J. N. L. Connor · Physical meaning of Regge poles • ReJn is related to the radius, R, of the...

transcript

Mathematical methods for understanding

reactive angular scattering

School of ChemistryThe University of Manchester

Manchester M13 9PLEngland

J. N. L. Connor

Quantum Days in Bilbao IVBasque Center for Applied mathematics, University of the Basque Country,

SpainJuly 15 -16th, 2014

Outline• Nearside-Farside (NF) theory of scattering.• Local Angular Momentum (LAM) Analysis.• Resummation of Partial Wave Series (PWS)• Glories in the angular scattering:

- Uniform semiclassical theory.• “Hidden” Rainbows in the angular scattering:

- Uniform semiclassical theory• Complex Angular Momentum (CAM) Theory:

- Regge poles, parametrized S matrix- Uniform semiclassical theory

Differential Cross Section (DCS)(or angular distribution)

( ) ( ) 2, ,i f i ffσ θ θ= R.,. θθ ≡bn

( ) ( ) ( ), ,0

12 1 cos

i f i f Ji J

f J S Pk

θ θ∞

== +∑ . .,n b S S≡

max 1J >> Localization Principle

differential cross section

PWS scattering amplitude

2008Experimental information:

Journal of Chemical Physics, 82, 3045-3066 (1985)

θ/deg

Experimental information:

Nearside-Farside picture of scattering

Nearside waves

Farside waves

Travelling angularwaves

J θ− +

J θ+ +

FULLER Nearside-Farside Decomposition

( ) ( ) ( )N Ff f fθ θ θ= +

( ) ( ) ( ) ( )2 i12 iN

12 1 cos cos2J J Jk

Jf J S P Qπθ θ θ

⎡ ⎤= + ⎢ ⎥⎣ ⎦+∑

( ) ( ) ( ) ( )2i12iF

12 1 cos cos2J J Jk

Jf J S P Qπθ θ θ

⎡ ⎤= + ⎢ ⎥⎣ ⎦−∑

( ) ( ) ( ){ }2i 1 12 41

cos cos exp iJ JJ

P Q Jπθ θ θ π>>

± ⎡ ⎤∝ + −⎣ ⎦∓n.b., Travelling angularwaves

Nearside-Farside analysis of the angular distribution for the F + H2 reaction (2008 expt)

( ) ( ) ( ) ( ) ( ) ( ) ( )2F F

2N N FN

2, ,f f f fσ θ θ θ σ θ θ σ θ θ= + = =

Phys. Chem. Chem. Phys., 20110 45 90 135 1803.5

θR / deg

σ (θ

sr _ 1 )

PWS/F/r=3

PWS/N/r=3

000 300

(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes

bright dark

θ/deg

( ) ( ) ( ) ( ) ( ) ( ) ( )2F F

2N N FN

2, ,f f f fσ θ θ θ σ θ θ σ θ θ= + = =

0 45 90 135 180qR ê deg

logsHqRLêHfi2sr

nearnear

farfar

F+H2 FH+H

(0,0,0) → (3,3,0)E = 0.3872 eVSW pes

Phys. Chem. Chem. Phys., 2004θ/deg

Nearside-Farside analysis of the angular distribution for H + D2 -> HD + D

(0,0,0) → (3,0,0)E= 2.00 eVBKMP2 pes

0 45 90 135 180qR ê deg

logsHqRLêHfi2sr

H+D2 HD+DPWS

farfar

nearnear

θ/deg

( ) ( ) ( ) ( ) ( ) ( ) ( )2F F

2N N FN

2, ,f f f fσ θ θ θ σ θ θ σ θ θ= + = =

Phys. Chem. Chem. Phys., 2004

Experimental information:

NEARSIDE

FARSIDEDef

b / angstrom

Nearside-Farside analysis at the University of Bordeaux

And more recent works

European Physical JJournal, 2006

Nearside-Farside theory for Local Angular Momentum (LAM)

( ) ( )θ

argdLAM f=

Full LAM:

Nearside and Farside LAMs:

( ) ( )NN

d argLAM

( ) ( )FF

d argLAM

Also ( ) ( ) ( ) ( )N FLIP LAM , LIP , LIPkθ θ θ θ=

(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes

Thiele rationalinterpolation

( )LAM θ

θ/deg

darkbright

Nearside-Farside LAM analysis of the angular distribution

for the F + H2 reaction (2008 expt)

Handout from Prof. Michael Polanyi’s1st year physical chemistry lectures at the

University of Manchester. About 1946.

