H in strong electric field

H in strong electric field

𝑝_!^" + 𝑝_#^"

2 − E u^" + v^" + 𝐹 𝑢^$ − 𝑣^$

2 Ψ = 2Ψ the semi-parabolic coordinates 𝑢 = 𝑟 + 𝑧, 𝑣 = 𝑟 − 𝑧

Hydrogen

Presuming 𝐵 along z-direction.

!^!

"# ⃗𝑟×𝐵 ^$ = _"&^%! B^$(x^$ + y^$)

With atomic units ℋ = ^'_$^! − ⁽₎ + ^*_"^! 𝑥^$ + 𝑦^$ + ^*_$ 𝐿₊ where, 𝛾 = _,^,

" with 𝐵_- = 2.35 ⋅ 10^.𝑇 . Effect negligible for n~1. But for n~50

Diamagnetism as 1^st order perturbation for 𝐿₊ = 0 𝑡𝑜 𝑚𝑎𝑘𝑒 𝑙𝑖𝑓𝑒 𝑠𝑖𝑚𝑝𝑙𝑒𝑟 Using Hellmann-Feynman theorem. ^/0_/1 = 𝜓 ^/ℋ_/1 𝜓

As _/*^/0_! ∼ 𝜓 𝑥^$ + 𝑦^$ 𝜓 the energy changes in a linear fashion ∝ 𝛾^$.

All levels upwards; states localized far from the Oz axis most affected With the observation that ℋ = ^'_$^! − ⁽₎ + ^*_"^! ⟨𝑥^$ + 𝑦^$⟩

Sph.(−_$3⁽_!) Cylindrical symmetry. ∼ 𝛾^$𝑛⁴

When both symmetries compete à the onset of Chaos.

Scaling possible ⃗𝑟^% = 𝜆 ⃗𝑟, ⃗𝑝^% = 𝜆^&!" ⃗𝑝, 𝐸^% → 𝜆^&'𝐸, 𝛾 → 𝜆^&#"𝛾 leads to:

ℋ = 𝜆 𝑝^%"

2 − 1

𝑟^% + 𝛾^"𝜆^&(

8 𝑥′^" + 𝑦′^"

With 𝜆^&(𝛾^" = 1,

E/ 𝜆 =E 𝛾&"/( = 𝜖 = ^*_"^" − ₊^'_$ + ^'_, 𝑥^" + 𝑦^"

𝜖

Poincare surface of section:

Basic atomic physics 2021

This classical transition to chaos manifests itself in the properties of quantum spectra

Multi-electron atoms: non-Coulombian potential (central)

• when?

some multi-electron atoms (e.g. alkali) have 1 electron

with average distance from the nucleus >> than the distances of other electrons,

F

has generally different valence el. trajectories.

with the shell unchanged

the electron "feels" the potential of el-stat. from the charge of the nucleus + Ze (Z = num. of protons) and from the charge of - (Z-1)e atomic shell

® the resulting effective potential from the shell has charge + e in the center, possible calculations as for a hydrogen atom

with potential V (r) = -(1/4pe ^{) (e / r)}

Possible situations:

- e + Ze -(Z-1) e

1) orbit not penetrating the shell

7/ 19

Penetration:

• small - circular orbits (large l)

• large - elongated orbits - elliptical (small l) (exception l= 0)

2) orbit penetrating

outside potential potential inside

constant is selected so as to match the potentials int. and ext. @ r =r

r V e

4 0

1

= pe 4 .

1

0

const r

V = Ze + pe

shell (valence electron penetrates the shell)

sodium

shell

|Y(r) |²

Additionally

u change with distance

® orbit precession

Basic atomic physics 2021

in Quantum Mech. no clas. orbits

→ description by Schrödinger eqn. using potential Energy

values from to

W (r) = qV = -eV

exact calculations difficult® model potentials ®calculate numerically simple, analytical model potential:

÷ø ç ö

èæ + -

r r

1 1 .

7 .

÷ø ç ö

èæ +

÷ = ø ç ö

èæ +

= -

r b r

C r

b r

r e

W 1 1

4 ) 1 (

2

pe0

r r Ze

W

2

4 0

) 1

( pe

= -

r r e

W

2

4 0

) 1

( pe

= -

V(r)

.2 .4

r

r - Ze r

-e

0

-100

-200

The choice of b allows one to

glue the potential in different

regions

9/ 19

Schrödinger Eqn. with model potential

potential V(r) still central - possible wave separation (as for hydrogen):

Y(r,q,j) = R (r) Y (q,j), substitute c(r) º r R (r)

0 )

) ( 1 1 (

2 2

) (

2 2

2 2

2 úû =

ê ù ë

é ÷ - +

ø ç ö

èæ + -

+ r

r l l r

b r

E C r

d r

d c µ µ c

!

