NUCLEAR PHYSICS Fall term 2017 Exercise set 9 … · NUCLEAR PHYSICS Exercise set 9 SOLUTIONS Fall...

NUCLEAR PHYSICSExercise set 9 SOLUTIONS

Fall term 2017Period 2

Theory of β decay: Energy relations. Spectrum shape and decay constant in Fermi theory. Gamow-Teller transitions. Electron capture. Literature: Lecture notes 9, Bertulani: Chapter 8 (B-8); Krane:Ch 6. Note that Bertulani uses Gaussian units. A recommended source of nuclear data to consult iswww.nndc.bnl.gov, or the handbooks listed in the previous Exercises.

Electromagnetic transitions, gamma decay, internal conversion and IPF, Mössbauer effect, CoulombExcitation Literature: Bertulani: Chapter 9 (B-9); Krane: Ch 10.

1. (B-8.11, Fermi-Kurie plot)

(a) Depict the Fermi-Kurie plot for the β− and β+ decays of the I = 1+ ground state of 64Cu,using points extracted from Fig. 8.1 (B-8) and values of the Fermi function from Fig 8.2 (B-8). Say, from the aspect of the figures, if they correspond to allowed (L = 0) or forbidden(L > 0) transitions.

(b) A non-relativistic expression for the Fermi function (in Gaussian units) is

F (Z,Ee) =2πη

1− exp (−2πη)where η± = ∓Ze

2

~ve= ∓Zα

β,

α is the fine-structure constant (= e2/~c in Gaussian units), and ve = βc is the velocity ofthe electron corresponding to the energy Ee. Calculate some (3) representative values andcompare with those that you used in (a). (Hint: Use reduced variables for momentum andenergy.)

(c) Use table (B-8.1) to determine the orbital angular momentum L of the emitted β particle ineach case. Are the results coherent with those obtained in (a)?

(d) How does one obtain the momentum distribution for the β particles in each case?

2. (B-8.12, Classification of β decay)

The Iπ = 0+ ground state of 50Mn decays by β+ to the ground state of the same spin and parityof 50Cr, with a half-life of 283.21±0.11 ms. Decay scheme is shown in the attached figure.

(a) Is the transition a Fermi transition or a Gamow-Teller transition?

(b) Determine the value of fT1/2 for this transition.

(c) Is the result compatible with table 8.1 and figure 8.5 (in Bertulani)?

3. (B-8.19, Electron capture, often denoted by ε in the literature)

(a) Try to xplain why the electron capture branch 50Mn→50Cr is very small compared to β+

branch (Fig 1).

(b) Use Eq. (B-8.42) and Fig. (B-8.4) or the corresponding expressions and figures given inthe lecture notes to find the proportion of electron captures with respect to the β+ decay of12555 Cs that emits positrons with maximum energy of 2.082 MeV.

(c) Compare the result with the value obtained from Fig (B-8.12) and with the experimentalresult shown in the attached decay scheme (Fig. 2).

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Figure 1: Decay scheme of 50Mn.

Figure 2: Decay scheme of 125Cs.

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4. Transition probabilities. The total transition probability (for Ji → Jf) is sometimes expressed inthe form

λfi =∑L,ν

|γ(νL, Ji → Jf)|2 , λi =1

τi=∑

f

λfi (1)

where the absolute transition amplitude is defined as

γ(νL, Ji → Jf) = iL+Λ(ν)

[8π

~[(2L+ 1)!!]2L+ 1

L

(Eγ~c

)2L+1

B(νL)

]1/2

. (2)

Here i2 = −1 and Λ(ν) = 0 for electric (ν=E) and Λ(ν) = 1 for magnetic (ν=M) transitions.The sum is over all possible multipoles allowed by the selection rules πiπf = (−1)L+Λ(ν) and∆(JiJfL). The reduced transition probabilities B(EL) and B(ML) are

B(νL; Ji → Jf ) =1

2Ji + 1

∣∣∣⟨f ||M̂(νL)||i⟩∣∣∣2

and they contain the essential nuclear structure information of the transition. Often two (some-times three) multipoles contribute to the total transition probability (1) and the (multipole) mixingratio with L > L′ is defined

δ(νL/ν ′L′) =γ(νL, Ji → Jf)

γ(ν ′L′, Ji → Jf)(3)

(a) Show that in the case of competing E2 and M1 multipoles, the transition probability can beexpressed in terms of B(E2) and δ(E2/M1).

(b) Show that B(µL; Jf → Ji) = 2Ji+12Jf+1B(µL; Ji → Jf ).

5. Single-particle units (Weisskopf units, W.u.)

The reduced transition probabilities are expressed in e2fm2L for EL multipoles and in µ2Nfm2L−2

for ML multipoles. An often used estimate is the Weisskopf unit, given

B(EL)W.u. =1

4π

(3

L+ 3

)2

(1.2A1/3)2L e2fm2L (4)

B(ML)W.u. =10

π

(3

L+ 3

)2

(1.2A1/3)2L−2 µ2Nfm2L−2 (5)

(a) For a light nucleus (A = 10), compute the ratio of the emission probabilities for electricquadrupole (E2) and magnetic dipole (M1) radiation according to the Weisskopf estimates.Consider all possible choices for the parities of the initial and final states.