( )cos 2b R θ=

( )( )

cos 2kR

Partial wave series

( ) ( )0

1 cos2i J J

Jf a P

kθ θ

== ∑

( ) ( )1 cos2i

m mJ J

J mf a P

kθ θ

== ∑

( ) ( ) ( )( )

( ), ,

f i f i

m m m mJ J

J m mf a d

kθ θ

== ∑

Jacobi function of the second kind

( ) ( ), cosnP α β θ ( ) ( ), cosnQ α β θ

( ) ( ) ( ),, cos

nm md P α βθ θ∝ ( ) ( ) ( ),, cos

nm me Q α βθ θ∝

f f i f in J m m m m mα β= − = − = +

Jet Wimp

This paper requires knowledge of:

( )2 1 ;, ,F a b c x

( )1 ,; ; ; ,F x yα β β γ′ ( )4 ,; ; ,;F x yα β γ γ ′

Casoratian identity Generating functions

θ/deg

Resummation of partial wave series (R=1)

( ) ( ) ( )

( )( ) ( ) ( )

1 cos 2 12i

1 1 cos2i 1 cos

J J J JJ

f a P a J Sk

β θβ θ

where( ) ( ) 1 11

1 1 11

2 1 2 3J J JJJ Ja a a a

J Jβ β β− +

+= + +− +

Choose

( ) ( ) 01 1

β β =

== = ⇒ = − Raimondo Anni

(died 29/05/2003)

Resummation of partial wave series (R=2)

( ) ( ) ( )

( )( )( ) ( ) ( )

1 21 2

1 cos 2 12i

1 1 , cos2i 1 cos 1 cos

J J J JJ

f a P a J Sk

β β θβ θ β θ

where( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 2 1 1 2 11 12 1 1 11,

2 1 2 3J JJ JJ Ja a a a

J Jβ β β β β β β− +

+= + +− +

Choose( ) ( ) ( ) ( )2

21 21, 0 , 0J Ja aβ β β β= == =

Advantages of Nearside-Farside (NF) Theory

• It is exact (although approximate NF decompositions can be usedwhen convenient).

• The input is exact (or approximate) S matrix elements as calculated by standard (or non-standard) computer programs.

• It is easily incorporated into existing computer programs.

• Semiclassical techniques such as stationary phase or saddle point integration are not invoked, although the semiclassical picture isstill evident.

• Resummation can be applied to the partial series, followed by a NFdecomposition. This can improve the physical usefulness of theNF decomposition.

• NF and resummation can be incorporated into LAM-LIP analysis

The following techniques need to be used with caution

•Time delay for a single partial wave, e.g., J = 0.

• varying Jmax in PWS from Jmax = 0 to convergence:

• contribution from Jth partial wave to DCS by varying Jmax:

( ) ( )σ θ=

= ∑max

R m x0

( ) ( )σ θ σ θ− −R Rmax max; ; 1J J

Recent comments by Nobel Laureates

(2010)

Glory seen from an airplane

Glory, or the “Spectre of the Brocken”, often seen in the Harz mountains, Germany

Chengkui Xiahou Dong Hui Zhang

(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes

σ (θ

θR / deg

0 15 30 45

000 300

θ/deg

Glory analysis for the F + H2 reaction (2008 expt)