!

0 )

) ( 1 (

) (

2 2

2 =

úûù êëé - - +

+ r

r l l r

A B r

d r

d c c

analogous to the hydrogen equation:

(

)

-

= _*

p l

E Rhc

n

(

)

En

= - +

+

= -

= -

n E Rhc

n

n^*= n - Dl -effective

main quantum num., Dl = l - l^* - quantum defect

Basic atomic physics 2021

Quantum defect D l = l - l

Coulomb potential non-Coulomb potential

(hydrogen atom) (alkaline atoms)

n2

E Rhc

n

= - _{2 2}

) (n l

Rhc n

E_n Rhc

D -

= -

= -

*

l^*(l^*+1) = (l - Dl) (l - Dl +1) º l (l + 1) - Bb

Dl² - 2 l Dl - Dl = - Bb, if b << 1, (D l)² << Dl then

2 0 1

1 1

2 a

b l

l l Bb

= +

» + D

2

21 0

, 1

÷÷ ø ö çç

è æ

- +

= -

l a n b E Rhc

l n

For potential C (1 + b / r) / r degeneracy in l removed

* it makes sense to label the energy levels. by a pair of numbers n, l,

* degeneracy of hydrogen levels due to l is accidental because it only occurs for Coulomb potential (related to the shape 1 / r, and not related to

more fundamental properties e.g. the spherical symmetry. of central potential.)

11/ 19 3s

3p

3d

1 2 3 4

l = 0

5s

5d 5f 5g

5p 4s

4p

4d 4f n =¥

-13.6 -3.4 -1.51 -0.85 0

E [eV] l = 0 ^{1 2} 3 4

n = 1 n = 2 n = 3 n = 4

hydrogen sodium

Sodium and hydrogen

(electrons from n = 1 and 2 form a closed hull)

Basic atomic physics 2021

Summary:

$ 3 important quantum num.

® full system characteristics º state of the system n, l, m_l, (we neglect m_s and nucleus)

- energy depends on n ® shell n2

E Rhc

n

= -

2

21 0

, 1

÷÷ø çç ö

è æ

- +

= -

l a n b

E_{n l} Rhc (n l)²

E Rhc

n - D

= -

for coulomb. pot. only – accidental degeneracy - non-coulomb pot. also depends on l - electron ang. momentum values

® designations of atomic states: set

(n, l) n= 1, 2, 3, ... l = s, p, d, f, ..., n-1

® subshell 1, 2, 3, 4, ...

- when there is no external fields, energies do not depend on m_l (degenerate)

- classic orbit ® probability distribution (orbital)

13/ 19

Quantum defects. in alkali:

l Dl

4 3 2 1 00

s 1

p

2 d

3 f Cs (55)

Rb (37) K (19) On (11)

Li (3) Basic atomic physics 2021

And the inner shells? _{log scale!}

15/ 19

Orders of magnitude:

the so-called atomic units:

- energy

m

c

- Compton length

l

= h / m

c

3.5x10

h n = m

c

typical values:

a

= (1/2 p ) l

/a = 137 l

/ 2 p

Rhc = a

m

c

/ 2

1 4

1 ²

0

÷ »

÷ ø ö çç

è

= æ

c e

pe

a

Fine structure constant

@ 13.6 eV

l

a

l

»1 /a »1 /a

electron speed:

u » a Zc << c

length

Basic atomic physics 2021

Orders of magnitude:

30 µsec - 3 msec 1-10 nsec

life time

10 meV

(cf. k_BT = 30 meV @ T = 300 K)

3 mm

» 10 eV

» 600 nm Energy Levels struc.:

- en. of binding el.

(ionization en. ) - freq. for

transition btwn. adjacent pos.

» 100 nm (0.1 µm)

» 0.1 nm (1 Å) (a₀= 0.5 Å) Radius of

el. orbits

n » 30 n » 1

-2

µ n E

-3

w

n

r Y µ Y !

w p

l

1 = Aµ n_-3

t

17/ 19

1. El-static: electrons - nucleus (M =¥) 2. El-static between electrons

3. magnetic spins and orbital angular mom. (result: $ el. spin and $ µ|| J) 4. magnetic between spins

5. Nuclear structure (Þ hyperfine and isotope ) a) magnetic and electric moments.

b) nuclear mass and size, charge distribution

Interactions in the atom

Start with considering el-stat. seperately (neglect 3.)