(b) Repeat for a heavy nucleus (A = 200).

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Solutions

1. (a) To avoid reading the data from the figures, you can scan them in good enough resolution(300 DPI) and use a decent digitiser (PlotDigitizer etc.) to extract the values. Or if you haveaccess to the electronic version of the book, take a screenshot to extract the data points.Because the intensity values (y axis) are given in arbitrary units, you may normalize thevertical scale to e.g. 100 or 1000.In the Fermi-Kurie plot, the values of√

N(pe)

p2eF (ZD, pe)

vs. Te = E −mec2

or √N(Ee)

F (ZD, Te)vs. Te = E −mec

2

are plotted. Because in β decay the electron and positron energies are in the MeV range,one must use relativistic formulas:

p2 =1

c2(E2 −m2

ec4) =

1

c2(E +mec

2)(E −mec2) =

1

c2(Te + 2mec

2)Te.

If done carefully, the FK-plots should show approximately linear shape towards the high-energy end of the spectrum, indicating that both are allowed transitions. Note that ZP = 29,so that ZD(β−) = 30 and ZD(β+) = 28.

(b) Calculation of the Fermi function from the given formula requires that you express it interms of the electron or positron kinetic energy. Just solve the velocity (or β = v/c) using

pe =mev√1− v2

c2

⇒ βe ≡ve

c=

√Te(Te + 2mec2)

Te +mec2

If calculated carefully, the given (non-relativistic) approximation of the Fermi function givesrather reasonable values!

(c) As both the transitions were found to be allowed, the orbital angular momentum of theleptons must be Lβ = 0. We can also try to estimate the possible values of the spin andparity of the final state, Iπf . Because Lβ = 0, the parity does not change and the totalangular momentum Jβ = Sβ where Sβ = 0 for a pure Fermi transition and Sβ = 1for a pure Gamow-Teller transition. Thus the final state is either 1+ (Fermi) or 0+ or 2+

(Gamow-Teller).

Our prediction is in agreement with the above decay scheme (β+). The β− decay to 64Znis energetically possible only to the ground state—which again happens to be 0+!

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(d) The conversion between momentum and energy spectra is done using EdE = c2pdp andthe relativistic formula

E = c√p2 +m2c2

You should convince yourself that the energy spectrum of Eq. (8.21) in Bertulani’s bookthen transforms to the momentum spectrum given in the lecture notes (with the mass of theneutrino→ 0).

2. (a) Trivially it is a Fermi transition, because the emitted leptons (positron and neutrino, bothwith spin 1

2~) can only couple to a spin singlet state in a 0+ → 0+ transition!

(b) The idea was to estimate the value of the Fermi integral, f(ZD, w0), for that transition andthen just calculate the product fT . (Note: because the transition is superallowed, the fTvalue should be about 3100 s, as shown in the lectures). The maximum energy w0 is 12.94(in units of mec

2) which corresponds to Fermi integral of 12094 (approximated from theasymptotic expression f(Z = 0, w0)

w0�1−−−−→ w50/30 given in the lecture notes). Together

with the given half-life we get fT = 3424 s, and log10 fT = 3.53. This is remarkablygood result, compared to the log10 fT value of 3.4846, shown in the decay scheme. (Asseen from Fig. B-8.4, the correct value of the Fermi integral (Z = 24) would be somewhatsmaller than we estimated and would bring the result closer to the literature value!)

(c) Oh yeah, it certainly is! What we in fact have accomplished is that the transition is (su-per)allowed already on the basis of the estimated value of the log fT , and we could predictwhat final state spin and parities were possible (if we didn’t know them already).

3. (a) We need to find a quantitative estimate for a quantity to show why the EC branch is so small.An obvious quantity to use is the ratio of the transition probabilities, i.e., the ratio λEC/λβ+.We can either use the curves in Fig. B-8.12. because it shows directly the required ratio,or calculate using Eq. B-8.42 with values of the Fermi integral from Fig. B-8.4. Let’s usethe first way: According to Bertulani ET = QEC + mc2 = 8145 keV = 15.94 mc2. Fromthe figure B-8.12 we have (for ZD = 24): log10(λK/λβ+) ≈ −3.9 so that the ratio ofprobabilities is about 1.25× 10−4.

(b) ZD = 54. Let’s try to calculate using the figures and formulas given in the lecture notes(there was a typo in Eq. 99, but it has been corrected so that the EC/β+ calculated using Eqs.98 and 99 is consistent with Bertulani’s 8.42). Maximum energy of positrons in reducedunits is w0 = 4.07, which corresponds to log10 f = −0.5 (from the Fig. after Eq. 99 in thelecture notes) giving the EC/β+ ratio of about 3:1.

