Scattering length and effective range of microscopic two-body potentials

Macêdo-Lima, Mathias; Madeira, Lucas

doi:10.1590/1806-9126-RBEF-2023-0079

Mathias Macêdo-Lima

Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brasil.

https://orcid.org/0009-0008-3276-6066

Lucas Madeira ^** Correspondence email address: madeira@ifsc.usp.br

Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brasil.

https://orcid.org/0000-0002-3219-6955

* Correspondence email address: madeira@ifsc.usp.br

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Figure 1
Scheme of a scattering process. We chose our coordinate system such that the incoming particle has its momentum in the

z

-direction,

k = k \hat{z}

. A plane wave

e^{i k \cdot r} = e^{i k z}

travels in space until it encounters a scattering potential. It produces an outgoing spherical wave

e^{i k r} / r

at large distances. The scattering angle

θ

defines the direction in which the momentum was transferred after the scattering process.

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Figure 2
Scattered radial solution

u_{0} (r)

and the free (

V = 0)

radial solution

g_{0} (r)

to highlight the phase shift. At low energies, an attractive potential (

V (r) < 0

) pulls the wave function, resulting in

δ_{0} (k) > 0

. The repulsive potential (

V (r) > 0

) pushes

u_{0} (r)

away, resulting in

δ_{0} (k) < 0

.

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Figure 3
Geometrical interpretation of the scattering length

a

based on three possibilities.

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Figure 4
Complex

k

-plane. A bound state is represented by a pole in the

S

-matrix at

k = i κ

, while the scattering continuum corresponds to a real value of

k > 0

.

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Figure 5
Behavior of the scattering length

a

as function of

\sqrt{2 ⁢ v_{0}}

for the spherical well, Eq. (). For

\sqrt{2 ⁢ v_{0}} = π / 2 + n ⁢ π

(

n = 0, 1, 2, \dots

),

a

diverges. This is related to the potential admitting an additional bound state at those points.

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Figure 6
Behavior of the effective range

r_{0}

as function of

\sqrt{2 ⁢ v_{0}}

for the spherical well, Eq. (). The vertical dashed lines denote where the scattering length diverges,

| a | \to \infty

, corresponding to an effective range equal to the range of the potential,

r_{0} / R = 1

.

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Figure 7
Spherical well, modified Pöschl-Teller, and Gaussian potentials in the unitary limit (

| a | \to \infty

). We make the axis dimensionless by rescaling the distance in effective range units, and the potential in

ℏ^{2} / (m_{r} ⁢ r_{0}^{2})

units, such that

\bar{V} \equiv m_{r} ⁢ r_{0}^{2} ⁢ V / ℏ^{2}

is dimensionless.

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Figure 8
Same as Fig., but for the Lennard-Jones potential. Differently from the other potentials, we observe a strong repulsive component near the origin.

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Figure 9
Reduced radial solutions,

u (r) = r A ⁢ (r)

, of Schrödinger’s equation with the proposed potentials for three different situations. The dashed curves correspond to the analytical solutions. Outside the range of the potential, large

r

, all solutions must be the same.

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Figure 10
Scattering length behavior as a function of the interaction strength for the spherical well, modified Pöschl-Teller, Gaussian, and Lennard-Jones potentials. The dashed curves in (a) are the analytical solutions for the spherical well, Eq. (), and for the modified Pöschl-Teller potential, Eq. ().

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Figure 11
Paths to calculate the integral (). We must choose the upper plane path

Γ_{R}^{+}

for

e^{+ i k^{'} | r - r^{'} |}

because

e^{+ i k^{'} | r - r^{'} |} \to 0

as

Im k^{'}

takes

+ \infty

values. Similarly, we choose the lower plane path

Γ_{R}^{-}

for

e^{- i k^{'} | r - r^{'} |}

because

e^{- i k^{'} | r - r^{'} |} \to 0

as

Im k^{'}

takes

- \infty

values. This makes the integration in

γ_{R}^{\pm}

go to

0

, and we are left with the path

C_{R}

, which lies in the axis

Re k^{'}

. That is, the closed path

Γ_{R}^{\pm}

is equal to the real integral ().

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Table 1.
Summary of the low-energy properties of two physical systems: the

^{4}

He dimer and the deuteron. The values of the scattering length

a

, effective range

r_{0}

, and bound-state energy

E

are taken from the indicated references. The zero-range and finite-range approximations,

E_{z ⁢ r}

and

E_{f ⁢ r}

, were calculated using Eqs. () and ().

