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o
« O <J> O te Ch ^-f-ELECTRON NUCLEAR DOUBLE RESONANCE INVESTIGATION OF THE MANGANESE-PROTON SUPERHYPERFINE INTERACTION IN
La2(Mg,Mn)3(N03)j2.24H20.
09 M
P R O E F S C H R I F T
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECH -NISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H.R. VAN NAUTA LEMKE, HOOGLERAAR IN DE AFDELING DER ELEKTROTECH-NIEK, VOOR EEN COMMISSIE UIT DE SENAAT
TE VERDEDIGEN OP WOENSDAG 30 JUNI 1971 TE 16.00 UUR DOOR RONALD DE BEER natuurkundig ingenieur geboren te Rijswijk (Z.H.)
/?
vx
Si^S
BIBLIOTHEEK TU Delft P 1942 2348 642958F. de Groot C A . van 't Hof G. de Jong H. Visser
hebben tijdens hun afstudeerperiode een bijdrage geleverd aan dit onderzoek. Hiervoor zeg ik hen hartelijk dank.
LIST OF ABBREVIATIONS 1 ENERGY CONVERSION TABLE I CHAPTER 1. INTRODUCTION
1.1. Aims of the investigation 2 1.2. Choice of the magnetic resonance technique 4
1.3. The crystal structure of La2Mg„(N0n) 2'24H20 7
CHAPTER 2. INSTRUMENTATION
2.1. Introduction II 2.2. The ligand ENDOR spectrometer 11
2.2.1. The microwave circuit 11 2.2.2. The RF equipment 13 2.2.3. The detection of the ENDOR signals 16
2.3. The cooling system 17 2.4. Preparation and mounting of the samples 20
CHAPTER 3. THEORY
3.1. The spin Hamiltonian of the manganese-proton system in
LMNtMn"^"*" 25 3.2. Calculation of the ENDOR frequencies. Evaluation of the
SHFI tensors 27 3.3. The ENDOR frequency as a function of the rotation of the
external magnetic field in the plane perpendicular to the
trigonal symmetry axis of LMN:Mn 37 3.4. Prediction of the angular dependence of the c j_ H_ ENDOR
spectra by means of the point dipole model 39 3.5. The effect of a small misorientation of the sample on the
c j_ H ENDOR frequencies 43
3.6. Splitting of the ENDOR lines due to the dipolar interaction
between the two protons within the same water molecule 44
CHAPTER 4. EXPERIMENTAL RESULTS
4.1. The EPR spectrum of M n in LMN at 20 K 48
4.2. The c // ^ ENDOR spectra 50 4.3. The angular dependence of the ö J_ ff ENDOR spectra 51
4.4. Evaluation of the SHFI tensors from the ENDOR data 55 4.4.1. The SHFI tensors belonging to the ligand protons
of site I 55 4.4.2. The SHFI tensors belonging to the ligand protons
CHAPTER 5. DISCUSSION
5.1. Interpretation of the anisotropic SHFI tensors 69 5.2. The influence of hydrogen bonding on the orientations of
the water molecules in LMN:Mn 78 5.3. The influence of the libration oscillations of the water
molecules in LMN:Mn on the anisotropic SHFI tensors 82
5.4. Interpretation of the isotropic SHFI 89
5.4.1. Introduction 89 5.4.2. Calculation of the isotropic SHFI 91
APPENDIX A 99 SAMENVATTING 103
LIST OF ABBREVIATIONS. AO CMN ENDOR EPR FM LCAO LMN LMN:Mn MO MSO IWR PDM RF SCF SHFI SHFS ++ ++ . : atomic orbital. : Ce2Mg3(N03)j2-24H20.
: electron nuclear double resonance. : electron paramagnetic resonance. : frequency modulation.
: linear combinations of atomic orbitals. : La2Mg2(N03)j2.24H20.
: LajUg^iW..) .2'2^]iJ^f contaminated with Mn ions.
: molecular orbital. : molecular spinorbital. : nuclear magnetic resonance. : point dipole model.
: radio frequency.
: self consistent field. : superhyperfine interaction. : superhyperfine structure.
ENERGY CONVERSION TABLE.
1 atomic unit '^ 2.1947 x lO^ cm~^ I kcal/mol 1 MHz 1 cm' - 1 2^ 3 . 4 9 8 4 X 102 cm"l 2i 3 . 3 3 5 6 X 1 0 - 5 cm-1 'V 1.9862 X 1 0 - 2 3 j o u l e
CHAPTER I. INTRODUCTION
Aims of the inoeatigation
This thesis will be concerived with the magnetic superhyper-fine interaction (SHFI) between the electronic spin of the para-magnetic transition ion. Mn and the nuclear spins of the protons within the complex [Mn(H20)g ] » incorporated in the crystalline hydrate La2Mg3(N03).j:2^i24H20 (LMN).
The salts formed from irongroup transition metals are pre-dominantly ionic. However the magnetic properties of transition metals are affected by the slight covalent character of the bonds with the surrounding ligands . Many of the experimental and
theoretical investigations concerning covalency in transition 2—8)
metal complexes, published in the past years , deal with
transition metals surrounded octahedrally by six negatively charged ligand ions with closed s and p shells such as, for example, F . Alternatively we are interested in the covalent bonding in octa-hedral transition metal complexes of which the ligands are water molecules. In our investigation the paramagnetic Mn acted as central ion. Since in a weak ligahd-field the manganese ion has
9)
five unpaired electrons in the outer 3d-shell i t s distribution
of magnetization is nearly spherically symmetric and therefore may be approximated by a magnetic point dipole located at the manganese nucleus . T h i s simplified picture of the magnetic charge distribution facilitated for a great deal the analyses of our experimental data.
From reduced Racah parameters as determined with optical
measurements on hydrated manganese salts Koide and Pryce could estimate a covalent intermixing of manganese and water orbitals by an amount of 30%. A more striking demonstration of covalency in the complex [Mn(H20), ] is given by the isotropic part of the SHFI between the manganese ion and the ligand protons. Using
12)
Watson's wave functions for a free manganese ion the theoretical magnitude of the isotropic SHFI is found to be of the order of 0.001 MHz for a proton at a distance of 2.80A. However because of covalency and overlap electrons are transferred partly from the water molecules into the 3d-holes of the manganese ion leaving a fraction of unpaired spin at the proton sites.
Thus the isotropic SHFI is enhanced as was confirmed by relaxation measurements on protons in a diluted solution of manganese sulphate
13) which yielded a value of 0.79 MHz
Recently Spence et.al. ' ' reported on the antiferro-magnetic ordering in some undiluted hydrated manganese salts as determined with nuclear magnetic resonance (NMR) measurements on protons and other magnetic nuclei. In these studies the magnetic charge distribution on the manganese ion was replaced by a magnetic point dipole whose direction was in accordance with the assumed magnetic space group. The experimental internal magnetic fields at the proton sites as evaluated from the proton NMR frequencies were compared with theoretical internal fields calculated from dipolar sums which extended over a large number of manganese ions. The discrepancies between the calculated and observed internal fields amounted to about 10% which was attributed to unpaired spin density transferred to the water molecules. Thus the know-ledge of the SHFI between a manganese ion and neighbouring
water protons is of considerable interest for workers in the field of hydrated antiferromagnetics.
After 1945 several magnetic resonance techniques have been developed, like for instance electron paramagnetic resonance (EPR) and electron nuclear double resonance (ENDOR) , which appeared to be very suitable for studying the magnetic properties of para-magnetic ions, incorporated in solid hosts. In 1967 Van Ormondt et.al. reported the results of EPR measurements on Mn in single crystals of LMN. When incorporated in LMN the manganese ion occupies two inequivalent magnesium sites both surrounded octahedrally by six waters of hydration (see section 1.3). To study the SHFI between the manganese ion and the protons within the complex [Mn(H20)g ] we investigated the same salt by means of ENDOR thus making use of valuable information provided by the EPR measurements mentioned above. In the next section the reasons for applying the ENDOR technique are mentioned.
1. Determination of the SHFI tensors of all protons within the
++ ++
two inequivalent [Mn(H20)g ] complexes in LMN:Mn . These tensors can be split in an isotropic and an anisotropic part. The existence of an isotropic SHFI is a direct consequence of covalent bonding in the [Mn(H20)^] complex.
