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5_10_2.tex
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%translator Savrov, date 23.03.13
%____________________________________________________________
\setcounter{Equation}{0} \setcounter{Figure}{0}
\vspace{-12pt}
\Work %10.2
{Hyperfine structure of electron paramagnetic resonance}
{Hyperfine structure of electron paramagnetic resonance}
{Hyperfine splitting of EPR line originated due to interaction of atomic electrons and nuclear magnetic moment is studied.}
Electron paramagnetic resonance was discovered by E.~Zavoisky in $1944$ who found that a paramagnetic sample placed in a constant magnetic field can absorb the energy of electromagnetic wave. The energy absorption is selective (resonant) because it occurs only when a certain relation between the field induction and the frequency of electromagnetic wave holds. For this reason the discovered phenomenon was named \emph{electron paramagnetic resonance} (EPR).
The discovery of EPR was preceded by several theoretical and experimental works. In $1922$ A.~Einstein and P.~Ehrenfest mentioned a possibility of radiation induced transition between magnetic levels of atoms. Using this assumption Ya.~Dorfman in~$1923$ predicted a resonant absorption of electromagnetic waves by a paramagnet. Dutch physicist K.~Gorter attempted to detect the resonance energy absorption by paramagnet using a calorimetric method based on measuring the heat effect. However the sensitivity of this method was insufficient, so it was E.~Zavoisky who detected the EPR signal first. The theory of the induced resonance transition between two neighboring Zeeman levels was proposed by Ya.~Frenkel.
A hyperfine splitting of EPR line (HS) was discovered in $1948$ by S.~Altshuler, B.~Kozyrev, and~S.~Salikhov in water solution of Mn$^{++}$ and Cu$^{++}$ salts. They experimented with electromagnetic waves of a frequency about $10^8$\,Hz and with a moderate concentration of paramagnetic ions; these conditions correspond to the Zeeman effect for weak field.
\vspace{-6pt}
\hFigure{Fine and hyperfine structures of $\textrm{Mn}^{2+}$ spectrum: \emph{a}~spectrum of $\textrm{Mn}^{2+}$ in manganese apatite, each of five lines of fine structure is split in six hyperfine components; \emph{b}~hyperfine structure of spectrum of $\textrm{Mn}^{2+}$ in water solution $\textrm{MnCl}_2$}10_1_1 {7.5cm}{3.2cm}{pic/L10_1_01.eps}
\vspace{-6pt}
\noindent
The hyperfine splitting is caused by interaction of electron magnetic moment and magnetic momenta $I$ of nuclei, such as hydrogen ($I=1/2$), nitrogen ($I=1$) etc.
A nuclear magnetic moment is three orders of magnitude less than that of electron. The corresponding splitting is therefore small. The splitting is due to the fact that electron magnetic moment interacts with a different field depending on possible orientation (one of $2I+1$) of nuclear magnetic moment in constant magnetic field. Therefore the number of lines of hyperfine structure is $2I+1$.
The diagram in~\refFigure{10_1_1}\emph{a} shows a fine and hyperfine structure of $\mathrm{Mn}^{2+}$ in manganese apatite at a frequency of $10^{10}\;\Hz$. Since the spin of mangenese nucleus is $I=5/2$ each of five lines of the fine structure splits in six lines of hyperfine components. The diagram in~\refFigure{10_1_1}\emph{b} shows the absorption spectrum observed in water solution of $\mathrm{MnCl}_{2}$ at $9345\;\MHz$ and in the magnetic field varied between $2900$ and \linebreak $3400\;\ersted$.
Let us consider the mechanism of hyperfine splitting using the interaction of uncoupled electron with a paramagnetic nitrogen nucleus as an example (see~\refFigure{10_3_1}). This interaction is observed in a molecule of NO and nitroxyl radicals which are widely used to study various biological systems.
\hFigure
{Energy diagram that illustrates the origin of hyperfine structure of EPR spectrum of paramagnetic molecule of NO}10_3_1
{7.09cm}{5.74cm}{PIC/l10_3_01.eps}
If a free electron is localized near a nitrogen nucleus, the magnetic field due to the nucleus magnetic moment ${\bf\mu}_N$ and the external magnetic field $\textbf{H}_0$ add up. A nitrogen nucleus has spin $I=1$, therefore three orientations of ${\bf\mu}_N$ are allowed: along, against, and perpendicular to $\textbf{H}_0$. These orientations of nuclear spin correspond to three values of magnetic quantum number, $I_z=+1,\,0,\,-1$. Hence, the interaction of a free electron and a nitrogen nucleus splits each Zeeman level of the electron into three ones as it is shown in~ \refFigure{10_3_1}. The transitions induced by microwave radiation between the energy levels must satisfy the selection rules: $\Delta S_z=\pm 1$ (orientation of electron spin changes) and $\Delta I_z=0$ (orientation of nuclear spin is preserved).
