CHINESE JOURNAL OF PHYSICS
VOL. 43, NO. 2
APRIL 2005
Scattering of Ultracold Potassium Atoms in the Triplet State
M. Kemal Öztürk∗ and Süleyman Özçelik†
Physics Department, University of Gazi, 06500 Teknikokullar, Ankara, Turkey
(Received May 31, 2004)
Elastic scattering properties of the ultracold interaction for the triplet state of 39 K have
been studied using a potential composed of short-range and long-range terms, where the
long-range term has a multiplier: “cutoff function”. For governing the decrease of the cutoff
function, the “cutoff radius” has been adjusted by comparing it to experimental energy
values. The scattering properties have then been found using the solutions of the radial
equation for two methods, the WKB and the Numerov method. The obtained results have
been compared in detail with the experimental and other theoretical results.
PACS numbers: 32.80.Cy, 34.20.Cf, 34.50.Pi
I. INTRODUCTION
Elastic collisions play a very important role on the road to the Bose-Einstein Condensation (BEC) of the same or mixed atoms. Collisions of the alkali atoms in dilute gases
have been recently studied in many experiments at ultracold temperatures [1–3]. Due to
the fast progress in the cooling and trapping of atoms in the last ten years, interest in the
detailed knowledge of the two-particle interactions has continued, as mentioned in Refs.
[4–15].
The character of the interaction between atoms of low-temperatures is determined by
the sign (±) and magnitude of the scattering length a: for bosonic atoms with a > 1, the
Bose condensate is stable, for a < 1 it is unstable [16]. Accurate calculations of the s-wave
scattering length and effective range for diatomic potentials are important, since the elastic
collisions at ultracold temperatures are dominated by s-wave scattering. In recent years,
these quantities have been calculated by several theoretical methods [17–21]. Collision
processes at near-zero temperatures (∼ µK) are sensitive to the details of the interaction
potentials between the colliding systems.
Previous theoretical and experimental calculations of the scattering length for the
potassium interaction have appeared in the literature [11–14]. Theoretical analysis enables
us to extract the shape of the collisional wave function near the outer turning point of
the diatom and hence determines the zero-energy scattering length, crucial for the understanding of the stability and shape of the Bose condensate of a weakly interacting gas of
alkali atoms [15]. In our work, the scattering properties including the scattering length,
effective
range, phase shift, and cross section for the collisions of 39 K+39 K in the state
P
+
3
u have been calculated, using the semiclassical method (WKB) [22] and the Numerov
method [23] at ultracold temperatures. The interaction of potassium atoms is described by
http://PSROC.phys.ntu.edu.tw/cjp
337
c 2005 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
SCATTERING OF ULTRACOLD POTASSIUM ATOMS . . .
338
VOL. 43
short and long range potentials, which include the cutoff functions introduced to truncate
the unphysical short-range contribution of the polarization potential near the origin [24].
Cutoff functions depending on a cutoff radius are adjusted by comparing the expressed potential form with the curve calculated, using perturbation facilitated optical-optical double
resonance Resolved Fluorescence spectroscopy, by L. Li [25]. Also, we asserted that if the
phase of two potentials in the interaction of potassium atoms are same, their scattering
properties actually gives the same results. By using this technique, a second potential, a
typical Lennard-Jones potential, has been determined for the scattering properties of potassium atoms in the triplet state. Then, for these potentials, the calculated results are also
compared with the experimental [1, 2, 11, 13, 14] and other theoretical [12] results.
In Sec. II, we discuss the computational process. The interaction potentials, which
play a crucial role in cooling by evaporating and also determine many properties of the
condensate are considered in Sec. III. The scattering length, effective range, and phase
shifts-cross section at ultracold temperatures are computed in Sec. IV and V. We end with
a brief conclusion.
II. PROCESSES
The scattering length is defined from the asymptotic behavior of the solution of the
radial Schrödinger equation at zero energy:
2
l(l + 1 )
d
2
− 2 µV (r ) + k +
yl (r) = 0 .
