Understanding Hyperfine Interactions in Atomic Physics

Nuclear probes in materials -
M
ö
ssbauer effect and
M
ö
ssbauer spectroscopy
External EM field - energy
Energ
y of a stationary charges and currents in external EM field
Odd electric moments = 0 = Even magnetic moments (
 conservation law
)
Higher orders can be neglected – 
E
 shifts weaker by a few orders of magnitude.
Charge;
Electric dipole 
moment (= 0)
;
Magnetic dipole moment;
Electric quadrupole moment;
Electrostatic potential
Intensity of electric field
Intensity of magnetic field
Tensor of electric
field gradient
In atoms/nuclei, 
hyperfine structure 
occurs due to the energy of the nuclear
magnetic dipole moment in the magnetic field generated by the electrons, and the
energy of the nuclear electric quadrupole moment in the electric field gradient
due to the distribution of charge within the atom/nuclei
.
Hyper-fine interactions (in atomic physics)
Schematic illustration of fine and hyperfine
structure in a neutral hydrogen atom –
electronic levels
Fine splitting is only due to interaction of e
-
hyperfine structure
 is a small shift in otherwise degenerate energy levels and
the resulting splitting in those levels of atoms (molecules) due to electromagnetic
multipole interaction between the nucleus and electron clouds
In atoms, HF structure arises from the energy of the nuclear 
 interacting with
B generated by the e- and the energy of the nuclear quadrupole moment in the
EFG due to the distribution of charge within the atom. 
The “influence” of magnetic moments leads to splitting (removing
degeneracy) of levels in the nucleus as well as for electronic levels.
In this course - hyperfine interaction “dominantly” deals only with splitting of
levels in the nucleus
However, HF interaction impacts also
(splitting of) levels of atomic nuclei
“Nuclear 
Zeeman 
effect”
Interaction of magnetic dipole moment with magnetic field
:
B
 is a sum of contribution of “external”
 
+ “surrounding e
-
 
complete
 
multiplet
 splitting
57
Fe
Allowed only
 
transitions with
 
m
I
 = -1,0,1
Energy of a magnetic dipole is then given by
Projection 
m
 can reach only discrete values:
m
 = 
-I, -I+1,…, I-1, I 
(in units of 
h
)
Actual energies are discrete
m
Ele
c
tric 
quadrupole in 
el
ectrostatic field
:
C
art
esian coordinate
system
S
pherical coordinate
system
After diagonalization in Cartesian CS
(
V
xy
,... = 0)
}
asymmetry
Interaction of quadrupole moment with the electric field gradient (at nucleus)
Only axial symmetry systems discussed below
Ele
c
tric 
qu
adrup
o
l
e
 v 
electrostat. field
:
nuclear
 Stark 
effect
eQ
 is electric quadrupole moment
(multipole moment is in general defined as zeroth moment
component for maximum projection 
m = I 
):
Only 
partial
multiplet
 splitting
 
}
 
for a nucleus with 
I = 0, ½ 
the quadrupole moment cannot be
“determined”
 does not exist
 only nuclear states with I>1/2 show a splitting
57
Fe
For cubic crystals, which are equivalent in all directions x, y and z (therefore V
zz
 = V
yy
= V
xx
) and also for axially symmetric crystals, which are equivalent in x and y (therefore
V
yy
 = V
xx
) the asymmetry parameter vanishes as η = 0. In cases of smaller symmetry η
takes values between zero and one.
for the energy difference can be written
For axially symmetric electric field gradients the energy
of the sublevels takes the values
The energy difference between two
sublevels m and m′ is given by
After introducing the quadrupole frequency
Ele
c
tric 
quadrupole in electrostatic field
Dominant application to 
57
Fe – only two different values of m
2
In contrast to the magnetic interaction the splitting of the sublevels of the nuclear state
through the electric quadrupole interaction depends on the angular moment I of the
sublevel and is therefore non-equidistant. The transitions frequencies ω
n
 between the
different sublevels are, in case of η = 0, integer multiples of the smallest frequency ω
0
.
It can be written
whereas ω
0
 = 3·ω
Q
 for integer nuclear spin I and ω
0
 = 6·ω
Q
 for half integer nuclear spin I.
As for the magnetic dipole interaction, also the electric quadrupole interaction induces a
precession of the angular correlation in time and this modulates the quadrupole
interaction frequency. This frequency is an overlapping of the different transition
frequencies ω
n
. The relative amplitudes of the different components depend on the
orientation of the electric field gradient relative to the detectors (symmetry axis) and on
the asymmetry parameter η.
Ele
c
tric 
quadrupole in electrostatic field
 
 
 
M
ö
ssbauer
 effect
Finite temperature impact
Doppler 
broadening is often not sufficient
to cause “significant” resonance absorption
(for “low” E
)
Nuclear 
re
s
onan
ce
 fluorescence 
does
not happen – difference to e
-
 transitions
emission
absorption
M
 ... 
Nucleus mass
Natural line width

 
 100 ns 
 
 = 10
-8
 eV)
M
ö
ssbauer 
effect
Recoil compensation (how can be compensated)
Doppler
 
broadening
Mechanic
al movement”
(Doppler)
Use of a cascade of 
 transitions
or nuclear reaction
1958 – M
ö
ssbauer (NC 1961) – 
nucleus bound in the crystal lattice and
there is no recoil of a single nucleus 
momentum is “transformed” to
the “phonons” (lattice vibrations)
, 
or – in a fraction of emissions – to the
whole crystal (mass is higher by ~10
23
 compared to a single nucleus)
emit
e
r
absorb
e
r
e.g
. 
for
 411 keV 
transition in
 
198
Hg
v = 6.7 x 10
2
 m/s
Angle
 
 
could be chosen to fulfill the
resonance condition for the 2
nd
 photon
M
ö
ssbauer 
effect
3 
different cases:
M
ö
ssbauer effect 
= (almost) recoil-free emission and/or absorption of 
 rays
B – atom binding energy in a crystal
 emission as from a free atom (typically for E
 
 
1 MeV)
Debay (Einstein) frequency 

D
 (
E
)
 Energy is transformed to phonons – for lower photon energies
(typically for E
 
 hundreds of
 keV)
„recoilless“ emission in a fraction of
cases (typically for E
 
 
10-100 keV)
- resonance absorption happens
“recoil” is absorbed by whole crystal –
almost all energy is carried by the
photon (crystal carries a negligible
energy … << linewidth)“
Recoil-free fraction 
f
In the first
approximation – in a
model with a single
ph
onon 
energy 
Recoil-free fraction
“Illustration” even within a “classical” approach
: 
oscillating nucleus
(
only an illustration
)
Fre
qu
enc
y
 
seen by an observer
(Doppler 
effect
):
The wave seen by an observer
:
Identity
:
Bessel func
tions
Probability of recoil-free emission is:
 - 
 + 
J
0
 
for a small argument
for
 harmon
ic
 
osci
l
l
a
tor
In a general case
 (
finite
 T) 
the probability is
where
 W (
~
Debye-Waller
 or Lamb-Mossbauer
 fa
c
tor)
in 
Deby
e
 model 
of a crystal 
(
f
 
decreases with 
T)
Speci
fically
, 
for
 T = 0
Corresponds to 
(
above-given
) 
Probability of recoil-free emission is:
D
 – a measure of the strength
of the bonds between the
Mössbauer atom and the lattice
Recoil-free fraction
Debye model, Einstein model
Debye model
 is a method for estimating the phonon contribution to the heat capacity in a
solid. It is a solid-state equivalent of Planck's law of black body radiation, where one
treats electromagnetic radiation as a gas of photons in a box. Debye treats the
vibrations of the atomic lattice (heat) as phonons in a box (the box being the solid).
Debye temperature
 is the temperature of a crystal's highest normal mode of vibration, i.e.,
the highest temperature that can be achieved due to a single normal vibration.
(
D
 (
α-Fe) = 464 K)
Einstein model
 treats the solid as many individual, non-interacting quantum harmonic
oscillators. It is based on three assumptions:
Each atom in the lattice is a 3D quantum
harmonic oscillator
Atoms do not interact with each another
Atoms vibrate with same frequency
(contrast with Debye model)
While the first assumption is quite accurate,
the second is not. If atoms did not interact
with one another, sound waves would not
propagate through solids.
Heat capacity
 (Debye 
gives
 
correct behavior at low 
T)
Line shape 
– Breit-Wigner
 shape
Line shape for emission
Line shape for absorption
Observed shape in absorption experiment
For a thin absorber
and if
then
Theoretical line-shape known 
 
very small change of
E
 
can be measured 
(
on a fraction of
 
)
 
 “energy resolution”
 (
depending on isotope
)
 is
 
The “best tool” to study 
hyperfine interaction
(change of energies of levels due to nucleus-
electron interaction)
s
M
ö
ssbauer 
effect
Typical experimental setup:
Energies of levels in absorber are checked/scanned
Can be shifted or split
Energy of “source photon” must be varied – usually via Doppler effect
(source is moving) – usually a periodic change
In 
57
Fe
 1mm/s 
 4.8x10
-8
 eV (

=11.6 MHz, 
E = h
)
Source (
57
Co
) must be in a material (crystal lattice) with a high 
D
necessary condition for recoilless emission
Lattice must be cubic non-magnetic metal (Rh, Cr, Pt, Pd, Cu) to have
no splitting (commercial sources have f 
 0.75)
Absorber must not be neither very thick to allow pass some radiation,
nor very thin to see an effect
If a thin absorber cannot be prepared, a "scattering" (instead
transmission) arrangement might be used
How to exploit M
ö
ssbauer effect?
Study of “hyperfine” interaction (i.e. interaction between nucleus and
electrons) that manifests itself due to shift and/or split of levels in the
nucleus – 
Mossbauer spectroscopy
Splitting of individual levels allows measurement/appears due to
Magnetic field at the nucleus
Gradient of electric field at nucleus
The ground and first excited states of 
57
Fe
split in a magnetic field.
Isomer shift and quadrupole splitting of the
nuclear levels in electric field gradient.
Energy difference due to splitting is on the order 10
-7
 eV (for 
B
 used in
NMR, a few Tesla)
actual fields in materials could be stronger - internal magnetic field of the
metallic iron is about 33 T, the splitting is slightly higher
Similar (could be much smaller or comparable in size) splitting observed
due to electric field gradient (EFG)
Recoil energy of a single nucleus is about 10
-3
 eV, i.e. much higher 
 no
chance to “observe” transitions via “standard” recoil
The Mössbauer effect cannot be observed for freely moving atoms or
molecules, i.e. in gaseous or liquid state
M
ö
ssbauer 
effect
M
ö
ssbauer 
spectroscopy
Nuclei of absorber re-emit photon (iso-tropically) after 
 10
-7
 s
If high (e
-
) conversion coefficient (

depends on E
), re-emission strongly suppressed
In the absorption measurement (narrow beam detected) can usually be fully neglected
Measurement of conversion electrons possible (CEMS) – only sensitive to sample
surface (hundred(s) of nm) as low-energy e
-
 are strongly absorbed
Requirements for “suitable” isotope for M
ö
ssbauer spectroscopy
Low E
 (10-150 keV) between 1
st
 excited and stable GS 
 minimization of E
R
The highest possible nucleus mass 
 minimization of E
R
 
 
 10
-7
-10
-9
 s (longer – too narrow line, difficult to observe; shorter – wide
“resonance” with a low resolution)
Stable isotope must be produced in a decay with relatively long lifetime (>1y)
Electric quadrupole and magnetic dipole moment of involved levels

 0
(for applications)
High 
D
 for wide range of T
 
Fe
: 
D
=470 K
, E
D
=0.04 eV
M
ö
ssbauer 
spectroscopy
Observed in about 8
0 i
sotopes (50 elements), 20 elements really used 
“The best nucleus”:
 
57
Fe, E
 = 14.4 keV
E
R
 = 1.95x10
-3
 eV,  
 = 0.45x10
-8
 eV, 2
 = 0.194 mm/s
high
 
 allows to exploit even c
onver
sion
 e
-
 (conversion electron Mossbauer spectroscopy)
Can be used for T significantly above “room T”
Energ
y
 r
esolution
 
 3x10
-13
57
Co preparation using cyclotrons
 
56
Fe(p,
)
The 2
nd
 most popular nucleus: 
119
Sn - 
2
=0.
626
 mm/s
 
(
worse resolution
), 
for chemistry
The lightest 
izotop
e
: 
40
K – 
difficult to get
 (
no accessible decay
)
The best resolution
: 
67
Zn,
(E
 = 93.26 keV, 
 = 9.4 
s, 2
 = 3.12x10
-4
 mm/s)
probl
e
m: vibra
tions disturb observability
Further
 