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Table 2.
Scattering length and effective range values that we want to reproduce with four forms of two-body potentials.

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Table 3.
Parameters for the spherical well, modified Pöschl-Teller, and Gaussian potentials that reproduce the desired scattering lengths and effective range.

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Table 4.
Numerical results for the Lennard-Jones potential.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

(75)

(76)

(77)

(78)

(79)

(80)

(81)

(82)

(83)

(84)

(85)

(86)

(87)

(88)

(89)

(90)

(91)

(92)

(93)

(94)

(95)

(96)

(97)

(98)

(99)

(100)

(101)

(102)

(103)

(104)

(105)

(106)

(107)

(108)

(109)

(110)

(111)

(112)

(113)

(114)

(115)

(116)

(117)

(118)

(119)

(120)

(121)

(122)

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

(A10)

(A11)

(A12)

(A13)

(A14)

(A15)

(A16)

(A17)

(A18)

(A19)

(A20)

(A21)

(A22)

(A23)

(A24)

(A25)

(A26)

(A27)

(A28)

(A29)

(A30)

(A31)

(A32)

(A33)

(A34)

(A35)

(A36)

(A37)

(A38)

(A39)

(A40)

(A41)

(A42)

(A43)

(A44)

(A45)

(A46)

(A47)

(A48)

(A49)

(A50)

(A51)

System	$a$	$r_{0}$	$E$	Ref.	$E_{z ⁢ r}$	$E_{f ⁢ r}$
$^{4}$ He dimer	90.4 Å	8.0 Å	$-$ 1.62 mK	[²²[22] W. Cencek, M. Przybytek, J. Komasa, J.B. Mehl, B. Jeziorski and K. Szalewicz, The Journal of Chemical Physics 136, 22 (2012).]	$-$ 1.48 mK	$-$ 1.63 mK
Deuteron	5.4112 fm	1.7436 fm	$-$ 2.224 MeV	[²³[23] R.W. Hackenburg, Physical Review C 73, 044002 (2006).]	$-$ 1.416 MeV	$-$ 2.223 MeV

System	$a$ (fm)	$r_{0}$ (fm)
Neutron-neutron	$- 18.5$	$2.7$
Unitarity	$\pm \infty$	$1.0$
Deuteron	$5.4$	$1.7$

Potential	$v$	$μ$ (fm $^{- 1}$ )	$a$ (fm)	$r_{0}$ (fm)
Neutron-neutron
Well	$1.1096$	$0.3918$	$- 18.52$	$2.7$
mPT	$0.9071$	$0.7991$	$- 18.51$	$2.7$
Gaussian	$1.2121$	$0.5672$	$- 18.55$	$2.7$
Unitarity
Well	$1.2337$	$1.0000$	$\sim - 10^{5}$	$1.0$
mPT	$1.0000$	$2.0000$	$\sim 10^{9}$	$1.0$
Gaussian	$1.3420$	$1.4349$	$\sim - 10^{5}$	$1.0$
Deuteron
Well	$1.7575$	$0.5000$	$5.4$	$1.70$
mPT	$1.4388$	$0.8631$	$5.4$	$1.73$
Gaussian	$1.9102$	$0.6754$	$5.4$	$1.70$

System	$C_{12}$ (fm $^{10}$ )	$C_{6}$ (fm $^{4}$ )	$a$ (fm)	$r_{0}$ (fm)
Neutron-neutron	$3.08836698$	$9.86668911$	$- 18.5$	$2.71$
Unitarity	$0.00034068$	$0.26462461$	$\sim - 10^{5}$	$1.00$
Deuteron	$0.90485319$	$6.81472000$	$5.4$	$1.70$

e^{i k r \cos θ} \overset{large r}{\to} \sum_{l = 0}^{\infty} \frac{(2 l + 1)}{2 i k r} [e^{i k r} - {(- 1)}^{l} e^{- i k r}] P_{l} (\cos θ) .

ψ (r, θ) \overset{large r}{\to}

𝒩 \sum_{l = 0}^{\infty} \frac{(2 l + 1)}{i k r} [c_{l}^{(1)} e^{i k r} - {(- 1)}^{l} c_{l}^{(2)} e^{- i k r}] P_{l} (\cos θ) .

ψ (r, θ) \overset{large r}{\to} 𝒩 \sum_{l = 0}^{\infty} \frac{(2 l + 1)}{i k r} c_{l}^{(2)} [e^{2 i δ_{l}} e^{i k r} - {(- 1)}^{l} e^{- i k r}] P_{l} (\cos θ) .

