Electron spin resonance

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Denis Phelps
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Elcton sonanc 00 Rlatd topics Zman ffct, ngy quantum, quantum numb, sonanc, g-facto, Landé facto. Pincipl With lcton sonanc (ESR) spctoscopy compounds having unpaid lctons can b studid.

Th g-facto of a DPPH (Diphnylpikylhydazyl) and th halfwidth of th absoption lin a dtmind, using th ESR appaatus.

1 Elcton sonanc 00 Rlatd topics Zman ffct, ngy quantum, quantum numb, sonanc, g-facto, Landé facto. Pincipl With lcton sonanc (ESR) spctoscopy compounds having unpaid lctons can b studid. Th physical backgound of ESR is simila to that of nucla magntic sonanc (NMR), but with this tchniqu lcton s a xcitd instad of s of atomic nucli. Th g-facto of a DPPH (Diphnylpikylhydazyl) and th halfwidth of th absoption lin a dtmind, using th ESR appaatus. Equipmnt 1 ESR sonato with fild coils ESR pow supply Pow supply, univsal MHz digital stoag oscilloscop DMM, auto ang, NiC-Ni thmocoupl Scnd cabl, NC, l = 750 mm Adapt, NC-sockt/4 mm plug pai Conncting cod, l = 500 mm, blu Conncting cod, l = 500 mm, d Conncting cod, l = 500 mm, yllow Options: 1 Tslamt, digital Hall pob, tangnt., pot. Cap Fig. 1: Expimntal st-up fo dtmining chaactistic cuvs. Tasks This xpimnt dals with th invstigation of th magntic momntum of th lcton. 1. Dtmin th g-facto (Landé-facto) of th DPPH (Diphnylpicylhydazyl) spcimn. Dtmin th FWHM (Full Width at Half Maximum) of th absoption lin P51100 PHYWE Systm GmbH & Co. KG All ights svd 1

2 000 Elcton sonanc Fig. 1 Fig. Stup St up th xpimnt accoding to th following instuctions and pictus: - Connct th pots fo altnating voltag of th univsal pow supply with th two yllow connct- ing cods to th ESR pow supply as shown in Fig. 1 and Fig. - Connct th two Hlmholtz coils of th ESR sonato in sis by conncting a d pot to a blu pot with a blu conncting cod (Fig. 3) Fig. 3 - Us a blu conncting cod to connct th minus pol of th dict voltag pots to th digital mul- timt and slct th 10 A d.c. stting (Fig. 4 and Fig. 5) Fig. 4 Fig. 5 PHYWE Systm GmbH & Co. KG All ights svd P51100

Elcton sonanc 00 - Th 10 A pot of th digital

th ESR sonato accoding to th following pictus:

7 - Aft that, connct th sonato to th altnating

conncting cod (Fig. 8 and Fig. 9) Fig. 8 Fig.

th upp pot of th altnating voltag with a d

3 Elcton sonanc 00 - Th 10 A pot of th digital multimt is thn connctd with a d conncting cod to th ESR sonato accoding to th following pictus: Fig. 6 Fig. 7 - Aft that, connct th sonato to th altnating voltag of th univsal pow supply with a blu conncting cod (Fig. 8 and Fig. 9) Fig. 8 Fig. 9 - Th plus pol of th dict voltag is applid to th upp pot of th altnating voltag with a d conncting cod as shown in Fig. 10 Fig P51100 PHYWE Systm GmbH & Co. KG All ights svd 3

000 Elcton sonanc - Thn, th ESR pow supply is connctd to th ESR sonato with two scnd NC cabls. Connct th HF Ausgang -pot of th pow supply to th HF Eingang -pot of th sonato ( Fig.

4 000 Elcton sonanc - Thn, th ESR pow supply is connctd to th ESR sonato with two scnd NC cabls. Connct th HF Ausgang -pot of th pow supply to th HF Eingang -pot of th sonato ( Fig. 11 and Fig. 1) Fig. 11 Fig. 1 - Th ESR Signal Eingang -pot is connctd to th ESR Ausgang -pot (Fig. 13 and Fig. 14) Fig. 13 Fig PHYWE Systm GmbH & Co. KG All ights svd P51100