(1)
r2
dr 2
The asymptotic behavior of the wave function is given by
y(r) = sr + s0
(2)
as r → ∞, where s and s0 are constants. The scattering length is given by
a=−
s
.
s0
(3)
Here, the coefficients are obtained by using the WKB method from semiclassical behavior,
and with the help of the exact solution at zero-energy. Eq. (3) is transformed to the form
[26]:
a = ā {1 − tan{ϕ(k = 0) − π/8}} ,
(4)
in atomic units, where ā is the “mean” or “typical” scattering length determined by the
√
asymptotic behavior of the potential through the parameter γ = 2µκ (µ is the reduced
mass, and κ = c6 is the Van der Waals constant) for atom-atom interactions:
ā =
p
2γ
Γ(3/4)
.
Γ(1/4)
(5)
VOL. 43
M. KEMAL ÖZTÜRK AND SÜLEYMAN ÖZÇELIK
339
The scattering length also depends on a semiclassical phase ϕ calculated at zero energy
from the classical turning point r0 , where y(r0 ) = 0, to infinity:
ϕ=
Z∞ p
r0
−2µV (r)dr .
(6)
It also determines the total number of vibrational levels with zero orbital angular momentum [26], ns = {ϕ/π − (n − 1)/(2(n − 1))}+ 1, where { } is the integer part, and n = 6 for an
atom-atom interaction. Unlike γ and ā, the phase factor ϕ depends strongly on the actual
shape of the interatomic potential well. When the phase is large, ϕ >> 1, the scattering
length is very sensitive to the slightest changes of the potential [16].
The effective range re can be written in terms of the zero-energy solutions of the
partial-wave equation, if u0 (r) is the solution of the partial-wave equation at k = 0 for
zero-potential, normalized by
u0 =
sin(kr + δ0 )
sin δ0
(7)
as k → 0, and if u0 (r) is the normalized solution of the radial Eq. (1) for the s-partial wave
at zero energy (for non-zero potential) with the boundary condition u0 (r → ∞) = 0 as
u(r) ∼ u0 (r) at r → ∞, then the effective range can be written as [22]
Z∞ n
o
2
u20 (r) − u (r) dr .
re = 2
(8)
0
This integral converges provided that u(r) approaches u0 (r) rapidly enough as r → ∞.
This requires V(r) to decrease faster than r −5 . For a −c6 /r n potential type, this expression
is adjusted by Flambaum [16] using the WKB method and the exact solution of the radial
equation at zero-energy to form:
re =
√
Γ(3/4) 4γ
2γ Γ (1/4)
2γ
−2
+
.
3
Γ(3/4)
a
Γ(1/4) a2
√
(9)
The low-energy scattering is dominated by the contribution l = 0. At values of k
close to zero, the l = 0 phase shifts δ0 can be represented by a power series expansion in k
[27]. The expression that is known as the Bethe equation is given by
1 1
k cot δ0 = − + re k2 + o k3 .
a 2
(10)
340
SCATTERING OF ULTRACOLD POTASSIUM ATOMS . . .
VOL. 43
TABLE I: Potential coefficients for potassium atom in the triplet state.
C
α
0.0031 5.07641
1
2
103 C6
3.8132
β
1.1341
105 C8
4.0962
107 C10
5.2482
Ref.[28]
Ref.[24]
III. INTERACTION POTENTIALS
Alkali-metal atoms have unpaired electron spin s = 1/2 in the ground state. Therefore, the interaction between two alkali atoms in the ground state results in the formation
of a diatomic molecule, which is described by two terms corresponding to the total spin
sab = sa + sb of the system equal to 1 or 0. The state with sab = 0 and 1 corresponds
to the singlet state and triplet state, respectively. The probability of the spin exchange is
determined by both the dependence of the potential curves on the internuclear distance r
and the splitting between the singlet and triplet terms. The exchange interaction has the
asymptotic form [28]
1
vexc. (r) = (vu − vg ) = Cr α exp(−β r) ,
2
(11)
where α, β and C are constants. For potassium these values, taken from [28], are given in
first three columns of Table I. The long-range part of the interaction potential is given by
even
vlong-range = − Σ
m
Cm
fc (r) ,
rm
(12)
which is the sum of Van der Waals terms, multiplied by the relevant cutoff function fc (r)
to cancel the 1/r n divergence at small distances. The Wan der Waals parameters C6 , C8 ,
and C10 were taken from Ref. [24], and they are given in the last three columns of Table I,
respectively.