127,129
I, 
125
Te – 
for
 chemi
stry
151,153
Eu, 
161
Dy, 6 Gd
 isotopes
- 
for
 magnetism 
of rare-earth nuclei
A
c
tinid
es
 (
237
Np, 
237
U, 
232
Th)
for physical constant measurements
 
Hyperfine Interactions
Electric monopole interaction (Coulombic)
between protons of the nucleus and s‐electrons “penetrating” the nuclear field.
Different shifts of nuclear levels A and S.
Isomer shift 
values give information on the oxidation state, spin state, and
bonding properties, such as covalency and electronegativity.
Electric quadrupole interaction
between the nuclear quadrupole moment (eQ ≠ 0, I > 1/2) and an
inhomogeneous electric field at the nucleus (EFG ≠ 0). Nuclear states split into
I + 1 sub-states.
The quadrupole splitting gives the information on oxidation state, spin state
and site symmetry.
Magnetic dipole interaction
between the nuclear magnetic dipole moment (μ≠ 0, I > 0) and a magnetic field
(H or B ≠ 0) at the nucleus. Nuclear states split into 2I+1 sub-states with m
I
 =
+I, +I‐1, .…,‐I
The magnetic splitting gives information on the magnetic properties of the
material under study ‐ ferromagnetism, anti-ferromagnetism.
Iso
mer(ic)
 shift
The nuclear radius in the excited state is usually different (in the case of 
57
Fe it is
smaller than that in the ground state, for 
119
Sn opposite). The fictitious energy levels of
the ground and excited states of a bare nucleus (no surrounding e
-
) are perturbed and
shifted by e
-
 interactions
. The shifts in the ground and excited states differ because of
the different nuclear radii in the two states, which cause different Coulombic
interactions (due to presence of s-electrons in nucleus).
The energy differences E
S
 and E
A
 also differ because of the different e
-
 densities in the
source (S) and absorber (A) material (chemical environment). The individual energy
differences E
S
 and E
A 
cannot be measured – only the difference of the transition
energies δ = E
A
 – E
S 
can be measured (this difference is called 
Isomer shift
).
 
spectrum in
The isomer shift is given by: 
δ
 = E
A
 – E
S 
= 
(2/5)πZe
2
 
A
 – ρ
S
)(R
exc
2
 – R
GS
2
) 
 …density of (s-)electrons (square of amplitude
of e
-
 wave function 
|Ψ(0)|
2
) “at nucleus”
I
s
omer 
shift
Measures the “energy difference” with respect to the source
Given by combination of nuclear (R) and chemical properties – R must be
known if chemical properties are to be measured
Also, electrons on higher shells (even valence) play a role – via “shielding
effect” (impact on the role of 
s
 electrons)
Direct impact – change of e
-
 population in s‐orbitals (mainly valence s‐orbitals)
changes directly |Ψ(0)|
2
Indirect impact – shielding by p‐, d‐, f‐electrons, increase of electron density in p‐,
d‐, f‐ orbitals increases shielding effect for s‐electrons from the nuclear charge
→ s‐electron cloud expands, |Ψ(0)|
2
 at nucleus decreases. 
Shift appears in the order of splitting …
“Isomer” because there is a difference
of R for different states (can be
“isomeric” but usually they are not
– “between different states”)
In principle, can be used to measure
the root-mean-square radius R
2
Effect on atomic spectral lines not
discussed … but similar
spectrum v
Ele
c
tric 
qu
adrup
ole in electrostatic field
The line intensities are given by the angular dependence of the quadrupole
interaction
In a randomly oriented polycrystalline material, the intensities are identical
In an anisotropic crystal, an angular
dependence is observed with respect to
the principal axis of the electric gradient
tensor 
(
); 
EFG is non‐zero in non‐cubic
valence e
-
 distribution and/or in non‐cubic
lattice site symmetry
Moreover, the percentage of recoil-free
emissions 
f
 
is dependent with respect to
the crystallographic axis with the highest
symmetry 
(vibrational anisotropy) – 
f(
)
If quadrupole electric moment know, electric
field 
gradient 
can be studied (and vice versa)
:
for
for
57
Fe

 IS
Nuclear quadrupole resonance
 (NQR)
“Analog to NMR” – the “splitting” of levels due to the interaction of the
electric field gradient (EFG) tensor with the quadrupole moment 
Q
ij
 of
nuclei is sometimes called NQR – “analogous features” (precession of the
main component of 
Q
ij
 around EFG direction)
EFG determined mainly by the valence e
-
 involved in the bond with
nearby nuclei
NQR is applicable only to solids and not liquids, because in liquids the
quadrupole moment “averages out”
Only nuclei with J > ½ have non-zero 
Q
ij
, which is a measure of the
degree to which the nuclear charge distribution deviates from that of a
sphere
The energy splitting (quadrupole frequency) is given by
 
One might think about realization of the technique in a similar way as NMR:
In principle, a specified EFG could be applied to influence 
Q 
as in NMR
However, in solids, the strength of the EFG is many kV/m
2
, making the
application of the “external” EFG's for NQR impractical
.
As a result, the NQR spectrum is specific to the substance (sometimes
called "chemical fingerprint“) and not “adjusted” by the external field –
only “internal” fields measured
As NQR frequencies are not chosen by the external field, they can be
difficult to find – making NQR a technically difficult technique – this is
not a problem if this effect is exploited as a part of the 
Mossbauer
spectroscopy
 or Perturbed angular correlation spectroscopy
Since NQR is done in an environment without a static (or DC) magnetic
field, it is sometimes called "zero field NMR".
Many NQR frequencies depend strongly upon temperature.
Nuclear quadrupole resonance
 (NQR)
“Nuclear 
Zeeman 
effect”
Interaction of magnetic dipole moment with magnetic field
:
B
 is a sum of contribution of “external”
 
+ “surrounding e
-
complete
 
multiplet
 splitting
57
Fe
Allowed only
 
transitions with
 
m
I
 = -1,0,1
Energy of a magnetic dipole is then given by
Projection 
m
 can reach only discrete values:
m
 = 
-I, -I+1,…, I-1, I 
(in units of 
h
)
Actual energies are discrete
m
 
Intensity ratios given by the direction of 
 
with respect to 
B
 
 
= 0     
      
 = 0
 
 
= 90
o
  
      
 = 4
 
randomly oriented field
  
  
 = 2
57
Fe
57
Fe
Both states are split 
in principle, 2 of 3 involved
parameters 
(
GS
, 
excit
, 
B
) can be determined
Very strong 
B
 can be applied
Nuclear
 Zeeman 
effect
in general:
Combined fields
Both 
E
M
 and E
Q
can be determined
M
ö
ssbauer 
effect
Mössbauer effect is one of the most important sources of information about
hyperfine interactions (together with NRM, PAC)
it is mainly used for studies of chemical properties and structure of substances
(nuclear properties are not the main “practical” interest, but can be used e.g. to
determine “parameters” of excited states – moments, spin) 
unlike NMR, it is used to study “
solid-state
 
substances
Isomeric shift provides information about the valence electrons
it is e.g. sensitive enough (sometimes in combination with quadrupole splitting)
to determine oxidation of Fe – determine whether Fe
II
 or Fe
III
Mossbauer spectroscopy - detectors
Detectors are usually scintillators (for
transmission measurement) – ideally transparent
to 122 keV transition, i.e. rather thin
However, there are other possibilities:
Gas-filled proportional counters are often
convenient and offer better energy resolution.
Solid-state detectors are excellent when the
count rate is not excessive.
For CEMS (conversion electrons) – proportional
counters
 
Measurement of “Gravitational red-shift”
The change (decrease) of potential energy with the “distance” from the
Earth surface should imply the change in the wavelength of a photon at
different distance
Measured (for the first time) using Fe for the “height difference” h 
 22 m
Expected shift in wave-length is still 2 orders of magnitude smaller 
(10
-11
eV) than the natural linewidth
Change in “absorption” (line profile) is largest for the "inflection point" on
the BW curve – this point was used and the wavelength change
corresponding to (0.859 +/- 0.085)x the value predicted by Einstein's theory
of relativity was obtained in the original work
E Cranshaw and J P Schiffer 1964 
Proc. Phys. Soc. 
84 
245
R.V. Pound and G.A. Rebka, Jr., Phys. Rev. Lett. 4(1960)337
R.V. Pound and J.L. Snider, Phys. Rev. B140(1965)788.
Hyperfine Interactions 72(1992)197-214
Measurement of “Gravitational red-shift”
 
 
 
 
Nuclear Probes in materials -
Perturbed angular correlations
Angular distribution of emitted photons
multipolarit
y
proje
ction
Observed angular distribution
:
Population of
initial states
 
|
J
i
m
i
Clebsch-Gordon
coef.
angular distribution
from individual
projections (square of
spherical harmonics)
For
                                          
yields
}
isotropy
const.
anisotropy for
For dipole:
Each individual m-state yield
s
 an anisotropic distribution, but with
equal
 
populations of each m-state, then by summing over all m-states
the angular distribution
 
becomes isotropic.
Angular distribution of emitted photons
When an excited nucleus decays via cascade 
 rays, an anisotropy is generally
found in the spatial distribution of the 2
nd
 

ray with respect to the 1
st
 one.
If the 1
st
 
 is defined to be the z-axis, and therefore defines the m-states, then the
intermediate state will, in general, have an unequal population of m-states.
For example, a J=0, m=0 initial state, the m=0 intermediate state cannot be
populated because 

must carry away at least one unit of angular momentum (L =
1) with m = 
±
1 (along the
 direction of propagation).
For a 0-2-0 cascade only the 
m = 1, 2 intermediate states are populated giving
Angular distribution of emitted photons 
- 
example
As higher and higher angular
momenta are considered, the more
complicated these scenarios
become. The general form for
angular correlations W(
) is:
Example of angular correlations for a variety of 
 cascades (involved
spins). Some correlations have stronger anisotropies than others.
If transitions are not purely dipole (L=1) or quadrupole (L=2), so-called
“mixing ratios” are considered and they change the angular correlation.
Angular distribution of emitted photons 
- 
example
Example
60
Co
W(θ) = 1/8 cos
2
(θ) + 1/24 cos
4
(θ)
Laboratories (3
rd
 year of Bc, summer term)
 
 
angular correlation of two photons emitted after 
 
decay of 
60
Co
T
1/2
=0.9 ps
T
1/2
=3.3 ps
T
1/2
=1925 d
Angular distribution of emitted photons
multipolarit
y
proje
ction
Observed angular distribution
:
Population of
initial states
 
|
J
i
m
i
Clebsch-Gordon
coef.
angular distribution
from individual
projections (square of
spherical harmonics)
For
                                          
yields
}
isotropy
const.
anisotropy for
How to get anisotropy
?
Using an external magnetic field
At room temperature, anisotropy is unmeasurable – very  low T must be used –
.B
~ k.T 
required
 – “system ” must be cooled to mK for visible effect
Anisotropy can be used as a thermometer (two detectors measure intensity –
temperature must be close to absolute zero)
Optical (laser) techniques
Polarization of "atomic spins" by absorption of circularly polarized light 
(m
z
=+1)
leads to strong magnetic field and thus polarization of the nucleus
By means of nuclear reactions
Projectile (with zero spin) brings a non-zero orbital angular momentum 
l
 to the
system (projection of 
l
 is perpendicular to the beam direction)
Ideally, if spins of both projectile and target nucleus are =0, only 
m=0 
states
Partial de-orientation occurs during de-excitation of the nucleus
Using a cascade of 
 transitions
In a coincidence measurement, the orientation
of the 1
st
 photon in a cascade defines “axis” and
“selects a subspace of events (initial m)”
… (Compton scattering - FEL,…)
Short comments on different methods
Low-temperature orientation
Idea 
from 
1948, 
for the first time measured in 
1950 
in
 paramagnetic 
crystals
containing
 
60
Co
Anisotropy can be used as a thermometer if the field (
B
) is well known
 (
two
detectors measure intensity at different angles with respect to 
B
  – temperature must
be close to absolute zero
)
Absorption of circularly-polarized light
Leads to a strong 
B
 at nucleus
and hence to orientation of nuclear
states
HF
 interaction couples nuclear
spin 
𝐼
 
and e
-
 spin 
𝐽
 
to a total
atomic spin 
𝐹
 = 
𝐼
 + 
𝐽
, which
splits a fine
-
structure level, into
several 
HF
 levels characterized
by the 
𝐹
 quantum numbers
Use of a nuclear reaction
Projectile with (
J=0
) brings (
l > 0
)
(“m
l
 =0” – vector of orbital momentum 
l 
is
perpendicular to beam direction)
If 
J=0 
for both projection and target
nucleus (ideal case), initial state has only
m=0
Decay of the initial states leads to partial
de-orientation – distribution of projection
is symmetric with respect to 0
If the number of emitted particles is not
extremely high, the 
m
 distribution is
strongly peaked around 
m=0
The more steps, the higher de-orientation
 
