{"id":335,"date":"2018-11-16T10:11:51","date_gmt":"2018-11-16T10:11:51","guid":{"rendered":"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=335"},"modified":"2019-04-30T08:37:16","modified_gmt":"2019-04-30T08:37:16","slug":"esr-nmr-and-mossbauer-spectroscopy","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/chapter\/esr-nmr-and-mossbauer-spectroscopy\/","title":{"rendered":"ESR , NMR and M\u00d6ssbauer Spectroscopy"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n \r\n\r\nContents<\/strong>\r\n\r\n \r\n\r\n1.\u00a0 <\/strong>Electron Spin Resonance (ESR)<\/strong>\r\n\r\n2.\u00a0 <\/strong>Nuclear Magnetic Resonance (NMR)<\/strong>\r\n\r\n3.\u00a0 <\/strong>M\u00d6ssbauer Spectroscopy<\/strong>\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\n1. Electron Spin Resonance (ESR)<\/strong>\r\n\r\n \r\n

Electron Spin Resonance (ESR) also known as Electron Paramagnetic Resonance (EPR) is an important method for obtaining information about paramagnetic substances. ESR absorption was first observed by Zavoisky in 1945 at Kazan and by Cummerow and Halliday in USA.<\/p>\r\n \r\n

This is high resolution spectroscopy that uses frequencies in the microwave region (~ 109<\/sup> \u2013 1011<\/sup> Hz) and is concerned with microwave induced transitions between magnetic energy levels of electron having a net angular momentum. ESR differs from simple microwave spectroscopy becauseit concern with paramagnetic materials only.<\/p>\r\n \r\n\r\nRegarding Substances for Investigation by ESR<\/strong>\r\n\r\n \r\n

As ESR requires the presence of the unpaired electrons in the sample to be studied; its range of applications is restricted to paramagnetic substances and to substances that can be converted to a paramagnetic form with sufficient stability for a spectrum to be observed. Para-magnetism occursin:<\/p>\r\n \r\n\r\n1.\u00a0<\/em>Atom and ions:<\/em>\r\n\r\n\u00a0<\/em>\r\n

All configurations with an odd number of electrons must possess angular momentum and therefore must be paramagnetic.<\/p>\r\n\u00a0<\/em>\r\n\r\n2.<\/em>Molecules and molecular ions:<\/em>\r\n\r\n\u00a0<\/em>\r\n

Molecules such as NO and NO2<\/sub> have odd number of electrons and are therefore paramagnetic. The molecules such as O2<\/sub> though having an even number of\u00a0electrons, but have a ground state with a partially filled molecular shell are thus paramagnetic.<\/span><\/p>\r\n \r\n

3.\u00a0<\/em>Transition group impurities:<\/em><\/p>\r\n \r\n

These are atoms of ions with incomplete 3d, 4d, 5d, 4f or 5f shell. However, not all the valence states of these transition metal ions are paramagnetic. The\u00a0<\/span>most commonly observed paramagnetic ions are V4<\/sup>+, VO2<\/sup>+, Ti3<\/sup>+, Cr3<\/sup>+, Mn2<\/sup>+, Fe3<\/sup>+, Fe2<\/sup>+, Co2<\/sup>+, Ni2<\/sup>+,Cu2<\/sup>+, Pd2<\/sup>+, Ru2<\/sup>+, Os2<\/sup>+, Gd3<\/sup>+, Eu2<\/sup>+, Mo5<\/sup>+\u2026.. .<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\u00a0<\/em>\r\n\r\n4.\u00a0\u00a0\u00a0\u00a0 Color centers (e.g. Vk, Fcentres)\r\n\r\n5.\u00a0\u00a0\u00a0\u00a0 Organic and inorganic radicals\r\n\r\n6.\u00a0\u00a0\u00a0\u00a0 Donors and acceptors in semiconductors such as phosphorous donor impurities in silicon\r\n\r\n7.\u00a0\u00a0\u00a0\u00a0 Activators and co-activators in phosphors, such as self-activated ZnS\r\n\r\n8.\u00a0\u00a0\u00a0\u00a0 Radiation damage centers\r\n\r\n9.\u00a0\u00a0\u00a0\u00a0 Conduction electrons\r\n\r\n \r\n\r\nResonance Condition<\/strong>\r\n\r\n \r\n

