Chapter 7

Nuclear Instability

◎ Nuclear decay and energy-level diagrams

● More on β-decay

◎ The stability of nuclei

● Spontaneous fission and transition rates

§ 7-1 Nuclear decay and energy-level diagrams

Three different types of nuclear decay:

1. α-decay

2. β-decay

3. γ-decay

The objective of this chapter is to investigate the kinematics of these decays and in particular to elucidate the role of α- and of β-decay in

determining which nuclei are stable and which are not.

1. In this figure the vertical distance between two levels is the energy difference and the level X having the higher energy, is drawn above the one, Y, having the lower energy.

2. A transition from X to Y normally involves the emission of a photon, the energy difference going into photon energy and energy of the recoil nucleus.

3. Photon energies involved in this type of transition usually are of about 10 keV up to 3 or 4 MeV and are all referred as γ-rays.

4. γ-rays are emitted when the nucleus makes a transition from an excited state to a state of lower energy.

γ-ray emitting transitions

The best way to study the existence of the heaviest elements, nucleosynthesis in exploding stars, and other phenomena peculiar to the atomic nucleus is to create customized nuclei in an accelerator like Berkeley Lab's 88-Inch Cyclotron, then capture and analyze the gamma rays these nuclei emit when they disintegrate. The Lab's Nuclear Science Division (NSD) has been a leader in building high-resolution gamma-ray detectors and was the original home of the Gammasphere, the world's most sensitive. Now NSD is leading a multi-institutional collaboration to build Gammasphere's successor, the proposed Gamm

a-Ray Energy Tracking Array, or GRETA.

http://www.lbl.gov/Science-Articles/Archive/sabl/2007/Feb/GRETINA.html

γ-ray spectrum of 257No

Quantum states in 257No and 253Fm

Energy spectrum observed through γ-ray emitting transitions

The energy-level diagram for two nucleic connected by α-decay

The energy-level diagram for the α-decay of 242Pu

He)4,2(),( 42 AZAZ 2)]4,2()4,2(),([ cMAZMAZMQ

α-particle decay

The available energy Qα goes into the kinetic energies of the α-particle and of the recoil of the daughter nucleus.

If Qα > 0, α-decay is energetically possible; however, it may not occur for other reasons.

The expected alpha-decay chain of new isotope 233Am and alpha-particle energy

spectrum. In the spectrum the decay chain is observed with the 6.78 MeV alpha-particle

originating from 233Am decay.

Intensity against alpha energy for four isotopes, note that the line width is wide and some of the fine details can not be seen. This is for liquid scintillation counting where random effects cause a variation in the number of visible photons generated per alpha decay

§ 7-2 More on β-decayThe apparent process in β-decay is he conversion of nucleus (Z,A) into nucleus (Z+1,A) and an electron (e-):

-e),1(),( AZAZ

It is actually the conversion of a bound neutron (n) into a bound proton (p).

epn

For some proton-rich nuclei they can frequently undergo β-decay in which a positron (e+) is emitted.

enp

β- - decay

β+ - decay

A proton-rich nucleus can capture an atomic electron and thereby change a proton into a neutron.

),1(),(e AZAZ or npe electron capture

The electron is captured usually from the K-shell but can be from the L, M, N, or even higher shell. In this process the electron in annihilated.

This experimental energy spectrum is from G. J. Neary, Proc. Phys. Soc. (L

ondon), A175, 71 (1940).

(38%)

(19%)

(43%)

To the right is a fit to the vetoed spectrum. The peaks from 99Tc and 100Ru don't help resolve the miniscule 100Mo peak. The spectrum would look much better if we could remove all the 99Tc contamination and veto more βs, which would also help by getting rid of the 100Ru x rays that come wit

h the βs.

Measurement of the beta-decay

Magnetic Trapping of Ultra-cold Neutrons

For more information, please see our publication P. R. Huffman et al., Nature, 403, 62 (2000).