(c) Bertulani: The curves should be read atET = Qε+mc2 = 5.01mc2, so that the log10-ratio

from figure B-8.12. is −0.2 yielding a EC to positron ratio of 0.63. It’s not far from theobserved ratio of 27:26 shown in the decay scheme. What if we calculate from Eq. B-8.42,it’s identical to Eq. 99 of the lecture notes, but in Gaussian units. Estimate f ≈ w5

0/30 =37.42, from the asymptotic behaviour, and use the Qε in the formula to get a ratio of 0.71.This approximation (w � 1)is perhaps too crude. Note that Eq. (97) gives f ≈ 27.9 forthe β+ decay and a log10 fT ≈ 5.48, not far from the value shown (5.60). Also note thatthe intensities (Iβ+, Iε) given in the figure are normalized to a total of 100% which meansthat the partial half-life for the β+ decay to the 125Xe ground state T1/2(1

2 1→ 1

2 1) = 46.7

min /0.26 = 10780 s.

4. (a) Assuming that only E2 and M1 multipoles contribute in the transition i→f,

λfi = |γ(E2)|2 + |γ(M1)|2 = |γ(E2)|2(

1 +|γ(M1)|2

|γ(E2)|2

)= |γ(E2)|2

(1 +

1

δ2(E2/M1)

),

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in terms of the squared transition amplitudes. Note that the phase factors for ML and E(L+1) are equal (those are often the multipoles that compete), although it seldom matters if weare dealing with the squared magnitudes of the absolute transition amplitudes. The sign ofthe mixing ratio depends on the chosen phase convention (Rose and Brink or Frauenfelderand Steffen), but the phases must be defined always so that δ2 ≥ 0, to make any sense!Write now explicitly

|γ(E2)|2 =8π

~[5 · 3 · 1]23

2

(Eγ~c

)5

B(E2) =4π

75~

(Eγ~c

)5

B(E2), (6)

we have the final expression

λfi =4π

75~

(Eγ~c

)5

B(E2)×(

1 +1

δ2(E2/M1)

). (7)

The units of the B(νL) deserve a short discussion:

[B(EL)] = 1 e2 fm2L = 102L e2b2

[B(ML)] = 1µ2N fm2L−2

where in gaussian units e2 = α~c ≈ 1.4400 MeV fm and the nuclear magneton µ2N =

e2(~c/2Mprotonc2) ≈ 0.01593 MeV fm. (In SI units, replace e2 → e2

4πε0.)

From Eqs. (1) and (2) it’s seen that the B(νL)/~ must have the unit of s−1.

(b) This is rather trivial to show, because it follows directly from the definition and the fact that

the matrix element∣∣∣⟨f ||M̂ ||i⟩∣∣∣2 =

∣∣∣|⟨i||M̂ ||f⟩∣∣∣25. (a) The Weisskopf estimate for B(E2) and B(M1)

BW(E2) =1

4π

(3

5

)2

(1.2A1/3)4 e2fm4 (8)

BW(M1) =10

π

(3

4

)2

(1.2A1/3)0 µ2Nfm0 =

90

16πµ2

N

=90

16πe2

(~c

2mpc2

)2

(9)

and from Problem 4.(a), Eq. (6), the Weisskopf estimate for the E2 transition probability

|γ(E2)|2 =4π

75~

(Eγ~c

)5

B(E2)⇒

|γW(E2)|2 =9× 1.24

625~A4/3

(Eγ~c

)5

e2fm4. (10)

Similarly,

|γ(M1)|2 =8π

~[3 · 1]22

1

(Eγ~c

)3

B(M1) =16π

9~

(Eγ~c

)3

B(M1)⇒

|γW(M1)|2 =10

~

(Eγ~c

)3

e2

(~c

2mpc2

)2

. (11)

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The Weisskopf estimate for the ratio of the transition probabilities is then∣∣∣∣ γW(E2)

γW(M1)

∣∣∣∣2 =1.24 · 90

625A4/3

(Eγ~c

)2(2mpc2

~c

)2

fm4

= 1.37× 10−3A4/3E2γ [MeV−2], (12)

after plugging in the constants in suitable units. In this formula, the transition energy is inMeV.Note that it doesn’t matter what the parity or spin of the initial or final states are, as long asboth multipoles are allowed. M1 and E2 are allowed only between states with same parity.Of course, in 0± → 1± or 1± → 0± only M1 multipole is the possible. If the spin changesby 2 units, then E2 is possible but not M1.The answer to part (c) is then about 3

100 × E2γ (0.0295 to be more exact).

(b) The factor A4/3 increases the ratio to about 64100 × E

2γ (rounded from 0.636).

In summary, we may say that the relative importance of the E2 multipole increases in heavynuclei and can surpass M1 if the transition energy is high enough. Remember, however, thatour conclusion is based on the crude Weisskopf estimate which certainly is not the wholestory.

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