Spin is the only magnetic moment that is associated with thefree electron. Classically the energy of interaction between the magnetic and magnetic field Bis<\/p>\r\n \r\n\r\nE=-\u00b5e<\/sub>B<\/strong>\r\n\r\n \r\n

The magnetic moment is a vector and is collinear with spin angular momentum vector S<\/strong>. The operators for the vectors are related by<\/p>\r\n \r\n\r\n\u00b5e<\/sub>=-\u03b3S<\/strong>\r\n\r\n \r\n

Here \u03b3 is the magnetogyric ratio that is the ratio between magnetic and mechanical moments. The value of \u03b3 is given by ge (e\/2me),\u00a0ge is positive dimensionless factor and its value is 2, (itsvalue is 2.0023 from quantum electrodynamical calculations).<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

\u00b5e<\/sub>is antiparallel to S because of the negative charge of the electron.<\/p>\r\n \r\n\r\n\u00b5e<\/sub>=-ge<\/sub>\u03b2e<\/sub>S\r\n\r\n \r\n\r\nS is dimensionless. Replace \u00b5e in above equationto obtain the quantum mechanical Hamiltonian,\r\n\r\n \r\n\r\nH=ge<\/sub>\u03b2e<\/sub>S.B<\/strong>\r\n\r\n \r\n\r\nAssumingthe magnetic field in z direction, Bx =By = 0 and Bz = B\r\n\r\n \r\n\r\nSo that\r\n\r\n \r\n\r\nH=ge<\/sub>\u03b2e<\/sub>Sz<\/sub>B\r\n\r\n \r\n

Here \u03b2e<\/sub> is Bohr magnetonand is e\u0452\/2meand the eigenvalues are the multiples of the eigenvalues of Sz\u00a0<\/sub>given by<\/p>\r\n \r\n\r\nE=ge\u03b2eBM\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (M=1\/2 and -1\/2)\r\n\r\n \r\n\r\nTherefore\r\n\r\n \r\n\r\nE=\u00b1(1\/2)ge\u03b2eB\r\n\r\n \r\n

The energy difference between the two states is<\/p>\r\n

\u0394E=h\u03bd=E2<\/sub>-E1<\/sub>=ge<\/sub>\u03b2e<\/sub>B<\/p>\r\n \r\n

There is an increase in separation between two levels with the magnetic field which is linear.<\/p>\r\n \r\n

The lowest state has the negative sign and corresponding to magnetic moment aligned parallel to the magnetic field and hence spin antiparallel to it.<\/p>\r\n \r\n

The figure depicts the energy level of an electron in a magnetic field where microwave radiation causes resonance at a field h\u03bd\/g\u03b2e<\/sub>.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

Transition between the two levels can be induced by microwave magnetic field B<\/strong>1<\/strong> at right angle to B. This establishes the magnetic dipole and the parity do not change. The selection rule for the magnetic quantum number M is \u2206M = \u00b1 1. This is in contrast to the electric dipole transitions observed in electronic spectra where the parity of the states differs.<\/p>\r\n \r\n

Substituting the values of constants in \u0394E=h\u03bd=E2<\/sub>-E1<\/sub>=g\u03b2e<\/sub>B<\/p>\r\n \r\n\r\nB (mT) = 35.724v (GHz)\r\n\r\n \r\n