Population of “low-lying states in
a reaction via compound nucleus
Evaporation 
of particles 
(p, n, d)
-ray cascade
110
Pd(
,2n
)
112
Cd
E
 
 25 MeV
Short comments on different methods
General description of angular correlation
Measured angular correlation is
(in general) given by
EL + 
ML´ +... 
S – 
“sample” reference
frame – statistical tensor is
expressed via 

 in this
frame
S´ - 
“detector” reference
frame – the frame
corresponding to the
response to individual
polarizations
Wigner 
D
-fun
ctions
Depends on
:
1. 
characteristics of radiation
    
(typ
e – E/M
,
 
multipolarity 
of both
    transitions types, mixing ratios
)
2. 
detector characteristics
    
(
response to individual
     polarizations
,
 geometrical
     “smearing coefficients” 
Q
)
Polarization not measured 
If (in addition) axial symmetry 
If (in addition) reflection symmetry 
 
 = even
     (typical case)
Angular correlation – special cases
 
cascades – two consecutive 
 rays
Angular distribution of photons in a cascade is usually anisotropic
Transition “1”
 
is not observed
 
 if 
P
m
(J)
 
isotropic (the same)
, 
no angular
dependence for transitions “2” is observed
Transition “
1“ 
 observed
 
 orient
ed
 
ensemble is formed
At (time)
 t = 0 
(
immediately after emission of 
1
)
 
Unperturbed angular correlations
...
original ensemble can be (in general)
oriented
...
 Population probability in
the reference frame given by
direction of
 
1
 
emission if
P
m
(J)
 is isotropic
...
W
2
 
~
 
angular dependence of 2
nd
 photon
Each individual m-state
yield
s
 an anisotropic
distribution, but with equal
populations of each m-
state, then by summing
over all m-states the
angular distribution
becomes isotropic.
Angular distribution of emitted photons
 
cascades – two consecutive 
 rays
Angular distribution of photons in a cascade is usually anisotropic
...
original ensemble can be (in general)
oriented
Transition “1“ observed 
 oriented ensemble formed
1.
At 
t = 0 
(immediately after emission of  
1
)
 
 Unperturbed angular correlations
2.
For (time after emission of “1”)
 t > 0
 –  evolution of statistical tensor
      “Periodic” modulation of coefficients 
a

due to precession in 
magnetic field 
B
 (or EFG)
 
Perturbed angular correlations
(measurement of internal fields – 
B
, EFG)
...
W
2
 
~
 
angular dependence (for the
2
nd
 photon)
Perturbed angular correlations (PAC)
Hyper
fine interactions lead
to precession of 
 and 
Q
ij
Number of coincidences depends on the angle
between detectors
:
Example – 
111
In
The level splitting
Due to the interaction of the intermediate state of the
nucleus with an electric field gradient, resulting from the
surrounding charge distribution, the degeneration of the
intermediate level is removed and the level splits into three
different levels. The three possible transitions between
these levels are visible as three frequencies in the PAC
(Perturbed angular correlation) Fourier transform spectrum
The time spectrum N(t)
The time spectrum is the raw data obtained from counting
-
 coincidences and sorting them after the time between
the start and the stop signal. The modulation, barely visible
in the example, is extracted into the R(t)-spectrum
The PAC-time spectrum
The PAC-time spectrum R(t) is extracted from the raw
data N(t). The x-scale is still the time between
coincidences, but the y-scale is changed. The exponential
decay function and the detector-geometry as well as their
sensitivities have been removed from the spectrum.
Fourier Transform Spectrum
The Fourier spectrum is obtained from R(t) via fast Fourier
transform. The F(
) spectrum yields a characteristic
frequency triplet for each electric field gradient.
PAC 
111
In
The most commonly-used radioactive probe nuclei are 
111
In and 
181
Hf.
The PAC isotope 
111
In has reached similar importance like 
57
Co for
Mößbauer spectroscopy. 
(
For 5/2- level: 
Q=0.8b, 
μ=-0.766
N
)
Produced via 
111
Cd(p,n), 
112
Cd(p,2n)
Radioactive isotopes must fulfil special requirements to use them for PAC
spectroscopy:
decay by 

 cascade
high anisotropy of angular correlation
of these transitions
existence of an intermediate state with 
adequate lifetime between ~10 ns and 1 
s
state should feature a suitable strong 
quadrupole moment 
Q
ij
 in terms of
a detectable electric quadrupole interaction
(suitable energy splitting) and in case 
of magnetic interaction a suitable strong 
magnetic dipole moment μ
N
easy handling of parent nuclide
 
Schematic of a typical online TDPAC spectroscopy setup at ISOLDE-CERN
for ion implantation. The radioactive beam produced by ISOLDE is delivered
to the beam line. The sample is placed inside the vacuum chamber, while the
TDPAC detection system is placed outside it. For isotopes that are not too
short lived, the measurements can be performed off-line.
Standard configuration is a four
detector setup in 90° symmetry.
Each detector is able to register
both different gammas and so
gives start and also stop signals.
This leads to eight 90°- and four
180°-coincidence spectra. With
this start-stop-principle lifetime
measurements of the
intermediate state are performed,
whose exponential decays are
temporally modulated due to
hyperfine interactions
N(
ϑ,
t) 
 = 
N
0
 e
−λt
 ·
(1 + A
22
P
2
(cosϑ) + A
44
P
4
(cosϑ))
PAC
If A
44
 << A
22
 it is sufficient to determine the angular correlation at two fixed
angles ϑ. Usually one uses the angles 90° and 180° because for this angles the
Legendre polynomials give simple values (P
2
(cos90°)=−1/2 and P
2
(cos180°)=1).
To eliminate the exponential decay, which does not contain any information about
the modulation, the ratio
 
R(t)
 = 2 (
N(180°,t) − N(90°,t)
) / (
N(180°,t) + 2N(90°,t)
)
is typically used (sometimes slightly different definition).
Supposed the decay is not perturbed by external magnetic or electric fields, then
the ratio R(t) is constant in time and equal to the coefficient A
22
.
PAC
10
7
-10
9
 atoms of a radioactive isotope per measurement is enough
Today only (or at least strongly dominantly) the time-differential
perturbed angular correlation (
TDPAC
) is used.
Advantages of PAC 
(compared to NMR, Mossbauer):
Any temperature 
– no signal attenuation at very high temperatures
appears
Any environment (solid, liquid, gas)
Low concentration of probe nuclei: 10
-12
 –10
-6
No restriction to low E
γ
Small samples: 1-100 mg
No external field or rf field required
The abbreviations are not “unique” – TDPAC or TDPAD, IPAC or
IPAD (“angular correlations”/”angular distribution”)
PAC
There are two different ways of
measuring the angular correlation,
the time integral and the time
differential mode. The integral
perturbed angular correlation
(IPAC) is used if the lifetime of
the intermediate state is smaller or
similar to the experimental time
resolution or for measuring strong
internal magnetic fields of ferro
magnets. The time differential
perturbed angular correlation
(TDPAC) measures the angular
correlation as fraction of the time
the nucleus spends in the
intermediate state.
Usually fast-slow coincidence unit:
time-to-amplitude converter (TAC)
constant franction discriminator (CFD)
IPAC, TDPAC
1.
t
L
 >> t* - IPAC (integrated PAC):
the measurement yields the time average and a small average
precession 

~ 
L
t* of the angular distribution is observed.
2.
t
L
 << t* - TDPAC (time differential PAC)
In reality, the time resolution of the measuring device plays a role…
t* - lifetime of an intermediate state
“Nuclear 
Zeeman 
effect”
Interaction of magnetic dipole moment with magnetic field
:
B
 is a sum of contribution of “external”
 
+ “surrounding e
-
complete
 
multiplet
 splitting
57
Fe
Allowed only
 
transitions with
 
m
I
 = -1,0,1
Energy of a magnetic dipole is then given by
Projection 
m
 can reach only discrete values:
m
 = 
-I, -I+1,…, I-1, I 
(in units of 
h
)
Actual energies are discrete
m
Example
TDPAD spectra for the 
γ
-decay of the 
I
π
 
= 29
/
2
, 
t
1
/
2
 = 9 ns isomeric 
rotational
bandhead in 193Pb, implanted respectively in a lead foil to measure its
 
magnetic
interaction (MI) and in cooled polycrystalline mercury to measure its
 quadrupole
interaction (QI).
Detectors are placed in a plane perpendicular to the magnetic field direction
(
θ 
= 90
) and at nearly 90
 
with respect to each other (
φ
1 
≈ φ
2 + 90)
 
PAC-spectrum of single crystal
ZnO with fit
 
TDPAC spectra of single-crystal
TiO
2
 measured at room temperature
using 
111m
Cd(
111
Cd) as probe nuclei,
after RTA at 873 K for 10 minutes in
vacuum (bottom) or in
O
2
 atmosphere (top). The least-
squares fits of the hyperfine
parameters are represented by the
blue solid curves.
 
Principle of the TDPAC technique: the
probe atom is placed inside the
material under research and emits
radiation in cascade, which is detected
by scintillators
Basic representation of the γ-γ TDPAC
technique: detection and coincidence of
the rotating emission gamma-cascade
pattern, elimination of the exponential
decay and generation of frequency-
modulated TDPAC curve of the
intermediate state.
PAC
 Applications
PAC is widely applied in condensed-matter and materials physics
studies. Applications include the following:
phase transformations; phase analysis
structural and magnetic phase transitions
static and dynamic critical behavior and exponents
magnetism: spin dynamics, stability of atomic magnetic moments in
different hosts, exchange interactions
diffusion and other atomic movement in solids, dynamical interactions
detected via nuclear relaxation
surfaces, interfaces and grain boundaries, thin films, nanocrystals
point defects: identification, thermodynamic properties, interactions
point defect production by plastic deformation, radiation and implantation
damage, and quenching
lattice location of impurities in compounds
http://tdpac.hiskp.uni-bonn.de/pac/tx-pac-en.html
http://pacweb.hiskp.uni-bonn.de/mediawiki/index.php?title=Main_Page
Constant Fraction Discrimination
In the CFD the pulses are timed from a point on the leading edge that is a
fixed fraction of the pulse height – used for precise “time determination”.
Each input signal - which exceeds the
adjusted threshold level - is split
(i) one portion is inverted and
delayed by a fixed time,
(ii)  giving a voltage magnitude that
the second pulse is attenuated to,
(iii) the two pulses are then combined
to give a bipolar constant fraction
signal with a zero crossing point.
This zero crossing point is then
detected and is virtually independent
of the input signal amplitude. It is
only the shape of the input signal
which is important and this is
assumed to be constant.
Signals have to have the same 
shape“
 
Today 
„digiti
z
ers“
 often used
 the current in a curcuit is determined at
defined times – with very high frequency 
(
hundreds of 
MHz 
to
 GHz) a
nd
processed 
off-line
 
END
 
A
d
v
a
n
t
a
g
e
s
 
o
f
 
P
A
C
 
(
c
o
m
p
a
r
e
d
 
t
o
 
N
M
R
,
 
M
o
s
s
b
a
u
e
r
)
:
Any temperature
Any environment (solid, liquid, gas)
Low concentration of probe nuclei: 10
-12
 
–10
-6
No restriction to low E
γ
Small samples: 1-100 mg
No external field or rf field required
P
říklady
TDPAD spectra for the 
γ
-decay of the 
I
π
 
= 29
/
2
, 
t
1
/
2
 = 9 ns isomeric 
rotational
bandhead in 193Pb, implanted respectively in a lead foil to measure its
 
magnetic
interaction (MI) and in cooled polycrystalline mercury to measure its
 quadrupole
interaction (QI).
D
etectors are placed in a plane perpendicular to the magnetic field
 
direction
(
θ 
= 90
) and at nearly 90
 
with respect to each other (
φ
1 
 
φ
2 + 90), the
R
(
t
) function in which the Larmor precession is reflected, is
 given by
Popis počátečního souboru
Statistický soubor lze v QM popsat pomocí matice
hustoty
 (
místo stavu se vezme projektor na stav,
je – v obecné bázi - nediagonální
)
Při axiální symetrii nejsou žádné smíšené stavy
Pokud je navíc „reflexní“ symetrie (z 
 -z)  
 
P
m
= P
-m
Ukazuje se jako výhodnější přejít k tzv. statistickému tenzoru
je to jen transformace matice hustoty
Sférický tenzor (vystupuje ve výrazech pro úhlové distribuce)
(
2
J+
1
)
2
-1 
nez
ávislých
reálných komponent
(normalizace)
Popis počátečního souboru
Axiální symetrie 
 odpovídá 
 = 0
Pokud je navíc „reflexní“ symetrie (z 
 -z)
Úplná izotropie
Statistický tenzor
 
 sudé
 
 = 0
M
ö
ssbauerův jev
Mössbauer effect is one of the most important sources of information about
hyperfine interactions (together with NRM, PAC)
it is mainly used for stud
ies
 of chemical properties and structure of substances
(nuclear properties are not the main interest, but can be used e.g. to determine
“parameters” of 
excit
ed
 
s
tates
 – moments, spin
) 
unlike NMR, it is used to study "solid" substances
Isomeric shift provides information about the valence electrons
it 
is e.g. sensitive enough (sometimes in combination with quadrupole splitting)
to determine 
oxidation of 
Fe 
– determine whether Fe
II
 or Fe
III
 