2. Interpretation of the anisotropic tensors by means of a model, which is based predominantly on the point dipolar interaction between the manganese ion and the surrounding water protons, but which accounts also for the effect of covalency. It will be shown that the proton positions in LMN:Mn can be calculated from the anisotropic SHFI with reasonable accuracy. The only other measurement of the proton positions in this type of
19) salt so far has been carried out by means of x-ray diffraction However the accuracy of the x-ray positions was not very
great. Knowledge of accurate proton positions in a crystalline hydrate like LMN is important for the study of hydrogen bonds.
3. Interpretation of the isotropic SHFI in terms of an ab initio calculation of the ground state of the [Mn(H20), ] complex.
4. Determination and interpretation of the splitting of ENDOR lines caused by the dipolar interaction between the protons within the same water molecule.
1.2.
Choice of the magnetic resonance technique.
As was mentioned in the preceeding section the first aim of our experiments was to determine the superhyperfine interaction
(SHFI) between the manganese ions and the surrounding protons in
++
the hydrated salt LMN:Mn . Our intention was to apply one of the known magnetic resonance techniques.
20 21 )
A number of investigators ' applied the electron para-magnetic resonance (EPR) method to study the SHFI. In their case
In case of Mn incorporated in LMN the SHFS is not resolved but 22)
causes an inhomogeneously broadened EPR line . Moreover if the external magnetic field is directed along the trigonal symmetry axis of LMN two small satellite lines are observable due to an additional effect of the SHFI'^'^'^^^
When undiluted paramagnetic salts are investigated the nuclear magnetic resonance (MIR) method can be applied . This
technique, although being less sensitive than EPR, possesses the advantage of having a high resolution, that is to say in many cases the shifts of the nuclear resonance frequencies due to the magnetic interaction with neighbouring paramagnetic centers are
large compared with the line widths of the NMR lines.
During the last ten years the electron nuclear double
25-30) resonance (ENDOR) technique became popular to study the SHFI
This technique was chosen by us since it combines the high sensitivity of EPR with the high resolution of NMR. In performing a ligand
ENDOR experiment the sample is placed in an external static magnetic field H_ and is irradiated at the same time with microwaves at a frequency of about 9000 MHz and with radiowaves at a frequency, which in case of the complex [Mn(H20), ] is lying in the range of a few MHz to about 40 MHz. In order to obtain an optimum ENDOR sensitivity
the static magnetic field H_ has to be perpendicular to the micro-wave field as well as to the RF field. By means of a proper adjustment of the magnetic field strength H the EPR part of the ENDOR spectro-meter is tuned to one of the EPR transitions of the paramagnetic
impurity. After that the EPR signal is partially saturated by means of an increase ofrthe microwave power. The frequency of a RF
oscillator, which generates the RF field at the position of the sample, is swept slowly through the expected resonance range of the nuclear transitions. At the moment a nuclear spin flips this
is detected via a variation of the strength of the saturated EPR signal, Apart from the higher sensitivity with respect to NMR the
ENDOR technique possesses the advantage of obtaining a less com-plicated spectrum of the nuclear transitions. Suppose that an EPR line due to a transition M -«-> M + 1 is saturated, M representing the magnetic quantum number of the electronic spin. In that case only transitions can be observed of nuclei interacting with para-magnetic ions in the electronic state M or M + 1. Moreover when
occupied by the paramagnetic ion the resonance transitions of nuclei surrounding different sites are measured separately in case the spectrometer is tuned to non overlapping EPR lines.
At present a number of ENDOR mechanisms have been proposed 31)
to explain the change of the saturated EPR signal. Feher suggested the "packet-shifting" ENDOR mechanism which satisfactorily explained the results of his ENDOR experiments. Consider an inhomogeneously broadened EPR line composed of a group of spin packets. The position of a spin packet within the EPR line is determined by the orientations of the spins of the nearby nuclei. When one increases the microwave power at a fixed magnetic field a group of spin packets whose resonance frequency corresponds to the applied microwave frequency is saturated which results in a decrease of the EPR signal. During the irradiation of the sample with radiowaves other unsaturated spin packets become resonant because of the transitions of nearby nuclei. Since these unsaturated spin packets can absorb energy the EPR signal should
increase.
32)
In 1961 Lambe et.al. reported ENDOR experiments on ruby
which they explained in terms of the so-called "distant-ENDOR" effect. In this model the change of the EPR signal is caused by the depolari-zation of distant nuclei. In case of ligand ENDOR the removal of the dynamic polarization of distant nuclei is induced (probably via spin diffusion) by the depolarization of nearby nuclei. Depending on whether the EPR spectrometer is tuned to the absorption or to the dispersion mode the model predicts a small increase or a large decrease of the EPR signal.
33)
Recently Davies and Reddy explained the low intensity or the disappearance of certain ENDOR lines, which some times occurs in ENDOR spectroscopy. They attributed this phenomenon to the
'X/ \
"RF enhancement mechanism" . When the electron magnetic moment adiabatically follows the RF-field ff. it provides an additional RF component ff.' at the nearby nucleus. Depending on the magnitude of ff J' and its direction with respect to H_. an enhancement of certain ENDOR lines is possible, while others even can disappear.
Finally we draw attention to a phenomenon which was encountered '>a.
26) . 35)
in our investigation as well as m an ENDOR study on substitutional hydrogen atoms in CaF2
At certain orientations of the sample with respect to the external magnetic field some of the ENDOR lines (due to nearest water protons
in our case) were found to have a reversed sign relative to
that of the other lines. However at other orientations of the sample the phase of all lines was found to be equal. Thus the intensities of the lines mentioned above have to become zero at that orientation where the sign reverses.
In general it might be possible that depending on the investigated compound and on the set of experimental circumstances several
mechanisms are involved at the same time. In case these mechanisms tend to cancel the detection of ENDOR signals is hindered.
The crystal structure of Lajig JW^)-r,.24HJD.
In 1963 the crystal structure of Ce2Mg„(NO3).2'24H2O (CMN) was determined by Zalkin et.al. . They obtained x-ray diffraction data for single crystals of CMN at room temperature which were of sufficient accuracy to locate all atoms including hydrogen. We assume the atomic co-ordinates in LMN to be the same as those in CMN since the rare earths lanthanum and cerium are neighbours in the periodical system. This assumption is reinforced by the
36^ results of an x-ray diffraction study on Nd2Mg„(N0o).„.24H2O In this investigation the heavy-atom frame work (the hydrogen atoms excluded) was established. The heavy-atom positions were found to be very nearly equal to those in CMN. The primitive cell of CMN consisting of one formula unit Ce2Mg3(NO-)i„•24H2O
is found to be rhombohedral. It contains two [Ce(NO~), ] complexes, three [Mg(H20)^ ] complexes and six waters of hydration not
belonging to a cation. The dimensions of the corresponding hexagonal cell containing three formula units are a = 11.004 A and a = 34.592 A. The space group is R3. Figure 1.1 shows the magnesium and cerium ions located in a plane parallel to the ö-axis and the [110 ] direc-tion, together with the water molecules and nitrate groups located near that plane. The vertical c-axis joining magnesium and cerium ions coincides with the trigonal symmetry axis of CMN. Two different types of magnesium sites denoted by I and II can be distinguished, site II occurring twice as often as site I. Site I and site II have the site symmetry C^^ and C^ respectively. Each kind of magnesium ion is surrounded by an octahedron of six water molecules. Thus four different waters of hydration occur, denoted by Wl surrounding site I, by W2 and W3 surrounding site II and by W4 which is not closely associated with any cation„ The two protons within the
9 CERIUM O OXYGEN
• HYDROGEN
h
(110)1
Figure 1.1. The crystal structure of CMN. The magnesium and cerium ions located in a plane parallel to the ö-axis and the [ 110 ] direction are shown together with the water molecules and nitrate groups ne^r that plane.
same water molecule are denoted by HI and H2.
When incorporated in LMN the manganese ions are located at the magnesium sites as was confirmed by our ENDOR investigation. The manganese ion appeared to have a slight preference to occupy
r37) site I .
