170
4 Two-Particle Systems in the Berggren Basis
10
-12
10
-10
10
-8
10
-6 (a)
10
-12
10
-10
10
-8
10
-6
Width (Ry)
(b)
-1
0
1
2
3
4
Energy (10
-2 Ry)
10
-12
10
-10
10
-8
10
-6 (c)
(0,2)
(8,10)
(8,8)
(6,8)
(8,6)
(6,6)
(4,6)
(6,4)
(4,4)
(2,4)
(4,2)
(2,2)+(0,2)
(2,0)
(7,8)
(7,6)
(5,6)
(1,0)
(5,4)
(3,4)
mixed: (1,2)+(3,2)
(8,8)
(6,6)
(4,4)
(2,2)
(0,0)
(0,0)
Fig. 4.8 (Color online) The distribution of J π = 0 + , 1 − , and 2 + resonant states in the complex
energy plane for the prolate configuration with Q +
zz = +6.88 ea 2
0 , s = 1.6a 0 , and I = 10 4 mea 0
calculated with max = 8. Bound states are marked by arrows. In most cases, families of
resonances are characterized by one dominant channel. The corresponding labels ((, j r ) are given.
Mixed groups are represented by empty symbols. Dashed lines indicate rotational energies of the
molecule, E jr = ¯
h
2 j r (j r + 1)/(2I) (from Ref. [94])
ranging from 10 −10 Ry to 10 −6 Ry corresponding to lifetimes in the range of
10 −7 − 10 −11 s. Each family of resonant states, marked by the same symbol and
color in Fig. 4.8, is characterized by one dominant channel ((, j r ) that represents
about 99% of the total wave function, whereas mixed groups where two dominant
channels exist are represented by empty symbols. Mixed groups in Fig. 4.8 are the
(1,2) and (3,2) families for J π = 1 − and the (0,2) and (2,2) families for J π = 2 + .
Within each family, narrow resonances have a very diffuse dominant-channel wave
function with a small number of nodes, while broader resonances tend to have
wave functions peaked closer to the origin and having larger numbers of nodes.
Dashed lines indicate rotational energies of the molecule, E j r = ¯
h 2 j r (j r + 1)/(2I ).
Overall, the resonance energies cluster close to the rotational states of the core,
except for higher excitations where significant deviations can be seen. Indeed,
higher-lying resonances have larger values of the orbital angular momentum in
their dominant channel, which results in larger centrifugal barriers.
4 Two-Particle Systems in the Berggren Basis
10
-12
10
-10
10
-8
10
-6 (a)
10
-12
10
-10
10
-8
10
-6
Width (Ry)
(b)
-1
0
1
2
3
4
Energy (10
-2 Ry)
10
-12
10
-10
10
-8
10
-6 (c)
(0,2)
(8,10)
(8,8)
(6,8)
(8,6)
(6,6)
(4,6)
(6,4)
(4,4)
(2,4)
(4,2)
(2,2)+(0,2)
(2,0)
(7,8)
(7,6)
(5,6)
(1,0)
(5,4)
(3,4)
mixed: (1,2)+(3,2)
(8,8)
(6,6)
(4,4)
(2,2)
(0,0)
(0,0)
Fig. 4.8 (Color online) The distribution of J π = 0 + , 1 − , and 2 + resonant states in the complex
energy plane for the prolate configuration with Q +
zz = +6.88 ea 2
0 , s = 1.6a 0 , and I = 10 4 mea 0
calculated with max = 8. Bound states are marked by arrows. In most cases, families of
resonances are characterized by one dominant channel. The corresponding labels ((, j r ) are given.
Mixed groups are represented by empty symbols. Dashed lines indicate rotational energies of the
molecule, E jr = ¯
h
2 j r (j r + 1)/(2I) (from Ref. [94])
ranging from 10 −10 Ry to 10 −6 Ry corresponding to lifetimes in the range of
10 −7 − 10 −11 s. Each family of resonant states, marked by the same symbol and
color in Fig. 4.8, is characterized by one dominant channel ((, j r ) that represents
about 99% of the total wave function, whereas mixed groups where two dominant
channels exist are represented by empty symbols. Mixed groups in Fig. 4.8 are the
(1,2) and (3,2) families for J π = 1 − and the (0,2) and (2,2) families for J π = 2 + .
Within each family, narrow resonances have a very diffuse dominant-channel wave
function with a small number of nodes, while broader resonances tend to have
wave functions peaked closer to the origin and having larger numbers of nodes.
Dashed lines indicate rotational energies of the molecule, E j r = ¯
h 2 j r (j r + 1)/(2I ).
Overall, the resonance energies cluster close to the rotational states of the core,
except for higher excitations where significant deviations can be seen. Indeed,
higher-lying resonances have larger values of the orbital angular momentum in
their dominant channel, which results in larger centrifugal barriers.