Thus the hyperfine splitting of EPR spectrum of a nitroxyl radical gives three lines corresponding to three possible orientations of nitrogen nucleus ($I_z=-1,\,0,\,+1$). Such a hyperfine splitting of EPR line is observed in a free stable radical ditretbutyldiphenylnitroxid (below it is referred to as \emph{F}) which chemical structure is shown in~\refFigure{10_3_2}.
\cFigure
{Chemical structure of ditretbutyldiphenylnitroxid}10_3_2
{7.4cm}{1.35cm}{PIC/l10_3_02.eps}
Notice that the EPR spectrum of a crystalline substance exhibits only a single narrow line. This is due to a strong exchange interaction between free electrons of neighboring molecules. The interaction turns out to be greater than the energy of hyperfine splitting. An averaging over many molecules, the typical feature of exchange interaction, results in disappearance of a hyperfine structure. A solution of a paramagnetic substance provides a possibility to observe the EPR hyperfine splitting; in this case the exchange interaction is much weaker because the distance between the molecular centers in the solution is significantly greater.
The distance between adjacent lines of hyperfine spectrum is determined by a hyperfine coupling~\emph{A} which significantly depends on electron density near the nucleus. If atomic nucleus or one of the atomic nuclei of a molecule has a spin~{\textbf{I}}, each level of hyperfine structure is specified by the net angular momentum $\textbf{F}=\textbf{J}+\textbf{I}$, where \textbf{J}~is the vector sum of the net angular momentum of electrons and the orbital angular momentum of nucleus. The quantum number~$F$ takes a value $F=|J-I|,\,|J-I|+1,\,\dots,\,J+I$ ($J$ and $I$ is the net angular momentum of electrons and nucleus, respectively). If $J\ge I$ the number of levels equals $2I+1$, for $J<I$ it is $2J+1$. The energy of a level is
$$
E_F=E_J+E_{M1},\,\eqno(1)
$$
where $E_J$~is the level energy in the absence of hyperfine splitting and $E_{M1}$~is the energy of magnetic dipole-dipole interaction (we do not take into account the interaction between the nucleus quadrupole moment and a nonuniform electric field in atom since it is usually much less than the magnetic dipole interaction). The interaction energy is
$$
E_{M1}={1\over 2}\hbar AC,\quad C=F(F+1)-I(I+1)-J(J+1), \eqno(2)
$$
where the hyperfine coupling $A$ has the dimension of frequency (Hz) and it is determined by the expectation value of the operator of magnetic interaction averaged over the state with angular momentum $F$.
\bigskip
\textbf{EPR spectrometer <<Ìèíñê--12Ì>>}
\vspace{4pt}
Detection of hyperfine splitting of EPR spectrum requires a sensitive spectrometer. In this experiment an EPR spectrometer <<Ìèíñê-12Ì>> is used (see~\refFigure{10_3_3}).
\hFigure
{The front panel of spectrometer: microammeter of a control unit (\emph{1});
control of magnetic induction (\emph{2});
control of high-frequency modulation (\emph{3});
shaft with ampule holder (\emph{4});
chuck for ampule with a sample (\emph{5});
switch of time constants ÔÍ× and ÓÍÒ (\emph{6});
control of signal amplifier (\emph{7});
control knob of frequency adjustment of UHF generator (\emph{8});
control of rate of magnetic induction sweeps (\emph{9});
interposition of direct and reverse sweeps (\emph{10});
control of range of magnetic induction sweeps (\emph{11});
switch of ranges of sweep rates (\emph{12});
switch of magnetic induction sweeps (\emph{13});
fine adjustment of magnetic induction (\emph{13});
control of detector current (\emph{15});
control of ÓÏÒ output (\emph{16});
starter button of UHF generator (\emph{17});
control of triangular sweep voltage (\emph{18});
control of magnet current (\emph{19})}10_3_3
{9.18cm}{7.48cm}{PIC/l10_3_03.eps}
The spectrometer operates at the frequency of 10\,GHz generated by a Gunn diode. The EPR spectrum is actually the intensity of absorption of UHF wave versus the magnetic induction. The power of UHF wave is measured by a detector based on a Schottky diode. The ampule with a sample is placed into a measuring bulk resonator located in the electromagnet air gap. The magnetic induction in the gap can be varied between $100$ and $4500$\,G using the knobs~\emph{2} (coarse) and~\emph{15} (fine).
\cFigure
{Triangular modulation of magnetic induction of spectrometer}10_3_4
{9.7cm}{4.8cm}{PIC/l10_3_04.eps}
\vspace{-8pt}
The magnetic induction is modulated by a triangular wave in order to obtain a resonant induction periodically (see~\refFigure{10_3_4}). The wave amplitude can be varied between $0$ and $500$\,G by a potentiometer <<\emph{Range of Í}>>~with a knob~\emph{12}. The sweep rate is controlled by a potentiometer <<\emph{Sweep time}>>~with a knob~\emph{10}. When the magnetic induction passes through the resonance value, the absorption of UHF wave by the sample increases sharply, which is registered by the detector.