0 10 20 30 40qR ê deg

sHqRLêHfi2sr

CSACSA USAUSAPWS

F+H2 FH+H

Glory analysis for the F + H2 reaction (1985 expt)

(0,0,0) → (3,3,0)Etrans = 0.119 eVE = 0.3872 eVSW pesθ/deg

Uniform Semiclassical Approximation (USA):

( ) ( ) ( ) ( )[ ] ( )( )

( ) ( )[ ] ( )( )⎭⎬⎫−+

⎩⎨⎧ +=

221212

θζθσθσ

θζθσθσθζπθ

( ) ( ) ( )[ ]θβθβθζ −+ −= 21

Semiclassical glory analysis for the H + D2 reaction

(0,0,0) → (3,0,0)Etrans = 1.81 eVE = 2.00 eVBKMP2 pes

10 20 30 40qR ê deg

0.0000

0.0008

0.0016sHqRLêHfi2sr

0 2 4 6 8 10qR ê deg

sHqRLêHfi2sr

USAUSA

CSACSA

H+D2 HD+D(a)

USAUSA

CSACSA

θ/deg

I + HI -> IH + I

0,0,0->0,2,0 Etrans = 21.3 meV

Ashley Totenhofer

Black: PWSRed: USA

θ/deg θ/deg

Rainbows in Elastic Scattering

From the book by

R.B. Bernstein,“Chemical Dynamics viaMolecular Beam and Laser Techniques”1982

θ/deg

bright dark

Journal of Chemical Physics, 1981

Hg - (H2)

θ/deg

bright dark( )( ) ( ) ( ) ( )

θ =′•• • + ••• •

( ) ( ) ( ) ( ) ( ) ( ) ( )2F F

2N N FN

2, ,f f f fσ θ θ θ σ θ θ σ θ θ= + = =

Phys. Chem. Chem. Phys., 20110 45 90 135 1803.5

θR / deg

σ (θ

sr _ 1 )

PWS/F/r=3

PWS/N/r=3

000 300

(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes

bright dark

θ/deg

Semiclassical theory of glory and rainbow scattering

( ) ( ) ( )θθ cos~120

k PSJf ∑∞

• Semiclassical glory theory applies to Legendre series

• Applies to the chemical reaction ( ) ( ) C0,,AB0,,BCA +=→=+ fffiii mjvmjvusing exact quantum mechanics, or to many approximate theories, e.g., rotating linear model and its many extensions, CSH, IOS, DW, etc

R.,. θθ ≡bn

• Key quantity is the quantum deflection function:

( ) ( )JJS

~argd~ ≡Θ

. .,n b S S≡

Quantum deflection function for F + H2 (2008 expt)

(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes

( )( ) ( ) ( ) ( )

θ =′•• • + ••• •

σ (θ

_ 1 ) σ

sr _ 1 )

θR / deg

PWSCSA

0 15 30 45

000 300

45 90 135 180

+SC/N/PSASC/N/PSA

SC/F/uAiry SC/F/tAiry

SC/F/uAiry

SC/N/PSA +

(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes

F + H2 (2008 expt)

θ/deg

bright dark

PWSθR

SC/full

0 45 90 135 180

σ (θ

sr _ 1 )

θR / deg

r(0,0,0) → (3,3,0)Etrans = 0.119 eVE = 0.3872 eVSW pes

Black: PWSGreen: Semiclassical rainbow

theory

θ/deg

bright dark

F + H2 (1985 expt)

J. Phys. Chem., A, 2009

Journal of Chemical Physics, volume 103, pages 5979-5998 (1995)

Complex Angular Momentum Theory of Scattering

• CAM theory is completely general.

• CAM theory describes both resonance and non-resonance scattering.

• CAM theory correctly describes scattering into angular regions that are classically allowed or classically forbidden.“Is a Regge rainbow all shadow? Answer: No!”

• The standard definition of a resonance in CAM theory is a pole in the first quadrant of the CAM plane as characterized by its position and residue at a fixed value of the total energy, E.

