Product Operators. Fundamentals of MR Alec Ricciuti 3 March 2011

Transcription

1 Produc Operaors Fundamenals of MR Alec Ricciui 3 March 2011

2 Ouline Review of he classical vecor model Operaors Mahemaical definiion Quanum mechanics Densiy operaors Produc operaors Spin sysems Single spin-1/2 Chemical shifs, RF pulses Spin-1/2 pair Spin-spin couplings Spin-1/2 riple Applicaions COSY

3 The Classical Vecor Approach Spinning of a precessing paricle in a magneic field described by a orque Bloch equaions connec he model o observable magneizaion vecors Semiclassical exensions are necessary o accoun for coherence and polarizaion ransfer

4 Operaors An operaor  is a ransformaion of operand funcions o new values A marix represenaion A can be generaed for a given se of basis funcions ψ n A produc of operaors is given by marix muliplicaion A 1  1 2  1 1  2 2  2 m m Ê n m Ê pp  n p  n * dxψ Âψ m n

5 Operaors Every quanum observable A has a Hermiian operaor, Â A is diagonal in is own eigenbasis wih real eigenvalues Eigenfuncions associaed wih nondegenerae eigenvalues are orhogonal A Â Â Âf ( ) λ λ 2 0 λf( ) 0 0 λ 3

6 Densiy Operaors Wihou relaxaion, he densiy operaor σ describes he ime evoluion of a spin sysem i σ( ) [ H, σ] According o he Schrodinger represenaion, any ime inerval τ over which a ime-independen Hamilonian H can be defined gives rise o he soluion σ( τ) e ihτ / σ( ) e ihτ /

7 Densiy Operaors M Given he ime evoluion of he densiy operaor σ, he observable magneizaion can be evaluaed ( ) NγTr{ I σ( )} x kx k ( ) NγTr{ I σ( )} where N is nuclei per uni volume, Tr is he diagonal race, and k represens a specific isoope In he eigenbasis of he unperurbed Hamilonian, diagonal elemens σ ii () represen populaion of energy level i whereas off-diagonal elemens σ ij represen coherence of he (i,j) ransiion M y ky k

8 The Produc Operaor Formalism Sørensen e al. proposed he formalism in 1983 o handle weakly-coupled spin sysems Finds a middle ground beween he simple, ye inuiive classical vecor models and he fully rigorous, ye uninuiive densiy operaor heory of spin sysems Two oher groups independenly developed similar operaor formalisms Laer work explored differen basis ses and srongly-coupled sysems

9 Produc Operaors Choice of orhogonal base operaors B s ha linearly combine o form he densiy operaor σ() is arbirary, bu paricular choices provide convenience and inuiion Effecs chemical shifs, weak scalar coupling, and RF pulses on he evoluion are hus described by a sequence of ransformaions e iφb r iφbr Bse bs ( r, ) φ B B s σ ( ) b ( ) s 2 ( q1) N k1 s B s ( I kv where N is he oal number of spin-1/2 nuclei; k is he index of he nucleus; v = x, y or z; q = number of operaors in he produc; a sk = 1 for he q nuclei, 0 for he N-q remaining nuclei ) a sk

10 Single Spin-1/2 Eigensaes: α, β iφb iφbr e Bse q = 0 ½E, E = uniy operaor q = 1 I kx : longiudinal magneizaion I ky : in-phase x-magneizaion I kz : in-phase y-magneizaion Chemical shifs: ϕb r = (Ω k τ)i kz RF pulse abou v-axis: ϕb r = βi kv B s 2 ( q1) N k1 ( I s kv ) a sk r b ( r, φ ) B

11 Spin-1/2 Pair ( q1) k1 r Possible eigensaes: s s αα, αβ, βα, ββ q = 0, q = 1 q = 2 2I kx I lz, 2I ky I lz : x,y-magneizaion of spin k aniphase wih respec o spin l 2I kx I lx, 2I ky I ly, 2I kx I ly, 2I ky I lx : wo-spin coherence of spins k and l 2I kz I lz : longiudinal wo-spin order of spins k and l B e s 2 N ( I kv iφb r iφb B e b ( r, ) ) a sk φ B

12 Spin-1/2 Pair

13 Spin-1/2 Pair B e s 2 ( q1) N k1 ( I kv iφb r iφbr Bse bs ( r, ) ) a sk φ B Spin-spin (J) coupling: ϕb r = (πj kl τ)2i kz I lz

14 Spin-1/2 Triple

15 Spin-1/2 Triple

16 COSY Transform z-magneizaion ino ransverse magneizaion Spin-spin coupling hen develops wih aniphase magneizaion of one nucleus wih respec o he oher nuclei A repea of he firs pulse causes he sysem o develop evoluions in chemical shif resonances and spin-spin coupling Idenifies coupling parners and neworks of coupled spins Impossible o describe wih Bloch equaions, cumbersome wih densiy operaors

17 Furher Discussion Composie roaions Muliple quanum coherence Quadrupolar coupling (S=1) Magneizaion ransfer Srongly coupled sysems Designing pulse sequences