Chapte. A Bef Revew of Electon Dffacton Theoy 8 Chapte. A Bef Revew of Electon Dffacton Theoy The theoy of gas phase electon dffacton s hadly a new topc. t s well establshed fo decades and has been thooughly descbed n the lteatue (see, fo example, Ref. ). Consequently, t wll not be dealt wth n such depth hee. The pupose of ths chapte s to seve as a pme such that the subsequent descptons of the ultafast gas-phase electon dffacton expemental and theoetcal methodology may be undestood n the context of the much moe famla feld of conventonal gas-phase electon dffacton. The followng sectons wll attempt to potay ts mpotant aspects n a smple and physcally ntutve way.
Chapte. A Bef Revew of Electon Dffacton Theoy 9. Momentum tansfe n electon dffacton, a beam of electons tavels an axs z and ntesects wth a beam of molecules. Although most electons pass though the beam unpetubed, some ae scatteed by the atomc and molecula electostatc potentals. f the molecules n the molecula beam ae andomly oented, and the electon beam s udged to be collmated, the electons scatte n a adal symmetc patten about z. The esultng pattens may then be descbed by an ntensty measue at an angle θ fom the cente poston. Typcally, these scatteng ntensty featues ae expessed n tems of the momentum tansfe paamete, s, the magntude of the change n the momentum vectos between the ncdent and scatteed electon. Fo elastc scatteng, the s value (wth unts of Å - ) at each pont fom the cente poston s gven by s k k k sn ( θ ) 4π sn( θ ), λ (-) whee k and k ae the momentum vectos fo the ntal and scatteed electon (fo a plane wave of the fom Ae k z ), espectvely. λ s the de Bogle wavelength of the electon. θ, the scatteng angle between k and k, s defned by smple tgonomety: θ actan( d L), (-) whee d s the tance on the detecto fom the cente poston and L s the tance fom the beam nteacton egon to the cente poston on the detecto, known as camea length. Fo a gven knetc enegy, T, the elatvstcally coected de Bogle
Chapte. A Bef Revew of Electon Dffacton Theoy wavelength s h hc λ, (-3) p T ( T + m c ) whee h s Planck s constant, m s the est mass of the patcle (hee, an electon), and c s the speed of lght. The elatvstc velocty of a patcle at a gven knetc enegy s gven by mc v c. (-4) T + mc Wth T at 3 kev fo an electon expeencng a 3 kv acceleaton voltage, λ s.6979 Å and v s 9.84 7 m/s. Compason between the elatvstc and nonelatvstc wavelengths and veloctes of an electon ae shown n Fg. -. t s clea that elatvstc coectons, although not wthout nfluence n the enegetc egme of UED, become much moe ctcal at the hghe electon acceleaton potentals used n some electon mcoscopes.. Atomc scatteng n a molecule, nteactons between the valence electons of neaby atoms hold the postvely chaged nucle at the bondng tance. Electons n the beam of an electon dffacton expement, as chaged patcles, scatte fom the electostatc potentals of both the nucle and the electon tbuton (compae ths wth the much weake scatteng of x-ays whch only nteact wth the electon tbuton). solated atoms, themselves a oughly sphecal postve nucleus suounded by a oughly sphecal shell
Chapte. A Bef Revew of Electon Dffacton Theoy of electons, become petubed once mplcated wth othe atoms n valence electon bondng to fom a molecule. As a standad pactce n electon dffacton, mattes ae smplfed by assumng that an atom s actually ust a sphecal potental and that a molecule s smply a collecton of these unpetubed atoms at the appopate ntenuclea tances. Ths appoxmaton s known as the ndependent atomc model (AM) and t hol emakably well acoss gas-phase electon dffacton. Ths s n geat pat due to the fact that scatteng fom the nuclea potental s so much moe ntense than the scatteng fom the bondng electon densty; even n a molecule, atoms stll ae oughly sphecal. nteactons between the electon beam and the bondng electon densty ae elatvely weak causng the AM to suffe some nadequacy only at small scatteng angles (s < 5 Å - ). Usng the AM, electon scatteng fom a molecule s sepaated nto the puely atomc contbuton and the nteatomc molecula ntefeence contbuton. The total scatteng ntensty, (s), then can be wtten as ( s) ( s) ( s). (-5) A + M The atomc scatteng, A (s), s a sum of elastc and nelastc components fo each atom; S ( s) f + 4, (-6) a s A 4 whee f and S ae the elastc and the nelastc scatteng ampltudes, espectvely, fo the th nucleus, and a s the Boh adus. The f fo a patcula scatteng cente (atom) s deved usng a smplfed expesson fo a wave scatteng fom a sphecal potental (the fst Bon appoxmaton). The elastc scatteng event may be thought of as
