Folie 1

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How do nuclei rotate?
energy spheroid
The nucleus rotates as a whole. (collective
degrees of freedom)
The nucleons move independently inside deformed
potential (intrinsic degrees of freedom)
The nucleonic motion is much faster than the
rotation (adiabatic approximation)
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Oblate and prolate quadrupole deformation
Choosing the vertical axis as the 3-axis one
obtains the oblate by R1 R2 gt R3 and the
prolate by R1 R2 lt R3 axially-symmetric
quadrupole deformations
prolate deformation (ßgt0)
oblate deformation (ßlt0)
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The Euler angles

• It is important to recognize that for nuclei the
  intrinsic reference frame can
• have any orientation with respect to the lab
  reference frame as we can hardly
• control orientation of nuclei (although it is
  possible in some cases).
• One way to specify the mutual orientation of two
  reference frames of the
• common origin is to use Euler angles.

• (x, y, z) axes of lab frame (red)
• (x,y,z) axes of intrinsic frame (blue)
• Rotation by angle a about the zaxis,
• ß about the xaxis and ? about the
• new zaxis brings the intrinsic frame
• onto the lab frame

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The Euler angles

• It is important to recognize that for nuclei the
  intrinsic reference frame can
• have any orientation with respect to the lab
  reference frame as we can hardly
• control orientation of nuclei (although it is
  possible in some cases).
• One way to specify the mutual orientation of two
  reference frames of the
• common origin is to use Euler angles.

• (x, y, z) axes of lab frame
• (1,2,3) axes of intrinsic frame
• The rotation from (x,y,z) to (x,y,z)
• can be decomposed into three parts
• a rotation by about the z axis to
• , a rotation of ? about the
• new y axis to , and
• finally a rotation of ? about the new
• z axis .

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Quantization
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Rotational motion of a deformed nucleus
For a deformed nucleus with axial symmetry
one obtains the same rotational frequency
for the 1- and 2-axis. The Hamiltonian is given by
The nucleus does not have an orientation degree
of freedom with respect to the symmetry axis
States with projections K and K are degenerated
This fact is expressed in the nuclear wave
function as a symmetric product
For K0, only even-J are allowed, so that the
wave function is given by a single term
If the total angular momentum results only from
the rotation (J R), one obtains for the
rotational energy of an axially symmetric nucleus
by
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Broad per spective on structural evolution
proton number
neutron number
Note the characteristic, repeated patterns
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?-rays from a superdeformed band in 152Dy
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Rotational motion of a deformed nucleus
axial symmetry rotational axis symmetry axis
kinematic moment of inertia
dynamic moment of inertia
rotational frequency
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Moment of inertia
Rigid body moment of inertia
Irrotational flow moment of inertia
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Moment of inertia
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Reduced transition probability
expectation value
wave function
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Reduced transition probability
Wigner-Eckart-Theorem (reduction of an
expectation value)
special case E2 transition I?I-2
reduced transition probability
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Reduced transition probability
Transition probability
half-life
Weisskopf estimate
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Hydrodynamical model
Reduced transition probability
Excitation energy
Moment of inertia
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Hydrodynamical model
Reduced transition probability
Excitation energy
first indication of a hexadecapole deformation
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Spherical harmonics