å å

÷ +

÷ ø ö çç

è

æ- D -

=

j

i i j

Z

i i

i r

K e r

e K Z

H m

2 1

2 2

2

!

4

1 º pe K

4

1 º pe K

Basic atomic physics 2021

I II III

) ... 2 ... 1

, 3 , 2 (

3, , 1

4 1 1

) 1 (

1

21 2

12

ïî ¥ ïí ì

= =

= -

= -

»

=

=

å

å

Z Z Z Z

Z r Z

r r

Z r II

III

i j i

i i

j

i ij

V º V

+ V

divide seperately

inter-atom.

between cental and non-central int.

å

å

+ -

=

j

i i j

i i r

K e r

e K Z

V

2 2

H = H

+ V = H

+ V

unsolvable when Z> 2, impossible perturbative calculation, because too big corrections from separate inter-electron interac.:

G

V r H

K e r

e K Z

H m

j

i i j

Z

i i

i

÷÷ + = +

ø çç ö

è

æ - D -

= å å

>

=

2 1

2 2

2

!

19/ 19

Central field approximation - cont

H = H

+V

å

å

û ê ù

ë

é- D +

=

i i

i i Vc ri h

H m ( )

2

2 0

! ¬ approx. independent

electrons in the central field

2 2

21 0

, 1 ^*

= -

÷÷ ø ö çç

è æ

- +

= -

i i

i l

n n

Rhc

l a n b

E Rhc

i i

* non-central correction:

å å

å

-

= -

=

> i c i

j

i i j

i i

c

nc V r

r K e

r e K Z

V V

V ( )

2 2

* self-consistent solution:

V

(r

)

spatial distribution

y

Z-1 electr. r=½y½²

* if V_c is large, and in comparison V_nc is a small correction - variational methods are effective

Start from a known solution e.g. this one:

Typically other numerical initial guesses

Basic atomic physics 2021

Central field approximation - energy levels

• for a given n,

E

,

l

,

• for small n, n uniquely defines energy;

all levels in n = 2 are below n = 3

F

å å ^{Þ =}

= h

E E

H

12 0

, 1

÷÷ ø ö çç

è æ

- +

= -

l a n b E Rhc

l n

G

3s

3p 3d

1 2 3 4

l = 0

5s

5d 5f 5g

5p 4s

4p 4d 4f

å å ^{Þ =}

= h

E E

H

12 0

, 1

÷÷ ø ö çç

è æ

- +

= -

l a n b E Rhc

l n

Typical properties of energies as in this formula

(it is however NOT valid except for some alkali atoms)

21/ 19

Sequence of filling the shells

energies 4s ȣ 3d, 5s ȣ 4d,

6s ȣ 5d, 4f

empirical rule:

energy ä if n + l ä

(Erwin Madelung) BUT!

exceptions when close to the energies of the subshells, e.g. ²⁴Cr and ²⁹Cu - almost

degenerate 4s and 3d)

E.

ä , if n ä

Basic atomic physics 2021

Summary the central field approx.

*

S

* eigenstates (wave fn.)

- sought in the form of a tensor product of single-electron functions:

å

=

=

º º

y

y

z g

b a z

g b a y

E E

E H

Z !

! , , ,

3 2 1

- definitions: shell = set of all electrons with a given n

subshell = set all electrons with a given (n, l) configuration = {(n_i, l_i)}

ground state = configuration with minimum energy

a a

h = a _, b _,!1-el. orthonorm states. a = n,l

23/ 19

* electrons = indistinguishable fermions

ÞIt is not possible to have a state in which 2 el. have the same quantum numbers

Summary the central field approx. - cont

0

, , ,

, , , ,

, ,

=

îí ì

= -

=

bb bb

y

b g b a

b g b b a

g b a

y

! ! ¬ identity

¬ antisymmetry

Pauli’s principle

* wave fn. satisfying ­ - Slater's determinant

( )

Z Z

Z Z

r r

z z

z

a a

a y

!

"

#

"

"

#

"

!

$

2 1

2 1

1 !

, = 1

* consequences of Pauli's principle:

•you can specify max. num. el. in the atom that have the same energy - filled shell

•max. num. el in subshell (n, l) = 2 (2l + 1)

•max. num. el. in the shell ^{1 2}

0

2 2 ) 1 4(

2 4

2 ) 1 2 (

2 n n n

n l

n l

n

l l

- = +

= +

=

å

å

•can determine degree of degeneracy = number of states correspond. to a config.=

•Periodic system of elements -determined by the order in which the shells are filled Fermions - particles with half spin

and antisymmetric wave function.

Basic atomic physics 2021