REFERENCES
1) J. Owen and J.H.M. Thornley, Rep.Prog.Phys., 29^ (1966) 675. 2) S. Sugano and R.G. Shulman, Phys.Rev., j_30 (1963) 517. 3) R.E. Watson and A.J. Freeman, Phys.Rev., 134 (1964) A1526. 4) 0. Matsuoka, J.Phys.Soc.Jap., 28 (1970) 1296.
5) J.W. Moskowitz, C. Hollister, C.J. Hornback and H. Basch, J.Chem.Phys., 53 (1970) 2570.
6) R.G. Schulman and K. Knox, Phys.Rev.Letters, 4^ (1960) 603. 7) F. Tsay and L. Helmholz, J.Chem.Phys., 5£ (1969) 2642.
8) R.K. J e c k , J . J . Krebs and V . J . F o l e n , J . A p p l . P h y s . , 4j_ (1970) 1116. 9) L.E. Orgel, An Introduction to Transition - Metal Chemistry.
Ligand-Field Theory, Methuen London 1960, page 44.
10) R.D. Spence, J.A. Casey and V. Nagarajan, Phys.Rev., 181 (1969) 488. 11) S. Koide and M.H.L. Pryce, Phil.Mag., 2 (1958) 607.
12) R.E. Watson, Phys.Rev., _n8 (1960) 1036. 13) H. Sprinz, Ann.Physik, 20 (1967) 168.
14) R.D. Spence, W.J.M. de Jonge and K.V.S. Rama Rao, J.Chem.Phys., 5J_ (1969) 4694.
15) R.D. Spence and K.V.S. Rama Rao, J.Chem.Phys., 52. (1970) 2740. 16) A. Abragam and B. Bleaney, Electron paramagnetic resonance of
transition ions. Clarendon Press Oxford 1970. 17) G. Feher, Phys.Rev., j_03 (1956) 834.
18) D. van Ormondt, T. Thalhammer, J. Holland and B.M.M. Brandt, Proc. of the XlVth Coll. Ampère, North Holland 1967, page 272. 19) A. Zalkin, J.D. Forrester and D.H. Templeton, J.Chem.Phys.,
39 (1963) 2881.
20) L. Helmholz, A.V. Guzzo and R.N. Sanders, J.Chem.Phys., 35 (1961) 1349.
21) T.L. Estle and W.C. Holton, Phys.Rev., L50 (1966) 159. 22) T.G. Castner, Phys.Rev., n_5 (1959) 1506.
23) D. van Ormondt and T. Thalhammer, Phys.Letters, j_4 (1965) 169. 24) V.F. Koryagin and B.N. Grechushnikov, Sov.Phys. (Solid State),
8^ (1966) 445.
26) R.G. Bessent, W. Hayes and J.W. Hodby, Proc.Roy.Soc., 297 (1967) 376.
27) C A . Hutchison and G.A. Pearson, J.Chem.Phys., 47 (1967) 520. 28) J.J.Davies, J.Phys. C (Proc.Phys.Soc), j_ (1968) 849.
29) J.M. Baker, E.R. Davies and J.P. Hurrell, Proc.Roy.Soc.A, 308 (1968) 403.
30) G.H. Rist and J.S. Hyde, J.Chem.Phys., 50 (1969) 4532. 31) G. Feher, Phys.Rev., 2_U (1959) 1219.
32) J. Lambe, N. Laurance, E.C Mcirvine and R.W. Terhune, Phys.Rev., J22 (1961) 1161.
33) E.R. Davies and T.Rs. Reddy, Phys.Letters, 3JA (1970) 398. 34) A.J. Freeman and R.B. Frankel, Hyperfine Interactions,
Academie Press New York 1967, page 235.
35) D. van Ormondt and H. Visser, Proc. of the XVth Coll. Ampère, North Holland 1969, page 475.
36) P.M. de Wolff and W. J.A.M. Peterse, Appl. Sci.Res. , BJ_0 (1963-64) 182.
CHAPTER 2. INSTRUMENTATION.
2.1.
Introduction.
The most difficult aspects of our ENDOR experiments were: 1) The generation of a high RF field for exciting the magnetic
resonance transitions of the protons.
2) The preparation and accurate mounting of the samples. The present state of art in our group is such that the most accurate positioning of the sample with respect to the static magnetic field is achieved by mounting it to a quartz rod which
is supported by one of the pole pieces of the electromagnet. With this method the deviation from the desired orientation can be kept within the range of +_ 0.2 . Our particular way of mounting the sample requires a special method of cooling. We do not immerse the cavity and the adjoining waveguide into the cooling liquid but rather pass cold gas through the centre region of the cavity via a double-walled quartz tube. Although this system has the obvious advantage that the cavity and adjoining waveguide remain at room temperature, thereby reducing the need of adjustments of the spectrometer during the measurements to a minimum, there is one
distinct drawback: temperatures below about 15K demand a considerable liquid helium consumption. At a sample temperature of about 20K
which we used in most ENDOR experiments the relaxation times of the manganese ions and protons are such, that an RF amplitude of the order of several Oersted (1 Oe = •; A/m) is required to
471
raise the ENDOR signals sufficiently above the noise. In view of this circumstance an RF circuit has been built, which is kept at resonance over almost the entire frequency region by means of a servo mechanism.
The author is indebted to Mr. J.R. de Haas and Mr. E.L. de Wild for developing and building the serv<j mechanism mentioned above.
The ligand ENDOR spectrometer.
The microwave circuit.
The microwave circuit of the ligand ENDOR spectrometer consists of a home made X-band homodyne EPR spectrometer with diode detection. Figure 2.1. shows a diagram of the EPR spectrometer together with additional RF equipment to induce the magnetic resonance transitions 2.2.
of the nearby protons. The EPR spectrometer can be tuned to the absorption as well as to the dispersion mode. When an absorption experiment is performed attenuator (a) in the bucking arm is set to maximal attenuation while the frequency of the signal clystron is locked to the resonance frequency of the sample cavity. The dispersion signal can be obtained by adjusting the amplitude and phase of the bucking signal by means of attenuator (a)
la)
and phase shifter (b) Moreover the bucking signal can provide a steady l e v e l of microwave drive a t the diode d e t e c t o r (d)
for an adequate d e t e c t o r efficiency . If frequency s t a b i l i z a t i o n during d i s p e r s i o n experiments i s required the wave_meter (c) can be used as reference c a v i t y . KLVSTRON
ï\^
b Af 'dB STABILIZA TION UNIT3 8 ^
-DETECTOR PRE AMPUFIER 80CMOBAND AMPLIFIER KlIcHi R F OSCILLATOR FM reorderFigure 2.1. Diagram of the ligand ENDOR Spectrometer,
The external magnetic field (VARIAN V-3401, with a fieldial
V-FR 2503) is modulated with 10 kHz. The static magnetic field strength can be measured with a home made magnetometer, based
2)
on NMR . For detailed information about the microwave part of the ligand ENDOR spectrometer we refer to the thesis of D. van Ormondt 3)
The EF equipment.
When a ligand ENDOR experiment is performed the magnetic resonance transitions of the nearby nuclei are detected via a variation of a partly saturated EPR signal. In fact the RF
equipment is only that part of an NMR spectrometer which generates the radiowaves at the position of the sample.
4)
According to Abragam and Bleaney ENDOR signals can be
observed only when the rates w of the nuclear transitions M, m •<->• W, m -1 and M+l, m -«-^AZ+l, m -1 are comparable with (or faster than)
n ' n ' n
"^
the relaxation rates \/T and \IT of the partially allowed
tran-sitions M, m -(-^ M+l, m -1 and M, m -1 •<->• M+] , m , where M and m represent the
' n ' n n n n '^
magnetic quantum number of the electronic spin and the nuclear spin respectively. Since in general T > T and T > T. the con-ditions mentioned above are sure to be fulfilled when
w„ '^ l/^j (2.1) where T. is the electronic spin-lattice relaxation time of the
allowed EPR transitions. Working out equation (2.1) we find for
the amplitude H. of the nuclear RF field at the sample
17 . 1 \/AV '
^ l " T V 2 7 (2.2)
where y is the nuclear gyromagnetic ratio and Av the width of the ENDOR line concerned. In our case, where protons surround manganese
10^
ions in LMN a RF field of approximately 2 Oe (1 Oe = A/m) was expected to be sufficient at about 80 K (the line width Av and the electronic spin-lattice relaxation time T^ were assumed to be 100 kHz and 4 x lO'^s respectively).