5 Elcton sonanc 00 - In od to display th ESR-signal on th oscilloscop, connct Phasnschib Ausgang to th X -pot of th oscilloscop (Fig. 15 and Fig. 17) and th Y -pot to th ESR-signal amplifi (Fig. 16 and Fig. 17) Fig. 15 Fig. 16 Fig You stup should now look lik th following pictu: Fig P51100 PHYWE Systm GmbH & Co. KG All ights svd 5

6 Elcton sonanc Pocdu A symmtically fd bidg cicuit (Fig. a) con- tains a vaiabl sisto R in on banch and a high-quality tund cicuit (sonato) in th oth. Th spcimn is locatd in th coil of th tund cicuit. Nomally, th bidg is balancd so that th complx impdanc of both banchs is th sam and consquntly th is no voltag btwn points a and b. If th xtnal magntic fild is now so adjustd thatt th sonanc absoption occus in th spcimn, th bidg bcoms unbalancd and th voltag st up btwn a and b ctifid and amplifid. If th magntic fild is modulatd with 50 Hz a. c. (voltag V), th sonanc point is passd though 100 tims a scond (Fig. 3), and th ab- soption signal can b displayd on an oscillo- scop, povidd th x-dflction is divn with th sam a.c. voltag in th coct phas. Fig. a: Masuing bidg of th ESR appaatus Dtminationn of th Landé-facto g - fo switching on th pow supply mak su, that th otating switch fo th dict volt- ag (lablld with V ) on th univsal pow supply is tund to 0 and th otating switch fo th cosponding cunt ( lablld with A ) is tund to th ight-hand stop (5 A) - Th altnating voltag should b st to V - It is impotant, that th dict voltag is sup- to display th ESR-signal on th oscilloscop (fo futh infomation about th functioning of th univsal pow supply plas ad its opat- ing instuctions) - Thn, switch on th univsal pow supply, th ESR pow supply and th oscilloscop imposd by an altnating voltag in od Fig. 3: Th magntic fild is compoundd fom a d.c. fild = and an altnating fild ~, so that Though l =, = is to b adjustd so thatt = =. = = + ~. 6 PHYWE Systm GmbH & Co. KG All ights svd P51100

7 Elcton sonanc 00 In th following, on has to do sval sttings on th ESR pow supply. Thfo, w will just f to th numbs givn in th following pictu (ths numbs cospond to th numbs that you can find in its opating instuctions): 8 9 Fig Push th idg balancing ( ückn Abglich ) button on th ESR pow supply (numb 8 ) - Th R otating switch of th ESR sonato should b in its middl position and th C otating switch should b bought to its lft-hand stop - At th oscilloscop, slct th X-Y-mod (Fig. 0) Fig. 0 - Slct th GND mod fo th X -channl and th d.c. mod fo th Y -channl - Th signal snsitivity fo both channls should b 1 V/cm (fo futh infomation about th sttings plas ad th opating instuctions of th oscilloscop) - You should s a singl point on th oscilloscop - Us th otating switchs Position to cnt th displayd point xactly in th middl of th coodinat systm - Aft that, push button 9 on th ESR pow supply (Fig. 19) and slct th "d.c." mod fo th "X"- channl at th oscilloscop - You should s a hoizontal lin on th oscilloscop - Incas th dict voltag on th univsal pow supply until th digital multimt shows about 1.3 A - Now, cafully tun C on th sonato to th ight until you s a signal appaing on th oscilloscop (it might b usful to incas th intnsitis of th X - and Y -channls to 0.5 V/cm o mo to gt a stong signal displayd) P51100 PHYWE Systm GmbH & Co. KG All ights svd 7

scal divisions fom th zo point to th positions, wh th signal and th x-axis int- sct (th mo symmtical th signal, th mo accuat this masumnt) - Th numb of scal divisions is dpndnt on th

points of int- sction in amps - Thfo, on has to switch off th altnating voltag and connct th sonato dictly to th di- ct voltag (i.

