1
Tunneling of atoms, nuclei and molecules Carlos Bertulani (Texas A&M University-Commerce, USA)
Collaborators A.B. Balantekin (Wisconsin) V. Flambaum (Sydney) M. Hussein (Sao Paulo) D. de Paula (Rio) V. Zelevinsky (Michigan)
Int. Workshop on Critical Stability, Santos, October 13, 2014
Resonant tunneling (point particle + double-hump barrier)
quasi-bound state
Well-known. Used in device technology. E.g. Resonant Diode Tunneling device.
V
GaAs
x
E
EF emitter collector EF
V
2
3 3
Resonant tunneling (with single barrier) (composite particle + single-step barrier)
quasi-bound state
Poorly known. Occurs in atomic, molecular and nuclear systems. E.g. fusion of loosely-bound nuclei.
(a)
(b)
4
Feshbach Resonances
5
€
H = −!2
2m∂2
∂x12 +
∂2
∂x22
$
% &
'
( ) +V(x1) +V(x2) +U x1 − x2( )
Step-barrier V + square-well U
€
V(x) =V0, − a /2 ≤ x ≤ a /20, otherwise
$ % &
€
U(x) =0, − d /2 ≤ x ≤ d /2∞, otherwise
% & '
barrier interaction between particles
U
V
Schroedinger equation
Step barrier
Square-well
CB, Flambaum, Zelevinsky, JPG 34, 1 (2007)
6
€
x =x1 + x22
"
# $
%
& ' , y = x1 − x2 +
d2
€
−!2
2m∂2
∂x2−!2
2m∂2
∂y2+U(y −
d2) +W(x,y)
$
% & &
'
( ) ) Ψ x,y( ) = EΨ x,y( )
(b) d/2 < a
change of variables
c.m. relative
barrier for c.m. motion
x
y
0
(a) d/2 > a
W=V0 W=V0
W= 0
x
y
0 W=2V0
7
Problem equivalent to a single particle (c.m. coordinate x) tunneling through 2 barriers.
Zakhariev, Sokolov, Ann. d. Phys. 14, 229 (1964) Saito, Kayanuma, J. Phys. Condens. Matter 6 (1994) 3759
x
y
0
8
Tunneling of Molecules ( )rVzZUzZU
mm+⎟⎠
⎞⎜⎝
⎛ ++⎟⎠
⎞⎜⎝
⎛ −++=224
22 pPH
€
p2
m+V r( )
"
# $
%
& ' φn r( ) = εnφn r( )
€
Ψ Z,r( ) = ψ Z( )φn r( )n=0
∞
∑
€
Unm Z( ) =4m!2
U Z +z2
"
# $
%
& ' +U Z − z
2"
# $
%
& '
)
* +
,
- . ∫ φn
* r( )φm r( )dr
€
d2
dZ2+ kn
2"
# $
%
& ' ψn Z( ) − Unm Z( )
m∑ ψm Z( ) = 0
€
kn2 =
4m!2
E − εn( )
€
ψnl Z( ) = eik nZδnl +12ikn
eik n Z−Z'( )Unm Z'( )ψml Z'( )dZ'−∞
∞
∫m=0
∞
∑
€
Rnl Z( ) =12ikn
eik nZ'Unm Z'( )ψml Z'( )dZ'−∞
∞
∫m=0
∞
∑
€
Tnl Z( ) = δnl +12ikn
e− ik nZ'Unm Z'( )ψml Z'( )dZ'−∞
∞
∫m=0
∞
∑
basis expansion
effective potential
effective equation for cm motion
general solution
= Z/a
effective potential
Um
i
U00
U11
U01
9
€
R l =knkln=0
∞
∑ Rnl2, Tl =
knkln=0
∞
∑ Tnl2
€
Pn→ l =knkl
Rnl2
+ Tnl2( )
€
U0a!c
=10, mU0
!2a = 6
Step barrier parameters
€
V0b!c
= 2, mV0!2
b = 4
Square-well particle-particle interaction
Reflection and transmission probabilities
Transition probability
Goodvin, Shegelski, PRA 72, 042713 (2005)
a
U0
b
V0
Transmission
Transition
k = ~
T1 P0 ! 1
10
Goodvin, Shegelski, PRA 72, 042713 (2005)
By moving to its ground state the excited molecule increases its chance of transmission.
Trans- mission
Transition
The opening of the higher channels decreases the probability of transmission.