The nuclear frequency power is provided by a home made
RF oscillator with an output of about ^W, which is frequency modulated by supplying a sinusoidal voltage at a frequency of 120 Hz to a
variable capacitance diode (varicap), connected across the resonance circuit of the oscillator. The output of the oscillator is raised to 3W by a broadband power amplifier (Instruments for Industry, type M 500), which operates between 0.2 and 220 MHz.
The ENDOR resonance circuit consists of the ENDOR coil in series with a mechanically variable condenser. The amplifier output is matched to this circuit by means of a transformer. To obtain an optimum current the ENDOR resonance circuit has to be kept at resonance when the RF oscillator is swept slowly through the
expected frequency range of the ENDOR transitions. For this purpose a feed-back system has been built (see figure 2.2). The resonance frequency of the ENDOR resonance circuit is modulated by means of a sinusoidal voltage of 400 Hz, supplied to a varicap (TRW Inc., type PC 117) which in series with a variable condenser C^ is connected across the variable condenser C, of the ENDOR resonance circuit. When the ENDOR circuit is out of resonance the resulting amplitude modulation of the RF voltage over C. is detected by a diode detector. After that the 400 Hz signal is amplified and fed into a servomotor, which is coupled mechanically to C. via a gearbox. At the same time the condenser Cj in series with the varicap is driven by the servomotor in drd,er to keep the magnitude of the frequency modulation constant over the entire frequency range. ENOOR COIL RF OSCIUATOR BR0A06AND AMPLIFIER n-C^
"Loc
ETECTOBl I L I M I T E R T ^ AOOINC AMPUFERy
POWER AMPLIFIER <OOHi OSCILLATORzic:
POWER AMPLIFIERL J L^J
I-..^701
.-/Figure 2.2. Diagram of the feed-back system for keeping the ENDOR resonance circuit at resonance.
Figure 2.3. shows the details of the ENDOR resonance circuit, the modulation unit and the diode detector. The condenser C. consists of two parallel connected film dielectric variable
capacitors, each with two sections (Philips, type 2222 807 10048). The capacitance swing of both sections is 385 pF, the zero
capacitance 5pF. The condenser C2 contains two FM sections with a capacitance swing of 12 pF and a zero capacitance of 7 pF. When the switch (see figure 2.3.) is in position 1 the ENDOR resonance circuit is suitable for the higher frequency range, up to about 35 MHz.
With the switch in position 2 the lower frequency range is covered, down to about 5 MHz. For the few ENDOR transitions below 5 MHz an external variable capacitor is connected parallel to C,. In that case the feed-back system is not used and 'the ENDOR resonance circuit has to be tuned for each individual ENDOR line. The
IS : BROADBAND \ AMPLIFIER
^ 1
MODULATION UNIT ilOOHz OSCILLATOR SELECTIVE AMPLIFLIERFigure 2.3. Details of the ENDOR resonance circuit, the modulation unit and thé diode idetectior.- '
The ENDOR coil is a three-turn- rectangular coil with a length of 45 mm and a width of 8 mm. The coil is glued (velpon,
two-components glue) to the outside of a quartz tube, which forms^the inner part of a double-walled dewar tube (see figure 2.4 and section 2.3). To cover the higher frequency range the supply wires of the ENDOR coil have to be kept as short as possible. To provide the RF field at 'the position of the sample the ENDOR coil is placed inside the sample cavity. Since its ends are located outside the sample cavity the ENDOR coil hardly affects the unloaded quality factor of the cavity, which is important for an optimum EPR sen-sitivity. For the same reason the narrow end of the double-walled dewar tube, which supports the ENDOR coil, is made of quartz.
U=-
•PUMP I I < I I I I I I • I IT
1
TWO-COMPONENTS 6LUE ENDOR COIL SAMPLE CAVITY SUPPY WIRESFigure 2.4. The ENDOR coil mounted in a double-walled dewar tube.
The RF current can be monitored on an oscilloscope by means of a current probe (Hewlett and Packard, type 1110 A) around one of the supply wires of the ENDOR coil. The amplitude of the RF current varies from about 3A at 2 MHz to about 0.7 A at 35 MHz. A BI current of lA corresponds with a RF field of about 3 Oe
(measured with a pick-up coil).
The detection of the ENDOR signals.
As is mentioned in the sections 2.2.1 and 2.2.2 two modulations are employed in the ligand ENDOR spectrometer. First the external magnetic field is modulated with 10 kHz, secondly the frequency of
the RF field with 120 Hz (see figure 2.1). To detect the EPR and ENDOR signals simultaneously two lock-in amplifiers are connected in series, the first one tuned to 10 kHz and the second one to
120 Hz. During an ENDOR transition the amplitude of the 10 kHz (EPR) signal is modulated with 120 Hz. In order to prevent suppression of the two sidebands the bandwidth of the first lock-in amplifier
(tuned to 10 kHz) has to be at least 240 Hz. In addition the time
constant at the output should not exceed 1/120 s. This double modulation scheme has the following advantages:
1. During the ENDOR experiments the EPR signal is monitored on an oscilloscope, so that the tuning of the EPR spectrometer can be
inspected.
2. If an error signal of 120 Hz is picked up at the microwave detector this signal is suppressed by the first lock-in amplifier which is tuned to 10 kHz.
3. A relatively high frequency of the magnetic field modulation can be chosen, which improves the signal-to-noise ratio of the EPR-signal.
The frequency modulation (as opposed to amplitude modulation) of the radiowaves has the advantage, that the first derivatives of the ENDOR absorption lines are recorded. Therefore the centre of the resonance can be determined with great accuracy. Moreover with FM the detection equipment is less sensitive to RF pick-up, than in case of amplitude modulation.
The cooling system.
In ENDOR spectroscopy it is convenient to perform the experiments at a low temperature of the investigated compound. Because of an increased electronic spin-lattice relaxation time T. the EPR signal can be saturated with a lower microwave power. Moreover a smaller RF field is required to achieve an optimum
ENDOR sensitivity (see section 2.2.2). To cool the samples a 7-9)
simple helium gas-flow system has been built which operates in the temperature range of 15 - 300 K. Figure 2.5 shows a schematic diagram of the cooling system. We draw attention to:
1. A standard helium storage dewar (a) from which helium is evaporated by electric heating.
2. A 1000 ^ radio resistor (b), acting as the heater.
3. A double-walled stainless steel siphon (c) which transports the cold helium gas to a double-walled dewar tube.
4. A double-walled dewar tube (d), mainly made of glass and ending in a narrow double-walled quartz tube which supports the ENDOR coil (see section 2.2.2.). To prevent extraneous heat leakage into the system it is important, that the ENDOR coil and its supply wires make no contact with the outer quartz tube. For the same reason the dewar tube is pumped continuously with a high vacuum pump (helium gas diffuses easily through quartz).
5. The quartz end of the dewar tube enters a pertinax chamber (e). The helium gas is released from this chamber via a tube at the side, which is connected to the helium return line with a silicone rubber tube (f).
6. The top of the pertinax chamber is connected vacuum tight to a thick-walled brass cylinder (g), which by means of a horizontal brass rod is attached to one of the pole pieces of the magnet. The cylinder acts as the support of a crystal rotator (sample holder), which closes the cooling system (see section 2.4).
7. At both ends (h) of the dewar tube the connections with the adjoining parts of the gas-flow system consist of a piece cut from a rubber balloon. Leakage of helium through the junction of the pertinax chamber (e) and the crystal rotator is prevented by silicone grease.
8. The specimen temperature is measured with a copper-constantan thermocouple with its reference junction (i) hanging into the helium bath and its signal junction glued to a small quartz tube, to which the sample is mounted (see section 2.4).
9. To prevent icing of the crystal rotator and freezing of the silicone grease a resistance wire (j), acting as a heater, is wound several times around the brass cylinder (g).
10. Close to the chamber outlet the silicone tube connecting the pertinax chamber (e) to the helium return line contains a piece of glass tubing, in which a heater (k) has been mounted. In this way icing and freezing of the silicone tube is prevented. Since the heater power is about 40 W protection against over-heating is needed in case of problems with the helium gas flow. Therefore a NTC resistor (1) forming a part of an electronic switching circuit is attached to the glass tube. Depending on the temperature of the NTC the heater current is controlled by switching on or off.
