Hyperspherical Theory of Three-Body Recombination in
Cold Collisions
James Sternberg∗ and J.H. Macek†
∗
†
Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996
Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996 and Oak Ridge National
Laboratory, Oak Ridge, TN
Abstract. Three-body recombination in cold atomic collisions has been successfully modeled using the hyperspherical closecoupling theory. This representation maps many-body fragmentation channels onto simple radial equations similar to those
for two-body processes. Computation of hyperspherical adiabatic energy eigenvalues is a major task using this theory. We
have developed asymptotic representations of the adiabatic eigenvalues based upon energy dependent zero-range potentials.
Our asymptotic expressions are compared with ab initio calculations as well as hyperspherical adiabatic calculations using
zero range potentials with constant M that have appeared in the literature.
INTRODUCTION
The theoretical study of cold three-body collisions has
become an important topic of study because of its application to Bose-Einstein condensation. In particular, the
experimental work of Inouye et al. has succeeded in observing recombination of a Bose condensate as a function of the two-body scattering length a[1]. Numerous
theory papers have dealt with the connection between
the scattering length and the recombination rate, which
has been the motivation for calculations using zero-range
potentials(ZRP)[2, 3] . A ZRP approximates the twobody scattering potential as a function of the scattering
length alone. In this work, we define an energy dependent zero-range potential, which not only depends on the
scattering length a, but also on the energy k[4].
The most successful methods for studying three-body
recombination of such systems have been hyperspherical adiabatic theories. Esry et al. have successfully calculated the hyperspherical potential curves for threebody recombination by directly numerically solving the
Schrödinger equation[5]. This calculation is essentially
exact, however it is computationally intensive. For small
values of the hyper-radius R, the calculations were quite
effective. They became more computationally expensive
for larger values of R, however. Furthermore, the calculation contains many parameters which must be chosen. As a result, it is more difficult to draw physical insight from that calculation than from a simpler one. The
theory of Nielsen et al. is both complete and accurate
within the constraints of the model they use[6]. Practical application of that theory can be difficult, however,
because of the generality of the theory. Simplified calculations have also been made. For example, the hyperspherical adiabatic theory has been applied to zero range
potentials (ZRP)[3]. The calculation of ref. [3] condenses
the physics the potential curves into knowledge of simple physical parameters such as the two-body scattering
length. Those calculations are only accurate for large values of the hyper-radius R, however. We have extended
this calculation to obtain potential curves for energy dependent zero range potentials (EDZRP). That is, instead
of assuming a constant M matrix M = −1/a, an energy
dependent M(k) is used. This method retains the simplicity of the hyperspherical adiabatic ZRP theory, while
improving on its range of validity.
METHOD
Hyperspherical adiabatic theory for
EDZRPs
In this work we will study three-body recombination
in the helium trimer. In order to calculate the hyperspherical adiabatic potential curves, we ignore explicit threebody interaction terms, and assume only pairwise interactions.
Several previous papers used the Aziz potential in
order to model the potential of the helium dimer [5,
7]. For convenience, we have chosen a potential which
can be made similar to the Aziz potential, but which
yields a Schrödinger equation with analytic solutions.
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
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133
We can thus write the M matrix
Comparison of the Aziz and Morse Potentials
M (k) = k cot δ (k) .
40
Aziz Potential
Morse Potential
30
Once this is obtained, the potential curves can be calculated by using the adiabatic hyperspherical zero-range
equation.
√ ν cos (νM π/2) − 8/ 3 sin (νM π/6)
= −RM (k) (7)
sin (νM π/2)
U(r)
20
10
0
with k = νM /R. The potential curves are given by
-10
-20
(6)
0
5
10
15
r
20
25
30
ε (R) =
FIGURE 1. The Aziz potential used by Esry et al. is compared with the Morse potential. These two potentials are visually almost identical to each other except for very small values
of R.
The potential we have chosen is the Morse potential[8].
The Morse potential can be expressed as,
V (r) = D exp [−2a (r − ro )] − 2D exp [−a (r − ro )] (1)
Here, D denotes the depth of the potential, ro is the location of the potential’s minimum and a is its width.