The cutoff function is analogous to that used for the H-H3 Σ+
u potential [29]
rc
2
fc (r) = ξ (r − rc ) + ξ(rc − r)e−( r −1) ,
(13)
where ξ(z) is the unit step function ξ(z) = 1(0) when z > (<) 0, and the only free parameter
for the potential sum is the cutoff radii rc , governing the decrease of the function fc (r) for
r < rc . The interacting potential of K+K in the K32 Σ+
u consisting of the above expressed
two terms, the sum of Eqs. (11) and (12), is given as
V (r) = V (r)exc. + V (r)long-range .
(14)
The magnitude of the cutoff parameters rc was adjusted by comparing this potential
form to the experimental curve in the triplet state calculated by L. Li et al. [25], using
VOL. 43
M. KEMAL ÖZTÜRK AND SÜLEYMAN ÖZÇELIK
341
Experimental values(x10 -3)
16
V(r)(hartree)
0.004
0.002
14
12
Regression curve
10
8
6
4
2
R2=0.9900
0
-2
-4 -2
0
2
4
6
8 10 12 14 16 18
Theoretical values(x10-3)
0.000
-0.002
Eq. (14)
Experimental
rc=18.75a.u.
-0.004
5
10
15
20
25
30
r(a.u.)
FIG. 1: The potential energy curve for the selected cutoff radius of 39 K−39 K in the triplet state, as
compared with the experimental potential curve versus the distance between the atoms in atomic
units. The inset shows the linear Regression curve and R2 value.
perturbation facilitated optical-optical double resonance resolved fluorescence spectroscopy
from r = 6.5 to 20 a.u.
The most relevant potential curve for the selected rc value is represented in Fig. 1,
together with the experimental calculation and regression curve (inset). As shown in this
figure, there is a good fit near the turning points of both potentials, but at the minimum
region of the curves, the obtained values have a weak fit with the experimental potential.
By taking into consideration this weakness, we have calculated five different cutoff radii.
We used another potential that is a Lennard-Jones type potential (LJ12,6 ) for determinations of the collision characters of potassium in the triplet state,
U (r) =
λ
η
−
.
r 12 r 6
(15)
This potential was adopted by Flambaum for the atom-atom interaction and the coefficients
were taken from Ref. [26]:
η = c6
3
√
5/6
.
0.42065463 2µc6
λ=
ϕ
(16)
Here, the U (r) potential depends wholly on ϕ and C6 , which are determined with
the V (r) potential. This means that if one considers two different interatomic potentials
characterized by the same asymptotic behavior (equal γ) and phases ϕ, such potentials will
produce very closely the scattering length, effective range, and s-wave function.
SCATTERING OF ULTRACOLD POTASSIUM ATOMS . . .
342
-0.03
0.10
Eq.(14)
LJ12,6
0.05
Eq. (14)
LJ12,6
-0.02
0.00
(normalized)
Triplet wave functions(a.u.)
VOL. 43
-0.01
-0.05
-0.10
0.00
-0.15
rc=18.75a.u.
rc=18.90a.u.
0.01
-0.20
0
20
40
60
80
Distance r (a.u.)
100
120
140
0
20
40
60
80
100
120
140
160
Distance r (a.u.)
FIG. 2: the normalized radial wave functions y(r) for the values selected of the cutoff radii as a
function of the effective range re .