  
Time-dependent relaxation effects can also be studied
characteristic time is given by the period of the Larmor precession 
(
 10
-7
-10
-8
 
s)
There is also in-beam Mossbauer spectroscopy
Fe n
uclei are excited 
via
 Coulomb excitation
Izomerní posun (Iso
mer
 shift)
Během emise 
 se mění efektivní poloměr jádra 
 ovlivňuje to interakci
(konečného) jádra s okolními 
e
-
 
(zejména s s elektrony) (mají nenulovou
hustotu pravděpodobnosti výskytu v oblasti jádra)
Energie 
 je posunuta vůči hypotetickému bodovému jádru s identickým
nábojem o
V experimentu jsou zdroj i absorbátor (v různých chemických prostředích)
pozorovaný posun 
 je
kde
 
spectrum v
 
The most positive isomer shift occurs with
iron(I)
 with spin S = 3/2.
The seven d‐electrons exert a very strong
shielding of the s‐electrons, this reduces the s
electron
density ρ
A 
giving a strongly negative
quantity (ρ
A 
– ρ
S
), as (R
e
2 
– R
g
2
)  0 for 
57
Fe, the
isomer shift becomes strongly positive.
Strongly negative for iron(VI) with spin S = 1.
There are only two d‐electrons, the shielding
effect for s‐electrons is very weak and the
selectron
density ρA at the nucleus becomes high.
Iron(II) high spin with S=2 can be easily
assigned.
In other cases with overlapping δ values
ambiguous assignment. Need to consider the
quadrupole splitting parameter.
 
 
Dvojité kaskády 
 přechodů
Jedna z 
„aplikací “ úhlových rozdělení
Přechod „1“ není pozorován 
 dochází k deorientace původně
orientovaného souboru
 
???!!!
(pokud 
P
m
(J)
 izotropní, nepozoruje se žádná
úhlová závislost pro přechod „2“)
Přechod „1“ je pozorován 
 vznik orientovaného souboru
V čase t = 0 (okamžitě po emisi 
1
)
 Neporušené úhlové korelace
...původní soubor je obecně orientován 
...Obsazovací psti v soustavě 
spojené s emisí 
1 
za 
předpokladu izotropie 
P
m
(J)
...
W
2
 
~
 úhlová závislost pro 2. foton 
 
First we observe that in the initial state II,> the nuclear spins are randomly oriented.
This can be stated formally by writing
where Pm,m’,(1) is an element of the density matrix that describes magnetic substate
populations of the initial state III> in any m representation. Now we
seek an expression for the density-matrix elements describing the subs tate
populations of the intermediate state /1> immediately after the emission of YI.
From angular momentum theory we know that when a magnetic substate |I
i
m
i
>
decays to the substates |Im> of state |I> , the fractional intensity decaying to each
substate |Im> is given by the square of the Clebsch-Gordan coefficient <ImL'(m
i
m) | I
i
m
i
>. Up to this point the quantization direction is arbitrary, although it must be
the same for the two states |Ii> and |I>.
 
Kernreaktionen III /
Nuclear Reactions III
Od autorů: D. E.
Alburger,I. Perlman,J. O.
Rasmussen,Earl K.
Hyde,Glenn T.
Seaborg,George R.
Bishop,Richard Wilson,S.
Devons,L. J. B.
Goldfarb,R. J. Blin-
Stoyle,M. A. Grace
 
 
 
Many PAC probes have an excited nuclear state
reached through a 
-
 cascade.  
Start
 and 
stop
 
rays signal creation and decay of an
intermediate state and (importantly) there is an
anisotropic angular correlation between
directions of emission of the two gamma rays.
Such is the case, e.g., following decay of 
60
Co,
for which the lifetime of the intermediate
nuclear state is very short. For longer-lived
states, the nuclear spins can precess through
appreciable angles and interaction frequencies
can be measured with good precision.
A small number of probes have long lifetimes
and other favorable nuclear properties for "table
top" PAC experiments, including 
111
In 

111
Cd, and 
181
Hf 

 
181
Ta.
Hyperfine interactions
 lead to frequencies of precession of probe nuclei that are
proportional to the internal fields and are characteristic of the probe's lattice location
 
 
Radioactive 
111
In
This is the probe atom, which is used for the PAC measurements.
Its decay initiates the emission of the detected 
-rays.
It is produced e.g. by 
109
Ag (
, 2n) 
111
In
This material is commercially available.
About 10
11
 atoms are needed for one measurement.
The detailed decay-scheme:
 
 
 
 
Statické momenty jader –
metody jejich měření
M
ěření statických momentů jader
Static moments of nuclei are measured via interaction of the nuclear charge
distribution and magnetism with electromagnetic fields in its immediate
surroundings. This can be the electromagnetic fields induced by the atomic
electrons or the fields induced by the bulk electrons and first neighboring
nuclei for nuclei implanted in a crystal, usually in combination with an
external magnetic field.
 
Atomic hyperfine structure
Not only the radial distribution of the nuclear charge (monopole
moment) but also the higher multipole electromagnetic moments of
nuclei with a spin 
I ≠
 0 influence the atomic energy levels. By
interacting with the multipole fields of the shell electrons they cause an
additional splitting called hyperfine structure. For all practical purposes
it is sufficient to consider only the magnetic dipole and the electric
quadrupole interaction of the nucleus with the shell electrons.
The shell electrons in states with a total angular momentum 
J ≠
 0
produce a magnetic field at the site of the nucleus. This gives a dipole
interaction energy 
E 
= 
−μ · B
. The spectroscopic quadrupole moment of
a nucleus with 
I ≥ 
1 interacts with an electric field gradient produced by
the shell electrons in a state with 
J ≥ 
1 according to 
E
 
= 
eQ 
(
2
V/∂z
2
).
 
Externally applied EM fields
When a nucleus with spin 
I 
is implanted into a solid (or liquid) material,
the interaction between the nuclear spin and its environment is no longer
governed by the atomic electrons. For an atom imbedded in a dense
medium, the interaction of the atomic nucleus with the electromagnetic
fields induced by the medium is much stronger than the interaction with
its atomic electrons.
The lattice structure of the medium now plays a determining role. This
“hyperfine interaction” is observed in the response of the nuclear spin
system to the internal electromagnetic fields of the medium, often in
combination with externally applied (static or radio-frequency) magnetic
fields.
 
Interakce jádra s vnějšími aplikovanými poli
Experimental techniques based on measuring the angular distribution of the
radioactive decay are often more sensitive than the atomic HF methods, and
in some cases also allow more precise measurements of the nuclear 
g 
factor
and quadrupole moment. This angular distribution is influenced by the
interaction of the nuclear moments with externally applied magnetic fields
and/or electric field gradients after implantation into a crystal
The radioactive decay intensity is measured as a function of time (TDPAD)
or as a function of an external variable, e.g., a static magnetic field or the
frequency of an applied radio-frequency magnetic field (
-NMR). The
former are called “time differential” measurements and the latter “time
integrated” measurements.
 
Metody
Velikost hyperjemného
pole nezávisí pro daný
prvek na izotopu
Lze změřit pole pomocí
jednoho izotopu a pak
měřit momenty u dalších
izotopů
Mößbauer
ův jev
Omezeno jen na izotopy a hladiny měřitelné pomocí Mossbauera
PAC
 (T
ime-
Differential Perturbed Angular Distribution 
-
 TDPAD)
NMR
β-NMR
 pro hladiny s “krátkou dobou života“
N
ízkoteplotní orientace
TDPAD
Spin-oriented isomeric states implanted into a suitable host will exhibit a
non-isotropic angular distribution pattern, provided the isomeric ensemble
orientation is maintained during its lifetime. If an electric field gradient
(EFG) is present at the implantation site of the nucleus, the nuclear
quadrupole interaction will reduce the spin orientation and thus the
measured anisotropy.
If the implantation host is placed into a strong static magnetic field (order of
0.1–1 Tesla), the anisotropy is maintained. If the field is applied parallel to
the symmetry axis of the spin orientation, the reaction-induced spin
orientation can be measured.
If a static magnetic field is placed perpendicular to the axial symmetry axis
of the spin orientation, the Larmor precession of the isomeric spins in the
applied field can be observed as a function of time [93], provided that the
precession period is of the same order as the isomeric lifetime (or shorter).
Can also be used to measure the quadrupole moments of these isomeric
states, by implantation into a single crystal or a polycrystalline material with
a non-cubic lattice structure providing a static electric
 field gradient.
P
říklady
TDPAD spectra for the 
γ
-decay of the 
I
π
 
= 29
/
2
, 
t
1
/
2
 = 9 ns isomeric 
rotational
bandhead in 193Pb, implanted respectively in a lead foil to measure its
 
magnetic
interaction (MI) and in cooled polycrystalline mercury to measure its
 quadrupole
interaction (QI).
D
etectors are placed in a plane perpendicular to the magnetic field
 
direction
(
θ 
= 90
) and at nearly 90
 
with respect to each other (
φ
1 
 
φ
2 + 90), the
R
(
t
) function in which the Larmor precession is reflected, is
 given by
Příklady
 
R
(
t
) curves obtained in the study of 
g
-factors of
 
= 9
/
2+ isomers in neutron-
rich isotopes of nickel and iron. The isomers,
 
with lifetimes of 13.3 μs and
250 ns, respectively, have been produced in a projectile
 
fragmentation
reaction at the LISE high-resolution in-flight separator
 at GANIL.
β-NMR
Time-differential measurements are
 
only suited for short-lived nuclear states,
mainly because of relaxation effects
 
causing a dephasing of the Larmor
precession frequencies with time (typically
 
in less than 100 μs). To measure
nuclear moments of longer-lived
 
isomeric states and also for ground states, a
time-integrated measurement is
 
required. Time integration of 
R
(t), taking
into account the nuclear decay time, will lead to a constant anisotropy.
Therefore, a time-integrated measurement of the angular distribution of this
system will not allow one to deduce information on the nuclear moments.
Hence a second interaction, which breaks the axial symmetry of the
Hamiltonian, needs to be added to the system.
One possibility to introduce a symmetry breaking in the system, is by adding
a radio-frequency (rf) magnetic field perpendicular to the static magnetic
field (and to the spin-orientation axis).
If the nuclei are implanted into a crystal with a cubic lattice symmetry or
with a noncubic crystal structure inducing an electric field gradient,
respectively, one can deduce the nuclear 
g
-factor or the quadrupole moment
from the resonances induced by the applied rf field between the 
nuclear
hyperfine levels.
β-NMR
Consider an ensemble of nuclei submitted to a static magnetic field 
B
0
 and
an rf magnetic field with frequency 
ν
 and rf field strength 
B
1
. If the applied
rf frequency matches the Larmor frequency the orientation of an initially
spin-oriented ensemble will be resonantly destroyed by the rf field. For 
β
-
decaying nuclei that are initially polarized, this resonant destruction of the
polarization can be measured via the change in the asymmetry of the 
β
-
decay.
For an ensemble of nuclei with the polarization axis parallel to the static
field direction, the angular distribution for allowed 
β
-decay can be written as
with the NMR perturbation factor 
G
10
11
 describing the NMR as a function
 
of
the rf frequency or as a function of the static field strength. At resonance, the
initial asymmetry is fully destroyed if sufficient
 
rf power is applied, which
corresponds to 
G
10
11 
= 0. Out of resonance
 
we observe the full initial
asymmetry and 
G
10
11 
= 1.
β-NMR
All forms of magnetic resonance require generation of nuclear spin
polarization out of equilibrium followed by a detection of how that
polarization evolves in time.
I
n conventional NMR a relatively small nuclear polarization is generated by
applying a large magnetic field after which it is tilted with a small RF
magnetic field. An inductive pickup coil is used to detect the resulting
precession of the nuclear magnetization. Typically one needs about 10
18
nuclear spins to generate a good NMR signal with stable nuclei. Consequently
conventional NMR is mostly a bulk probe of matter. On the other hand, in
related nuclear methods such as muon spin rotation (μSR) or β-detected NMR
(β-NMR) a beam of highly polarized radioactive nuclei (or muons) is
generated and then implanted into the material. The polarization tends to be
much higher – between 10% and 100%. Most importantly, the time evolution
of the spin polarization is monitored through the anisotropic decay properties
of the nucleus or muon which requires about 10 orders of magnitude fewer
spins. For this reason nuclear methods are well suited to studies of dilute
impurities, small structures or interfaces where there are few nuclear spins.
Příklad
NMR curve for 11Be implanted in metallic Be at 
T 
= 50K. At this
 
temperature the
spin-lattice relaxation time 
T
1
 is of the order of the nuclear lifetime
 
τ 
= 20 
s.
 