These parameters were varied in order to best fit the Aziz
potential which was used in ref. [5]. A plot comparing
these two potentials can be seen in Fig. 1. The advantage of this potential over the Aziz potential is that the
Schrödinger equation can easily be transformed into a
confluent hypergeometic equation with known, analytic
solutions. The solutions to the Schrödinger equation are
1
Ψ1 =
exp −β a r − r0 − γx (r)
(2)
r 1
× 1 F1 β + − γ; 1 + 2β ; 2γx (r)
2
and
1
exp β a r − r0 − γx (r)
(3)
r 1
× 1 F1 −β + − γ; 1 + 2β ; 2γx (r) .
2
√
Here we use the definitions γ = 2mD/a, x(r) =
exp −a r − r0 and β = ik/a.
From these solutions, the S matrix can be determined
by
Ψ (0)
S= 1
,
(4)
Ψ2 (0)
and from the S-matrix, the phase shift is found to be
"
!#
√
ik 1
ik 2 2mD
δ = arg 1 F1
+ − γ; 1 + 2 ;
(5)
a 2
a
a
Ψ2
=
134
2 − 1/4
νM
.
2µR2
(8)
It should be noted that the significant difference between the expression in equation (7) and previous expressions is a more complicated dependence on k = ν/R
on the right-hand side of the equation. Ordinarily it is assumed that M(k) = −1/a, leading to a directly solvable
equation, rather than the transcendental equation shown
above.
Connection with the complete theory of
Nielsen
The EDZRP theory presented here can be derived
from the complete theory of Nielsen et al.[6]. Starting
from Nielsen’s equation (71), we can define
1
1
1
1
ω
= sinlx +1 α cosly +1 αPν(lx + 2 ,ly + 2 ) (cos 2α)
υ
= sinlx +1 α cosly +1 αPν(ly + 2 ,lx + 2 ) (− cos 2α) (9)
where Pν(a,b) are the Jacobi functions defined in Appendix
A of ref. [6]. Throughout this section, ν will be the
defined as it is used by Nielsen in his paper. It is related
to the νM used throughout the rest of this report by the
ν −2
equation ν = M2 .
The logarithmic derivative is written as
D=
∂ log [sin α cos αu]
.
∂α
(10)
Where u is defined in eq. 14. We can then rewrite
Nielsen’s equation (71) as
∂υ
∂ω
Dυ −
A = Dω −
RA
(11)
∂α
∂α
where R are Nielsen’s Ri j written in matrix form and A
are expansion coefficients. If we define
M=
Dυ − ∂∂ αυ
Dω − ∂∂ ωα
(12)
then Eq. 11 requires
and
det (M − 2R) = 0.
(13)
The functions ω, υ and u are solutions to the
Schrödinger equation
lx (lx + 1) ly (ly + 1)
∂2
+
(14)
− 2+
∂α
cos2 α
sin2 α
i
+ K ∗ 2mR2V µ −1
R
sin
α
−
λ
ψ =0
N
jk
where K = 0 for ω and υ and K = 1 for u. One can show
that to order α 4 that equation (14) becomes the usual
radial equation for potential scattering as α → 0, with
r = Rα µ −1
jk ,
λN − 31 lx (lx + 1) − ly (ly + 1)
2mk =
.
R2
2
(15)
and λN = (2ν + lx + ly )(2ν + lx + ly + 2).
In the that limit, ω and υ become linear combinations of spherical Bessel functions. ω goes to the regular Bessel function kr jlx (kr) in this limit, but υ goes to
a linear combination of the regular Bessel function and
the irregular Bessel function krylx (kr). We can therefore
write
υ = Aω + Bζ ,
(16)
where to order (kr)4 relative to the lowest non-vanishing
term
(17)
ω → Cω kr jlx (kr) 1 + O(kr)4
4
ζ → Cζ krylx (kr) 1 + O(kr) .
Here A,B,Cω and Cζ are constants which are to be determined.