Figure 2.5. Diagram of the gas-flow cooling system.
Control of temperature is achieved by varying the evaporation rate. Starting slowly at room temperature in about 15 minutes the required temperature, for instance 20 K, can be achieved. At this temperature the liquid helium consumption is approximately 2 1 h The temperature stability is affected mainly by the helium return line pressure , which depends on other helium experiments performed
at the same time. The observed stability in the temperature range of 15-25 K was about 1 K in most cases which is adequate for our ENDOR experiments. Therefore a feed-back system for correcting temperature variations was not needed.
One of the main reasons for choosing the helium gas-flow system is the fact that with this method only the sample is cooled. Bacause of this a very accurate crystal rotator can be used (see section 2.4), which determines for a great deal the accuracy of the ENDOR experiments. A disadvantage of the system is the poor helium economy. In addition temperatures below 4.2 K can not be achieved. In that case a liquid helium cryostat is required.
Preparation and mounting of the samples.
Single crystals of La2Mg3(N03) 2-24H20 (LMN) contaminated with Mn ions were grown from a saturated solution of LMN, to which some manganese nitrate was added. The crystals were grown at a temperature of 5 C In most cases the Mn to Mg ratio was about 1 : 3000 in the solution. Usually the crystals grew in the form of a flat hexagonal plate with a thickness of 3 to 8 mm, depending on the length of the growing period and the desiccation rate of the saturated solution. The flat sides of the hexagonal plates are perpendicular to the trigonal sjmunetry axis (c-axis), while the six sides of the hexagon are parallel to the a-axes. Using the outer geometry of a single crystal as a reference a plane parallel to an a^ a plane is ground. Out of such a machined crystal samples of rectangular shape were cut (dimensions approximately 8 x 3 x 3 mm^). During the cutting care was taken not to affect the ground a^ o- plane nor one of the natural growing planes
perpendicular to the <3-axis. A sample thus contains two reference planes, one parallel to the a, c plane and one perpendicular to the c-axis. The cutting was performed with a wet wire machine.
The samples were mounted (see figure 2.6) to a small quartz tube (a) with a polished end (b) perpendicular to its longitudinal axis. Parallel to the axis two surfaces (c) are polished to which two thin quartz plates (d) are glued (velpon, two-components glue). For reinforcement a small quartz block (e) is mounted between the
two plates. Depending on the desired direction of the external magnetic field with respect to the crystalline axes, the natural
growing plane is glued (velpon hard, cetabever) either to one of the quartz plates or to the polished end of the quartz tube. A quartz plate is used when the c-axis has to be parallel to the magnetic field and the polished end when rotation of the magnetic field in the plane perpendicular to the c-axis is desired. In each case glue is only applied between the natural growing plane and the crystal mount, while in the latter case the a, a reference plane is just pressed against one of the quartz plates. This
procedure reduces the chances of formation of cracks in the sample since the coefficient of thermal expansion parallel to the c-axis is five times larger than perpendicular to this axis .•10)
SAMPLE
I
d e
Figure 2.6. A sample mounted to a quartz tube.
By means of a polarization microscope the required direction of the c-axis with respect to the longitudinal axis of the quartz tube is inspected. As a rule only samples with a deviation smaller than 0.5 were used.
The small quartz tube with the sample is glued (velpon, two-components glue) to a crystal rotator which, supported by the brass cylinder (g) (see figure 2.5), can rotate in the external magnetic field H. When considering the orientation of the sample with respect to H two types of misalignments can occur.
First, depending on the desired direction of _ff with respect to the c-axis, the angle between the rotation axis of the
crystal rotator and the c-axis can deviate from 0 or 90 . Secondly the angle between E_ and the rotation axis can deviate from 90 . Hereafter the former and latter misalignments are called a and 3 respectively. During the experiments typical values of the angles a and 6 were found to be 0.4 and 0.15 respectively.
By means of special facilities of the crystal rotator the misalignments mentioned above could be reduced (see figure 2.7): A narrow brass rod (f) with a rounded lower end can move inside a stainless steel tube (j). The rounded end pushes a small brass rod (h) in the direction of the inner side of the tube (j). The brass rod (f) is held in position by means of a spring of which the tension can be adjusted with the screw (a). The tube (j) can
rotate inside a second stainless steel tube (k) which contains a square hole (i) at its lower end. A pyramidal part (with a square cross-section) of the brass rod (h) fits into the square hole (i). Furthermore the brass rod (h) fits into a hole (g) somewhat out of the middle of the cross-section of the tube (j). The small quartz tube (see figure 2.6) containing the sample
is glued to the end of the brass rod (h) with two-components glue. When the outer stainless steel tube (k) is rotated around its longitudinal axis the brass rod (h) describes a conical surface with a very small vertical angle. The magnitude of this angle can be varied from approximately 0.1 to 0.6 by changing the height the inner tube (j). This is done by adjusting the disc (c) which via the disc (b) determines the height of the tube (j).
By means of a proper adjustment of the vertical angle mentioned above the misalignment a can be reduced.
The outer tube (k) rotates inside a brass cylinder (e) which closes the cooling system (see section 2.3). The longitudinal hole in this cylinder makes an angle of aboutO.1 with the cylinder axis. By means of a proper adjustment of the cylinder (e) with respect to the support of the crystal rotator (the thick-walled brass cylinder (g), drawn in figure 2.5) the misalignment 3 can be reduced with about 0.1 .
The disc (d) contains a scale of 360 . Using a nonius the rotation angle of the sample can be determined with an accuracy of ±0.1°.
Figure 2.7. Crystal rotator with adjustment mechanism to reduce a misorientation of the sample.
In case an ENDOR experiment is performed with the c-axis parallel to the magnetic field (c//_ff) the proper alignment of the sample is achieved when, because of the trigonal sjrmmetry around the c-axis, three ENDOR peaks coincide. In the perpendicular case
(o j_ H_ ) the frequency of a proton transition should be the same
at six equivalent positions of the sample (at angles which are multiples of 60 ) with respect to the static magnetic field
(see section 3.5). The crystal rotator is adjusted in such a way that this situation holds as well as possible. The effect of a small remaining misorientation is eliminated by averaging the six ENDOR frequencies mentioned above. This procedure also enables us to determine the actual misalignments (see section 3.5 and 4.4.1).
In case of a olIH^ experiment the values of the ENDOR frequencies do not depend on the orientation of the Oj c plane. However, when a c J_ ff experiment is performed a calibration of the rotator scale with respect to the orientation of the a, c plane is required. This can be done as follows. First a sample is attached to the quartz tube with the c-axis perpendicular to one of the quartz plates. Secondly the proper orientation of the sample is performed for a cllH_ experiment. After that we know the value of the rotator scale at which one of the quartz plates is perpendicular to the magnetic field. Finally the sample is removed and a second sample is attached to the same quartz plate with its a, c reference plane parallel to it. It is not necessary to perform this calibration each time a new quartz tube is mounted because after a number of rotation experiments enough ENDOR data are available to use as reference In this way the direction of the crystallographic a-axis with respect to the static magnetic field could be determined with an accuracy of approximately ± 0.5 .
REFERENCES
1) T.H. Wilmshurst, Electron spin resonance spectrometers, Adam Hilger London 1967. a) page 46, b) page 48.
2) J.R. de Haas, D. van Ormondt and C Slottje, J.Sci.Instr., 44 (1967) 471.
3) D. van Ormondt, Thesis, Delft 1968.
4) A. Abragam and B. Bleaney, Electron paramagnetic resonance of transition ions. Clarendon Press Oxford 1970, page 246. 5) D. van Ormondt, Thesis, Delft 1968, page 119.
6) J.M. Baker and F.I.B. Williams, Proc.Roy.Soc., A 267 (1962) 283.
7) I.e. Morris, D.A. Read and B.K. Temple, J. Phys. E (Sci. Instr.), 2 (1970) 343.
8) A. Hausmann, W. Sander and P. Schreiber, Cryogenics, _1_0 (1970) 70. 9) J. Haupt, Z. Ang. Physik, 23 (1967) 377.
