Rydberg atoms Tobias Thiele References • T. Gallagher: Rydberg atoms Content • Part 1: Rydberg atoms • Part 2: A typical beam experiment Introduction – What is „Rydberg“? • Rydberg atoms are (any) atoms in state with high principal quantum number n. Introduction – What is „Rydberg“? • Rydberg atoms are (any) atoms in state with high principal quantum number n. • Rydberg atoms are (any) atoms with exaggerated properties Introduction – What is „Rydberg“? • Rydberg atoms are (any) atoms in state with high principal quantum number n. • Rydberg atoms are (any) atoms with exaggerated properties equivalent! Introduction – How was it found? • In 1885: Balmer series: – Visible absorption wavelengths of H: bn2 n2 4 – Other series discovered by Lyman, Brackett, Paschen, ... – Summarized by Johannes Rydberg: Introduction – How was it found? • In 1885: Balmer series: – Visible absorption wavelengths of H: bn2 n2 4 – Other series discovered by Lyman, Brackett, Paschen, ... – Summarized by Johannes Rydberg: Introduction – How was it found? • In 1885: Balmer series: – Visible absorption wavelengths of H: bn2 n2 4 – Other series discovered by Lyman, Brackett, Paschen, ... Ry – Summarized by Johannes Rydberg: ~ ~ n2 Introduction – Generalization • In 1885: Balmer series: – Visible absorption wavelengths of H: bn2 n2 4 – Other series discovered by Lyman, Brackett, Paschen, ... ~ ~ Ry – Quantum Defect was found for other atoms: 2 (n l ) Introduction – Rydberg formula? • Energy follows Rydberg formula: hRy E E 2 (n l ) Introduction – Rydberg formula? • Energy follows Rydberg formula: hRy 13.6 eV E E 2 (n l ) Introduction – Hydrogen? • Energy follows Rydberg formula: 13.6 eV 0 hRy hRy E E 2 2 (n l ) n 0 0 Energy Quantum Defect? • Energy follows Rydberg formula: hRy E E 2 (n l ) Quantum Defect 0 Energy Rydberg Atom Theory e • Rydberg Atom A • Almost like Hydrogen – Core with one positive charge – One electron • What is the difference? Rydberg Atom Theory e • Rydberg Atom A • Almost like Hydrogen – Core with one positive charge – One electron • What is the difference? Rydberg Atom Theory e • Rydberg Atom A • Almost like Hydrogen – Core with one positive charge – One electron • What is the difference? Rydberg Atom Theory e • Rydberg Atom A • Almost like Hydrogen – Core with one positive charge – One electron • What is the difference? – No difference in angular momentum states Radial parts-Interesting regions W 1 V (r) r 0 r Radial parts-Interesting regions W 1 1 V (r) V (r) V (r) r r core 0 r Radial parts-Interesting regions W r r0 Ion core 1 1 V (r) V (r) V (r) r r core 0 r Radial parts-Interesting regions W r r0 Ion core 1 1 V (r) V (r) V (r) r r core 0 r Interesting Region For Rydberg Atoms Radial parts-Interesting regions W r r0 Ion core 1 1 V (r) V (r) V (r) r r core 0 r H Radial parts-Interesting regions W r r0 Ion core 1 1 V (r) V (r) V (r) r r core 0 r H nH Radial parts-Interesting regions W r r0 Ion core 1 1 V (r) V (r) V (r) r r core 0 r δl H nH 1 (Helium) Energy Structure W 2 2(n l ) • l usually measured – Only large for low l (s,p,d,f) • He level structure • is big for s,p 1 (Helium) Energy Structure W 2 2(n l ) • l usually measured – Only large for low l (s,p,d,f) • He level structure • is big for s,p 1 (Helium) Energy Structure W 2 2(n l ) • l usually measured – Only large for low l (s,p,d,f) • He level structure • l is big for s,p 1 (Helium) Energy Structure W 2 2(n l ) • l usually measured – Only large for low l (s,p,d,f) • He level structure • l is big for s,p Excentric orbits penetrate into core. Large deviation from Coulomb. Large phase shift-> large quantum defect 1 (Helium) Energy Structure W 2 2(n l ) • l usually measured – Only large for low l (s,p,d,f) • He level structure • l is big for s,p dW 1 • 3 dn (n l ) 1 W Electric Dipole Moment 2 2(n l ) • Electron most of the time far away from core – Strong electric dipole: – Proportional to transition matrix element dW 1 3 dn (n l ) 1 W Electric Dipole Moment 2 2(n l ) • Electron most of the time far away from core – Strong electric dipole: d er – Proportional to transition matrix element dW 1 3 dn (n l ) 1 W Electric Dipole Moment 2 2(n l ) • Electron most of the time far away from core – Strong electric dipole: d er – Proportional to transition matrix element f d i e f r i e f r cos() i dW 1 3 dn (n l ) 1 W Electric Dipole Moment 2 2(n l ) • Electron most of the time far away from core – Strong electric dipole: d er – Proportional to transition matrix element f d i e f r i e f r cos() i • We find electric Dipole Moment 2 – f d i r l 1 cos()l n • dWCross Section:1 3 dn (n l ) 2 d a0n 1 W Electric Dipole Moment 2 2(n l ) • Electron most of the time far away from core – Strong electric dipole: d er – Proportional to transition matrix element f d i e f r i e f r cos() i • We find electric Dipole Moment 2 – f d i r l 1 cos()l n • dWCross Section:1 r 2 n4 3 dn (n l ) 2 4 d a0n n 1 W Stark Effect H H0 dF E2 2(n l ) • For non-Hydrogenic Atom (e.g. Helium) – „Exact“ solution by numeric diagonalization of f H i f H0 i f d i F in undisturbed (standard) basis ( n~ ,l,m) dW 1 3 dn (n l ) 2 4 d a0n n 1 W Stark Effect H H0 dF E2 2(n l ) • For non-Hydrogenic Atom (e.g. Helium) – „Exact“ solution by numeric diagonalization of f H i f H0 i f d i F in undisturbed (standard) basis ( n~ ,l,m) 1 W dW 1 2 2(n l ) 3 dn (n l ) 2 4 d a0n n 1 W Stark Effect H H0 dF E2 2(n l ) • For non-Hydrogenic Atom (e.g. Helium) – „Exact“ solution by numeric diagonalization of f H i f H0 i f d i F in undisturbed (standard) basis ( n~ ,l,m) 1 W dW 1 2 Numerov 2(n l ) 3 dn (n l ) 2 4 d a0n n Stark Map Hydrogen n=13 n=12 n=11 Stark Map Hydrogen n=13 Levels degenerate n=12 n=11 Stark Map Hydrogen n=13 k=-11 blue n=12 k=11 red n=11 Stark Map Hydrogen n=13 Levels degenerate n=12 Cross perfect Runge-Lenz vector conserved n=11 Stark Map Helium n=13 n=12 n=11 Stark Map Helium n=13 Levels not degenerate n=12 n=11 Stark Map Helium n=13 k=-11 blue n=12 k=11 red n=11 Stark Map Helium n=13 k=-11 blue n=12 s-type k=11 red n=11 Stark Map Helium n=13 Levels degenerate n=12 Do not cross! No coulomb-potential n=11 Stark Map Helium