P
říklad
Nuclear magnetic resonances
for 8Li (
I 
= 2) implanted into
different non-cubic crystals.
This illustrates the influence
of the implantation host on
the quadrupole frequency as
well as on the resonance line
widths. The nuclear level
splitting for a nucleus with
spin 
I 
= 2, submitted to a
magnetic field and an EFG,
and the corresponding
transition frequencies are
shown for one- and two-
photon transitions. The five
levels are non-equidistant,
resulting in four equidistant
one-photon resonances in the
NMR spectrum
β-NMR
At radioactive ion beam facilities such as 
ISOLDE
 and 
ISAC
 it is possible
to generate intense (>10
8
/s) highly polarized (80%) beams of low energy
radioactive nuclei.
Furthermore one has the added possibility to control the depth of
implantation on an interesting length scale (6–400 nm).
Although in principle any beta emitting isotope can be studied with β-NMR
the number of isotopes suitable for use as a probe in condensed matter is much
smaller. The most essential requirements are:
(1) a high production efficiency
(2) a method to efficiently polarize the nuclear spins and
(3) a high β decay asymmetry.
Other desirable features are:
(4) small 
Z
 to reduce radiation damage on implantation,
(5) a small value of spin so that the β-NMR spectra are relatively simple and
(6) a radioactive lifetime that is not much longer than a few seconds.
 
Table gives a short list of the isotopes we have identified as suitable for
development at ISAC. Production rates of 10
6
/s are easily obtainable at ISAC.
8
Li is the easiest to polarize and therefore was selected as the first one to
develop as a probe at ISAC
 
 
 
 
 
 
LMR
Another possibility
 to NMR is 
B
eta
-R
ay
 Detected Level Mixing Resonance
(
-LMR)
Here, the axial symmetry is broken via combining a quadrupole and a dipole
interaction with their symmetry axes non-collinear. This gives rise to
resonant changes in the angular distribution at the magnetic field values
where the nuclear hyperfine levels are mixing.
The resonances observed in a LMR experiment are not induced by the
interaction
 
with a rf field, but by misaligning the magnetic dipole and
electric
 
quadrupole interactions. This experimental technique does not need
an
 
additional rf field to induce changes of the spin orientation. The change of
the spin orientation is induced by the quantum mechanical “anti-crossing” or
mixing of levels, which occurs in quantum ensembles where the axial
symmetry
 is broken.
 
Nuclear 
HF
 levels of a
nucleus with spin 
I 
= 3
/
2
submitted to a
 
combined
static magnetic
interaction and an
axially symmetric
quadrupole interaction:
(
a
) for collinear
interactions, 
β 
= 0
; 
(
b
) and (
c
) for non-
collinear interactions
with 
β 
= 5
and 
β 
= 20
,
respectively.
Crossing or mixing of hyperfine levels occurs
 
at well-defined values for the ratio of
the involved interactions frequencies, i
f 
 
(
d
) At these positions, resonances are observed in the decay
 
angular distribution of
oriented radioactive nuclei, from which the nuclear spin and
 moments can be deduced
Atomic hyperfine structure
For a particular atomic level characterized by the angular momentum 
J
,
the coupling with the nuclear spin 
I 
gives a new total angular momentum
F,
 
F 
= 
I 
+ 
J
, 
|I − J| ≤ F ≤ I 
+ 
J
. The HF interaction removes the
degeneracy of the different 
F 
levels and produces a splitting into 2
J 
+ 1 or
2
I
+1 hyperfine structure levels for
 
J < I 
and
 
J > I
, respectively
.
Example of the atomic fine
and hyperfine structure of
8
Li. For free atoms the
electron angular momentum
J 
couples to the nuclear spin
I
, giving rise to the HF
structure levels 
F
. The
atomic transitions between
the 
2
S
1
/
2
 ground state to the
first excited 
2
P 
states of the
Li atom are called the D1
and D2 lines
 
Using vector coupling rules the HF structure energies of all 
F 
levels
The determination of nuclear moments from hyperfine structure is
particularly appropriate for radioactive isotopes, because the electronic
parts 
Be
(0) and 
Vzz
(0) are usually known from independent measurements
of moments and hyperfine structure on the stable isotope(s) of the same
element.
Optical pumping
Polarization of a fast beam by optical pumping was introduced for the
 
β
-
asymmetry detection of optical resonance in collinear laser spectroscopy.
M
ost applications took advantage of the
 
additional option to perform nuclear
magnetic resonance spectroscopy with
 
β
-asymmetry detection (
β
-NMR) on a
sample obtained by implantation of
 
the polarized beam into a suitable crystal
lattice. Whatever is the particular
 
goal of such an experiment, it is important to
achieve a high degree of nuclear
 polarization.
Optical pumping within the hyperfine structure Zeeman levels for polarization
of the nuclear spin. The example shows the case of 
I 
= 1 for the case of
 28Na
Repeated absorption 
and
spontaneous emission of photons
results in an accumulation of the
atoms
 
in one of the extreme 
MF
states for which the total angular
momentum
 
F 
= 
J 
+
I
, for an 
S 
state
just composed of the electron spin
and the nuclear
 spin, is polarized.
 
 
 
Optical pumping
If a weak magnetic 
field defines the quantization axis in the direction of the
atomic and the laser
 
beam, each absorption of a circularly polarized photon
introduces one unit
 
of angular momentum in the atomic system. This can be
expressed by the
 selection rule 
Δ
M
F
 
= 
±
1 for 
σ
±
 
light
,
 
with 
σ
+ and 
σ− 
being
the conventional notations for the circular polarization
 
of the light with
respect to the direction of the magnetic field.
Repeated absorption 
and
spontaneous emission of photons
results in an accumulation of the
atoms
 
in one of the extreme 
MF
states for which the total angular
momentum
 
F 
= 
J 
+
I
, for an 
S 
state
just composed of the electron spin
and the nuclear
 spin, is polarized.
Collinear Laser Spectroscopy
 
 
 
 
 
 
 
 
Measurement of nuclear radius
Four methods outlined for charge matter radius:
Diffraction scattering
Atomic x-rays
Muonic x-rays
Mirror Nuclides
Measurement of nuclear radius
Three methods outlined for nuclear matter radius:
Rutherford scattering
Alpha particle decay
-mesic x-rays
Diffraction scattering
The charge
 distribution does not have a sharp boundary
Edge of nucleus is diffuse - “
skin
Depth of the skin ≈ 
2.3 f
RMS radius
 is calculated from the charge distribution and,
neglecting the skin, it is easy to show
q = momentum transfer
 
Measure the scattering intensity as a function of 
 to infer
the 
distribution of charge in the nucleus
, 
Atomic X-rays
Assume
 
the nucleus is uniform charged sphere
.
Potential 
V
 is obtained in two regions:
Inside
 the sphere
Outside
 the sphere
Nuclear Mean Square Charge Radius
I
nformation on isotopic differences between nuclear
 
mean square charge
radii is contained in the isotope shifts of o
 
spectral lines. Let 
A
, 
A 
and 
m
A
,
m
A
 
be the mass numbers and atomic
 
masses of the isotopes involved. Then
for an atomic transition 
i 
the isotope
 
shift, i.e. the difference between the
optical transition frequencies of both
 isotopes, is given by
This means that both the field shift (first term) and the mass shift (second
term) are factorized into an electronic and a nuclear part. The knowledge
 
of
the electronic factors 
Fi 
(field shift constant) and 
Mi 
(mass shift
 
constant)
allows one to extract the quantity 
δr
2
 
of the nuclear charge distribution.
These atomic parameters have to be calculated theoretically or
 semi-
empirically.
 
For unstable isotopes high-resolution optical spectroscopy is a unique
approach
 
to get precise information on the nuclear charge radii, because it
is
 
sensitive enough to be performed on the minute quantities of (short-
lived)
 
radioactive atoms produced at accelerator facilities. 
Other techniques are
 
suitable only for stable isotopes of which massive
targets are available. 
Elastic
 
electron scattering [27] even gives details of the charge
distribution, and
 
X-ray spectroscopy on muonic atoms [28] is dealing with
systems for which
 
the absolute shifts with respect to a point nucleus can be
calculated. Thus
 
both methods give absolute values of 
r
2
 
and not only
differences. Eventually,
 
the combination of absolute radii for stable
isotopes and differences of
 
radii for radioactive isotopes provides absolute
radii for nuclei all over the
 
range that is accessible to optical spectroscopy.
 
 
 
 
On-Line NMR/ON
On-Line NMR/ON
Nuclear Magnetic Resonance on Oriented Nuclei is done at ~10 mK temperatures.
Polarised radioactive nuclei are exposed to an RF field of variable frequency.
When the Zeeman splitting frequency is found         resonant absorption changes the
sublevel populations and hence also the observed anisotropy        a resonance in the
anisotropy versus frequency plot.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
COLlinear LAser
SPectroscopy
 
68
C
u
Δ
E=const=
δ
(
1
/
2
mv
2
)≈mv
δ
v
Resolution ~1  MHz
,
 resulting from
the velocity compression of the line
shape through energy increase.
I
n
 
S
o
u
r
c
e
,
 
D
o
p
p
l
e
r
 
w
i
d
t
h
 
r
e
s
o
l
u
t
i
o
n
 
~
 
2
5
0
 
M
H
z
C
o
l
l
i
n
e
a
r
 
C
o
n
c
e
p
t
 
-
 
a
d
d
 
c
o
n
s
t
a
n
t
 
e
n
e
r
g
y
 
t
o
 
i
o
n
s
 
On-Line Laser spectroscopy 
Collinear and In-Source
Methods
:            Atomic Hyperfine Structure splitting
With nuclear spin I these each form a doublet  with F (= I + J) = I
+1/2 and I - 1/2.
Transitions between these doublets give four lines in two pairs
with related splittings.
- poor resolution (
In Source
) only for the A  (large magnetic
dipole) splitting
- good resolution (
Collinear
) for both A and B (smaller electric
quadrupole splitting)
In Cu+ ion, electron states involved are s
1/2
and p
1/2
.
The NSCL Fragment Separator, MSU
Fragmentation 
-NMR
Fragments are polarised in their creation.
Implanted in cubic materials, their polarisation can be detected by
measurement of the asymmetry of their beta decay. Application of a
magnetic field creates a Zeeman splitting which is deduced from
resonant destruction of the asymmetry, yielding the nuclear g-factor.
 
 
 
 
 
 
The spectroscopic quadrupole moment can be related to an
intrinsic
 
quadrupole moment 
Q
0 reflecting the nuclear
deformation 
β
, only if certain
 
assumptions about the nuclear
structure are made. An assumption that
 
is often made (but is
not always valid!), is that the nuclear deformation is
 
axially
symmetric with the nuclear spin having a well-defined
direction with
 
respect to the symmetry axis of the deformation
(strong coupling). In this
 
case, the intrinsic and the
spectroscopic quadrupole moment are related as
 follows:
 
 
Měření hmot jader
An ideal 
Penning trap
 consists of a strong homogenous magnetic field and a
weak quadrupolar electrostatic potential. In contrast to a Paul trap, full
confinement is achieved with static trapping fields. As a Paul trap, a Penning
trap also consists of ring and endcap electrodes. Quite often so-called guard
or correction electrodes are placed between endcaps and the ring to
compensate for the truncation of the hyperbolical electrodes. Two types of
geometry configurations are commonly used: hyperbolic and cylindrical.
Both constructs have their own benefits although in precision experiments
usually hyperbolical are favored due to better production of quadrupolar
electric field. On the other hand, cylindrical electrodes are easier to
manufacture and sometimes more open geometry offer other benefits such as
better conductance of gas.
 
 
Ion motion in a Penning trap
A charged particle in a Penning trap exhibits three different eigenmotions.
One is in the direction of the magnetic field lines (
axial motion
)) and two of
them (
magnetron
 and 
cyclotron
) are perpendicular to the magnetic field.
Ideally the axial 
oscillation
 is independent of magnetic field and has a
frequency
ν z =1 2π qU 0 md 2 − − − − √ ,
where q and m are the charge and mass of the particle, U 0 the voltage
across the ring and endcap electrodes and d the characteristic trap parameter
defined as d=1 2 (z 2 0 +ρ 2 0 2 ) − − − − − − − − − − √ . The frequencies of
the radial modes -- denoted ν − for magnetron and ν + for trap-modified
cyclotron frequency areν ± =1 4π (ν c ±ν 2 c −2ν 2 z − − − − − − − √ ),
where ν c =1 2π q m B is the so-called free-space cyclotron frequency
which gives the frequency of ion in absence of any electric fields.In first
order, the magnetron frequency is independent of the ions mass-over-charge
ratio m/q and only depends on the trap geometry d and the applied voltage
U 0 .
 