Electrons emitted through a process called the internal conversion

Internal conversion is a process by which a nuclear excited state decays by the direct ejection of one of its atomic electrons; it occurs normally in competition with photon emission.

Early mysteries concerning the β- decay:

1. The kinetic energy spectrum of the emitted electrons is continuous.

2. Consider the following decay. It seems to violate the law of angular momentum conservation.

-147

146 eNC

0 1 1/2 ?

There is no way that angular momentum can be

conserved in the decay.

In 1930 Pauli made a hypothesis that provided a solution to these difficulties and that has satisfied all experimental tests.

Wolfgang Ernst Pauli

Austrian–Swiss physicist (1900–1958)

He proposed that an electrically neutral particle of spin 1/2 is created

and emitted at the same time as the electron (or positron) in β-decay.

The particle is called a neutrino (symbol, ν) and it can take a share of the energy because in β-decay there is now a three-body configuration of the final state.

It turns out that nature has three kinds of neutrinos each with its own antiparticle and mass.

) ,( ee ) ,( μμ ) ,( ττ

Three flavors of neutrinos

We need to label the neutrinos in nuclear β-decay in the following fashion:

ee),1(),( AZAZ

ee),1(),( AZAZ

e),1(),(e AZAZ

β- - decay

β+ - decay

Electron capture

In order to consider the energetics of β-decay, we need to know the mass of the neutrino emitted in β-decay. It is known to be less than 18 eV and since this is small compared to the total energy released in most β-decays we shall assume that the mass is zero. In fact the neutrino mass would be of considerable significanc

e in cosmology and in theories of elementary particles.

universe-review.ca/R15-13-neutrino.htm

Neutrino’s mass is not zero.

Free nucleon decays:

(1). A free neutron undergoes β-decay with a mean life time τ = 898 seconds.

eνepn Qβ = [Mn – (Mp+me)]c2 = [939.573 – (938.791 + 0.511)] MeV = 0.782 MeV > 0

This process is certainly energetically possible since Qβ = 0.782 MeV which is larger than zero.

(2). Consider the case for a free proton:

eνenp Qβ = [Mp – (Mn+me)]c2 = [938.791 – (939.573 + 0.511)] MeV

= – 1.293 MeV < 0

This process is energetically impossible since Qβ = – 1.293 MeV which is smaller than zero.

This is a fortunate situation as the stability of protons (on the time scale of >> 1014 years) is essential to the existence of the universe and of ourselves.

The IMB Proton Decay Detector in the Fermi Lab The IMB detector was a 60-foot (18.3 m)cube of ultra-pure water constructed in a salt mine underneath Lake Erie. The water was surrounded by 2000 light-sensitive phototubes, designed to detect proton decay. The experiment became famous for the observation of the neut

rino burst emitted by a nearby Supernova (exploding star).

A diver takes a swim in the IMB detector.

http://www.foster08.com/about/?id=science

Super-K is located 1,000 m underground in the Mozumi Mine (Kamioka Mining and Smelting Co.) in Hida city (formerly Kamioka town), Gifu, Japan. It consists of 50,000 tons (1 ton = 103 kg) of pure water surrounded by about 11,200 photom

ultiplier tubes. The cylindrical structure is 41.4 m tall and 39.3 m across.

http://www.arch102-07.form-ula.com/?p=1537

(3). Electron capture in a hydrogen atom

e- νnpe

Qβ = [(Mp+me) – Mn]c2 = [(938.791+0.511) – 939.573] MeV

= – 0.271 MeV < 0

This process is unlikely to happen as its Qβ value is seen smaller than zero.

Furthermore the safety of the proton against electron capture follows immediately from the existence of free neutr

on decay.

Energy conditions in β-decay and electron capture in terms of nuclear masses, M(Z, A):

eνe)1() ( : ,AZAZ,

eνe)1() ( : ,AZAZ,

eν)1(e) ( :EC ,AZAZ,

Qβ = [M(Z,A) – M(Z + 1, A) – me]c2 > 0

Qβ = [M(Z,A) – M(Z – 1, A) – me]c2 > 0

QEC = [M(Z,A) – M(Z – 1, A )+ me]c2 > 0

If a condition is satisfied, then the appropriate decay is possible and the excess energy available is shared as kinetic energy among the products in a manner which conserves linear momentum.

