content_image/377521.jpg
because the three-body computation can not place the resonance at precisely the correct measured energy, and because the width is strongly dependent on the height and thickness of the barrier at the correct energy. 

<img src='content_image/377521.jpg'>

In Fig.4 we show the width as a function of the res- onance energy. The solid line shows the WKB estimate for $^{9}$Be. For an energy just above the threshold energy of the $^{8}$Be ground state (0 . 0918 MeV) the width is about 0.1 MeV. This agrees with the result obtained by fitting the measured peak in the photodissociation cross section [11] (triangle down). The width increases slowly up to about 0 . 7 MeV at the top of the barrier. At smaller energies the width is vanishingly small due to the thick barrier provided by the $^{8}$Be structure. 

The complex scaling computations for $^{9}$Be present nu- merical difficulties due to the fast increase of the width for energies above 0 . 0918 MeV. To get an estimate we have instead attached a unit charge to the neutron, which immediately leads to the results shown in Fig.4 for the 1 / 2 + analog state in $^{9}$B. A continuous decrease of the charge leads us back to $^{9}$Be which is approached and finally reached by extrapolation. Unfortunately the ac- curacy only allows the conclusion that the width at an energy of 0 . 11 MeV is larger than 0 . 06 MeV and most likely around 0 . 1 MeV but 0 . 2 MeV is not numerically excluded. The inaccuracy is due to the very large width compared to the distance to the threshold. Then the poles computed by the complex scaling method are very 

difficult to distinguish from the continuum background. This becomes increasingly worse as the charge of the neu- tron is decreased from unity to zero where our present techniques prohibits a clean result. 

Comparison to widths obtained in previous works re- quires precision in the definitions of a resonance. The ambiguities arise when the resonance is broad and then necessarily asymmetric. To account for the energy de- pendence of the decay probability the R -matrix theory employs an energy dependent width. This immediately implies that any number claimed to describe the width must be an average of some kind. The “observed” width in R -matrix theory is in [22] most directly related to the full width at half maximum, or the S -matrix pole, and then adopted as the width in tables of resonance proper- ties [17]. The results quoted in [11] and [22] are shown in Fig.4 by the square and the circle, respectively. 

Our computed three-body resonance width is from the WKB tunneling and crude extrapolations from the imag- inary value of the pole of the S -matrix. Our estimate is very inaccurate but we expect a value of about 0 . 1 MeV which is only about half the “observed” table value from R -matrix theory. This rather large discrepancy can be due to inaccuracies in the complex scaling extrapolation to zero charge of the neutron, or the WKB approxima- tion of only one adiabatic potential combined with the uncertainty in the knocking rate estimate. However, we believe that the main reason is that our three-body bar- rier, which entirely is responsible for the width, arises from three-body restructuring effects inherently impossi- ble to include in the R -matrix analysis. An unambiguous settlement of this accuracy issue would require the use of another method dedicated to precise width computations near threshold. 

Attempts to settle the issue experimentally face the problems inherent to relatively very broad resonances. Population in a reaction or beta-decay provide informa- tion about lifetime, which necessarily is an average, or de- tails about decay probability as function of energy, which requires a model for analyzing the data. The direct mea- surements in photo dissociation [11] reveal an asymmet- ric peak with a width of about 0 . 1 MeV. This is con- sistent with corresponding computations in the present model with a method to compute strength functions by discretization of the three-body continuum [23]. 

Three-body results for $^{9}$B( 1 / 2 $^{+}$). The isobaric analog state should exist in $^{9}$B but so far it has never been found. The influence of the additional Coulomb interaction in the present fragile case could be substantial and perhaps even destructive. The corresponding low energy three- body and two-body levels are shown in Fig.1b. To discuss this problem we also show in Fig.2 the corresponding low- est adiabatic potential for $^{9}$B. The additional Coulomb repulsion due to the proton in $^{9}$B has a relatively small effect. At small distances the pocket is as pronounced but almost entirely above zero energy. The barrier is higher 