Ion motion excitation
Any of the three eigenmotions can be modified by using time-varying
electric fields. Usually the ring electrode (or one of the correction electrodes
between is radially split in order to create electric fields that have azimuthal
and/or axial component. Using a time-varying dipolar electric field with a
frequency of one of the eigenmodes will increase or decrease the radius of
the corresponding motion.
Motion excitation with quadrupole fields is another commonly used
technique. With a quadrupole field one can couple two motions like
cyclotron and magnetron motion with a quadrupole field having frequency ν
+ +ν − .With ion motion excitations, mass-over-charge ratio of trapped
particles can be linked to frequency, which can be easily determined with
high accuracy.
 
Mass separation with a Penning trap
Penning traps have been used to separate ions having different mass-over-charge
ratio for years. In short, unwanted ions are driven to large orbits and rejected,
typically by letting the contaminants hit the electrode surface.
One well established separation is technique is the so-called 
sideband cooling
technique
 [Sav1991], which allows separation of ions down to parts-per-million
level. This technique was pioneered at ISOLTRAP and later adopted to other trap
setups such as SHIPTRAP and JYFLTRAP. In order to perform mass separation,
the trap needs to be filled with dilute gas to allow fast ion motions (axial and
cyclotron) to be cooled. Another requirement is a small extraction aperture for the
ions which is usually accommodated by having a small hole in one of the endcap
electrodes.
Once a bunch of ions is captured to the trap, the ions are let to cool down. This
typically takes 10-100 ms. Next, a dipolar RF field is switched on at magnetron
frequency in order to radially drive all ions to such a large magnetron orbit that ions
would, if extracted towards the extraction side, hit the electrode. After establishing
a large magnetron orbit for the ions, a quadrupole electric field with frequency ν +
+ν − is switched on to convert the established magnetron motion to cyclotron
motion. Since the sideband frequency ν + +ν − ≈ν c =1 2π q m B is highly mass
selective, only ions near the 
resonance
 frequency get their magnetron motion
converted to cyclotron motion.
 
In a Paul trap the trapping effect is achieved solely with electric fields. They
consist of a ring electrode and two endcap electrodes that in ideal case are
hyperboles of revolution. Confinement of ions is achieved by using both DC
and AC electric fields. Motion of ions is described with 
Mathieu equations
which in short describes the suitable combinations of frequency and
amplitude of the electric field for storing ions with certain mass-over-charge
ratio. In nuclear physics Paul traps are used mainly for storing and cooling
ions. Some trap structures are prepared so that the center of the trap is
exposed for example for 
lasers
 and particle detectors.
A special type of Paul trap is a so-called linear Paul trap, also known as
radiofrequency quadrupole or RFQ
. In such a device the ring electrode is
significantly longer compared to a Paul trap.
 
 
 
 
 
SCATTERING OF HIGH-ENERGY
NEUTRONS BY NUCLEI:
cross section of the very fast neutrons
(usually 14 and 25 MeV neutrons
used) reaches the value 2
R
2 
THE YIELD OF NUCLEAR
REACTIONS INITIATED BY
PROTONS OR 
-PARTICLES:
Comparison of excitation functions
with theory can give information
about nuclear radius
Scattering of e- of high energy (200
MeV)
Diffraction pattern is expected if the
charge is expected to be uniformly
distributed around the nucleus (not
point-like)
Assuming different values of R and b,
one can try to find the best fit
observed angular distribution
 
 
 
radioactive 111In
This is the probe atom, which is used for
the PAC measurements. Its decay initiates
the emission of the detected g-rays.
It is produced e.g. by 109Ag (a, 2n) 111In
This material is commercially available.
About 1011 atoms are needed for one
measurement.
The detailed decay-scheme:
 
 
 
„1“ není pozorován 
 dochází
 
nuclear hyperfine interactions
 lead to frequencies of precession of probe nuclei that are
proportional to the internal fields and are characteristic of the probe's lattice location
Many PAC probes have an excited nuclear state reached through a gamma-gamma
cascade.  
Start
 and 
stop
 gamma rays signal creation and decay of an intermediate state and
(importantly) there is an anisotropic angular correlation between directions of emission of the
two gamma rays.  Such is the case, for example, following decay of 60Co, for which the
lifetime of the intermediate nuclear state is very short.  A classic experiment in advanced
teaching laboratories involves measuring the anisotropy of the angular correlation of radiations
from 60Co by measuring the coincidence counting rate of the two gammas as a function of the
angle subtended by the two detectors.  For longer-lived states, the nuclear spins can precess
through appreciable angles and interaction frequencies can be measured with good
precision.  A small number of probes have long lifetimes and other favorable nuclear
properties for "table top" PAC experiments, including 111In, decaying into 111Cd, and 181Hf,
decaying into 181Ta. 
Internal fields in solids are produced mostly by charges and spins within the first few atomic
shells, with more distant charges and spins only contributing to inhomogeneous signal
broadening.  As a result, 
interaction frequencies can be used to characterize the local
atomic environments in which probe atoms find themselves
. Once a frequency has been
identified with an underlying environment, it can be monitored in measurements made
following diffferent methods of sample preparation, or for changing conditions and
temperature.
 
 
 
In a PAC-experiment one measures the characteristic
frequencies 
L, 
 0 associated with the hyperfine interaction.
We use for this the 111In (I=5/2) probe. The probe decays to
111Cd through some intermediate level with a lifetime of about
122ns. Activity is introduced into the sample by diffusion or
implantation. The detector set-up consists of 4 BaF2-
scintillation detectors. One essentially measures the nuclear
decay of the intermediate I=5/2 level (Q=0.8b, m =-0.766m N).
In the presence of hyperfine interactions oscillations appear
superimposed on the exponential decay. By performing a
mathematical operation, the PAC-time spectrum or R(t)-
function is calculated. Apart from the size of the MHF and
EFG one can in principle deduce the angles between these two,
the angles with respect to the detector set-up and the
divergence of axial symmetry for the EFG (h).
 
The PAC technique
PAC measurements rely on a radioactive probe atom like 
111In 
. The quantity measured is the
electric field gradient (efg), which arises from the immediate lattice surrounding at the site of
the probe atom. The observed effect results from the hyperfine interaction between the efg and
the 
quadrupole moment
 of the isotope 
111Cd
, which which is a product of the radioactive
decay of the probe 
111In
. A non zero efg, which is the result of a deviation of the electronic
charge distribution around the probe from spherical symmetry, is observed in case of a non-
cubic lattice structure or a defect in the immediate vicinity of the probe atom. The efg is the
second spatial derivative of the electrostatic potential and, therefore, can be described by a
second rank, traceless tensor. In its principal axis system this tensor is completely described by
two quantities, usually its largest component Vzz, expressed via the quadrupole coupling
constant n
Q
 = e
Q
V
zz
/h, and the asymmetry parameter h = (Vxx-Vyy)/Vzz . By 
detecting
 two g
-
quanta
 in coincidence, which are emitted following the radioactive decay, a 
time spectrum
 is
obtained, which is described by a function
 
,
which holds for the case of a single efg. The frequencies wn are extracted from the 
Fourier
transform
 F(w), yielding nQ, w1 and h, which is determined by the ratio w2/w1 .
 
 
 
Ferrite
 or 
alpha iron
 (
α-Fe
) is a 
materials science
 term for iron, or a solid
solution with iron as the main constituent, with a 
body centred cubic
 crystal
structure. It is the component which gives 
steel
 and 
cast iron
 their magnetic
properties, and is the classic example of a 
ferromagnetic
 material.
It can be considered as pure 
iron
 practically (strength = 280N/mm2). Ferrite
can be strictly defined as a solid solution of iron in body-centered cubic
(
BCC
) containing a maximum of 0.03% carbon at 723oC and 0.006%
carbon at room temperature.
Fázový diagram Fe-C
 
 
 
 
Rezonanční jaderná absorbce - Mössbauerův jev
Pokud mají kvanta záření g energii přesně rovnou energii určité vzbuzené jaderné hladiny, může být takový g-foton 
pohlcen
jádrem
, čímž se jádro 
excituje
 na příslušnou energetickou hladinu. Vzápětí nastane 
deexcitace
 za vyzáření fotonu g téže
energie (který však samozřejmě vyletí obecně jiným směrem, než foton dopadající); jedná se tedy o 
jadernou fluorescenci
.
Zatímco v atomovém obalu je buzení rezonančního fluorescenčního záření běžným jevem, rezonanční jaderná absorbce se za
normálních okolností nevyskytuje a je ji možno uskutečnit pouze ve 
speciálním experimentálním uspořádání
. Důvodem je
to, že jaderné absorbční čáry jsou 
velice úzké
 (cca 10-5-10-9 eV), takže je velmi obtížné se energeticky "trefit" do tak úzkého
rozmezí. Dokonce i tehdy, když ozařujeme zářením g, emitovaným určitými (dceřinnými) excitovanými jádry, látku složenou
z téhož isotopu prvku, který záření g emituje, nedochází k jaderné absorbci, neboť skutečná energie kvanta se poněkud liší od
energie jaderné hladiny z následujícího důvodu: Při přechodu jádra z excitovaného stavu se vyzáří energie Eexcit daná
rozdílem počátečního (excitovaného) a konečného (základního) stavu. Celou tuto energie však neodnese foton g, ale podle
zákona akce a reakce se tato energie rozdělí mezi energii Eg vyslaného kvanta g a kinetickou energii Ek 
odraženého jádra
, tj.
Eg = Eexcit - Ek < Eexcit. Obráceně, při dopadu fotonu a jeho absorbci se rovněž část jeho energie přemění na kinetickou
energii odraženého jádra, takže na vybuzení jádra na energii Eexcit je potřeba vyšší energie Eg dopadajícího záření (právě o
tuto hodnotu Ek). Energie emitovaného a absorbovaného záření g jsou tedy vůči sobě 
posunuty
 o hodnotu DE = 2.Ek.
Kinetická energie odraženého jádra plyne ze zákona zachování hybnosti, tj. když je původní jádro v klidu, musí se hybnost pg
fotonu g rovnat hybnost pnucl odraženého jádra, takže Ek = p2nucl/2mnucl = p2g /2mnucl = E2g /(2mnucl.c2) »
E2excit/(2mnucl.c2). Pro přechody o energii Eexcit cca 100 keV ve středně těžkých jádrech hodnota Ek činí několik setin eV,
což podstatně převyšuje energetickou šířku jaderné hladiny. Aby mohlo dojít k rezonanční jaderné absorbci, je nutno tuto
ztracenou energii odraženého emitujícího a absorbujícího jádra ve vhodném experimentálním uspořádání nahradit -
kompenzovat Dopplerovým jevem
. Toho je možno dosáhnout třemi způsoby:
1. 
Mechanickým pohybem
 zdroje směrem k absorbátoru, např. umístěním zdroje na rotující kotouč, aby se Dopplerovým
efektem zvýšila frekvence (a tedy i energie) emitovaného záření. Potřebná rychlost 
v
 tohoto pohybu je dána vztahem v/c =
Ek/Eg.Tímto způsobem se v r.1951 podařilo P.B.Moonovi pozorovat 
rezonanční rozptyl
 záření g energie 412keV emitovaného
dceřinným jádrem 198Hg radioisotopu 198Au. V r.1958 pak R.L.Mössbauer prokázal 
rezonanční absorbci
 záření g energie
129keV emitovaného dceřinným jádrem 191Ir radioisotopu 198Os *); podle něj byl pak příslušný jev nazván Mössbauerův.
*) V Mössbauerově experimentu byl navíc zdroj i absorbátor chlazen na teplotu 88°K (nižší než tzv. Debyeova teplota
materiálu), čímž byla zvýšena pravděpodobnost emise záření g bez odrazu emitujícího jádra, a podobně i absorbce. Není-li
totiž energie odraženého jádra Ek dostatečná k tomu, aby vybudila krystalovou mřížku, v níž je atom vázán, ze základního
stavu do prvého vzbuzeného stavu, nemůže být žádná energie přenesena na krystalovou mřížku a emise záření g nastává bez
odrazu jádra. Jaderná rezonanční absorbce má pak své maximum kolem nulové rychlosti pohybu zdroje vzhledem k
absorbátoru.
2. Zahříváním radioaktivního zdroje, aby se Dopplerovým jevem "rozmítala" frekvence a zvýšila se tak šířka čáry
emitovaného záření g.
3. Využitím předcházejícího rozpadu b či jaderné reakce k získání vhodné rychlosti emitujícího jádra pro dosažení Dopplerova
frekvenčního posuvu.
Na Mössbauerově jevu rezonanční jaderné absorbce záření g s využitím způsobu 1. je založena metoda 
Mössbauerovy
spektroskopie
, zmíněná v §3.3.
Nejvhodnější podmínky pro realizaci Mössbauerova jevu jsou u jader isotopu železa 57Fe. Jako zdroj záření g se použije
radionuklid 57Co, který se s poločasem 270 dní elektronovým záchytem rozpadá na excitované jádro 57Fe. Toto jádro při
deexcitaci emituje záření g o energiích 692keV (0,14%), 136keV (11%), 122keV (87%) a 14,4keV (9%). Právě záření g
dílčího přechodu z excitované hladiny o energii 14,4keV je díky nízké energii vhodné pro buzení rezonanční jaderné absorbce.
Vzhledem k vysoké Debyeově teplotě Fe (360°K) probíhá Mössbauerův jev i za normální laboratorní teploty, přičemž
Dopplerův frekvenční posun, potřebný pro kompenzaci odrazu jádra 57Fe, se dosahuje mechanickým posuvem zdroje
rychlostí pouze řádu 1mm/s.
Pozn.: Vysokou citlivost Mössbauerova jevu rezonanční jaderné absorbce g-záření 14,4keV 57Fe využili v r.1960 R.V.Pound
a G.A.Rebka ke změření 
gravitačního frekvenčního posuvu
 v tíhovém poli Země, což bylo významným testem správnosti
Einsteinovy obecné teorie relativity jakožto fyziky gravitace a prostoročasu - viz §2.4 "
Fyzikální zákony v zakřiveném
prostoročase
", pasáž "
Gravitační frekvenční posun
" v knize "
Gravitace, černé díry a fyzika prostoročasu
".
 