If we substitute the above expressions into equations
(11) and (12) and take only diagonal elements of R we
find that
(M + A) B−1Cω Cζ−1 = cot δlx
(18)
In the calculations presented in this paper, we are
interested in S-wave scattering. In this case lx = ly = 0.
We can therefore write the functions ω and υ as
1 1
ω
= sin α cos αPν( 2 , 2 ) (cos(2α))
υ
= sin α cos αPν( 2 , 2 ) (− cos(2α)).
(19)
υ
=
+
Γ ν + 32
Γ(ν + 1)Γ
Γ ν + 32
Γ(ν + 1)Γ
3
2
3
2
sin(πν) cos[(2ν + 2)α]
(21)
2(ν + 1)
cos(πν) sin[(2ν + 2)α]
.
2(ν + 1)
One can easily see from equations (20) and (21) that
A = cos(πν), B = sin(πν) and Cω = Cζ .
We now substitute the constants A and B into Eq. (18)
to obtain
(M + cos(πν)) (sin(πν))−1 = cot δ .
(22)
Multiplying both sides of Eq. (22) by k and using the
relation k = (2ν + 2)/R (please note that we are still
using Nielsen’s definition of ν) on the left-hand side
gives,
2ν + 2
(M + cos(πν)) (sin(πν))−1 = k cot δ = M(k).
R
(23)
Solving for M , we find that
M=
sin(πν)RM(k)
− cos(πν),
2ν + 2
(24)
which is substituted into Eq. (13) to obtain
RM(k) =
2(ν + 1) cos(πν) + √83 sin
sin(πν)
π
3 (ν
+ 1)
. (25)
This agrees with the result given in Eq. (100) of ref. [6].
Substituting ν = (νM − 2)/2 we exactly obtain equation
(7). For k = 0, this yields exactly the ZRP result. It is
important to note that in deriving this expressions that
we never make the assumption that k = 0 and that M(k)
is constant. Equation(25) can be used for arbitrary values of k, and is expected to improve of the standard ZRP
model in cases where the energy is significantly different from zero. This treatment eliminates the constant M
approximation from the ZRP model, in contrast to more
traditional treatments such as that of Baz’ et al.[9].
BOUND STATES
1 1
Using the definitions of the Jacobi functions, we rewrite
these as
Γ ν + 32
sin [(2ν + 2) α]
ω=
(20)
2(ν + 1)
Γ(ν + 1)Γ 32
135
The Aziz potential cited in ref. [5] has a single bound
state. The Morse potential with the parameters we have
chosen also supports a single bound state. The hyperspherical adiabatic potential curves ε(R) are plotted in
Fig. 2. Direct comparison between the more extensive
calculation and the EDZRP calculation is presented here
here. As expected, the two results match almost exactly
for large values of R. This is to be expected, since in
Comparison of EDZRP, Exact and ZRP Calculations
Comparison of EDZRP, ZRP and Exact Calculations
Single Bound State
0
40
EDZRP Morse Potential Data
Aziz Potential ref.[5]
ZRP calculation
EDZRP Calcluation
Exact. ref.[9]
ZRP Calculation
35
-0.1
λ(λ+4)
Epsilon(R)
30
-0.2
25
20
-0.3
15
-0.4
0
100
200
300
400
10
500
R (a.u.)
FIGURE 2. Comparison of EDZRP calculation with the
exact calculation of ref. [5] and the ZRP calculation of
Gassaneo[3]. This figure compares the exact calculation using
the Aziz potential to the EDZRP calculation using the Morse
potential and the ZRP calculation of ref. [3] for a bound state.
the zero energy limit, M(k → 0) = −1/a. The advantage over the standard ZRP calculation can be seen in
the small R region. The ZRP calculation in reference [3]
reports that for R/a ≈ 0.25, the value of ε(R) agrees with
the Esry’s calculation within 17%. The EDZRP calculation improves on this and gives a value within 6.5%.