)) - As soon as a signal appas, th two lins a mad to coincid with th Phas otating switch of th ESR pow supply - Adjust th signal with C until you s a symmtical figu with a minimum (ty to

8 000 Elcton sonanc. Dtminationn of th FWHM - Thn mov th signal with th hlp of th Position otating switchs of th oscilloscop in such a way, that th x-axis gos xactly though th half of th signal s hight - Count th scal divisions fom th zo point to th positions, wh th signal and th x-axis int- sct (th mo symmtical th signal, th mo accuat this masumnt) - Th numb of scal divisions is dpndnt on th snsitivity that you hav slctd; thfo, n- su that you do not chang th signal snsitivity duing th st of th xpimnt - Not you sults - In od to dtmin th FWHM, on has to masu th distanc btwn th two points of int- sction in amps - Thfo, on has to switch off th altnating voltag and connct th sonato dictly to th di- ct voltag (i.. mov th d conncting cod that connctd a dict voltag pot with an alt- nating voltag pot and connct th blu conncting cod of th sonato to th now f dict volt- ag pot (Fig. )) - As soon as a signal appas, th two lins a mad to coincid with th Phas otating switch of th ESR pow supply - Adjust th signal with C until you s a symmtical figu with a minimum (ty to adjust th signal as symmtical as possibl) - y lowing th dict voltag on th univsal pow supply, th minimum of th figu should b bought xactly onto th y-axis of th oscilloscop (again us C to gt a symmtical figu) - You hav found a good sonanc signal whn you figu looks lik th following pictu: Fig. 1 - Th cunt that is now flowing and that is displayd on th digital multimt, is th sonanc cu- nt I - Not this valu Fig. 8 PHYWE Systm GmbH & Co. KG All ights svd P51100

Elcton sonanc 00 - Rmov th NC cabl fom th ESR pow supply that is connctd to th X -channl of th oscilloscop - Connct th nd of this NC cabl to th cosponding adapt as shown in Fig. 4 Fig.

4 - Pay attntion to th fact, that you do not chang th snsitivity of th X - and Y -channls duing this masumnt - Vay th dict voltag as long as a singl point appas on th oscilloscop - Whn th point

9 Elcton sonanc 00 - Rmov th NC cabl fom th ESR pow supply that is connctd to th X -channl of th oscilloscop - Connct th nd of this NC cabl to th cosponding adapt as shown in Fig. 4 Fig. 4 - Connct th adapt to th dict voltag pots of th univsal pow supply (Fig. 4) Fig. 4 - Pay attntion to th fact, that you do not chang th snsitivity of th X - and Y -channls duing this masumnt - Vay th dict voltag as long as a singl point appas on th oscilloscop - Whn th point appas, you can chang its position with th Position otating switchs of th oscilloscop until it lis on th x-axis - Mov th point by vaying th dict voltag on th univsal pow supply to on of th two points of intsction that you dtmind bfo - Not th cunt I that is now displayd on th digital multimt - Thn, mov th point to th oth point of intsction and not th valu of th cunt. Thoy and valuation In gnal, th phnomna in this xpimnt can b xplaind with th Zman ffct and th tansitions btwn Zman-lvls. Thfo, w will hav to shotly discuss th Zman-ffct itslf and also hav to talk about th basics of th atomic physics and quantum mchanics. Fist of all, th a two diffnt atomic magntic dipols: on th on hand th cicula cunts that psnt th lctons on thi volution aound th atomic nuclus, and on th oth hand th magntic momntums that a dpndnt on th lcton. P51100 PHYWE Systm GmbH & Co. KG All ights svd 9

10 000 Elcton sonanc Th magntic momntum of a cicula cunt is dfind as: obital = I A = π R ω = R T wh A = A R = π th aa aound which th lcton movs, R th adius of th obit, and π th unit vcto, which is ppndicula to th aa A; T = ω is th volution tim and ω th angula v- locity. This magntic momntum is popotional to th obital angula momntum L, which is dfindd as: ( 1) L = m v = m ω R () wh m th mass of an lcton. It follows: obital = m g obital L = γ obital g obital L (3) wh g obital th Landé-facto and γ obital = th so calld gyo magntic atio. As you can con- m clud fom th compaison of quations (1) and (), th Landé-facto fo a pu obital momntum is = 1. Thfo: g obita al obital = γ L. obital (4) In gnal, on can imagin th of an lcton as a slf-otatiopopotional to this angula momntum S, so thatt th following is with a angula momntum S. Th magntic momntum of th lcton is valid: = m g S = γ S (5) wh γ = g = m g γ. obital On aim of this xpimnt is th dtmination of th Landé-facto g fo th lcton. In addition, th atomic nucli also hav a, but sinc th mass of a nuclus is much lag than th masss of an lcton, th cosponding gyo magntic atio γ is vy small. That is th ason why on can disgad th magntic momntum of th atomic nucli. nuclus 10 PHYWE Systm GmbH & Co. KG All ights svd P51100