Note that electron capture can sometimes occur when

β+-decay is impossible.

§ 7- 3 The stability of nuclei

In this figure the stable nuclei lie on or near a curve of N against Z.

For those nuclei which are not in this region of stability β-decay can occur.

Beta decay process while keeping A constant can step Z to bring the nucleus onto a stable position in the chart.

For an unstable proton-rich nucleus a positron is emitted so as to step down its atomic number Z by 1 while keeping its mass number A a

constant and moving toward the stable region.

For an unstable neutron-rich nucleus an electron is emitted so as to step up its atomic number Z by 1 while keeping its mass number A a constant

and moving toward the stable region.

We are therefore interested in the nuclear mass of isobars (fixed A) as a function of Z.

In this figure the nuclear mass of A = 101 is shown:

The masses are calculated using the semi-empirical mass formula and a smooth curve of no physical significance is drawn through the points. The actual atomic masses are given by open points.

Since N > Z for most nuclei, an increase in Z decreases the asymmetry contribution to the mass. However, the increase in Z increases the Coulomb energy contribution. On the left of the minimum in the mass the decrease wins over the increase and the atomic mass decreases

with Z. On the right the increase in Coulomb energy wins and the mass increases.

1. On the left limb as Z changes from 41→42→43→44 the atomic mass decreases at every step so that β- decay can occur at each step.

2. The mass changes on the right are also greater than 2mec2, so β+ decay or electron capture can occur at each step. Actually 45→44 can go by electron capture only but not the β+ decay.

Stable isobar of A = 101

22c),( ZZAZAM (4. 14)

For odd-A isobars, δ = 0, and equation (14) gives a single parabola, which is shown in the figure (a) for a typical case. We will see later that if

M(A,Z) > M(A, Z+1)

beta (electron) decay takes place from Z to Z+1

M(A,Z) > M(A, Z-1)

electron capture and perhaps positron decay takes place from Z to Z - 1

It is clear from the figure (a) that for odd-A nuclides there can be only one stable isobar.

(4. 15)

β-decay in odd A and even A isobars ― from chapter 4

22c),( ZZAZAM (4. 14)

For even-A isobars, two parabolas are generated by the equation (14), differing in mass by 2δ. A typical case is given in the figure (b). Depending on the curvature of the parabolas and the separation 2δ, there can be several stable even-even isobars. Figure (b) shows that for certain odd-odd nuclides both conditions (15) are met so that electron and positron decay from the identical nuclide are possible and do indeed occur.

In this figure nucleus Z = 43 can decay by electron capture to Z = 42 or by

β –- decay to Z = 44. The prediction is that there are two stable isobars for A = 100, namely Z = 42 and 44, which i

s true.

odd Z odd N

even Z even N

odd Z odd N

even Z even N

In the case of even-A nuclei even-Z nuclei have a binding energy

advantage arising from the pairing term, whereas the odd-Z nuclei have a

lower binding energy due to the opposite contribution from this term. Thus there are two curves of isobar

atomic mass against Z and alternate Z lie on different curves.

Another example

This situation in even-A nuclei sometimes leaves an odd-odd isobar energetically able to decay by all modes. There is

indeed an example of 40K19 which is able to decay in three different modes with different branching ratios.

K4019

A4018

Ca4020

Conclusion 1 is almost true; there are four real exceptions among the light nuclei, namely 2H1, 6Li3, 10B5, and 14N7. In the case of 2H1, it is the only bound system with A = 2. At the other A, the masses are changing rapidly with Z and the two parabolas of A = 100 as shown in the figure become, at this low Z, narrower, and have steeper sides. The result is that each of these values of A, the nuclei one place away in Z from each of these odd-odd nuclei are heavier and there can be no decay to these neighbors.