Chapte. A Bef Revew of Electon Dffacton Theoy ϕ m e ϕ k z m snce the no enegy s tansfeed. 3 The f ae popotonal to s and have a lnea dependency on Z (see Fg. -). The nelastc S factos take nto account exctatons of the electons wthn each atom caused by the scatteng event and ae essentally a sum of coss tems of the fom k z ϕ m e ϕn to evaluate all possble tanstons. Both elastc and nelastc scatteng dop off shaply wth nceasng s, wth nelastc events mpotant at small scatteng angles (see Fg. -3). Accuate theoetcal values of scatteng factos ae calculated usng the method of patal waves and ae avalable n the lteatue. 4.3 Molecula scatteng The othe man contbuton to the total scatteed ntensty (Eq. -5) s the molecula tem, M (s) the ntefeences fomed as coheent electon waves scatte off pas of nucle n a molecule. The sphecal wave ntefeence ntoduces weak snusodal oscllatons nto the dffacton sgnal (see Fg. -3). These oscllatons ae the famla ngs vsble n a gas electon dffacton patten. Fo an sotopc spatal tbuton of molecules the dffacton patten s adally symmetc about the cente postons and M (s) s defned as M ( s ) sn ( s) f f cos( η η ), (-7) s, whee η s the phase facto fo the th nucleus, s the tance between th and th vb
Chapte. A Bef Revew of Electon Dffacton Theoy 3 nucle, and the backet denotes an aveage ove all vbatonal moton of the nucle. The η tems, added to the molecula scatteng fomula by the second Bon appoxmaton (see Ref. ) take nto account the non-zeo phase shft of a wave scatteed off two nucle of dffeent Z. The dffeence, η η, s essentally neglgble except when the Z ae vey dffeent (see Fg. -4). The ntegal aveage ove all vbatonal moton n Eq. (-7) can be evaluated usng the hamonc appoxmaton, by whch t becomes M ( ) ( ) sn( ) s a, ( s) f f cos η η exp l h s, (-8) s, e, whee e, and a, ae the equlbum and the effectve ntenuclea tances between th and th nucle, espectvely, and l h s the hamonc mean vbatonal ampltude. The dampng tem models the educton of sgnal when vbatons ae ntense. By expandng on the hamonc appoxmaton to model the anhamoncty of a Mose potental, the effectve tance can be expessed by a lh g, (-9) e h 3 + al, (-) g e whee a s the anhamoncty constant. g coespon to the tance between centes of gavty at tempeatue T. Although thee ae some tabulated values fo a, 5 t s often smply set at fo dect bon and at fo non-bonded tances. a s the electon dffacton opeatonal ntenuclea tance, whch must be conveted to e fo compason wth othe metho (tances epoted n subsequent chaptes ae e values).
Chapte. A Bef Revew of Electon Dffacton Theoy 4 To vsualze the molecula scatteng ntensty and emphasze the damped oscllatoy behavo, the modfed molecula scatteng functon, sm (s), s ceated: sm ( s) f f M ( s) s, (-) whee f and f ae atomc elastc scatteng factos of two selected atoms (the scalng atoms). The sm (s) s the standad fom n whch dffacton data ae pesented and on whch the theoetcal model s efned and molecula stuctues extacted. Chapte 4 wll descbe the pocess by whch the sm (s) and the molecula stuctues ae obtaned fom the dffacton patten..4 The adal tbuton cuve n the eal space of dffacton data, the sgnal s epesented by the sum of pobabltes of two nucle beng sepaated by tance. Plotted vesus, ths s known as the adal tbuton cuve, D (), often used by dffactonsts to show the molecula scatteng n a moe ntutve way. The conveson fom sm (s) n ecpocal space to the D () n eal space s made though sne tansfom: D ( ) sm ( s)sn( s) (-) Howeve, due to the fnte s ange of data detectos, the ntegal to nfnty s not possble and f (), a modfed adal tbuton, s needed. n ode to account fo ths cut-off of sgnal and to stem the spuous ngng that t ntoduces, a Gaussan wndow functon, exp ( ks ), s ncluded and the ntegal pefomed to s max.