11 Elcton sonanc 00 y th laws of th quantum mchanics, th angula momntums a quantisd, i.. thy can only ach ctain valus. Thfo, th following valus a possibl fo th obital angula momntum: L z = mh ( m = l, l 1, l,..., 1 l, l ) (6) 34 wh L z is th z-componnt of th obital angula momntum, h = Js is Planck s quantum h of action with h, and l = 0,1,,... a quantum numb; m is calld th magntic quantum numb. π Th angula momntum of an lcton only has th magntic quantum numb 1. Thfo, th following is valid: S = ± 1 h. (7) z Fom this, it is asonabl, that th magntic momntums a also quantisd. Thy a xpssd in units of th oh magnton : 4 = h = Am. (8) m It follows: and = γ L = m z, obital obital z ( m l, l 1, l,..., 1 l, l z = γ S z = g γ obital S z = = ) (9) 1 ( ± ) g,. (10) If an lcton has an obital angula momntum L as wll as a angula momntum S, th sulting total angula momntum J will b: J = L + S J = L S, L S + 1, L S +,..., L + S. Th cosponding magntic momntum is thn: P PHYWE Systm GmbH & Co. KG All ights svd

12 000 Elcton sonanc = j obital = m = h + L L + g ( m g S ). S (11) If on knows th valus fo to th following quation: L and S, on can dictly calculat th thotically Landé-facto accoding g j = 1+ + J ( J ) +1 + S( S + 1) L( L +1). J J + 1 ( ) (1) g j Sinc th aim of this xpimnt is th dtmination of th magntic momntumm of th lcton, and thfo th Landé-facto g, on consids an unpaid lcton: In od to discuss th chaactisticss of an atom o molcul spctivly, on has to consid all its lctons, i.. th lctons in th shlls of th sval nucli as wll as th lctons, which caus a chmical bond. Th total obital angula momntum of th lctons in filld shlls is zo. Sinc th s of two lctons that caus a chmical bond a always anti-paalll to ach oth, th total angula momntum is zo, too. In this cas, th molcul is diamagntic. ut th a also substancs, which hav lctons that do not hav a cosponding - compnsating patn. Ths lctons a calld unpaid lctons. Substancs that hav such an lcton a paamagntic. Ou DPPH spcimn has xactly on unpaid lcton. Its obital magntic momntum is azd (i... L = 0 ) and that is th ason why its total magntic momntum is only givn by its. Thfo, its Landé-facto g = is naly th sam as th Landé-facto of a f lcton. Th thotically x- pctd valu fo th Landé-facto of th DPPH spcimn can b obtaind whn nsting th valus fo 1 L ( L = 0 ) and S ( S = ) in quation (1). On gts: g =. j Howv, th al valu fo g is b slightly high than bcaus of oth intactions, which dpndd on th magntic displacmnt. Zman ffct and magntic sonanc In od to undstand th pincipl and th functioning of this xpimnt, w now hav to talk about th Zman ffct and hav to alis what magntic sonanc is. At fist, th Zman ffct dscibs th splitting of an atomic spctal lin into sval lins whn apply- ing an xtnal magntic fild to an atom o molcul spctivly. This ffct is attibutd to th intac- tion btwn th xtnal magntic fild and th magntic momnts of th atom o molcul spctiv- ly. Fo th potntial ngy in th magntic fild is valid: E =. (13) 1 PHYWE Systm GmbH & Co. KG All ights svd P51100

13 Elcton sonanc 00 On diffs btwn two diffnt Zman ffcts: th nomal (without lcton : S = 0 ) and th anomalous (with lcton : S 0 ). Whn daling with th nomal Zman ffct, only th obital angula momntum is xistnt and thfo th xtnal magntic fild only intacts with th obital magntic momntum. Whn th xtnal magntic fild is applid, th ngy lvls within th atom a split into qually spacd ngy lvls: ΔE = Δobital = ΔLz m = Δm. (14) This splitting into sval ngy lvls is calld th Zman ffct. ut nomally, on has to consid th lcton as wll, as it is th cas in ou xpimnt. Thn, th xtnal magntic fild also intacts with th magntic momntum and th Zman intaction taks th fom ΔE = g Δm. (15) j j Th slction ul fo magntic tansitions is: Δm = ±1. So th distanc btwn th two Zman-lvls is thn: j ΔE = g. (16) j A tansition fom a low to an high ngy lvl is achivd by absobing a adiant quantum whos absolut valu is qual to th ngy diffnc btwn th two ngy lvls. This adiant ngy coms fom th applid lctomagntic wav of fquncy f within th ESR-sonato. Th absolut valu of th ngy thn sults in: E = h f. (17) This pocss is calld magntic sonanc. Fom this w gt th condition of sonanc that on has to adjust duing th xpimnt by vaying th magntic sonanc displacmnt (by vaying th dict voltag) and thn on can calculat th Landé-facto accoding to th following quation: g g j j = h f h f = (18) Am. Aft insting ths val- 34 wh h = Js, f 6 = Hz = 146 MHz, us on gts: = P PHYWE Systm GmbH & Co. KG All ights svd

