Grigorenko-MIFI-Lecture-6

1.

Леонид Григоренко
Лаборатория Ядерных Реакций
им. Г.Н. Флерова, ОИЯИ, Дубна
Ядерная физика
Лекция 6. Резонансные явления.
1. Что такое резонанс.
2. От дискретного спектра к непрерывному.
3. Альфа распад. Теория Гамова.
4. Резонансы в упругом рассеянии.
5. Квазистационарные состояния.
6. Время задержки.
7. R-матричный формализм.
8. Теория компаунд-состояния Бора.
9. Состояния распада с комплексной и действительной энергией.
МИФИ, весна 2026

2.

Что такое резонанс?

3.

Что такое резонанс?
In elastic scattering – phase shift passing 90 dgr
In analitical properties of s-matrix – poles trajectories
In observables – peaks in various cross sections
In nuclear dynamics – quasistationary state concept
In decays – resonance width is lifetime
In reactions – resonance is time delay

4.

От дискретного спектра к непрерывному
REVIEWS OF MODERN PHYSICS, 84 (2012) 567
Radioactive decays at limits of nuclear stability
M. Pfutzner, M. Karny, L. V. Grigorenko, K. Riisager

5.

Nuclear dynamics vs. excitation energy
“Phase volume”
dynamics – only initial
state is important
E*
Transition region –
resonances are broad
Resonance phenomena
r
Radioactivity
Haloes
“Normal nuclei”

6.

Nuclear dynamics vs. excitation energy
E*
From formal dynamics
point of view there is no
clear borderline between
resonance phenomena
and radioactivity
r
From formal structure
point of view there is
no clear borderline
between stationary and
quasistationary states
(radioactivity)

7.

8.

~ 10 MeV
~ 2 MeV

9.

Кулон. Парциальные ВФ
Пороговое поведение сечений в Кулоновском случае

10.

Resonances in elastic scattering
Elastic scattering formulation
is not comfortable for
radioactivity studies

11.

Виртуальные состояния
Теория эффективного радиуса
l=0
s(E) ~ 4p / k2 (1+ctg2(d)) -1
k ctg(d) ~ - 1 / a
s(E ~ 0) ~ 4p a2
k ctg(d) ~ - 1 /a + k2 r0 /2
s(E ~ 0) ~ 4p a2 / (1 - k2(r0/a-1) ) ~ 4p a2 / (1 - cE)
y (r) ~ Exp(- gr)
r0 = 2(ga-1) / g2a

12.

Resonance in elastic
in elastic scattering
Lorentian
profile peak
in the
internal
normalization
(red)
Lorentian
profile peak
in the cross
section
(yellow)
“Normal
resonance”
under barrier
Phase shift pass 90o (magenta)

13.

Joke 1
In reality –
test for
internal
structure
Peaks above
barrier
Two expressed peaks
WF phase shift is not
passing 90o from below
There is WF
concentration in the
interior ONLY for
the first peak

14.

Joke 2
Cross section
dropdown to
zero.
No scattering
at all instead of
very active
scattering
Total “transparency”

15.

Joke 3
“Shallow water
resonance”
WF concentration is provided by
(i) small velocity above the “step” and
(ii) Interference of the waves reflected from origin
and from the right part of the step

16.

R-matrix formalism
Preexponent
Exponent
Elastic scattering formulation is not
comfortable for radioactivity studies
Gamow approach has problems (i) only
probabilities no amplitudes (ii) width
behave wrong on top of the barrier
Strong Coulomb, l = 0
Dimension energy
limiting width – Wigner
limit
Dimensionless
structure factor
No Coulomb, l = 0
G ~ 1/T inverse flight time
through nuclear interior
G ~ exp[-2ph] - the same as
for Gamow approach

17.

R-matrix phenomenology
Description of
elastic/inelastic scattering
Effects of broad
levels
Effects of
structure
In contrast with
normalizations of WFs
spectroscopic factors are
overlaps and their norm
could be larger than 1

18.