11
€
k0 +2mU0a!2
tan k0 r2( ) = 0
For d + d ! 4He
With a and U0 simulating Coulomb barrier For large Z’s, A’s
! Many resonances possible, if <r2> large (loosely-bound)
Problem: strong interactions are too strong!
Tunneling of a (two-particle) nucleus
€
U0a→δ Z( )valid for
delta barrier T 0
12 Electron screening in fusion reactions
€
ΔE = E'−EAdiabatic model:
Rolfs, 1995
€
σlabfusion ~ σbare(E + ΔE)
~ exp πη(E) ΔEE
&
' ( )
* + σbare(E)
Electron screening enhancement
Small effects
Vacuum polarization
Balantekin, CB, Hussein, NPA 627 (1997)324
Z1Z2 = 1
• Thermal motion, lattice vibrations, beam energy spread
• Nuclear breakup channels (in weakly-bound nuclei)
• Dynamics of tunneling
all ≤ 1%
Not a solution! (we need ~ 100%) 13
Bang, PRC 53 (1996) R18 Langanke, PLB 369 (1996) 211
H + He
Golser, Semrad, PRL 14 (1991) 1831
Mainly charge-exchange
E’ = E – Sp.∆x
Wrong extrapolation of stopping power
€
Sp = −dEdx
Data has to be corrected for stopping power:
Very few data on stopping at ultra-low energies:
14
Simplest test
CB, de Paula, PRC 62, 045802 (2000) PLB 585, 35 (2004)
Charge exchange (pickup) Projectile slows down to carry electron with
p + H
P + D
e-
Elliptic coordinates
€
ξ =r1 + r2R
; η =r1 − r2R
; φ
Ψ = F(ξ)G(η)eimφ
€
ddξ
ξ2 −1( ) dFdξ$
% &
'
( ) +
R2ξ2
2E + 2Rξ − m2
ξ2 −1$
% &
'
( ) F ξ( ) = 0
ddη
1−η2( ) dGdη$
% &
'
( ) −
R2ξ2
2E + 2Rξ +
m2
η2 −1$
% &
'
( ) G η( ) = 0
change of variables
Two-center wfs (molecular orbitals):
15
Expansion basis: molecular orbitals for p+H
He+ H+H+
!,3210φδπσ
λ
letterCodeofValue
€
lzΦs = ±λΦs
16
(Hellman, Feynmann relation)
For Ep < 10 keV, only 1sσ and 2pσ 2-level problem - resonant exchange
Coupled-channels calculation
b (a.u.)
€
i! ddtam t( ) = Em t( )am t( ) − i! an t( ) m d
dtn
n∑
€
m ddtn =
mdVp /dt nEn t( ) −Em t( )
,
€
Pexch ≈12
+12cos
1!
E2p t( ) −E1σ t( )[ ]dt−∞
∞
∫' ( )
* + ,
17
H+ + He collisions (two-active electrons)
Slater-type orbitals
F . C = S . C . E
Two-center basis for two-electrons Hartree-Fock equations
t. d. coupled-channels equations
€
φ =Nrn−1 e−ξr Ylm θ,φ( )
€
Φi = c jiAφ i
A + c jiBφ i
B[ ]i=1
n
∑
€
Fµν =Hµν + Pλρλρ
∑ µν | λρ( ) − 12 µρ | λν( )'
( ) *
+ ,
€
Hµν = φµ
* 1( ) −12∇12 −
1r1AA
∑'
( )
*
+ , φν
* 1( ) dτ1∫∫ , Pλ ρ = 2 cλ icρ ii=1
occ
∑
€
µν | λρ( ) = φµ1( ) φ
ν1( ) 1r12
∫∫ φλ2( ) φ
ρ2( )dτ1dτ2 , Sµν = φ
µ1( ) φ
ν1( )∫ dτ1
18
Damping of resonant exchange H(1s) ⇔ He(1s2s)
Landau-Zener effect + dissipation
€
P = e−Γ Δt coll cos2H12 a2v
%
& ' (
) * 19
Stopping power at very low energies
p + H
P + D
e-
Threshold effect
He: 1s2 ! 1s2s: 19.8 eV
€
EP ≥µ2
4MPme
ΔE ≥ 8 keVCB, PLB 585, 35 (2004)
CB, de Paula, PRC 62, 045802 (2000)
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21
Baron Muenchhausen escaping from a swamp by pulling himself up by his own hair. G.A. Buerger (1786).