IV. SCATTERING LENGTH AND EFFECTIVE RANGE
In order to determine the scattering length and the effective range, we must examine
the wave function at large range. The normalized radial wave functions at large positive
r values for the two potentials, U (r) and V (r), are presented in Fig. 2. As shown in this
figure, the s-wave functions change with increasing cutoff radius. At the same time, the
changing of the cutoff radius has increased the difference between the wave functions, and
the s-wave function for rc = 18.75 a.u. provides a good agreement between the s-wave
functions of the U (r) and V (r) potentials
Table II presents the values of the scattering lengths obtained by the different methods
and potentials. The zero-energy semiclassical phases ϕ presented in the second column of
Table II have been calculated from Eq. (6), and the values of the asymptotic parameter
and mean scattering length, where the semiclassical scattering length asc is calculated from
these parameters, have been found as γ = 1.7 × 104 a.u. and ā = 62.322 a.u., respectively.
The values of the semiclassical scattering length asc calculated from Eq. (4) using V (r) are
given in the third column of Table II as a function of the cutoff radius that is given in first
column of this table. In order to calculate numerically the scattering length, the s-wave
phase shift δ0 was obtained for five different cutoff radii; then we solved the radial equation,
using the Numerov algorithm at e = k2 /2µ in atomic units (µ = 35802 a.u.), and found δ0
from the asymptotic behavior of the wave function y(r) = sin(kr + δ0 ). The phases at small
momenta k are used to extract the scattering length numerically from a = limk→0 (tan δ0 /k).
The numerical scattering lengths obtained via this method for the two potentials are given
in the fourth and fifth column in Table II. These scattering lengths given in the third,
forth, and fifth column of Table II have an excellent agreement for the different cutoff radii.
Also the parameters of U (r) may be easily obtained from the expressions in Eq. (16) if ϕ
is known, and then the scattering length for U (r) are calculated numerically, which shows
that the calculations for both U (r) and V (r) give the same numerical results
VOL. 43
M. KEMAL ÖZTÜRK AND SÜLEYMAN ÖZÇELIK
343
TABLE II: Scattering lengths a (in atomic units) and semiclassical phase shifts for potassium atom
scattering depending on five different values of cutoff radii.
rc (a.u.)
18.75
18.80
18.85
18.90
18.95
a
ϕa
Scattering length(a.u.)
aisc (a.u.) aii (a.u.)
aiii (a.u.)
73.402
3.884
3.9411
3.964
72.166
93.510
93.540
93.360
71.039
-1489.0
-1483.4
-1490.8
69.900
35.954
35.718
36.099
68.913
102.800
102.700
102.260
−17 ± 25a − 60 < a < 15b − 1200 < a < −60d
81.1 ± 2.4c − 80 < a < −28e − 33 ± 5f
Calculated from Eq. (6).
Calculated using the semiclassical method for V (r).
ii
Calculated using the Numerov method for V (r).
iii
Calculated using the Numerov method for the LJ12,6 potential.
Experimental results taken from a Ref.[2] , b Ref. [1],
d
Ref.[14], e Ref. [13], f Ref. [11].
c
Numerical result taken from Ref. [12].
i
Scattering lengths, marked with the “a-f” letter in the bottom of Table II, which have
been derived from photoassociative spectroscopy [1, 2, 11, 13, 14] and numerical calculation
[12] are presented to compare with our results. The scattering length marked with the letter
“c” is calculated numerically as 81.1±2.4 a.u. [12] for a different interaction potential in the
K2 3 Σ +
u state. This result is in good agreement with our values in the Table II. As shown
in that table, the scattering lengths calculated for the collision of 39 K in related references
have a large interval (−1200,83.5). Our results are approximately in this range. On the
other hand, the variation ranges of the obtained scattering lengths are at the larger interval
(−1489,102.3).
Although our results show that the stability of large condensates requires repulsive
interactions (positive a), the hot atoms in the trap are removed from the trap via elastic
collision, whereas for attractive interactions (negative a) it is unstable, only a finite number
of atoms can be found in the condensate state in a trap. Therefore, it may be said that
the percentages of the variation of the positive and negative scattering length are 6.4% and
96.6%, respectively, from which we predict a stable condensation for bosonic potassium
isotope due to the 6.4% ratio for the positive scattering range.