ψ (r, θ) \overset{large r}{\to} 𝒩 {[\sum_{l = 0}^{\infty} \frac{(2 l + 1)}{2 i k r} (e^{i k r} - {(- 1)}^{l} e^{- i k r}) \times P_{l} (\cos θ)] + f (θ) \frac{e^{i k r}}{r}} .

β_{l} = {[r \frac{u_{l}^{'} (r)}{u_{l} (r)}]}_{r = R^{+}} = 1 + k R [\frac{\cos δ_{l} j_{l}^{'} (k R) - \sin δ_{l} n_{l}^{'} (k R)}{\cos δ_{l} j_{l} (k R) - \sin δ_{l} n_{l} (k R)}],

\cot δ_{l} (k) = \frac{k R n_{l}^{'} (k R) - (β_{l} - 1) n_{l} (k R)}{k R j_{l}^{'} (k R) - (β_{l} - 1) j_{l} (k R)} .

A_{0} (r) = \frac{u_{0} (r)}{r} = e^{i δ_{0}} (\cos δ_{0} j_{0} (k r) - \sin δ_{0} n_{0} (k r)) = e^{i δ_{0}} [\frac{1}{k r} \sin (k r + δ_{0})],

u_{k_{2}}^{''} (r) - U (r) u_{k_{2}} (r) + k_{2}^{2} u_{k_{2}} (r) = 0 .

u_{k_{2}}^{''} (r) - U (r) u_{k_{2}} (r) + k_{2}^{2} u_{k_{2}} (r) = 0 .

u_{k_{1}}^{''} (r) u_{k_{2}} (r) - u_{k_{1}} (r) u_{k_{2}}^{''} (r)

= \frac{d}{d r} [u_{k_{1}}^{'} (r) u_{k_{2}} (r) - u_{k_{2}}^{'} (r) u_{k_{1}} (r)] .

{[u_{k_{2}}^{'} (r) u_{k_{1}} (r) - u_{k_{1}}^{'} (r) u_{k_{2}} (r)]}_{0}^{R}

= (k_{2}^{2} - k_{1}^{2}) \int_{0}^{R} d r u_{k_{1}} (r) u_{k_{2}} (r) .

{[g_{k_{2}}^{'} (r) g_{k_{1}} (r) - g_{k_{1}}^{'} (r) g_{k_{2}} (r)]}_{0}^{R}

- {[u_{k_{2}}^{'} (r) u_{k_{1}} (r) - u_{k_{1}}^{'} (r) u_{k_{2}} (r)]}_{0}^{R}

= (k_{2}^{2} - k_{1}^{2}) \int_{0}^{R} d r [g_{k_{1}} (r) g_{k_{2}} (r) - u_{k_{1}} (r) u_{k_{2}} (r)] .

k_{2} \cot δ_{0} (k_{2}) - k_{1} \cot δ_{0} (k_{1})

= (k_{2}^{2} - k_{1}^{2}) \int_{0}^{R} d r [g_{k_{1}} (r) g_{k_{2}} (r) - u_{k_{1}} (r) u_{k_{2}} (r)] .

k \cot δ_{0} (k) = - \frac{1}{a} + k^{2} \int_{0}^{R} d r [g_{0} (r) g_{k} (r) - u_{0} (r) u_{k} (r)],

ψ (r, θ) \overset{large r}{\to} \frac{1}{{(2 π)}^{3 / 2}} \sum_{l = 0}^{\infty} \frac{(2 l + 1)}{2 i k} P_{l} (\cos θ)

\times [S_{l} (k) \frac{e^{i k r}}{r} - \frac{e^{- i (k r - l π)}}{r}] .

u (r) = {\begin{matrix} A \sin (\sqrt{k^{2} + k_{0}^{2}} r) & for r < R, \\ \cot δ_{0} (k) \sin (k r) + \cos (k r) & for r > R, \end{matrix}

\frac{\sqrt{k^{2} + k_{0}^{2}} \cos (\sqrt{k^{2} + k_{0}^{2}} R)}{\sin (\sqrt{k^{2} + k_{0}^{2}} R)}

= \frac{k \cot δ_{0} (k) \cos (k R) - k \sin (k R)}{\cot δ_{0} (k) \sin (k R) + \cos (k R)} .

u (r) = {\begin{matrix} A^{'} \sin (\sqrt{k_{0}^{2} - κ^{2}} r) & for r < R, \\ B^{'} e^{- κ r} & for r > R, \end{matrix}