POSITIVE ENERGY STATES
We have shown the the previous section that the EDZRP
calculation is an improvement over the standard ZRP
calculation for small values of R. This is the region
over which the value of k is significantly different from
zero. A challenge for ZRP calculations is positive energy
states. In these cases, the k ranges over a much wider
range and the approximation that k = 0 leads to inaccurate results. By allowing the value of k to change in the
EDZRP calculation, the results are accurate for a much
larger range of the value R. Just as in the bound state calculation, we find that in the limit as R → ∞, the value
of k goes to zero. In fact, we confirm that the EDZRP
and ZRP calculations give identical results. This correct
asymptotic behavior is one of the main motivations for
using the hyperspherical adiabatic method. Incidentally,
it is for these large values of R that the numerical calculation of ref. [5] becomes more difficult. The advantage of
the EDZRP method over the standard treatment is seen
for smaller values of R. In Fig. 3, we plot λ (λ + 4) for R
between approximately 20 and 300 atomic units, where
λ is related to ν(R) by the relation ν(R) = λ + 2, and is
directly related to the adiabatic energy eigenvalues ε(R)
shown in equation (8). The results are compared directly
136
0
100
200
R (a.u.)
300
400
FIGURE 3. Comparison of EDZRP calculation with the exact calculation of ref. [10] and the ZRP calculation of ref. [3].
The EDZRP calculation closely reproduces the behavior of the
direct numerical calculation.
to the ZRP calculation of reference [3] and the exact numerical calculation of reference [10]. For the larger values of R, all three calculations are converge to the same
value. The asymptotic limit in this case is zero. For R less
than approximately 200, the behavior the of the ZRP and
the exact numerical calculations are quite different, however. The ZRP calculation has an almost linear dependence on R, and in fact goes to a value of λ (λ + 4) ≈ 17
at R = 0[3]. The exact calculation rises steeply as R goes
to zero[10]. Our EDZRP calculation reproduces the behavior of the exact calculation. In fact, it is highly accurate, even to R ≈ 25.
POSITIVE ENERGY STATES WITH A
RESONANCE
The ability to calculate potential curves with arbitrary k
affords us the opportunity to examine phenomena which
can not be calculated in the standard ZRP model. For
example, positive energy states in the presence of a resonance can be examined. For a resonance, it can be shown
that M(k) can be written in the form
M(k) = −
k2 − Eres
(γ 2 /2)
(26)
Here Eres is the energy where the resonance occurs and
γ is reduced width[11]. Using the M(k) above, we calculate the potential curves for several values of λ . We chose
Eres = .01 and a width of γ 2 /2 = .001. These curves can
be seen in FIGURE 4. As one might expect, we see a
series of avoided crossings. In the simple model M(k)
which we have chosen above, we can analytically predict
ACKNOWLEDGMENTS
4
The authors would like to thank Dr. Gustavo Gassaneo for supplying data from both his calculation[3], and
that of B. Esry[5]. We also would like to thank the National Science Foundation for support under grant number PHY0140321.
Epsilon(R)
3
2
REFERENCES
1
1.
0
0
50
FIGURE 4.
100
R
150
200
Scattering with a resonance
the positions of each of
√ these avoided crossings. These
should occur at r = λ / Eres . In this case we should find
them at R = 20, 60, 80, 100....
CONCLUSIONS
We have derived an energy dependent extension of the
standard hyperspherical adiabatic ZRP model for threebody recombination in cold collisions. In this new model,
the term which is the inverse of the scattering length a is
replaced by a function M(k), which depends on energy.
We have related this theory with the more general theory
of Nielsen et al. [6], which depends independently on
both k and R.
We then used this EDZRP model with the Morse
potential to calculate hyperspherical adiabatic potential
curves for both bound states and positive energy states.
These calculations were compared with both a direct
numerical solution to the Schrödinger equation[5] and
a calculation using the adiabatic hyperspherical ZRP
model[3]. In all cases, the EDZRP model reproduced the
numerical solution more accurately than the ZRP model.
This was especially true for smaller values of R because
the energy is significantly different from zero, and the
zero energy assumption of the ZRP model is no longer
valid.
Finally, we used the EDZRP model to examine positive energy states with a resonance included. This calculation produced hyperspherical adiabatic potential curves
with a series of avoided crossings.
137
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