The effective ranges for the two potentials are presented in Table III. The close
agreement between the calculations of re from Eqs. (8) and (10) confirms the accuracy of the
numerical integration of the partial-wave equation. The effective ranges have also changed
as a function of the semiclassical phase and cutoff radius. The effective range depends
wholly on the s-wave functions for the triplet state. The larger values of the effective
range in Table III for the three different calculations comes from the larger oscillation of
344
SCATTERING OF ULTRACOLD POTASSIUM ATOMS . . .
VOL. 43
TABLE III: Effective ranges re (in atomic units) for the potassium atom scattering depending on
five different values of the cutoff radii.
rc (a.u.)
18.80
18.85
18.90
18.95
19.00
Effective range(a.u.)
rei (a.u.) reii (a.u.)
reiii (a.u.)
98.157
194.285
609.668
92.804
175.410
2083a
98.155
194.317
608.977
92.809
175.403
98.175
184.291
609.396
92.810
175.446
i
Calculated semiclassically for V (r).
Calculated numerically for V (r).
iii
Calculated numerically for LJ12,6
a
Fermi Pseudopotential method Ref.[30].
ii
the s-wave function y0 (r). The obtained results have a large interval, (92.804, 609.668).
The effective range calculated from an expression derived by Gao for the energy-dependent
pseudopotential in the last row of Table III, is a higher value than the interval given above.
The size of the scattering lengths and effective ranges are closely related to the position
of the last vibrational bound states of the energy curves, as can be anticipated by inspection
of Eq. (3), and the number of vibrational levels ns (ϕ/π), which is consistent with Levinson’s
theorem, showing that as the binding energy of the highest level tends to zero, the scattering
length tends to ±infinity [31].
V. PHASE SHIFTS AND CROSS SECTION IN TRIPLET STATE
The scattering length describes the behavior of the atom scattering at low energy,
and phase shifts are also an important parameter for the scattering length. Furthermore,
at a low energy, E, of relative motion the phase-shift is Nbd π + δ, where Nbd is the number
of bound states. The phase shift can be calculated by fitting the solution y(r) to sin(kr)
and cos(kr) which are the asymptotic solutions of Eq. (1). Let Si and Ci denote sin(kri )
and cos(kri ), respectively. When we fit at ri and rj , we find a small shift [23, 32],
δ ≈ tan(δ) = −
Si y j − Sj y i
,
Ci yj − Cj yi
(17)
where ri,j = r0 + (i , j)h (h is the step-length ). Phase shifts calculated numerically for the
two potentials are given in Fig. 3. As the momenta go to zero, for l = 0 and five different
cutoff radii, the s-phase shifts for both potentials are almost indistinguishable, as shown in
Fig. 3. Also, in this case the low energy scattering is always dominated by the l = 0 partial
VOL. 43
M. KEMAL ÖZTÜRK AND SÜLEYMAN ÖZÇELIK
345
0.6
Eq. (14)
LJ12,6
0.4
δ0=ka
Phase shift
0.2
0.0
-0.2
-0.4
-0.6
0.000
0.001
0.002
0.003
0.004
0.005
0.006
k(a.u.)
FIG. 3: The calculated s-wave phase shifts for the potential in Eq. (14) with different cutoff radii
(dashed lines) and the LJ12,6 potential with equal phases ϕ (triangles). Solid circles are δ0 = -ak.
wave, the corresponding phase shift being given by
δ0 = −ka ,
(18)
which is given in Fig. 3; as seen in this figure there is a good agreement with our other
results calculated numerically.