10) H.J.M. Lebesque, C.W. de Boom and J. Carp-Kappen, unpublished results.
CHAPTER 3. THEORY.
3.1.
The spin Hamiltonian of the manganese-proton system in LMN:Mn .
Among the irongroup transition ions the manganese ion occupies a special place because of its half-filled 3d-shell. According to the rules of Hund the ground state of the free
ion has no orbital momentum while the electronic spin S = 5/2 (^S state). The distance between the nearest exited level (^G state) and the ground state is 26800 cm"-^
When the manganese ions are incorporated in LMN at the magnesium sites all levels are split by the electric field due to the surrounding lattice, the splitting of the ground state
9 "1
being of the order of 0.1 cm" . Since our EPR spectrometer operates at X-band the sample is irradiated with microwave quanta of the order of about 0.3 cm~^. In order to observe EPR transitions within the ground manifold the splitting of the ground state has to be raised to the magnitude of the microwave quanta by means of a proper adjustment of the external magnetic field strength.
The distances between the ground state levels of the man-ganese ion are mainly determined by the interaction of the un-paired electrons with the external magnetic field H_ as well as with the local electric crystal field. In addition there are a number of smaller effects such as the hyperfine interaction of the unpaired electrons with the manganese nucleus, the SHFI of the unpaired electrons with the surrounding protons, the Zeeman
interaction of these protons with H_, the interaction of the manganese nucleus with the surrounding protons, the Zeeman interaction of
the manganese nucleus with H_ and the quadrupole interaction of the axial electric crystal field with the quadrupole moment of the manganese nucleus. In magnetic resonance spectroscopy the influence of all these interactions on the ground state levels is described usually by a spin Hamiltonian. To analyse the ENDOR frequencies of the j neighbouring water proton the following spin Hamiltonian was used
where S = 5/2, J = 5/2 is the spin of the manganese nucleus and i^ = 1/2 is the spin of the j nearby proton. The first three terms in (3.1) represent the electronic Zeeman interaction, the second degree interaction of the unpaired electrons with the
electric crystal field and the hyperfine interaction respectively. The fourth and fifth term describe the SHFI with the j nearby proton and the proton Zeeman interaction respectively while the last term stands for the interaction of the manganese nucleus
th '^ '^ '\]' 'W
with the J proton. The symbols D^ A , A and B'^ represent second rank tensors. The spin Hamiltonian (3.1) does not account for the fourth degree interaction of the unpaired electrons with the electric crystal field, the Zeeman interaction of the
manganese nucleus and the quadrupole interaction, since these interactions were found to have a negligible effect on the ENDOR frequencies. Moreover only the interactions with one
nearby proton are considered, thus neglecting the influence of the surrounding protons on the ENDOR frequencies of that proton as
3)
a first approximation (independent bonding model ). However, in section 3.6 the influence of the nearest proton is described. It is found that the dipolar interaction between the two protons within the same water molecule causes a small splitting of the ENDOR lines.
From now on we shall drop the superscript j on J^, A and E^ . In case the proton belongs to a nearest neighbour of the manganese ion the fourth and fifth term are of the same order of magnitude and about three orders of magnitude smaller than the
electronic Zeeman interaction. Moreover the last term in (3.1) is three orders of magnitude smaller than the fourth and fifth term. Therefore the terms in J.B.I containing J and J
(sup-X y
pose the external magnetic field to be directed along the z-axis of the reference frame) were neglected in our calculations. Furthermore the remaining terms were approximated by
(0,0,J^).S.I = {0,0,- JLIL J^) 2.1 (3.2)
where g stands for the spectroscopic splitting factor of the manganese nucleus and 3 for the nuclear magneton. Equation
(3.2) is correct in case the electronic spin can be considered as a magnetic point dipole located at the position of the manganese nucleus (point dipole model). The ratio g 3 Ig^ is 0.377 x IQ 3 -3 ^)
Calculation of the ENDOR frequencies. Evaluation of the SHFI tensors.
The theoretical ENDOR frequency of a nearby proton is equal to
V = /z~l \E(,M,m,m = -1/2) - E{M^m,m = 1/2)] (3.3)
where E(M,m,m ) is an eigenvalue of the spin Hamiltonian (3.1). Afj m and m represent the magnetic quantum numbers of the electronic
spin, the manganese nucleus and the proton respectively. One way of evaluating the SHFI tensor A is to calculate the theoretical ENDOR frequencies (3.3) with experimental values of the static magnetic field strength by means of an exact diagonalization of the 72 X 72 matrix ((M^m^m i3f|M',m',m' ) ) and to fit the elements of A to the corresponding measured ENDOR frequencies with least squares adjustment. A second way is to approximate the solution of the secular equation of the spin Hamiltonian by means of perturbation theory. The perturbation treatment has the advantage of obtaining
a better understanding of the influence of the various spin Hamiltonian parameters on the ENDOR frequencies. Since diagonalization of the
spin Hamiltonian matrix in combination with least squares adjustment required a rather complicated computer program, which became
available not until very recently, we calculated the ENDOR fre-quencies with perturbation theory up to second order. In the following of this section an example is given of the applied procedure.
To evaluate the elements of the SHFI tensors it was found to be sufficient to measure the ENDOR frequencies for three mutually perpendicular orientations of the sample with respect to the external magnetic field. For each proton the z-axis of the experimental reference frame was taken along the c-axis. Thus H_ was directed either parallel or perpendicular to the trigonal symmetry axis of LMN. These directions coincide with the principal directions of the tensors D and A which simplified the analysis of the ENDOR frequencies.
In the following we suppose H_ to be directed along the z-axis of the experimental reference frame. In that case the second term S.D.S_ in (3.1) is written usually as
DiS"^ - •;rS{S^\)} since the principal elements D , D and D ^ of D
are equal to -]/3D, -1/3P and 2/3D respectively . In figure (3.1) a schematic diagram of a part of the energy matrix belonging to the spin Hamiltonian (3.1) is drawn. The symbols AA and BE indicate matrix elements of the order of magnitude of g^H and A respectively, where A represents one of the principal elements of the tensor A (A =A because of the axial syimnetry
XX yy
about the c-axis). Furthermore the symbols CC and DD denote elements of the order of A .. (ij Q = x, y, z), A . • being one of the elements of the SHFT tensor A. Thus we can write for these symbols AA = 0 (g^H), BB = 0 ( / ^ ) , CC = O(A^j) and DD = 0(^^^0. 5 2 5 2 3 2 1 2 1 4. 3
" T
5 ~ 2 5 2 % 1i
1 "2 1 2 1 2 1 2 1 " 27
~T'7-f 7
1I
1r.7
7
- = f
M.4
1 " 2 A A C C DD D D 1 2 C C A A DD DD 3 2 1 " 2 D 0 D D AA C C D 0 D D BQ1
2 D D DD CC A A DD DD BB 1 2 1 0 D DD A A CC DD 0 0 1 2 0 D OD CC AA D D DD 1 2 1 " 2 OD DD AA CC 0 0 OD 1 2 0 0 DD CC A A' DO DD 3 " 2 1 ~ 2 D 0 D D AA CC 1Li.,-• OD DD C C A A Doic b 1 DD 1 DD 1 5 1 " 2
kli
DD DD AA CC DD DD CC A A 37
5
2 1 ~ 2 B B AA CC 11 ^
BO C c AAi
1 2 BS D D OD 1 "^ 1 - 1 .1 ^
j _...J 1 BB 1 ! 11
1 1 1 1 OD DDFigure 3.1. Schematic diagram of a part of the energy matrix belonging to the spin Hamiltonian (3.1) in case the external magnetic field is directed along the c-axis of LMN; AA = 0(g gi), CC = 0(A..), DD = 0(/4..)
T t-J ''t/
As a first step in the evaluation of the SHFI tensor from the ENDOR frequencies we ignore the matrix elements BB and DD
(see figure 3.1) due to the terms in (3.1) containing S and S .