 
 
 
 
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Hyperfine interactions play a crucial role in atomic physics, leading to small energy shifts and splitting of degenerate levels in atoms and molecules. These interactions involve the electromagnetic multipole interactions between the nucleus and electron clouds, resulting in the splitting of energy levels due to the interaction of nuclear magnetic and electric moments with external fields. Hyperfine structure impacts both atomic nuclei and electronic levels, removing degeneracy and causing discrete energy values. This overview covers the Nuclear Zeeman effect, interactions of magnetic moments, and the Electric quadrupole in an electrostatic field.


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  1. Nuclear probes in materials - M ssbauer effect and M ssbauer spectroscopy

  2. External EM field - energy Energy of a stationary charges and currents in external EM field Charge; Electrostatic potential Electric dipole moment (= 0); Intensity of electric field Intensity of magnetic field Magnetic dipole moment; Tensor of electric field gradient Electric quadrupole moment; Odd electric moments = 0 = Even magnetic moments (p conservation law) Higher orders can be neglected E shifts weaker by a few orders of magnitude. In atoms/nuclei, hyperfine structure occurs due to the energy of the nuclear magnetic dipole moment in the magnetic field generated by the electrons, and the energy of the nuclear electric quadrupole moment in the electric field gradient due to the distribution of charge within the atom/nuclei.

  3. Hyper-fine interactions (in atomic physics) hyperfine structure is a small shift in otherwise degenerate energy levels and the resulting splitting in those levels of atoms (molecules) due to electromagnetic multipole interaction between the nucleus and electron clouds In atoms, HF structure arises from the energy of the nuclear m interacting with B generated by the e- and the energy of the nuclear quadrupole moment in the EFG due to the distribution of charge within the atom. The influence of magnetic moments leads to splitting (removing degeneracy) of levels in the nucleus as well as for electronic levels. In this course - hyperfine interaction dominantly deals only with splitting of levels in the nucleus Schematic illustration of fine and hyperfine structure in a neutral hydrogen atom electronic levels Fine splitting is only due to interaction of e- However, HF interaction impacts also (splitting of) levels of atomic nuclei

  4. Nuclear Zeeman effect Interaction of magnetic dipole moment with magnetic field: Bis a sum of contribution of external + surrounding e- Energy of a magnetic dipole is then given by Projection m can reach only discrete values: m = -I, -I+1, , I-1, I (in units of h) Actual energies are discrete m complete multiplet splitting hyper-fine interaction (can cause shift in E) 57Fe E DE Ip (shifted due to HI) + split multiplet of states (2I +1) x degenerated level 0 Allowed only transitions with DmI= -1,0,1

  5. Electric quadrupole in electrostatic field: Interaction of quadrupole moment with the electric field gradient (at nucleus) Cartesian coordinate system Spherical coordinate system After diagonalization in Cartesian CS (Vxy,... = 0) If axial symmetry: = 0: } asymmetry Only axial symmetry systems discussed below

  6. axial symetry = 0: Electric quadrupole v electrostat. field: nuclear Stark effect Only partial multiplet splitting 57Fe eQ is electric quadrupole moment (multipole moment is in general defined as zeroth moment component for maximum projection m = I ): } for a nucleus with I = 0, the quadrupole moment cannot be determined does not exist only nuclear states with I>1/2 show a splitting

  7. Electric quadrupole in electrostatic field For cubic crystals, which are equivalent in all directions x, y and z (therefore Vzz = Vyy = Vxx) and also for axially symmetric crystals, which are equivalent in x and y (therefore Vyy = Vxx) the asymmetry parameter vanishes as = 0. In cases of smaller symmetry takes values between zero and one. For axially symmetric electric field gradients the energy of the sublevels takes the values The energy difference between two sublevels m and m is given by After introducing the quadrupole frequency for the energy difference can be written Dominant application to 57Fe only two different values of m2

  8. Electric quadrupole in electrostatic field In contrast to the magnetic interaction the splitting of the sublevels of the nuclear state through the electric quadrupole interaction depends on the angular moment I of the sublevel and is therefore non-equidistant. The transitions frequencies n between the different sublevels are, in case of = 0, integer multiples of the smallest frequency 0. It can be written whereas 0= 3 Qfor integer nuclear spin I and 0= 6 Q for half integer nuclear spin I. As for the magnetic dipole interaction, also the electric quadrupole interaction induces a precession of the angular correlation in time and this modulates the quadrupole interaction frequency. This frequency is an overlapping of the different transition frequencies n. The relative amplitudes of the different components depend on the orientation of the electric field gradient relative to the detectors (symmetry axis) and on the asymmetry parameter .

  9. Mssbauer effect Finite temperature impact Nuclear resonance fluorescence does not happen difference to e- transitions emission absorption M ... Nucleus mass Doppler broadening is often not sufficient to cause significant resonance absorption (for low Eg) Natural line width (t 100 ns G = 10-8eV)

  10. Mssbauer effect Recoil compensation (how can be compensated) Doppler broadening Mechanical movement (Doppler) absorber emiter e.g. for 411 keV transition in198Hg v = 6.7 x 102m/s Use of a cascade of g transitions or nuclear reaction Angle a could be chosen to fulfill the resonance condition for the 2nd photon 1958 M ssbauer (NC 1961) nucleus bound in the crystal lattice and there is no recoil of a single nucleus momentum is transformed to the phonons (lattice vibrations), or in a fraction of emissions to the whole crystal (mass is higher by ~1023 compared to a single nucleus)

  11. Mssbauer effect 3 different cases: B atom binding energy in a crystal emission as from a free atom (typically for Eg 1 MeV) Debay (Einstein) frequency D ( E) Energy is transformed to phonons for lower photon energies (typically for Eg hundreds of keV) recoilless emission in a fraction of cases (typically for Eg 10-100 keV) - resonance absorption happens Recoil-free fraction f In the first approximation in a model with a single phonon energy recoil is absorbed by whole crystal almost all energy is carried by the photon (crystal carries a negligible energy << linewidth) M ssbauer effect = (almost) recoil-free emission and/or absorption of g rays

  12. Recoil-free fraction Illustration even within a classical approach: oscillating nucleus (only an illustration) Frequency seen by an observer (Doppler effect): The wave seen by an observer: - + Bessel functions Identity: Probability of recoil-free emission is:

  13. for harmonic oscillator Recoil-free fraction Probability of recoil-free emission is: J0for a small argument In a general case (finite T) the probability is where W (~Debye-Waller or Lamb-Mossbauer factor) in Debye model of a crystal (f decreases with T) QD a measure of the strength of the bonds between the M ssbauer atom and the lattice Specifically, for T = 0 To calculate one needs adequate model of a crystal e.g. Debye model - model of phonon gas with frequencies Corresponds to (above-given)

  14. Debye model, Einstein model Debye model is a method for estimating the phonon contribution to the heat capacity in a solid. It is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a gas of photons in a box. Debye treats the vibrations of the atomic lattice (heat) as phonons in a box (the box being the solid). Debye temperature is the temperature of a crystal's highest normal mode of vibration, i.e., the highest temperature that can be achieved due to a single normal vibration. (QD ( -Fe) = 464 K) Einstein model treats the solid as many individual, non-interacting quantum harmonic oscillators. It is based on three assumptions: Each atom in the lattice is a 3D quantum harmonic oscillator Atoms do not interact with each another Atoms vibrate with same frequency (contrast with Debye model) While the first assumption is quite accurate, the second is not. If atoms did not interact with one another, sound waves would not propagate through solids. Heat capacity (Debye gives correct behavior at low T)

  15. Line shape Breit-Wigner shape Line shape for emission detector Line shape for absorption s source absorber Observed shape in absorption experiment For a thin absorber and if then Theoretical line-shape known very small change of Egcan be measured (on a fraction of G) energy resolution (depending on isotope) is The best tool to study hyperfine interaction (change of energies of levels due to nucleus- electron interaction)

  16. Mssbauer effect detector Typical experimental setup: source absorber Energies of levels in absorber are checked/scanned Can be shifted or split Energy of source photon must be varied usually via Doppler effect (source is moving) usually a periodic change In 57Fe 1mm/s 4.8x10-8 eV (n =11.6 MHz, E = hn) Source (57Co) must be in a material (crystal lattice) with a high QD necessary condition for recoilless emission Lattice must be cubic non-magnetic metal (Rh, Cr, Pt, Pd, Cu) to have no splitting (commercial sources have f 0.75) Absorber must not be neither very thick to allow pass some radiation, nor very thin to see an effect If a thin absorber cannot be prepared, a "scattering" (instead transmission) arrangement might be used

  17. How to exploit Mssbauer effect? Study of hyperfine interaction (i.e. interaction between nucleus and electrons) that manifests itself due to shift and/or split of levels in the nucleus Mossbauer spectroscopy Splitting of individual levels allows measurement/appears due to Magnetic field at the nucleus Gradient of electric field at nucleus The ground and first excited states of 57Fe split in a magnetic field. Isomer shift and quadrupole splitting of the nuclear levels in electric field gradient.

  18. Mssbauer effect Energy difference due to splitting is on the order 10-7 eV (for B used in NMR, a few Tesla) actual fields in materials could be stronger - internal magnetic field of the metallic iron is about 33 T, the splitting is slightly higher Similar (could be much smaller or comparable in size) splitting observed due to electric field gradient (EFG) Recoil energy of a single nucleus is about 10-3 eV, i.e. much higher no chance to observe transitions via standard recoil The M ssbauer effect cannot be observed for freely moving atoms or molecules, i.e. in gaseous or liquid state

  19. Mssbauer spectroscopy Nuclei of absorber re-emit photon (iso-tropically) after 10-7 s If high (e-) conversion coefficient (a, depends on Eg), re-emission strongly suppressed In the absorption measurement (narrow beam detected) can usually be fully neglected Measurement of conversion electrons possible (CEMS) only sensitive to sample surface (hundred(s) of nm) as low-energy e- are strongly absorbed Requirements for suitable isotope for M ssbauer spectroscopy Low Eg (10-150 keV) between 1st excited and stable GS minimization of ER The highest possible nucleus mass minimization of ER t 10-7-10-9 s (longer too narrow line, difficult to observe; shorter wide resonance with a low resolution) Stable isotope must be produced in a decay with relatively long lifetime (>1y) Electric quadrupole and magnetic dipole moment of involved levels 0 (for applications) High qD for wide range of T Fe: qD=470 K, ED=0.04 eV

  20. Mssbauer spectroscopy Observed in about 80 isotopes (50 elements), 20 elements really used The best nucleus :57Fe, Eg= 14.4 keV ER= 1.95x10-3eV, G = 0.45x10-8eV, 2G = 0.194 mm/s high a allows to exploit even conversion e- (conversion electron Mossbauer spectroscopy) Can be used for T significantly above room T Energy resolution 3x10-13 10% 90% a = 8.56 57Co preparation using cyclotrons 56Fe(p,g) The 2nd most popular nucleus: 119Sn - 2G=0.626 mm/s (worse resolution), for chemistry The lightest izotope: 40K difficult to get (no accessible decay) The best resolution: 67Zn, (Eg= 93.26 keV, t = 9.4 ms, 2G = 3.12x10-4mm/s) problem: vibrations disturb observability Further127,129I, 125Te for chemistry 151,153Eu, 161Dy, 6 Gd isotopes - for magnetism of rare-earth nuclei Actinides (237Np, 237U, 232Th) for physical constant measurements

  21. Hyperfine Interactions Electric monopole interaction (Coulombic) between protons of the nucleus and s electrons penetrating the nuclear field. Different shifts of nuclear levels A and S. Isomer shift values give information on the oxidation state, spin state, and bonding properties, such as covalency and electronegativity. Electric quadrupole interaction between the nuclear quadrupole moment (eQ 0, I > 1/2) and an inhomogeneous electric field at the nucleus (EFG 0). Nuclear states split into I + 1 sub-states. The quadrupole splitting gives the information on oxidation state, spin state and site symmetry. Magnetic dipole interaction between the nuclear magnetic dipole moment ( 0, I > 0) and a magnetic field (H or B 0) at the nucleus. Nuclear states split into 2I+1 sub-states with mI = +I, +I 1, . , I The magnetic splitting gives information on the magnetic properties of the material under study ferromagnetism, anti-ferromagnetism.