The amplitude for scattering a particle in the direction specified by the polar angle θ
is f (θ) ± f (π − θ), and the differential cross section is
dσ
= |f (θ) ± f (π − θ)|2 ,
dΩ
(19)
the plus sign is applied to bosons and the minus sign to fermions. The physical content of
this equation is the sum or difference of the amplitude for one of the particles to be scattered
through an angle θ and the amplitude for the other particle to be scattered through an angle
π − θ. The total cross section is obtained by integrating the differential cross section over
all distinct final states. As a result, the total elastic cross sections can be defined by
t
σel.
∞
4π X
(2l + 1) sin2 δlt ,
= 2
k
(20)
l=0
in the triplet state, where l = 0, sin δ0 ∼
= δ0 in the limit of zero-energy and Eq. (20) is
t
2
transformed to σel. = 4πa using Eq. (10). If the scattering is purely s-wave, the total cross
section is given by
σtot = 8πa2
(21)
346
SCATTERING OF ULTRACOLD POTASSIUM ATOMS . . .
VOL. 43
TABLE IV: Phase shifts (δ` ) and cross sections (σ` ) for the collision 39 K+39 K calculated for U (r)
and V (r) using numerical (Numerov) and semicalssical (WKB) methods; and the total cross sections
σtot. .
Log10 E(a.u.)→
l→
δ` (×10−13 )
U (r)
(mod π )
V (r)
σ`
U (r)
(µm2 )
V (r)
U (r)N um.
σtot.
V (r)N um.
(µm2 ) V (r)W KB
-11.1950
2
4
20.680 177.2
20.680 177.2
0.485 0.527
0.485 0.527
3.157
3.126
3.150
-9.7518
2
4
2.394 7.217
2.394 7.217
0.990 0.919
0.990 0.919
-
for identical bosons [22].
We have found the phase shifts (δl ) and the cross sections (σ` ) for the collisions
39 K+39 K described in the two different potentials with l = 2, 4 angular momentum values
and two different energies using the Numerov method. They are listed in Table IV. This
table shows that the two potentials give very close phase shifts.
Total cross sections (σtot ) have been calculated using the Numerov method for two
different potentials at the five different cutoff radii, and their averages are given by comparing the seventh and eighth row of the second column in Table IV, respectively. Also, the
mean calculated total cross section for V (r) using the WKB method is given in the last row
of that table. There is an excellent agreement between the obtained total cross sections.
Fig. 4 shows the differential cross sections (DCS) for the triplet state of elastic colliding
potassium atoms. Due to the fact that integer hyperfine atomic quantum number are
bosons, only even partial waves should contribute to the DCS [33]. In Fig. 4, DCSs are
given with the dependence on temperature and cutoff radius for the two potentials and
dominant partial waves l = 2, 4. At low temperature, DCSs for the two potentials show
a good agreement. Also at low energy, the peaks of the DCSs are sufficiently deep, since
the dominant partial waves contribute excessively to the scattering of potassium. At large
temperatures, we observe the weak peaks of the DCS in Fig. 4. Also, the graphic curves
are more regular at low angular momentum.
VI. CONCLUSIONS
The elastic scattering properties for the collisions of the two potassium atoms in a
limited range of momentum (k < ā−1 = 0.141 a.u. for potassium atom) at low temperatures
are very sensitive to the details of the interaction potentials. Scattering properties such as
the scattering length, effective range, phase shift, and cross section have been computed
using semiclassical and numeric methods for LJ12−6 and V (r) with the dependence on cutoff
VOL. 43
M. KEMAL ÖZTÜRK AND SÜLEYMAN ÖZÇELIK
1e+8
1e+7
Log(dσ/dΩ)
1e+7
Eq. (14)
4µK
1e+8
LJ12,6
111µK
1e+6
1e+9
Eq. (14)
4µK
347
LJ12,6
111µK
444µK
444µK
1e+6
1e+5
1e+5
44mK
1e+4
1e+4
1e+3
1e+3
44mK
1e+2
1e+2
L=2
L=4
1e+1
1e+1
0
20
40
60
80
100
120
140
Scattering Angle(degrees)
160
180 1e+0
0
20
40
60
80
100
120
140
160
180
Scattering Angle(degrees)
FIG. 4: Differential cross-section for the scattering of potassium atom at different temperatures and
increasing values of angular momentum; they are only given for the selected cutoff radii rc =18.75
a.u.
radius adjusted by comparing with the experimental potential energy. We have shown that
there is a strong relation between the s-wave function and effective range. We have obtained
the average cross sections as ∼1.5 µm2 at low energy. The phase shifts, an intermission
parameter for scattering length, and effective range have a linear behavior at small momenta
k < ā−1 , and they have been determined by computing both numerically and semiclassically
for the two potentials.