X y
Because of this the spin Hamiltonian reduces to
r g P Jf' = gms^ . DiSl - 1^(5.1)} . A'^^S^J^ - g^^m^ . \o,o,s- - ^ J
z
.2'.J(3.4)
The first three terms in ( 3 . 4 ) , which do not depend on the protonspin J, are diagonal. Therefore we can restrict ourselves (see equation (3.3)) to the simplified Hamiltonian
3f' = - a 3 ffJ + p ^p n z 0 , 0 , 5 - ^ V ^ J ' ' z g^ z A A A XX xy xz A A A xy yy yz A A A xz yz zz
'^x]
'y I Z^ ( 3 . 5 ) '\,We suppose A to be symmetric which is true in case of an isotropic 6)
electronic g'-factor we use the notation
For the energy matrix belonging to (3.5)
E = 1
r*
r
t,
(3.6) where t = <M,m,-\l2 \K' I M^m,-\/2 > t^ = <M^m, 1/2 |3C' I M,m 1/2 > r = {M,m,-\/2 |7f' I M,m, 1/2 > r* = {M,m, 1/2 |3C'| M,m,-1/2 ) = ^g e> H - loM'. '^p n ^ ^^p n ^ = W ' ( X + i Y ) . = M ' ( X - i Y ) . a = A zz X = u xzY = M
yz (3.7) M' = M - - ^ m 9^By reducing £" to a diagonal form it is found that
= h-^\EAM,m,-\l2) -E {M,m,\l2)\ =
/2 = h-^ 9p&^{(H-M^H//^)^ + (M')2 H y \
where
H„ = A Ig ^ and H\ = (^2 +^2 )/(^ g )2 (3.8b)
^2
zz^p n
\z xz yz ^pn
The eigenstates \Mjmj-\/2 )P and \Mjmj\/2 >" belonging to the eigenvalues E, (Af,mj-l/2) and ff, (Mjmjl/2) are equal to
|M,m^-l/2 >P = C^j |M,m,-l/2 > + C^2 |W^'"^l/2 >, IMJW^ 1/2 >^ = C^J 1^^^3-1/2 > + C^2 \^3rn,]/2 > , (3.9) where
^ r ^ ^
(^r^^
'^ /r-r* + (ff,-*,)^ ^"^ /rr* + (£j-tj)^^ r ^ =_Ü2l!li
21 /..„ . .„ , ..; 22 /rr* + (E^-t)'^ /rr* + (^-t.)^The next step is to account for the ignored matrix elements BB (see figure 3.1) due to the components containing 5 and 5 in
X y
the third term of (3.1). Because of these matrix elements the eigenstates {M^m > of the simplified Hamiltonian (3.4) are mixed which results in a small change of the expectation value of the operator S . Corrected expectation values were calculated from
'<M,m|5 IA/JW ) ', where \Mjm > ' represents one of the eigenstates of the Hamiltonian
(3.10) due to the first three terms in (3.1). The eigenstates \Mjm > '
were determined by means of an exact diagonalization of the energy matrix belonging to (3.10) with the aid of an electronic computer. The corrected equation for the ENDOR frequency becomes
where M" = '{M,m\S \M,m >'
g
3
°n
ng^
m.
(3.11b)In appendix A it is shown that this procedure is allowed since only small matrix elements with a negligible influence on the
eigenvalues have been ignored. In table 3.1 a number of corrected expectation values belonging to site I are listed in illustration of the order of magnitude of the correction as well as the dependence on the state of the manganese nucleus.
Table 3.1. Corrected expectation values of S as cal-culated for site I in case ff is parallel to the c-axis. m 5/2 3/2 1/2 - 1/2 - 3/2 - 5/2 '< 5/2,ml5^|5/2,m > ' 2.5000 2.4950 2.4920 2.4910 2.4920 2.4950 '<3/2,m|5 |3/2,m > ' 1.5047 1.5011 1.4983 1.4961 1.4945 1.4936
The principal elements of the tensor A , the parameter D and the electronic spectroscopic splitting factor g were determined with EPR by van Ormondt et.al. . Table 3.2 shows the values of these parameters used for the calcualtion of '(M^mlS |Af,m > '.
Table 3.2. The spin Hamiltonian parameters D (in 10 '^cm ^ ) ,
.J
A''., (in lO'^cm"-'; i = x^ y^ z) and the electronic
^^ . . . ++ .
spectroscopic splitting factor g of Mn in LMN as determined with EPR at T = 20K.
D A'^
^xx
4
\ < .
g
Site I - 220.0 - 90.2 - 90.2 - 90.4 2.001 Site II - 49.0 - 89.7 - 89.7 - 90.0 2.001From equation (3.11a) it is straightforward to derive, that
the quantities H^ and ^? can be evaluated from two ENDOR fre-quencies V. and Vj, belonging to A/'.' and M'^ and measured at H = H. and H = Hj respectively:
H«
2M''M"
M^'ffj+M'^H^-h^{(M")^v^
- (MV)2v2}
g^&^(M'^,
-
M'^H^)
(3.12a)H
'i-1
4(My)2 (w'p2
(M'^H^
- M^5j)2 +{h^{(M")^v^ - (M':)^v^
4^n
(^2^1 - ^ï^2>
h^
gh^
°p n
2(M'p2 2(M^)2 (3.12b)For the orientations of the sample with H_ parallel to the x- or y-axis of the experimental reference frame formulas analogous to
(3.8), (3.11) and (3.12) are found by cyclic permutation of Xj y and z. Thus from the corresponding ENDOR frequencies the quantities
H,
HiH„
and ff|2 can be evaluated. The values of M" havetz' ''Ix' ""lly
'^"'^
"iy
The diagonal tensor elements A . A and A follow straight-away from H,, , H,, and H,, respectively while the absolute values of the non diagonal tensor elements are derived from:
Ky\ - ^p^^^^ix ' % -
^lP/2
\A \ = g Q /THT^~+~H7^~-~HT^T/2
The signs of the non diagonal elements follow from a comparison with the tensor calculated on the basis of the point dipole model using approximate proton positions (see section 3.4).
The final step in the evaluation of the SHFI tensors is to account for the ignored matrix elements DD due to the fourth term in (3.1) containing 5 and S . The contribution of these matrix
X y
elements to the eigenvalues of (3.1) is of the order of A'^, ./g^H which amounts to as little as about 1 kHz when expressed in frequency units. Having such a small effect the influence of the matrix
elements DD is approximated by a second order perturbation cal-culation using the approximate SHFI tensor elements obtained with the preceeding procedure. For the unperturbed Hamiltonian and the perturbation Hamiltonian we chose
V/? = ^^^^2
'
^^^3 -
^^'"-'^^ ' Kz^'^z
(3.14a)
and
JC = -g B HI + S.A.I (3.14b)
p ^p n z — — ^ •'
respectively, thus neglecting the influence of the substitution of M by M" (see equation (3.11)) on the second order shift of the theoretical ENDOR frequency. According to the perturbation theory of Condon and Shortley the matrix elements of the type (MjWjW |5f \M,mjm^ ) have to be zero unless m = m'. In our case this
condition is not fulfilled because of the terms in S_.A.I_ containing S I+. Therefore the eigenstates iM^m^m ) ^ (see (3.9)) of the
p
simplified Hamiltonian (3.5) have to be used instead of the states iMjWjm > . The second order shift of the theoretical ENDOR frequency is equal to
AV2 = h-^ {E^iM^m^-\/2)-E^{M^my\l2)} ( 3 . 1 5 ) where P<M,m,m„\K \MjmJm' > ? ? < M » ' |3C |M,m,m > ^
E (M.m.m )
= ^ c^—2
1^
^-—t
1
P Mlm;m^'^M,m,m E^iM.m.m^) - E^{Mlmlmpand
ffQ(W,m,mp) =P<
M,m,m^
|Jf^p^Iw^m^m^ > ^
The second order perturbation energy is denoted by
E^{M,m,m ) .