  22. Isomer(ic) shift The nuclear radius in the excited state is usually different (in the case of 57Fe it is smaller than that in the ground state, for 119Sn opposite). The fictitious energy levels of the ground and excited states of a bare nucleus (no surrounding e-) are perturbed and shifted by e- interactions. The shifts in the ground and excited states differ because of the different nuclear radii in the two states, which cause different Coulombic interactions (due to presence of s-electrons in nucleus). The energy differences ES and EA also differ because of the different e- densities in the source (S) and absorber (A) material (chemical environment). The individual energy differences ES and EA cannot be measured only the difference of the transition energies = EA ES can be measured (this difference is called Isomer shift). The isomer shift is given by: = EA ES = (2/5) Ze2( A S)(Rexc2 RGS2) r density of (s-)electrons (square of amplitude of e- wave function | (0)|2) at nucleus spectrum in

  23. Isomer shift Measures the energy difference with respect to the source Given by combination of nuclear (R) and chemical properties R must be known if chemical properties are to be measured Also, electrons on higher shells (even valence) play a role via shielding effect (impact on the role of s electrons) Direct impact change of e-population in s orbitals (mainly valence s orbitals) changes directly | (0)|2 Indirect impact shielding by p , d , f electrons, increase of electron density in p , d , f orbitals increases shielding effect for s electrons from the nuclear charge s electron cloud expands, | (0)|2 at nucleus decreases. Shift appears in the order of splitting Isomer because there is a difference of R for different states (can be isomeric but usually they are not between different states ) In principle, can be used to measure the root-mean-square radius R2 Effect on atomic spectral lines not discussed but similar spectrum v

  24. Electric quadrupole in electrostatic field The line intensities are given by the angular dependence of the quadrupole interaction In a randomly oriented polycrystalline material, the intensities are identical In an anisotropic crystal, an angular dependence is observed with respect to the principal axis of the electric gradient tensor (Q); EFG is non zero in non cubic valence e-distribution and/or in non cubic lattice site symmetry Moreover, the percentage of recoil-free emissions f is dependent with respect to the crystallographic axis with the highest symmetry (vibrational anisotropy) f(q) d - IS 57Fe : for for If quadrupole electric moment know, electric field gradient can be studied (and vice versa)

  25. Nuclear quadrupole resonance (NQR) Analog to NMR the splitting of levels due to the interaction of the electric field gradient (EFG) tensor with the quadrupole moment Qij of nuclei is sometimes called NQR analogous features (precession of the main component of Qij around EFG direction) EFG determined mainly by the valence e- involved in the bond with nearby nuclei NQR is applicable only to solids and not liquids, because in liquids the quadrupole moment averages out Only nuclei with J > have non-zero Qij, which is a measure of the degree to which the nuclear charge distribution deviates from that of a sphere The energy splitting (quadrupole frequency) is given by

  26. Nuclear quadrupole resonance (NQR) One might think about realization of the technique in a similar way as NMR: In principle, a specified EFG could be applied to influence Q as in NMR However, in solids, the strength of the EFG is many kV/m2, making the application of the external EFG's for NQR impractical. As a result, the NQR spectrum is specific to the substance (sometimes called "chemical fingerprint ) and not adjusted by the external field only internal fields measured As NQR frequencies are not chosen by the external field, they can be difficult to find making NQR a technically difficult technique this is not a problem if this effect is exploited as a part of the Mossbauer spectroscopy or Perturbed angular correlation spectroscopy Since NQR is done in an environment without a static (or DC) magnetic field, it is sometimes called "zero field NMR". Many NQR frequencies depend strongly upon temperature.

  27. Nuclear Zeeman effect Interaction of magnetic dipole moment with magnetic field: Bis a sum of contribution of external + surrounding e- Energy of a magnetic dipole is then given by Projection m can reach only discrete values: m = -I, -I+1, , I-1, I (in units of h) Actual energies are discrete m complete multiplet splitting hyper-fine interaction (can cause shift in E) 57Fe E DE Ip (shifted due to HI) + split multiplet of states (2I +1) x degenerated level 0 Allowed only transitions with DmI= -1,0,1

  28. Nuclear Zeeman effect 57Fe Intensity ratios given by the direction of g with respect to B in general: Q = 0 Q = 90o randomly oriented field a = 2 a = 0 a = 4 57Fe Both states are split in principle, 2 of 3 involved parameters (mGS, mexcit, B) can be determined Very strong B can be applied

  29. Combined fields Both DEM and EQ can be determined

  30. Mssbauer effect M ssbauer effect is one of the most important sources of information about hyperfine interactions (together with NRM, PAC) it is mainly used for studies of chemical properties and structure of substances (nuclear properties are not the main practical interest, but can be used e.g. to determine parameters of excited states moments, spin) unlike NMR, it is used to study solid-state substances Isomeric shift provides information about the valence electrons it is e.g. sensitive enough (sometimes in combination with quadrupole splitting) to determine oxidation of Fe determine whether FeII or FeIII

  31. Mossbauer spectroscopy - detectors Detectors are usually scintillators (for transmission measurement) ideally transparent to 122 keV transition, i.e. rather thin However, there are other possibilities: Gas-filled proportional counters are often convenient and offer better energy resolution. Solid-state detectors are excellent when the count rate is not excessive. 10% 90% a = 8.56 For CEMS (conversion electrons) proportional counters

  32. Measurement of Gravitational red-shift The change (decrease) of potential energy with the distance from the Earth surface should imply the change in the wavelength of a photon at different distance Measured (for the first time) using Fe for the height difference h 22 m Expected shift in wave-length is still 2 orders of magnitude smaller (10-11 eV) than the natural linewidth Change in absorption (line profile) is largest for the "inflection point" on the BW curve this point was used and the wavelength change corresponding to (0.859 +/- 0.085)x the value predicted by Einstein's theory of relativity was obtained in the original work E Cranshaw and J P Schiffer 1964 Proc. Phys. Soc. 84 245 R.V. Pound and G.A. Rebka, Jr., Phys. Rev. Lett. 4(1960)337 R.V. Pound and J.L. Snider, Phys. Rev. B140(1965)788.

  33. Measurement of Gravitational red-shift Hyperfine Interactions 72(1992)197-214

  34. Nuclear Probes in materials - Perturbed angular correlations

  35. Angular distribution of emitted photons multipolarity projection Observed angular distribution: angular distribution from individual projections (square of spherical harmonics) Clebsch-Gordon coef. Population of initial states |Jimi For yields For dipole: } const. isotropy anisotropy for

  36. Angular distribution of emitted photons Each individual m-state yields an anisotropic distribution, but with equal populations of each m-state, then by summing over all m-states the angular distribution becomes isotropic.

  37. Angular distribution of emitted photons - example When an excited nucleus decays via cascade g rays, an anisotropy is generally found in the spatial distribution of the 2ndg ray with respect to the 1st one. If the 1stg is defined to be the z-axis, and therefore defines the m-states, then the intermediate state will, in general, have an unequal population of m-states. For example, a J=0, m=0 initial state, the m=0 intermediate state cannot be populated because g must carry away at least one unit of angular momentum (L = 1) with m = 1 (along the direction of propagation). For a 0-2-0 cascade only the Dm = 1, 2 intermediate states are populated giving As higher and higher angular momenta are considered, the more complicated these scenarios become. The general form for angular correlations W(q) is:

  38. Angular distribution of emitted photons - example Example of angular correlations for a variety of g cascades (involved spins). Some correlations have stronger anisotropies than others. If transitions are not purely dipole (L=1) or quadrupole (L=2), so-called mixing ratios are considered and they change the angular correlation.

  39. Example 60Co Laboratories (3rd year of Bc, summer term) angular correlation of two photons emitted after b decay of 60Co T1/2=1925 d T1/2=3.3 ps T1/2=0.9 ps W( ) = 1/8 cos2( ) + 1/24 cos4( )

  40. Angular distribution of emitted photons multipolarity projection Observed angular distribution: angular distribution from individual projections (square of spherical harmonics) Clebsch-Gordon coef. Population of initial states |Jimi For yields } const. isotropy anisotropy for

  41. How to get anisotropy? Using an external magnetic field At room temperature, anisotropy is unmeasurable very low T must be used m.B ~ k.T required system must be cooled to mK for visible effect Anisotropy can be used as a thermometer (two detectors measure intensity temperature must be close to absolute zero) Optical (laser) techniques Polarization of "atomic spins" by absorption of circularly polarized light (mz=+1) leads to strong magnetic field and thus polarization of the nucleus By means of nuclear reactions Projectile (with zero spin) brings a non-zero orbital angular momentum l to the system (projection of l is perpendicular to the beam direction) Ideally, if spins of both projectile and target nucleus are =0, only m=0 states Partial de-orientation occurs during de-excitation of the nucleus Using a cascade of g transitions In a coincidence measurement, the orientation of the 1stphoton in a cascade defines axis and selects a subspace of events (initial m) (Compton scattering - FEL, )

  42. Short comments on different methods Low-temperature orientation Idea from 1948, for the first time measured in 1950 in paramagnetic crystals containing60Co Anisotropy can be used as a thermometer if the field (B) is well known (two detectors measure intensity at different angles with respect to B temperature must be close to absolute zero) Absorption of circularly-polarized light Leads to a strong B at nucleus and hence to orientation of nuclear states HF interaction couples nuclear spin ? ? and e-spin ? ? to a total atomic spin ? ? = ? ? + ? ?, which splits a fine-structure level, into several HF levels characterized by the ? ? quantum numbers

  43. Short comments on different methods Use of a nuclear reaction Projectile with (J=0) brings (l > 0) ( ml=0 vector of orbital momentum l is perpendicular to beam direction) If J=0 for both projection and target nucleus (ideal case), initial state has only m=0 Decay of the initial states leads to partial de-orientation distribution of projection is symmetric with respect to 0 If the number of emitted particles is not extremely high, the m distribution is strongly peaked around m=0 The more steps, the higher de-orientation Evaporation of particles (p, n, d) g-ray cascade 110Pd(a,2ng)112Cd Ea 25 MeV Population of low-lying states in a reaction via compound nucleus

  44. General description of angular correlation EL + dML +... Measured angular correlation is (in general) given by S - detector reference frame the frame corresponding to the response to individual polarizations Wigner D-functions Depends on: 1. characteristics of radiation (type E/M, multipolarity of both transitions types, mixing ratios) 2. detector characteristics (response to individual polarizations, geometrical smearing coefficients Qlm) S sample reference frame statistical tensor is expressed via rlm in this frame

  45. Angular correlation special cases Polarization not measured If (in addition) axial symmetry If (in addition) reflection symmetry l = even (typical case)

  46. g g cascades two consecutive g g rays Angular distribution of photons in a cascade is usually anisotropic ...original ensemble can be (in general) oriented Transition 1 is not observed if Pm(J) isotropic (the same), no angular dependence for transitions 2 is observed Transition 1 observed oriented ensemble is formed At (time) t = 0 (immediately after emission of g1) ... Population probability in the reference frame given by direction of g1 emission if Pm(J) is isotropic Unperturbed angular correlations ...W2~ angular dependence of 2nd photon

  47. Angular distribution of emitted photons Each individual m-state yields an anisotropic distribution, but with equal populations of each m- state, then by summing over all m-states the angular distribution becomes isotropic.

  48. g g cascades two consecutive g g rays Angular distribution of photons in a cascade is usually anisotropic ...original ensemble can be (in general) oriented Transition 1 observed oriented ensemble formed 1. At t = 0 (immediately after emission of g1) Unperturbed angular correlations 2. For (time after emission of 1 ) t > 0 evolution of statistical tensor Periodic modulation of coefficients al due to precession in magnetic field B (or EFG) Perturbed angular correlations (measurement of internal fields B, EFG) ...W2~ angular dependence (for the 2nd photon)

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