14 000 Elcton sonanc g = T (19) Whn insting th valus and calculating th Landé-facto, on has to pay attntion to th units: g = = = = Js Hz 1 Am 1 Js s 1 Am Nm 1 Am N 1 Am = [TT ]. (0) Within th ESR-appaatus w = 41 and th adius R = m. Fo th magntic displacmnt is thn th magntic fild is poducd by a pai of Hlmholtz coils with th winding numb valid: 8 I w = 0 15 R I w = R (1) wh = 4 π Tm and I th cunt flowing though th coils. Fo gomtically asons, howv- A, th coils within th ESR-sonato a no idal Hlmholtz coils. If on masus th al magntic displacmnt btwn th two coils, on will gt th following quation: I ω = R () Aft insting th spctiv valus, on gts: = T A I. (3) Now, on can xpss th Landé-facto g with th hlp of th sonanc cunt I (3) in quation (0) as : by insting quation 14 PHYWE Systm GmbH & Co. KG All ights svd P51100

15 Elcton sonanc A g =. (4) I Evaluation 1. Dtmination of th Landé-facto g Th Landé-facto g can b xpssd with th hlp of th sonanc cunt I (s appndix):.565 A g =. I In ou sampl masumnt, w got th following sult ( I = 1. 4 A): g = Th litatu valu is givn as: g = Dtmination of th FWHM In ou sampl masumnt, w slctd a snsitivity fo th X -channl of 0.5 V/cm and fo th Y - channl of 0 mv/cm. In od to dtmin th full width at half maximum us you masumnt sults fo th cunts I 1 and I and calculat th diffnc. Not you sult blow: ΔI = I 1 I = Thn, calculat th sulting magntic displacmnt in th appndix): 3 T Δ = ΔI A Δ by insting Δ I following quation (fomula 3 This Δ cosponds to th FWHM that was sought. In ou sampl masumnt, w got th following sults: ΔI = 0.1 A 4 Δ = T. Th litatu valu is givn by: P PHYWE Systm GmbH & Co. KG All ights svd

16 000 Elcton sonanc 4 Δ =.8 10 T. This xpimntal sult is not vy accuat compad to th litatu valu. Th ason fo this is th gomtical stup of th ESR-sonato. Th two coils do not function as idal Hlmholtz coils. Fo a mo accuat sult on must us much lag coils in od to gt an almost idal pai of Hlmholtz coils, which would lad to an unasonabl lag incas in costs of th xpimnt, which st-up is al- A sonanc cunt of 1.45 A was masud, cosponding to a g-facto of.0, and a half-width val- u of ady vy good fo dtming th Landé-facto g.. Nvthlss, this will b topic duing a dsign. T. (Litatu valus fo DPPH: g =.0037, half-width T). Not Th analyzd spcimn DPPH (Diphnylpicylhydazyl) is a oganic, paamagntic matial with on stabl adical. Th magntic momnt of th molcul is dtmind only by th momnt of th va- lnc in th N -bidg. 16 PHYWE Systm GmbH & Co. KG All ights svd P51100

Hydrogen atom. Energy levels and wave functions Orbital momentum, electron spin and nuclear spin Fine and hyperfine interaction Hydrogen orbitals

Hydrogen atom. Energy levels and wave functions Orbital momentum, electron spin and nuclear spin Fine and hyperfine interaction Hydrogen orbitals Hydogn atom Engy lvls and wav functions Obital momntum, lcton spin and nucla spin Fin and hypfin intaction Hydogn obitals Hydogn atom A finmnt of th Rydbg constant: R ~ 109 737.3156841 cm -1 A hydogn mas