phgn 422: nuclear physics - today at minesinside.mines.edu/~kleach/phgn422/lectures/lecture3.pdf ·...

P H G N 4 2 2 : N U C L E A R P H Y S I C S

PHGN 422: Nuclear PhysicsLecture 3: Nuclear Radii, Masses, and Binding Energies

Prof. Kyle Leach

September 3, 2019

Last Week.....

• The atomic nucleus is a very dense, positively charged objectcomposed of protons and neutrons

• Nuclei are organized according to their Z and N values on theNuclear Chart (or Chart of the Nuclides)

• Nuclei are held together by the strong interaction, and thenuclear force is attractive at short range, but repulsive at veryshort distances (we will talk about why today)

• So...back to our electron scattering experiments!

— Prof. Kyle Leach — PHGN 422: Nuclear Physics

Electron Scattering on Nuclei

Source: Fig. 3.1 (pg. 46) – Introductory Nuclear Physics, Ken Krane

Light Scattering on an Opaque Object

Source: Department of Physics, Brock University

What Can This Tell Us About the Nucleus?

Opaque Object Nuclear Matter

The Nuclear Charge Distribution

Source: Fig. 3.4 (pg. 49) – Introductory Nuclear Physics, Ken Krane

What Does This Tell Us About The Nucleus?1 The boundary of the nucleus is not sharp, but displays a

probability distribution

• The angular distributions from elastic scattering of electrons fromnuclei do not show sharp minima

• These minima become even less sharp with increasing Z

2 The central nuclear charge density is nearly the same for allnuclei

• There is no dependence on the density of charge as a function of Z• Nucleons do not seem to preferentially organize based on type (ie.

protons or neutrons)

3 The overall matter density of all nucleons in the nucleus musttherefore be constant as well? (number of nucleons per unitvolume)

• If this is true, we should be able to determine what the density ofnuclear matter is

• Also, can we find a generic way of obtaining the matter radius of agiven nucleus?

probability distribution• The angular distributions from elastic scattering of electrons from

nuclei do not show sharp minima

• These minima become even less sharp with increasing Z

nuclei do not show sharp minima• These minima become even less sharp with increasing Z

• There is no dependence on the density of charge as a function of Z

• Nucleons do not seem to preferentially organize based on type (ie.protons or neutrons)

Density of Nuclear MatterWell, to start...let’s assume that the nucleus is a perfect sphere. Fromhere, we can estimate the volume and perhaps the density...

+Proton (π)

Neutron (ν)

V =43πR3

The Nuclear Matter Radius

If the nuclear matter density is also indeed constant for all nuclei:

V =43πR3 ≈ constant

Then, we can relate the radius of a nucleus to the number ofnucleons A:

R ∝ A1/3

To determine this proportionality constant, we can relate the totalnuclear matter radius R to the matter radius of the individual nucleonsR0

The Nuclear Matter RadiusThe nucleons can also be considered spherical:

Therefore:

43πR3 = A · 4

=⇒ R = R0 · A1/3

Experimentally we know that R0 ≈ 1.2 fm. So, the nuclear matterradius is R = 1.2 · A1/3! Further detailed discussion on this topic canbe found in Chapter 3.1 of Krane.

The Nature of Nuclear MatterOne of the most remarkable conclusions from all of this is thatnuclear matter does not seem to change density regardless of thesize of the nucleus!! In other words, the number of nucleons per unitof volume is roughly constant for all nuclei.

How dense is nuclear matter (comparatively speaking). Well....

• Sea Water: 1.0× 103 kg/m3

• Tin Oxide: 1.6× 103 kg/m3

• Steel: 1.1× 104 kg/m3

• Lead: 2.5× 104 kg/m3

• Core of the Sun: 1.5× 105 kg/m3

• Nuclear Matter: 2.3× 1017 kg/m3

• Steel: 1.1× 104 kg/m3

• Lead: 2.5× 104 kg/m3

• Steel: 1.1× 104 kg/m3

• Lead: 2.5× 104 kg/m3

• Steel: 1.1× 104 kg/m3

• Lead: 2.5× 104 kg/m3

• Steel: 1.1× 104 kg/m3

• Lead: 2.5× 104 kg/m3

• Steel: 1.1× 104 kg/m3

• Lead: 2.5× 104 kg/m3

Question:What if the nucleus were nearly 20 orders of magnitude larger?