\frac{\sqrt{k_{0}^{2} - κ^{2}} \cos (\sqrt{k_{0}^{2} - κ^{2}} R)}{\sin (\sqrt{k_{0}^{2} - κ^{2}} R)} = \frac{- κ e^{- κ R}}{e^{- κ R}} .

u (r) = {\begin{array}{l} \frac{\cot δ_{0} (k) \sin (k R) + \cos (k R)}{\sin (\sqrt{k^{2} + k_{0}^{2}} R)} \sin (\sqrt{k^{2} + k_{0}^{2}} r) & for r < R, \\ \cot δ_{0} (k) \sin (k r) + \cos (k r) & for r > R . \end{array}

u ⁢ (r) = {\begin{matrix} \frac{(1 - R / a)}{\sin (k_{0} ⁢ R)} ⁢ \sin (k_{0} ⁢ r) & for ⁢ r < R, \\ 1 - r / a & for ⁢ r > R . \end{matrix}

r_{0} = 2 ⁢ \int_{0}^{R} d r ⁢ [{(1 - \frac{r}{a})}^{2} - {(1 - \frac{R}{a})}^{2} ⁢ \frac{\sin^{2} (k_{0} ⁢ r)}{\sin^{2} (k_{0} ⁢ R)}] .

u (r + Δ r) = u (r) + (Δ r) u^{'} (r) + \frac{{(Δ r)}^{2}}{2} u^{''} (r)

+ \frac{{(Δ r)}^{3}}{6} u^{'''} (r) + \dots,

u (r - Δ r) = u (r) - (Δ r) u^{'} (r) + \frac{{(Δ r)}^{2}}{2} u^{''} (r)

- \frac{{(Δ r)}^{3}}{6} u^{'''} (r) + \dots

y_{i + 1} = \frac{1}{(1 + \frac{{(Δ x)}^{2}}{12} ξ_{i + 1})} {2 y_{i} (1 - \frac{5 {(Δ x)}^{2}}{12} ξ_{i})

- y_{i - 1} (1 + \frac{{(Δ x)}^{2}}{12} ξ_{i - 1})

+ \frac{{(Δ x)}^{2}}{12} (s_{i + 1} + 10 s_{i} + s_{i - 1})} + 𝒪 [{(Δ x)}^{6}],

\int_{x_{1}}^{x_{N}} f (x) d x = Δ x [\frac{1}{2} f_{1} + f_{2} + \dots + f_{N - 1} + \frac{1}{2} f_{N}]

+ 𝒪 (\frac{{(x_{N} - x_{1})}^{3} f^{''}}{N^{2}}) .

\int_{x_{1}}^{x_{N}} f (x) d x = Δ x [\frac{1}{3} f_{1} + \frac{4}{3} f_{2} + \frac{2}{3} f_{3} + \frac{4}{3} f_{4} + \dots

+ \frac{2}{3} f_{N - 2} + \frac{4}{3} f_{N - 1} + \frac{1}{3} f_{N}]

+ 𝒪 (\frac{{(x_{N} - x_{1})}^{5} f^{(4)}}{N^{4}}) .

r_{0} = 2 \int_{0}^{R} d r [1 - \frac{\tanh^{2} (μ_{PT} r)}{\tanh^{2} (μ_{PT} R)}]

= 2 [R - \frac{R}{\tanh^{2} (μ_{PT} R)} + \frac{1}{μ_{PT} \tanh (μ_{PT} R)}] .

U_{I} (t, - \infty) = lim_{t_{0} \to - \infty} U_{I} (t, t_{0}) = lim_{ϵ \to 0} ϵ \int_{- \infty}^{0} d t^{'} e^{ϵ t^{'}} U_{I} (t, t^{'}),

U_{I} (0, - \infty) = lim_{ϵ \to 0} ϵ \int_{- \infty}^{0} d t^{'} e^{ϵ t^{'}} e^{i H t^{'} / ℏ} e^{- i H_{0} t^{'} / ℏ} .

| ϕ {(t = 0)〉}_{I} = lim_{ϵ \to 0} ϵ \int_{- \infty}^{0} d t^{'} e^{ϵ t^{'}} e^{i (H - E_{i}) t^{'} / ℏ} | i 〉

= lim_{ϵ \to 0} \frac{i ϵ}{E_{i} - H + i ϵ} | i 〉 .