References
∗
†
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
Electronic address: [email protected]
Electronic address: [email protected]
C. J. Williams, E. Tiesinga, et al., Phys. Rev. A 60, 4427 (1999).
John L. Bohn et al., Phys. Rev. A 59, 3360 (1999).
G. Modugno, Science 294, 1320 (2001).
M. H. Anderson et al., Science 269, 198 (1995).
D. G. Fried et al., Phys. Rev. Lett. 81, 3811 (1998).
A. Robert et al., Science 292, 461 (2001).
K. B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995).
S. L. Cornish et al., Phys Rev. Lett. 85, 1795 (2000).
C. C. Bradley et al., Phys. Rev. Lett. 75 , 1687 (1995).
D. Guéry-Odelin, J. Söding, P. Desbiolles, and J. Dalibard, Euro. Phys. Lett. 44, 25 (1998).
H. Wang, A. N. Nikolov et al., Phys. Rev. A62, 052704 (2000).
R. Côté, A. Dalgarno, H. Wang, and W. C. Stwalley, Phys. Rev. A 57, R4118 (1998).
B. DeMarco et al., Phys. Rev. Lett. 82, 4208 (1999). (1999).
H. M. J. M. Boesten, J. M. Vogels, et al., Phys. Rev. A 54, R3726 (1996).
Eite Tiesinga, Carl J. Williams, et al., J. Res. Natl. Inst. Stand. Technol. 101, 505 (1996).
V. V. Flambaum, G. F. Gribakin, and C. Harabati, Phys. Rev. A 59, 1998 (1999).
N. Balakrishnan, R. C. Forrey, and A. Dalgarno, Phys. Rev. Lett. 80, 3224 (1998).
348
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
SCATTERING OF ULTRACOLD POTASSIUM ATOMS . . .
VOL. 43
R. Côté, A. Dalgarno, and M. J. Jamieson, Phys. Rev. A 50, 399 (1994).
M. Marinescu, Phys. Rev. A 50, 3177 (1994).
Zbigniew Idziaszek et al., J. Phys. B: At. Mol Opt. Phys. 32, L205 (1999).
Bo Gao, Phys. Rev. A 58, 4222 (1998).
Charles J. Joachain, Quantum Collision Theory (North-Holland, Amsterdam, 1975).
R. Côté, M. J. Jamieson, Journal of Computational Physics 118, 388 (1995).
M. Marinescu, H. R. Sadeghpour, and A. Dalgarno, Phys. Rev. A 49, 982 (1994).
L. Li, A. M. Lyyra, W. T. Luh, and W. C. Stwalley, J. Chem. Phys. 93, 8452 (1990).
G. F. Gribakin and V. V. Flambaum, Phys. Rev. A 48, 546 (1993).
H. A. Bethe, Phys. Rev. A 26, 38 (1949).
B.M. Smirnov and M. I. Chibisov, Soviet Physics JETP 21, 624 (1965).
D. G. Friend and R. D.Etters, J. Low Temp. Phys. 39, 409 (1980).
D. Blume and Chris H. Greene, Phys. Rev. A 65, 043613 (2002).
R. Côté and A. Dalgarno, Phys. Rev. A 50, 4827 (1994).
J. J. Sakurai, Modern Quantum Mechanics (Benjamin Cummings Publishing, 1985).
P. Westpal, S. Koch, et al., Phys. Rev. A 56, 2784 (1997).