Question:What if the nucleus were nearly 20 orders of magnitude larger?Well, this is not hypothetical....these are known as neutron stars

Source: NASA.gov

Question:What if the nucleus were nearly 20 orders of magnitude larger?Well, this is not hypothetical....these are known as neutron stars

Source: NASA.govSlide 12 — Prof. Kyle Leach — PHGN 422: Nuclear Physics

Bound Nuclear SystemsLimits of Nuclear Existence

Putting aside neutron stars for now, let us take a look at the limits ofwhat nuclei can exist, and how we define it.

The Atomic Mass and Nuclear Binding Energy

As we briefly mentioned last week, the mass of a given atom is notsimply the sum of neutron, proton, and electron masses, ie:

M(AZXN)c2 6= Z · mpc2 + N · mnc2 −

(Z · mec2 −

Z∑i=1

For a nucleus to exist (ie. be a bound system), the followingconstraint must be satisfied (neglecting the electrons for a moment):

M(AZXN)c2 < Z · mpc2 + N · mnc2

For the nucleons to be bound inside of the nucleus, there needs to besome energy difference. We call this the Binding Energy.We’ll define what we mean on the chalkboard....

Mass Excess

Since the atomic mass in MeV/c2 can become a cumbersome way ofdealing with larger nuclei (ie. m(208Pb) = 193 733 MeV/c2)

We can define a useful experimental mass value relative to ourdefinition of the atomic mass unit in Lecture 1 (1u = 931.502 MeV/c2).

m(AZXN)c2 = (A · u)c2 + ∆c2

=⇒ ∆c2 = m(AZXN)c2 − A) · u

Where ∆ is referred to as the Mass Excess or Mass Defect, andhelps us to quantify how much a specific nucleus deviates from ourapproximation of the atomic mass unit.

• It can be either positive or negative, as long as we satisfyM(A

ZXN)c2 < Z · mpc2 + N · mnc2.

Example: What is the Mass Excess (∆) for 16O in MeV?

First we’ll start with the experimentally measured mass of 16O in u:

• Remember, we say mass 16 for 16O, but this is not exactly true.

m(AZXN)c2 = 15.994915 u

Now solve for the mass excess ∆, (recall 1 u = 931.505 MeV/c2)

∆ = m(AZXN − A) · u

= (15.994915− 16) · 931.505 MeV= −4.737 MeV

Characteristics of Nuclear BindingThe Proton and Neutron Separation Energies (Sp and Sn)

Analogous to atomic ionization energies, these separation energiescan tell us about the binding strength for an individual nucleon. Wecan define these on the chalkboard:

Characteristics of Nuclear BindingThe Proton and Neutron Separation Energies (Sp and Sn)

Analogous to atomic ionization energies, these separation energiescan tell us about the binding strength for an individual nucleon. Wecan define these on the chalkboard:

We can also look at the trends of how nuclear binding changes as afunction of the mass number A

Characteristics of Nuclear BindingBinding Energy per Nucleon (BE/A)

Characteristics of Nuclear BindingBinding Energy per Nucleon (BE/A)This brings us to some other revelations about the way nuclei behave:

1 Most nuclei have almost exactly the same BE/A, which isroughly 8 MeV/A. This means the nuclear force saturates suchthat only each nucleon can interact with a few of itsneighbours. Recall that the nuclear force is strongly attractiveONLY at short distances (∼ 1 fm).

2 The most bound nuclei are in the region of A ∼ 56− 62

3 Some structure in this curve also exists (particularly for 4He) thatresults from quantum effects of the nucleus. We will discuss theshell structure of nuclei in a couple of weeks.

4 Nuclei on the left of the peak can release energy by joiningtogether (Nuclear Fusion)

5 Nuclei on the right of the peak can release energy by breakingapart (Nuclear Fission)

Characteristics of Nuclear BindingBinding Energy per Nucleon (BE/A)

Source: The Open University

Nuclear Fusion in Stars

Source: A.C. Phillips, The Physics of Stars, 2nd Edition (Wiley, 1999)

Nuclear Fission in Reactors

Source: Department of Physics, UC Davis

Next Class...

Reading Before Next Class

• Sections 3.2 and 3.3 (first part) in Krane

Next Class Topics

• More on binding energy

• Ways to release of energy in a nuclear decay or reaction

• The experimental determination of atomic masses...and why dowe care?

phgn 422: nuclear physics - today at minesinside.mines.edu/~kleach/phgn422/lectures/lecture3.pdf ·...

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