| ϕ {(t = 0)〉}_{I} = lim_{ϵ \to 0} \frac{i ϵ}{E_{i} - H_{0} + i ϵ} | i 〉

+ \frac{1}{E_{i} - H_{0} + i ϵ} V \frac{i ϵ}{E_{i} - H + i ϵ} | i 〉

= lim_{ϵ \to 0} \frac{i ϵ}{E_{i} - H_{0} + i ϵ} | i 〉

+ \frac{1}{E_{i} - H_{0} + i ϵ} V | ϕ {(t = 0)〉}_{I}

G_{+} (r, r^{'}) = \frac{ℏ^{2}}{2 m} \sum_{k^{'}, k^{''}} 〈 r | k^{'} 〉 ⟨ k^{'} | \frac{1}{E - H_{0} + i ϵ} | k^{''} ⟩ 〈 k^{''} | r^{'} 〉 .

G_{+} (r, r^{'}) = \frac{1}{L^{3}} \sum_{k^{'}} \frac{e^{i k^{'} \cdot (r - r^{'})}}{k^{2} - k^{', 2} + i ϵ},

G_{+} (r, r^{'}) = \frac{1}{{(2 π)}^{3}} \int d^{3} k^{'} \frac{e^{i k^{'} \cdot (r - r^{'})}}{k^{2} - k^{', 2} + i ϵ} .

G_{+} (r, r^{'}) = \frac{1}{{(2 π)}^{2}} \int_{0}^{\infty} d k^{'} k^{', 2} \int_{0}^{π} d θ \frac{e^{i k^{'} | r - r^{'} | \cos θ}}{k^{2} - k^{', 2} + i ϵ}

= \frac{1}{4 π^{2} | r - r^{'} |} \int_{- \infty}^{\infty} d k^{'} \frac{k^{'} \sin (k^{'} | r - r^{'} |)}{k^{2} - k^{', 2} + i ϵ},

G_{+} (r, r^{'}) = \frac{1}{8 i π^{2} | r - r^{'} |} \int_{- \infty}^{\infty} d k^{'} \frac{k^{'} (e^{- i k^{'} | r - r^{'} |} - e^{i k^{'} | r - r^{'} |})}{(k^{'} - k - i ϵ) (k^{'} + k + i ϵ)} .

Res {k^{'}; k + i ϵ}

= lim_{\begin{matrix} k^{'} \to k + i ϵ \\ ϵ \to 0 \end{matrix}} (k^{'} - k - i ϵ) \frac{k^{'} e^{i k^{'} | r - r^{'} |}}{(k^{'} - k - i ϵ) (k^{'} + k + i ϵ)}

= \frac{e^{i k | r - r^{'} |}}{2},

Res {k^{'}; - k - i ϵ}

= lim_{\begin{matrix} k^{'} \to - k - i ϵ \\ ϵ \to 0 \end{matrix}} (k^{'} + k + i ϵ) \frac{k^{'} e^{- i k^{'} | r - r^{'} |}}{(k^{'} - k - i ϵ) (k^{'} + k + i ϵ)}

= \frac{e^{i k | r - r^{'} |}}{2} .

lim_{R \to \infty} \int_{- R}^{R} d k^{'} \frac{k^{'} (e^{- i k^{'} | r - r^{'} |} - e^{i k^{'} | r - r^{'} |})}{(k^{'} - k - i ϵ) (k^{'} + k + i ϵ)} = 2 π i \times e^{i k | r - r^{'} |} .

〈 r | ψ 〉 = 〈 r | i 〉 - \frac{2 m}{ℏ^{2}} \int d^{3} r^{'} \frac{1}{4 π} \frac{e^{i k | r - r^{'} |}}{| r - r^{'} |} 〈 r^{'} | V | ψ 〉 .

〈 r | ψ 〉 = 〈 r | i 〉 - \frac{2 m}{ℏ^{2}} \int d^{3} r^{'} \frac{1}{4 π} \frac{e^{i k | r - r^{'} |}}{| r - r^{'} |} V (r^{'}) 〈 r^{'} | ψ 〉 .

lim_{ϵ \to 0} ϵ \int_{- \infty}^{0} d t^{'} e^{ϵ t^{'}} f (t^{'}) =

{[f (t^{'}) e^{ϵ t^{'}}]}_{- \infty}^{0} - lim_{ϵ \to 0} \int_{- \infty}^{0} d t^{'} \frac{d f}{d t^{'}} e^{ϵ t^{'}}

= f (0) - \int_{- \infty}^{0} d t^{'} \frac{d f}{d t^{'}} = f (- \infty) .

[1] * Correspondence email address: madeira@ifsc.usp.br