The spectrometer sensitivity is increased by means of a high-frequency amplitude modulation of UHF wave and a subsequent synchronous detection of the UHF signal. The modulation amplitude is controlled by a potentiometer <<\emph{Modulation level}>>~with a knob~\emph{3}. As a result the spectrometer measures the first derivative of the EPR absorption spectrum.
The spectrometer UHF generator uses an electronic starter. By pressing the button~\emph{19} the power voltage of the Gunn diode is modulated by a triangular waveform. This initiates search and capture of the generator frequency by the natural frequency of the measuring resonator that is detected by a decrease of the detector current.
Before the experiment one should calibrate the magnetic induction sweeps. To do this adjust the optimal conditions (see the spectrometer operation manual above) and obtain the EPR spectrum of a DPPH polycrystalline sample (its structure is described in~5.10.1).
\hFigure
{Typical EPR spectra (the first derivative of absorption spectrum): \emph{a}~without hyperfine splitting, a singlet; \emph{b}~hyperfine splitting, triplet. \emph{A}~hyperfine coupling}10_3_5
{6.1cm}{5.5cm}{PIC/l10_3_05.eps}
It is known that the width of the derivative of absorption signal of DPPH is $\Delta Í=2{.}3$\,Oe at $T=295$\,Ê and $\nu=10$\,ÃÃö (see~\,\refFigure{10_3_5}\emph{a}). When working with the spectrometer it is advised to compare the results with the readings of a divider of magnetic induction sweeps on the front panel.
\Task
\textbf{Observation of fine and hyperfine structure of EPR spectra.}
Plug the oscilloscope to the mains of 220\:V.
Buttons <<{2\:X--Y}>> in the chanell block and <<{X--Y}>> in the sweeps block must be pressed, the rest must be released. The <<X>> channel of oscilloscope is controlled by magnetic induction sweeps. The optimal attenuator range is $0{.}06$--$0{.}2$\,V/div. The <<{Y}>> channel of oscilloscope amplifies the absorption signal. The optimal attenuator range is $1$--$20$\,mV/div.
Place the ampule with the studied substance using the chuck, first, with DPPH (sample $1$). The sample presence in the resonator is detected by a decrease of the detector current at the pressed button~\emph{17} <<{I}>>. The ampule should be installed so that the color stripe on the ampule is at the level of chuck butt.
Obtain the EPR signal. To this end set the switch~\emph{12} (amplitude of magnetic induction sweeps) to the middle position. The switches~\emph{3} and~\emph{7} (amplitude of HF-modulation and signal amplification) must be at the rightmost position, the switch~\emph{14} (magnetic induction sweep) must be at <<\emph{start}>>, set the switch~\emph{10} (range of sweep rates) at~\emph{100}, set the switch~\emph{13} (sweep range <<{ÌÑ-Ñ}>>) to the down position <<{ÌÑ}>>. Button~\emph{21} <<H$_0$>> (electromagnet current) must be pressed. By rotating the control knobs of magnetic induction~\emph{2} <<H$_0$>> (coarse) and~\emph{15} (fine) obtain the EPR signal of DPPH (see\:Fig.\,5).
Minimize distortions of the EPR spectrum on the oscilloscope screen by adjusting the sweeps rate, the modulation level, and the signal amplification. Align the signals of direct and backward sweeps by using the knob~\emph{11} <<\emph{align}>>. To reduce the noise set the knob~\emph{6} <<\emph{time constant}>> at <<$0{.}10$>>.
Replace the DPPH sample with a sample of radical <<F>> and obtain the EPR signal of the samples $2$, $3$, $4$, and $5$; each successive sample has a lower concentration of radical <<F>> up to the complete hyperfine structure of the EPR line (see Fig.\:6\emph{b}). Sample $2$ is the radical <<F>>, sample$3$~is its solution of $2$--$10$\,mol/l, sample $4$~has $10\cdot 10^{-3}$\,mol/l, and sample $5$~has $50\cdot10^{-3}$\,mole/l. Explain why the spectrum structure changes along with dilution of radical <<F>>. Evaluate the resonance width by comparison with the EPR signal of DPPH.
Evaluate the hyperfine coupling from the obtained hyperfine structure of EPR spectrum.
\Literat
{\small
1.\;\emph{Öèïåíþê\;Þ.\,Ì.} Êâàíòîâàÿ ìèêðî- è ìàêðîôèçèêà.~--- Ì.: Ôèçìàòêíèãà, 2006.
2.\;\emph{Áëþìåíôåëüä\;Ë.\,À., Òèõîíîâ\;À.\,Í.} Ñîðîñîâñêèé îáðàçîâàòåëüíûé æóðíàë. 1997. ¹\,9.
C.\,91.
3.\;\emph{Âîíñîâñêèé\;Ñ.\,Â.} Ìàãíåòèçì.~--- Ì.: Íàóêà, 1971.
}