Inglis-Teller Limit n-5 n=13 n=12 n=11 Hydrogen Atom inH an electricH0 dF Field E • Rydberg Atoms very sensitive to electric fields – Solve: in parabolic coordinates • Energy-Field dependence: Perturbation-Theory 1 3 F 4 2 2 2 5 W F(n1 n2 )n n 17n 3(n1 n2 ) 9m 19 O(n ) 2n2 2 16 n1 n2 k 1 W Hydrogen Atom in an electric Field 2 2(n l ) • When do Rydberg atoms ionize? – No field applied – Electric Field applied – Classical ionization: dW– Valid only1 for • Non-H atoms3 if F is dn (n l ) Increased slowly 2 4 5 d a0n n FIT n 1 W Hydrogen Atom in an electric Field 2 2(n l ) • When do Rydberg atoms ionize? – No field applied – Electric Field applied – Classical ionization: dW– Valid only1 for • Non-H atoms3 if F is dn (n l ) Increased slowly 2 4 5 d a0n n FIT n 1 W Hydrogen Atom in an electric Field 2 2(n l ) • When do Rydberg atoms ionize? – No field applied – Electric Field applied 1 – Classical ionization: F 1 cl 16n4 V Fz r W 2 1 F cl 4 16n4 dW– Valid only1 for • Non-H atoms3 if F is dn (n l ) Increased slowly 2 4 5 F 1 n4 d a0n n FIT n cl 16 1 W Hydrogen Atom in an electric Field 2 2(n l ) • When do Rydberg atoms ionize? – No field applied – Electric Field applied 1 – Classical ionization: F 1 cl 16n4 V Fz r W 2 1 F cl 4 16n4 dW– Valid only1 for • Non-H atoms3 if F is dn (n l ) Increased slowly 2 4 5 F 1 n4 d a0n n FIT n cl 16 1 W Hydrogen Atom in an electric Field 2 2(n l ) • When do Rydberg atoms ionize? – No field applied – Electric Field applied 1 – Quasi-Classical ioniz.: F cl 16n4 Z m2 1 F V () 2 2 2 4 4 m0 W 2 dW F 1 3 4Z2 dn (n l ) 2 4 5 F 1 n4 d a0n n FIT n cl 16 1 W Hydrogen Atom in an electric Field 2 2(n l ) • When do Rydberg atoms ionize? – No field applied – Electric Field applied 1 – Quasi-Classical ioniz.: F cl 16n4 2 1 Z2 m 1 F F V () 2 r 9n4 2 4 4 2 1 dWm0 W 1 F 4 red 9n3 4Z2 dn (n l ) 2 4 5 F 1 n4 d a0n n FIT n cl 16 1 W Hydrogen Atom in an electric Field 2 2(n l ) • When do Rydberg atoms ionize? – No field applied – Electric Field applied 2 Fb 9n4 1 – Quasi-Classical ioniz.: F cl 16n4 2 1 Z2 m 1 F Fr V () 2 9n4 2 4 4 1 2 dW W 1 4 red F 9n 2 3 4Z2 dn (n l )4 blue 9n 2 4 5 F 1 n4 d a0n n FIT n cl 16 Hydrogen Atom in an electric Field Blue states Red states classic Inglis-Teller 1 W Lifetime 2 2(n l ) • From Fermis golden rule – Einstein A coefficient for two states n,l n,l 2 3 4e n,l,n,l max(l,l) 2 An,l,n,l nl r nl 3c3 2l 1 – Lifetime dW 1 3 dn (n l ) 2 4 5 F 1 n4 d a0n n FIT n cl 16 1 W Lifetime 2 2(n l ) • From Fermis golden rule – Einstein A coefficient for two states n,l n,l 2 3 4e max(l,l) 2 n,l,n,l An,l,n,l 3 n l r nl 3c 2l 1 1 – Lifetime n,l An,l,n,l n,ln,l dW 1 3 dn (n l ) 2 4 5 F 1 n4 d a0n n FIT n cl 16 1 W Lifetime 2 2(n l ) • From Fermis golden rule – Einstein A coefficient for two states n,l n,l 2 3 4e max(l,l) 2 n,l,n,l An,l,n,l 3 n l r nl 3c 2l 1 1 – Lifetime n,l An,l,n,l n,ln,l dW 1 3 dn (n l ) 2 4 5 F 1 n4 d a0n n FIT n cl 16 1 W Lifetime 2 2(n l ) • From Fermis golden rule – Einstein A coefficient for two states n,l n,l 2 3 4e max(l,l) 2 n,l,n,l An,l,n,l 3 n l r nl 3c 2l 1 1 3 For l0: n 2