Virtual particles enhance tunneling
QFT: a "physical" particle consists of a "naked" particle "dressed" in a cloud of short-lived "virtual" particles.
22
€
Π r,r ';E( ) = gkk ,λ∑
2 r ˆ p ⋅ ekλ( )eik ⋅r n n ˆ p ⋅ ekλ*( )e− ik ⋅r r '
E −En −ωk − i0n∑
= M r,r ';E( ) +Γ r,r';E( )
€
HΨ r( ) + Π r,r';E( )Ψ r '( )d3r∫ = EΨ r( )
Quantum Muenchhausen
Non-relativistic reduction:
Flambaum , Zelevinsky, PRL 83, 3108 (1999)
€
M +Γ
€
M
Hagino, Balantekin PRC 66, 055801 (2002) Small effect for tunneling of stable nuclei. Effect for loosely-bound nuclei (and molecules) unexplored.
M = self-energy, mass renormalization, virtual photons Γ = decay width, real photons
Self-energy:
23
Neutron stars and neutron-rich nuclei
Neutron-rich nucleus (light with N/Z ~ 1.5)
Stable nucleus N/Z ~ 1.5
24
Loosely-bound nuclei
Tetraneutron?
10/26/2002
Marques et al, PRC 65, 044006 (2002)
halo
25
Tetraneutron as a dineutron–dineutron molecule CB, Zelevinsky, J. Phys. G 29 (2003) 2431
€
Ψ r1,r2,r3,r4( ) = A ψ R( )φa r1,r2( )φb r3,r4( ){ }
€
H = TR + Ta + Tb + viji< j∑ (rij)
€
P2
2mN
+U1 R( ) +U2 R( )"
# $
%
& ' ψ R( ) = Eψ R( )
R 1
2
3
4
- Effective potential repulsive - No margin for pocket or state - no tetraneutron in singlet, or triplet state! - Confirmed by Pieper, PRL 90 252501 (2003)
With realistic NN potentials vij
Antissimetrization (Pauli-principle)
Effective wave equation
26
n
n 9Li
11Li
Fusion of halo nuclei
€
H = −!2
2m∇12 −!2
2m∇22 +VA r1A( ) +VB r1B( ) +VA r2A( ) +VB r2B( )
€
Ψ± ≅N ΨA r1A( ) ± c±ΨB r1B( )[ ]
€
I = ΨA VB r1B( ) ΨA
€
L = ΨA VB r1B( ) ΨB
€
O = ΨA |ΨB
€
E(R) =ΨHΨΨ |Ψ
=S2n 1+ O2( ) + 2OJ+ 2I
1+ O2
Model Hamiltonian
Landau approximation
Covalent bond
Loosely bound nuclei are like Rydberg states in atoms
CB, Balantekin, PLB 314, 275 (1993)
27
€
9Li+11Li
€
V(R) = E(R) −S2n
€
σ Ecm( ) =πk2
2l+1( )Tll∑ Ecm( )
CB, Balantekin, PLB 314, 275 (1993)
Effective covalent potential Fusion cross section
More on fusion with weakly-bound nuclei :
Canto, Gomes, Donangelo, Hussein, Phys. Rep. 424, 1 (2005)
28
Data: Kasagi, PRL 79, 371 (1997)
Bremsstrahlung in α-decay
Interference between photons emitted within and outside the barrier tunneling time
γ
29
Time-dependent analysis
€
dE ω( ) =8πω2
3m2c3Z2e2 Ψ r,t( ) −i!∇Ψ r,t( )[ ]d3r∫
2
€
dPdE γ
=1
E γ
dE ω( )dE γ
C(out)
QM (out) QM (all space)
Eα » 0
Eα ~ 0
C(out)
QM (out)
QM (all space)
CB, de Paula, Zelevinsky, PRC 60, 031602(R) (1999)
Power emitted:
Solving for t.d. w.f. ! power emitted:
Not well understood. Not verified experimentally.
