NUCLEAR PHYSICS
Exercise set 10
Fall term 2015
Period 2
Turn in your solutions by
WEDNESDAY, Nov. 18th
Electromagnetic transitions, gamma decay, internal conversion and IPF, Mössbauer effect, Coulomb
Excitation
Literature: Bertulani: Chapter 9 (B-9); Krane: Ch 10; Heyde: Ch 5. For experimental data see
www.nndc.bnl.gov or consult the handbooks listed in the previous Exercises.
1. Kinematics of photon emission.
(a) Each of the following nuclei emits a photon in a γ transition between an excited state and
the ground state. Given the energy of the photon, find the energy of the excited state and
comment on the relationship between the nuclear recoil energy and the experimental uncertainty in the photon energy: (i) 320.08240±0.00042 keV in 51 V; (ii) 1475.7792±0.0023
keV in 110 Cd; (iii) 1274.537±0.007 keV in 22 Ne; (iv) 8578.696±0.010 keV in 36 Cl; (v)
884.54174±0.00074 keV in 192 lr.
(b) The 1476-keV excited level of 110 Cd in (a) decays also to a lower excited state emitting a
γ ray with energy of 818.0244±0.0018 keV. Determine (i) the energy of that excited state
and (ii) the energy of the corresponding de-exciting γ ray (ground-state transition).
2. Transition probabilities. The total transition probability (for Ji → Jf ) is sometimes expressed in
the form
X
X
1
λfi =
|γ(νL, Ji → Jf )|2 , λi = =
λfi
(1)
τi
L,ν
f
where the absolute transition amplitude is defined as
#1/2
"
8π
L + 1 Eγ 2L+1
L+Λ(ν)
γ(νL, Ji → Jf ) = i
B(νL)
.
~[(2L + 1)!!]2 L
~c
(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 πi πf = (−1)L+Λ(ν) and
∆(Ji Jf L). The reduced transition probabilities B(EL) and B(ML) are
D
E2
1
B(νL; Ji → Jf ) =
f
||
M̂
(νL)||i
2Ji + 1
and they contain the essential nuclear structure information of the transition. Often two (sometimes three) multipoles contribute to the total transition probability (1) and the (multipole) mixing
ratio with L > L0 is defined
δ(νL/ν 0 L0 ) =
γ(νL, Ji → Jf )
γ(ν 0 L0 , Ji → Jf )
(3)
(a) Show that in the case of competing E2 and M1 multipoles, the transition probability can be
expressed in terms of B(E2) and δ(E2/M1).
2Ji +1
2Jf +1 B(µL; Ji → Jf ).
experimental data1 , excluding those
(b) Show that B(µL; Jf → Ji ) =
(c) A recent analysis of
for closed-shell nuclei, leaded to
the global best fit prediction for the partial E2 mean lifetime (in ps) of the first-excited 2+
state of even-even nuclei
τγ (E2) = τ (1 + α) = (1.59 ± 0.28) × 1014 Eγ−4 Z −2 A2/3 ,
where the energy of the transition, Eγ , is in keV. Write down the corresponding prediction
for the reduced electric quadrupole transition probability, B(E2) ↑, from the ground state to
the first-excited 2+ state.
1
S. Raman, C.W. Nestor, Jr., and P. Tikkanen, At. Data Nucl. Data Tables 78, 1–128 (2001)
NUCLEAR PHYSICS
Exercise set 10
Fall term 2015
Period 2
Turn in your solutions by
WEDNESDAY, Nov. 18th
3. Single-particle units (Weisskopf units, W.u.)
The reduced transition probabilities are expressed in e2 fm2L for EL multipoles and in µ2N fm2L−2
for ML multipoles. An often used estimate is the Weisskopf unit, given
1
4π
B(EL)W.u. =
10
π
B(ML)W.u. =
3
L+3
2
3
L+3
2
(1.2A1/3 )2L e2 fm2L
(4)
(1.2A1/3 )2L−2 µ2N fm2L−2
(5)
(a) For the following γ transitions, give all permitted multipoles and indicate which multipole
might be the most intense in the emitted radiation.
−
+
−
−
−
3+
(a) 29 → 72
(b) 12 → 72
(c) 1− → 2+ (d) 4+ → 2+ (e) 11
(f) 3+ → 3+ .
2 →2
+
+
(b) A nucleus has the following sequence of states beginning with the ground state: 32 , 27 ,
5+ 1−
3−
2 , 2 , and 2 . Draw a level scheme showing the intense γ transitions likely to be emitted
and indicate their multipole assignment.
(c) For a light nucleus (A = 10), compute the ratio of the emission probabilities for electric
quadrupole (E2) and magnetic dipole (M1) radiation according to the Weisskopf estimates.
Consider all possible choices for the parities of the initial and final states.
(d) Repeat for a heavy nucleus (A = 200).
4. Internal conversion, a decay process competing with (and sometimes dominating over) γ emission. The total conversion coefficient for a transition Ei → Ef is defined as the ratio
αtot =
Ne
=
Nγ
X
αk
(6)
k=K, L1, L2. . .
The isomeric 2+ state of 60 Co at 58.6 keV decays to the 5+ ground state. Internal conversion
competes with γ emission: the observed internal conversion coefficients are αK = 41, αL = 7,
and αM = 1.
(a) Compute the expected half-life of the 2+ state if the transition multipolarity is assumed to
be M3, and compare with the observed half-life of 10.467 min.
(b) If the transition also contained a small component of E4 radiation how would your estimate
for the half-life be affected?
(c) The 2+ state also decays by direct β − emission to 60 Ni. The maximum β energy is 1.55
MeV and the log10 f T is 7.2. The 2+ state decays 0.25% by β emission and 99.75% by γ
emission and internal conversion. What is the effect on the calculated half-life of including
the β emission?