''B''(''A'')/''A'' with respect to ''A'' gives the nucleus which is most strongly bound, i.e. most stable. The value we get is ''A'' = 63 ([[copper]]), close to the [[Iron#Isotopes|measured values]] of ''A'' = 62 ([[nickel]]) and ''A'' = 58 ([[iron]]).
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==References==
| |
| *{{cite book
| |
| |last1=Freedman |first1=R.
| |
| |last2=Young |first2=H.
| |
| |year=2004
| |
| |title=Sears and Zemanskey's University Physics with Modern Physics
| |
| |edition=11th
| |
| |pages=1633–1634
| |
| |isbn=0-8053-8768-4
| |
| }}
| |
| *{{cite book
| |
| |last=Liverhant |first=S. E.
| |
| |year=1960
| |
| |title=Elementary Introduction to Nuclear Reactor Physics
| |
| |publisher=[[John Wiley & Sons]]
| |
| |page=58-62
| |
| |lccn=60011725
| |
| }}
| |
| *{{cite book
| |
| |last1=Choppin |first1=G.
| |
| |last2=Liljenzin |first2=J.-O.
| |
| |last3=Rydberg |first3=J.
| |
| |year=2002
| |
| |chapter=Nuclear Mass and Stability
| |
| |chapterurl=http://jol.liljenzin.se/KAPITEL/CH03NY3.PDF
| |
| |title=Radiochemistry and Nuclear Chemistry
| |
| |url=http://jol.liljenzin.se/BOOK-3.HTM
| |
| |edition=3rd |pages=41–57
| |
| |publisher=[[Butterworth-Heinemann]]
| |
| |isbn=978-0-7506-7463-8
| |
| }}
| |
| | |
| ==External links==
| |
| * [http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/liqdrop.html Nuclear liquid drop model]
| |
| * [http://www.phy.uct.ac.za/courses/phy300w/np/ch1/node22.html The semi-empirical mass formula]
| |
| * [http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/liqdrop.html Liquid drop model] in the [http://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.html hyperphysics] online reference at [[Georgia State University]].
| |
| * [http://www.phys.jyu.fi/research/gamma/publications/akthesis/node4.html Liquid drop model with parameter fit] from ''First Observations of Excited States in the Neutron Deficient Nuclei <sup>160,161</sup>W and <sup>159</sup>Ta'', Alex Keenan, PhD thesis, [[University of Liverpool]], 1999 ([http://www.phys.jyu.fi/research/gamma/publications/akthesis/thesis.html HTML version]).
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| [[Category:Nuclear physics]]
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| [[Category:Nuclear chemistry]]
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| [[Category:Subatomic particles]]
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| [[Category:Radiochemistry]]
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| [[Category:Mass]]
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