By
means of (3.9), (3.14a,b) and (3,15) it is straightforward to derive:
£:2(W,m,-l/2) = F,(Af,m) x
\ r M M+\ ,^,
. W M+l , 1, r M M+l
M
M+1,,2 ^ 2^•l^•
X L'^i, ^22 ( '*'^^ * ^12 ^^21
^
^^12 ^22 " '^ll ^21 ^ "^
"
^ Ü
2 + , r M M+l , . , , M M+l I r ^ W+1 M M + l , , 2 ^ 2 M 1 ^ [ _ a , , <3:22 (Y<5 - XTI) - a , 2 ^ 2 1 '^^ " ^'^ ^ ^ 1 2 ^^22 " ^ 1 1 ^ 2 1 ^ Y ' ' ) > J + T M M+l , , . ^ , M+l M , ^ , , , M M+l M M+l . , , 2 ^ 1 ^ ^ [_ajj a j 2 (^<5 + Y n ) + a^^ a^^zX + i ^ { a , 2 « j 2 " '^11 '^ll ^ Y^)}J +r M M+l , . , , M+l M , , M M+l M M+l , , 0 2 M ~ 1 ^
| _ a , , a j 2 (YÖ - Xn) - C j j aj2YE - iY{a,2 «12 " ^ n '^i 1 ^^ •*" ^ ^ U I* + a . , a
+ F^(M,m) X
f f M A^-1 , , r ^ . M M-1 , ^ , w M M-1 M M-1 , , 2 ^ 2 A I 1 ^
|_aj2 a.2\ ^ '^'^^ ^\\ "^22 i^^^i2 ^^^22 ~ ^11 ^^21 ^ Y'^)>J +
T M M-1 f. ,- M M-1 , . M M-1 M M-1 , , 2 ^ 2 ^ ^ 1 ^ ^ l_a,2 a 2 , (Xn - Y<5) + Cj j 022 Ye + iY{aj2 «22 " '^ll '^21 ^ Y ' ' ) } j +
T M M-1 , , - . M M-1 , , , . M M-1 M M-1 , , 2 ^ 2 N - I 1 ^ ^
|_aj2 a , ) (^«5 + yn) + a^^ a^^ eX + iX{aj2 a.^2 ~ ^\\ ^11 ^ Y'')>J +
F M M - 1 , , ,v M M-1 , r W M-1 M M-1 , , 2 ^ 2\-i~l^l |_a';2 a:,, (XTI - y6) + a , , a , 2 Ye + ^ { « 1 2 «12 " '^ll ' ^ n ^^ "^ Y ^ l J J
E2(M,m,\t2) = Fj(M,m) x
f f M + l M , , ^ , M+l M , , , . M+l M M+l M , , 2 o. 2Nil X [_aj2 ^25 (X6 + yn) + a^^ a^^ eX + èX{aj2 ^22 " '^jj <^2\ ^ Y''))J
r M+1 M . ^ , . M+1 M , , M+1 M W+1 W / -, 2 ^ 2 N \ n + |_a,2 ^21 ("^ ~ "^^ ~ '^ll ^22 '^^ ~ ^'^^^\2 ^22 ~ 11 21 ^ ï )>J T M M+l , , - , M M+l ., , , , M M+l M M+l , . o ^ 2.^-r\ ^ + [_a2i ^^22 ( "^ "^'"^^ '^ ^22 ^^21 2^T^'3;22 0:22 ~ ^^21 ^^21 ^ Y )>J
T M M+l , r , . M M+l , . M M+l M M+l . , 2 ^ 2A \ 1 ^ "
+ [_a2j «22 (YÖ - Xn) - a22 a^^ ye - ^ { « 2 2 ^22 ~ ^^21 ^^21 ^ Y'^))J z
+
+ F^iM.m) X
f r ^ - 1 ^ / , X N M-1 M , ^ ,, , M-1 M Af-1 M ,., 2 - 2 M 1 ^ • X |_^ii «22 ( "^ ^"^^ ^^12 ^^21 ^ !^^'2i2 '^22 ~ '^l 1 ^21 ^ ^ Y'^)iJ "^ ^ r M-1 M ,, .- ^ M-1 M ^ , j. M-1 M M-1 M ,,^2 ^ 2 x i l L ^ n ^^22 ( ~ "^ ^^^12 ^^21 '^^ 5Y{aj2 «22 ~ '^ll ^21 ^ » ) > J T M M-1 , , . , M M-1 , ,,f M M-1 M M-1 , , 2 2 M 1 L'^22 '^21 ( "^ ^'^^ "^ '^21 ^22 * 2A{a22 ^^22 ~ ^^21 ^21 ^ "^ Y^)>J T M M-1 , , ^, M M-1 , , M M-1 M M-1 , , 2 2M~1' + L^22 ^21 ( ^ ~ "^^ "*" ^^21 ^^22 '^^ * ^"^^^22 ^22 ~ ^21 ^^21 ^ '*' '^ U 2 + 2 + ( 3 . 1 6 b ) w h e r e M '11 M ^21 > = ^ j / ( X + i y ) ; = C ^ , / ( X + i y ) ; = i(A -A ) * ' XX yy' M a , 2 M «22 6 j 2 , ^ 2 2 ' F,(M,m) = i^2(^^'") = (5-M)(5+M+l) -g^H-{.m+\)D-mA'' (S+M)iS-M+1) g&H+(,2M--l)D+mA'l^ E = i(A +A ) * XX yy
n = M
In the second column of table 3.3. a number of second order shifts AV2 are listed, as calculated from the formulas (3.16a,b). To check these values the 1 2 x 1 2 matrix belonging to the Hamiltonian
"^EPR ^ "^n "^^ diagonal ized exactly for the same values of M and rrj^
From the ENDOR frequencies calculated in this way the first order contributions (3.8a) were subtracted. The results listed in table 3.3 are denoted by Av'. The agreement with the second order shift is satisfactory.
Table 3.3. Some of the second order shifts Av2 (in kHz) belonging to proton HlWl (c^_g), calculated by means of (3.16a,b). For the shifts denoted by Avi see text.
MjW
1
5/2, -5/2
3 / 2 , - 3 / 2 1/2, 3/2 - 1 / 2 , 3/2 - 3 / 2 , - 5 / 2 - 5 / 2 , - 3 / 2 AV2 - 2.29 - 4 . 7 0 - 3.05 - 1.23 0.38 0.52AV'
1
- 1.84 - 4 . 5 0 - 3.16 - 1.34 0.40 0.52The s'econd order shift is subtracted from the experimental ENDOR frequency and the result is substituted again in the equations
(3.12a) and (3.12b). Hereafter the final values of the tensor elements are evaluated from the corrected values of H« . and H\^. (i = x, y, z).
In the interpretation of the SHFI two different types of
9) magnetic interactions with the nearby protons can be distinguished First the Fermi contact term which gives an isotropic contribution to the SHFI tensor. Secondly the dipolar interaction, which causes the anisotropic part of the SHFI tensor. Since the anisotropic tensor is traceless the isotropic SHFI A , follows from
1
^i "l^^xx- ^y - ^zz^'
(3.17)The anisotropic SHFI tensor is obtained by subtracting A . from the experimental diagonal tensor elements. In general the experimen-tal reference frame chosen for the investigated proton is not
coinciding with the principal reference frame of the anisotropic part of the SHFI tensor. By means of an exact diagonalization
the three principal elements are determined as well as the direction cosines of the principal axes, expressed in the experimental reference frame. As a result of the mounting method of the samples (see
2. Because the angle 6 depends on the magnetic quantum number M ENDOR lines of the same proton but belonging to a
different quantum number may be maximal or minimal at dif-ferent values of c)), unless c = 0 (see section 3.4).
3. At ^ = 0 equation (3.21) reduces to
^ = h-^ g^^^i^H-MH^^)^ ^ M^H^^ )K
which is the same result as is obtained from equation (3.11) by replacing the quantity M" by M and the subscript z by x. Analogous the equation for the y-direction is obtained for (j) = 90°.
Finally the ignored terms in the spin Hamiltonian (3.1), the second order shift excluded, can be accounted for by replacing the magnetic quantum number M in equation (3.22) by M", which can be calculated after (3.1) has been transformed to the x", y", z" reference frame.
Prediction of the angular dependence of the c \_ H_ ENDOR spectra
by means of the point dipole model.
In order to facilitate the assignment of the o \_ H_ spectra the angular dependence of the ENDOR frequencies was predicted by means of a SHFI tensor calculated on the basis of the point dipolar
interaction, using approximate proton positions.
In the point dipole model the electronic spin is considered as a magnetic point dipole located at the position of the manganese nucleus. Suppose the proton to have the co-ordinates (ic_, 2/ » s )
hr £r tr
with respect to the manganese nucleus. The dipolar interaction between the electronic spin and the proton can be written in the