Conclusion 2 is true without any exception.

An examination of the even-A nuclei suggests two conclusions:

1. there is no stable odd-odd nucleus;2. many even-even nuclei can have more than one stable isobar.

Away from the line of stability β-decay is a certainty. This conclusion becomes evident when we consider regions fairly close to the line of stability.

Nuclei that are very rich in protons, or in neutrons, may be beyond the appropriate drip line, where it is energetically possible to emit a proton (or neutron) as a direct relief of the richness.

Such a process occurs relatively very rapidly and so these nuclei have very short mean lives, in fact so short in some cases that the nucleus may not have a distinct existence before the neutron is emitted and the producing process and decay become a nuclear reaction.

The best prediction of the neutron drip line for

heavier elements

Experimentally determined

neutron drip line

Nature 499(2007)992

Neutron drip line:The line on the Z, N plane where the neutron separation energy is zero.

We now have to apply the energy conditions for α-decay to occur in real nuclei and to find where in the periodic table it is expected to occur.

Rewrite the definition of Qα in terms of the nuclear binding energies.

),()4,2()4,2( AZBBAZBQ

He)4,2(),( 42 AZAZ

Thus α-decay is energetically allowed if

A

B

A

ABA

A

B

AZBAZBB

d

)/(d4

d

d4

)4,2(),()4,2(

A

B

A

B

AA

A

A

A

B

A

B

AA

A

BA

AA

B

d

d

d

d

d

d

d

d

d

d

(1)

(2)

A

A

B 3107.743.28 which is AA

B 3107.7075.7

Above A ≈ 120, d(B/A)/dA is about −7.7×10−3 MeV. Now B(2,4), the helium nuclear binding energy, is 28.3 MeV, so the critical A must satisfy the following relation:

A = 151

A

B

A

ABA

A

B

AZBAZBB

d

)/(d4

d

d4

)4,2(),()4,2(

(3)

Above this A the inequality of equation (3) is satisfied by most nuclei and α-decay becomes, in principle, energetically possible. In fact from A = 144 to A = 206, 7 α-emitters are known amongst the naturally occurring nuclides.

From A = 144 to A =206, there are 7 α-emitters of naturally occurring nuclides. When α-emitters are found in this range of A, the energies of the emitted α-particle are normally less than 3 MeV. It is known that the lower the energy release the greater is the lifetime. Their existence implies mean lifetimes comparable to or greater than the age of the earth (about 4 × 109 years). Most nuclei in this range on the line stability may be energetically able to decay by α-emission. They do not do so at a detectable level because the transition rate is too small.

From A = 144 to A =206

Most of the heavy nuclei to be found on earth were probably produced in one or more supernova explosions of early massive stars. Such explosions can produce very heavy nuclei including trans-uranic elements (Z > 92) and their subsequent decay by α-emission will take them down the periodic table in steps of ΔA = −4. Each α-decay increases the ration N/Z until a β- decay intervenes to restore the nucleus closer to the line of stability.

Above Z = 82 many naturally occurring α-emitters are found, many with short lives.

Why are they to be found when their lifetime is so short?

Above Z = 82 (A > 206)

Very long lifetime comparable to the age of the earth

Relatively long lifetime

Fast-decaying daughter nuclei are in secular equilibrium.

Large binding energy per nucleon (7.08 MeV) of the α particle makes α-decay possible for heavy unstable nuclei. It is interesting to note that the nucleon in 12C6 are even more strongly bound (7.6 MeV) than are those in 4He2. As a result decay by 12C6 nucleus emission is energetically possible in some heavy nuclei. Decay by 14C6 emission has been found but it is very rare.

CPbRa 14209223

The probability for this process is very small, about 10-9 relative to the α-decay.

However, all these processes are a part of a spectrum of possibilities in which a heavy nucleus breaks into two (or more) parts. These are called “spontaneous fission”.

n3BrLaU 9035

14557

23892

Sunset on The Mars

~ The End ~