30
€
H =px2
2mx
+py2
2my
+12k x − y( )2 +
12qx2
Composite particles seeing different potentials
Normal modes: 0,:0 22 →+=→ −+ ωµ
ωx
x mqkm
xy m
qkm →=→ −+22 ,:0 ω
µω
q < 0
Equal masses and k >> q:
€
ω− =ω0 1+q8k
$
% &
'
( ) , ω0
2 =q2m
Tunneling probability:
€
P = exp2πE!ω−
%
& '
(
) * = exp −
2πE!ω0
%
& '
(
) * × exp
πEmω0
2!k%
& '
(
) *
enhancement
€
mx /my <<1: ω− ≈q
mx +my
1+q8kmy
mx
%
& '
(
) *
Zelevinsky, Flambaum, JPG 34, 355 (2005)
Only one particle sees barrier:
Finite size always enhance tunneling probability
31
Tunneling of Cooper pairs
€
H =px2 + py
2
2m+12k x − y( )2 +
12q x2 + y2( )
Normal modes:
€
ω+2 =
2k + qm
, ω−2 =
qm
Does not depend on k ! no finite size effect
€
E = E ∞( ) +! 2k /m
2−! 2k + q( ) /m
2
Energy transfer from internal to c.m. motion
Adiabatic approx. not valid for tunneling through a Josephson junction (more complicated)
Zelevinsky, Flambaum, JPG 34, 355 (2005)
Both particles see barrier:
32
€
H = −px2
2mx
+py2
2my
+V x − y( ) +U x( ) +U y( )
€
x,y→r,R
€
Ψ r,R( ) =ψ(R)φ(r,R)
€
α R( ) = φ∂φ∂R
€
ψ R( ) = u R( )exp − α R'( )dR'R∫&
' ( ) * +
ϕ normalized
€
u" R( ) +2M!2 E − ˜ U R( )[ ] u R( ) = 0
€
˜ U R( ) = ε R( ) −E0 +!2
2Mα2 R( ) + α' R( ) +β R( )[ ]
€
β R( ) = φ∂2φ∂R2
Composite particle fusion enhancement CB, Flambaum, Zelevinsky, JPG 34, 1 (2007)
Particles see different barriers:
change of variables Adiabatic approximation
33 33
€
E0 = −2.225 MeVr0 = 2 fm
CB, Flambaum, Zelevinsky, JPG 34, 1 (2007)
Features probably seen in experiments. But not disentangled from uncertainties in potentials, polarization effects, etc.
Deuteron tunneling through barrier step.
34
1D
2D
Laser
Interference
Optical Lattices
35
€
a(B) = aB=0 1−Δ
B −B0( )$
% & &
'
( ) )
B0
Δ = B-field width at the resonance
Feshbach Resonances
36
Probability distribution of strongly bound (Eb=10 kHz) rubidium molecules in optical lattice with size D = 2 µm. From top to bottom the time is t = 0, t = 10 ms, t = 25 ms and t = 100 ms.
Diffusion of Strongly Bound Molecules
T. Bailey, CB, E. Timmermans, PR A 85, 033627 (2012)
37
Escape probability of strongly bound rubidium molecules in optical lattice with D=2 µm, for increasing potential barrier heights. Barrier heights increase by 1.5 from the solid to dashed-dotted, from dashed-dotted to long-dashed, and from long-dashed to dotted curve.
Diffusion of Strongly Bound Molecules
38
Integrity probability of loosely bound rubidium molecules in an optical lattice D = 2 µm, and potential barrier height V0=5 kHz. The binding energies were parametrized in terms of the barrier height, with E4=V0/20, E3=V0/5, E2=V0/2, and E1=V0/1.2.
Diffusion of Loosely Bound Molecules
39
Blue histograms are the relative probability of finding a molecule in its ground state at a given position along the lattice. Red histograms give the relative probability of finding individual atoms after the dissociation.
Diffusion of Loosely Bound Molecules
t1 = 200 ms
t2 = 400 ms
40
Spreading position width of bound molecules, σM(t), shown by solid line and of dissociated atoms, σA(t), shown by dotted line. The dashed curve is a fit to the asymptotic time dependence σM(t) ~ t1/2. No clear asymptotic dependence is found for the dissociated atoms.
Diffusion of Loosely Bound Molecules
€
D = 0.156 !2
mbσ02
Compared to Einstein diffusion coefficient
€
D =kTb
€
T ~ !2
mσ02
41
Conclusions:
“Although a [large] number of theoretical works have studied tunneling phenomena in various situations, quantum tunneling of a composite particle, in which the particle itself has an internal structure, has yet to be clarified.” Saito and Kayanuma, J. Phys.: Condens. Matter 6 (1994) 3759 • 20 years after: still remains open to imagination and creativity.