Задача
Расчитать ширины основных состояний для 5Не 3/2-, 7Не 1/2-,25O 5/2+, 15Nе 3/2-.
(i) Методом Гамова, используя потенциала Вуда-Саксона
V(r) = V0 / (1 + exp[(r - r0)/a]), V0 ~ - 50 МэВ, а = 0.5 fm, r0 из систематики.
(ii) Используя R-матричное выражение , rch из систематики
(iii) Сравнить с экспериментом

19.

Bohr’s compound nucleus theory
Kinematical
limit. QM cross
section for
spinless particles
cannot be larger
Compound state
resonance is like a pool
with attached pipes.
Each pipe is a decay
channel. Pool can be
filled via one selected
pipe, but the water is
coming out via all
opened pipes.
Statistical factor –
QM mantra: sum
over the final states,
average over initial
a
Lorentian (BreitWigner) profile
peak in the cross
section
Ingoing and
outgoing
channels
Total resonance
width – sum of
partial widths
ON RESONANCE
g
b
d
etc…

20.

Decay states with complex energy
Time dependent WF
with probability
exponentially
decreasing with time
Complex energy
Hamiltonian
Green’s procedure for
complex conjugate
“Natural” definition of
width: for WF with
pure outgoing
asymptotic width is
outgoing flux devided
by normalization
(“number of particles”)
in the internal region
Строго: время жизни в отличие
от ширины содержит не вполне
Фиксируемый параметр радиуса

21.

Asymptotic of the decay WF
Decay WF with complex
energy shows unphysical
exponetial growth at large
energy

22.

Asymptotic of the decay WF
Decay WF with complex
energy shows unphysical
exponetial growth at large
energy
Reason – simple Ansatz
above do not work in all time
domain
Nuclear
interior
Exponential
growth
Reaction
mechanism
for resonance
population
For decay of Radium with
Qa ~ 10 MeV and T1/2~ 5000
years the integral of the WF
in the outer part become
comparable with inner part
for r > 100 light years
Computation to astronomical
radial scale are needed to see
decay WF in all its complexity

23.

Quasistationary state
on resonance
Quasistaionary WF is
normalized in internal region
Near resonance the radial
and energy degrees of
freedom are factorized
WF in this form combines properties of bound and scattering WFs and thus
demonstrates how transition from discrete to continuous spectrum happens

24.

Integral formula for width
Formulation
K. Harada and E. A. Rauscher, 1968.
S. G. Kadmenskii and V. E. Kalechits, 1970.
Real Hamiltonian
Auxilliary Hamiltonian
Green’s procedure
Wronskian after partial
integration

25.

Integral formula for width
Square both sides
Here flux is velocity and width is flux devided by internal
normalization
If we take point like Coulomb potential for auxiliary
Hamiltonian especially simple expression is obtained
Analogous for expression for T matrix
- <plane wave|potential|real WF>
Useful technique. Works when integral is converged on the upper bound. This is
guaranteed if asymptotic behavior of real and auxilliary Hamiltonian are the
same.

26.

Time delay
For radioactivity
scale widths G
ON resonance
T~1/G
time is exponentially large
OF resonance
T ~ G/ (E-Er)2
time is exponentially small

27.

Decay states with real energy
In AB channel we have BOTH
in and outgoing waves
In FR channel we have ONLY
outgoing waves
>>
For weak channel coupling we
can use static “source” function
in inelastic channel
Example – sudden removal
approximation
Remove particle r from WF
Instead of vector r we get vector q of
transferred momentum in the source

28.

Different facets of resonance phenomenon
Generic idea
Decays
Elastic scattering
Lorentian (Breit-Wigner) profile peak in the cross section
Lifetime
Exponentially growing WF long-range tail
Phase shift pass 90o
Delay time
Reactions
WF concentration in the interior
S-matrix
Pole
Quasistationary WF
Separation of energy and radial degrees of freedom
Resonance is not
necessarily peak
Resonance
phenomena
vs
Peak is not necessarily
resonance
Excitation modes