Chapte. A Bef Revew of Electon Dffacton Theoy 5 f ( ) D( ) f ( ) s s max max sm ( s)sn( s)exp ( ks ) sm ( s)sn( s)exp( ks ) Δs, (-3) Typcally, k.5 Å n UED. n addton to the cut-off of the data at hgh s, thee s also a cut-off at low s whee the sgnal fom multple scatteng and nelastc effects s geatest the egon of beakdown of the AM and often the egon coveed by a beam stop n electon dffacton expements. Hee, a pece of theoetcally deved sm (s) s appended such that the ntegaton s contnuous fom. t s to be noted that fo a pa of nucle and, the aea unde ts coespondng peak n the f () s popotonal to nzz whee n s the multplcty of the tance n the molecule. Ths can be seen by combnng Eqs. (-8), (-), and (-3) and pefomng the ntegaton. f ( ),, e, e s f f, e,, ZZ Z Z ZZ e Z Z Z Z f f f f f f cos( Δη )exp( cos( Δη ) exp[ exp[ π 8(k + l ( ) a, exp, (k + l ) s ( l l s s ( l sn( s ) s + k)]sn( s ( a ) exp ) (k + l + k)]sn( s a e ) sn( s)exp ) sn( s) a ) a ( ks ) ) sn( s) (-4) whee the Gaussan functon contans the convoluton of both the k-dampng and l-
Chapte. A Bef Revew of Electon Dffacton Theoy 6 vbatonal ampltude. t s to be noted that although the above teatment s fo an sotopc tbuton, t has been shown elsewhee 6 that oentatonal effects n dffacton can be quanttatvely expessed..5 Refeences 3 4 5 6 Steeochemcal Applcatons of Gas-Phase Electon Dffacton. Pat A: The Electon Dffacton Technque, edted by. Hagtta and M. Hagtta (VCH, New Yok, 988).. Hagtta, n Steeochemcal Applcatons of Gas-Phase Electon Dffacton. Pat A: The Electon Dffacton Technque, edted by. Hagtta and M. Hagtta (VCH, New Yok, 988), pp.. R. A. Bonham and M. Fnk, Hgh Enegy Electon Scatteng. (Van Nostand Renhold Co., New Yok, 974); G. F. Dukaev, Collsons of Electons wth Atoms and Molecules. (Plenum, New Yok, 987). A. W. Ross, M. Fnk, R. Hldebandt,. Wang, and V. H. Smth., n ntenatonal Tables fo Cystallogaphy, edted by A.. C. Wlson and E. Pnce (Kluwe Academc Publshes, Boston, 999), Vol. C, p. 6. K. Kuchtsu, M. Nakata, and S. Yamamoto, n Steeochemcal Applcatons of Gas-Phase Electon Dffacton. Pat A. The Electon Dffacton Technque, edted by. Hagtta and M. Hagtta (VCH, New Yok, 988), p. 7.. S. Baskn and A. H. Zewal, ChemPhysChem 6, 6 (5);. C. Wllamson and A. H. Zewal,. Phys. Chem. 98 (), 766 (994); K. Hoshna, K. Yamanouch, T. Ohshma, Y. Ose, and H. Todokoo,. Chem. Phys. 8 (4), 6 (3); S. Ryu, R. M. Statt, K. K. Baeck, and P. M. Webe,. Phys. Chem. A 8, 89 (4); S. Ryu, R. M. Statt, and P. M. Webe,. Phys. Chem. A 7, 66 (3).