Physical meaning of Regge poles

• ReJn is related to the radius, R, of the interaction zone by ReJn ≈ k R.

• A Regge state is a short- or long lived “quasi-molecule” formed from the colliding partners. It corresponds to a pair of decaying surface

waves that propagate around the interaction region.

• 1/(2 ImJn) determines the life-angle of the system.

• rn is a measure of the probability of exciting the nth Regge state.

, 0,1, 2,...n

r nJ J

• The surface waves decay like exp(- ImJn θ ).

Pade reconstruction of S(J) including Regge poles near the Re J axis

D. Sokolovski, E. Akhmatskaya and S. K. Sen, Comput. Phys. Commun. (2011)-a FORTRAN code.

( ) a polynomial ina (different) polynomial in

I have also used Thiele Rational Interpolation,implemented in Mathematica.

The QP decomposition

( )max

J J Jnn

aS Q iJ J

⎛ ⎞⎜ ⎟= +⎜ ⎟−⎝ ⎠

2J a J b J cφ = + +where

( ) ( ) ( ) ( ) ( )Q PR R Rf f fθ θ θ= +

Plot ( ) 2Rf θ ( ) ( )R

Q 2f θ ( ) ( )R

P 2f θ versus Rθ

DCSs for QmodPmod decomposition

θR/deg

F + H2 (1985 expt)

Linear plot

(0,0,0) → (3,3,0)Etrans = 0.119 eVE = 0.3872 eVSW pes

( ) ( )2 40

param1 exp exp polynomial up toJQ A A J i Jα⎡ ⎤ ⎡ ⎤= + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

S matrix parameterization

( )maxparam ram

pa expn

aS Q iJ J

⎛ ⎞⎜ ⎟= +⎜ ⎟−⎝ ⎠

Used to test the uniform CAM theory

F + H2 (1985 expt)

Parameterized S matrix

Numerical S matrix

Comparison of numerical and parameterized S matrix elements

modulus

F + H2 (1985 expt)

θR/deg

DCSParameterized PWS

Numerical PWS

PWS DCSs

Linear plot

Watson transformation (1918) for PWS

12Jλ = +

Contour for background integral

Complex J plane

( ) ( ) ( ) ( )direct erfR R R R

reba sck cidue~f f f fθ θ θ θ+ +

Asymptotically: refc

( ) ( ) ( )pol a kR

b ceR Rf f fθ θ θ= +

( ) ( )

( )( ) ( )

pole pole

i coscos n

n n nJ

π θπ

+= − −

⎡ ⎤+⎣ ⎦

( ) ( ) ( ) ( ) ( )( )N1 1R R 2

back i cosJf k S J Q J d Jθ θΓ

−= +∫

Uniform CAM Theory

Linear plot

θR/deg

F + H2 (1985 expt)

Uniform CAM

Uniform CAM and PWS DCSs

(0,0,0) → (3,0,0)E = 0.3112 eVFXZ pes

F + H2 (2008 expt) Uniform CAM and PWS DCSs

Log plot

θR/deg

Uniform CAM

Xiao Shan

For the 0,0,0 -> 3,0,0 transition of the F + H2 reaction at E =

0.3112 eV on the FXZ pes, the S matrix has a pole in the

complex angular momentum J plane, given by

J J≈

J0 = 10.43 + 0.53i Implies a life-angle = 1/( 2 ImJ0) = 0.95 rad = 54o

r0 = – 0.013 – 0.029i

0J J≈for

Important Result F + H2 (2008 expt)

H + CD4 → HD + CD3

G. Nyman, Goteborg, Sweden

S matrix elements calculated by the rotating line umbrellamethod

Semiclassical CAM

θ/deg

Thank you for listening!

UK Funding:

Overseas Research Students Awards Scheme

School of Chemistry, The University of Manchester

Leverhulme Emeritus Fellowship

J. N. L. Connor · Physical meaning of Regge poles • ReJn is related to the radius, R, of the...

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