The different frequency bands and magnetic field for resonance (g = 2)for conventional ESR spectrometer are<\/p>\r\n \r\n\r\n\r\n\r\n\r\n\r\n
Band<\/td>\r\n\u03bb(cm)<\/td>\r\nv(GHz)<\/td>\r\nB(T)<\/td>\r\n<\/tr>\r\n
S<\/td>\r\n9.0<\/td>\r\n3<\/td>\r\n0.11<\/td>\r\n<\/tr>\r\n
X<\/td>\r\n3.0<\/td>\r\n9<\/td>\r\n0.33<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n
K<\/td>\r\n1.2<\/td>\r\n24<\/td>\r\n0.85<\/td>\r\n<\/tr>\r\n
Q<\/td>\r\n0.8<\/td>\r\n35<\/td>\r\n1.25<\/td>\r\n<\/tr>\r\n
E<\/td>\r\n0.4<\/td>\r\n70<\/td>\r\n2.50<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nThe above equation establishes the dependence on\r\n\r\n \r\n

1.\u00a0\u00a0\u00a0\u00a0 There is the appearance of spin subleveldue to Magnetic field B<\/strong>and the energy difference between themcan be determined.<\/p>\r\n \r\n

2.\u00a0\u00a0\u00a0\u00a0 The microwave frequency corresponding to energy quantum hv<\/em> causes transitions from M = -1\/2 to M = +1\/2 statesthat produces an absorption signal.<\/p>\r\n \r\n

3.\u00a0\u00a0\u00a0\u00a0 The g-value defines the change in the position of the absorption line in the spectrum under given hv<\/em> and B depending on the features particular to the state of paramagnetic electron in the reference sample.<\/p>\r\n \r\n\r\nESR by Precession<\/strong>\r\n\r\n \r\n\r\nThe torque N acting on magnetic dipole in a magnetic field is\r\n\r\n \r\n\r\nN <\/strong>= \u00b5e<\/sub>\u00d7B<\/strong>\r\n\r\n \r\n

Let S<\/strong> is the angular momentum and the rate of change of angular momentum will give torque, therefore<\/p>\r\n\"\"\r\n\r\nUsing the above equations \u00b5e=-\u03b3S<\/strong>\r\n\r\n\"\"\r\n\r\n \r\n\r\nEvidently,the change in \u00b5eis perpendicular to both \u00b5eand B.\r\n\r\n \r\n\r\nAs the magnetic field is considered in z- direction i.e. <\/span>B<\/strong> = B<\/span>k<\/strong><\/sub> and<\/span>\u00b5<\/strong>e<\/sub> =\u00b5x<\/sub><\/span>i<\/strong>+\u00b5y<\/sub><\/span>j<\/strong>+\u00b5z<\/sub><\/span>k<\/strong>So<\/span>\r\n\r\n \r\n\r\nAs this relates to two vectors, the components on each side will be identical. Therefore,<\/span>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\nThese infers\r\n\r\n \r\n\r\n\u00b5z<\/sub>=Constant;\u00b5x=cos(\u03c9L<\/sub>t);\u00b5y<\/sub>=sin(\u03c9L<\/sub>t)\r\n\r\n \r\n\r\nTaking \u03c9L=\u03b3B\r\n\r\n \r\n

and \u00b5 (magnetic Moment) precess about B (magnetic field) with a constant frequency\u03c9L making a fixed angle with the direction of static magnetic field B as depicted in the figure. The frequency \u03c9Lis referred asLarmor frequency. Now considering a second coordinate system (x\u2019,y\u2019,z\u2019) rotating about z axis at an angular velocity \u03c9.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

The figure mentions the fixed laboratory coordinates x, y, z and rotating coordinates x\u2019, y\u2019, z\u2019 for the angular velocity along z-axis. The axes marked primed and unprimed are the same at t = 0<\/p>\r\n \r\n\r\nThe connection between the laboratory and rotating frames is\r\n\r\n\"\"\r\n

The rate of change of \u00b5 measured in the rotating system can be calculated, as it appears that the rotating system was stationary having an effective magnetic field B\u2019 <\/strong>along z-axis. If B\u2019 <\/strong>is constant and \u03c9= \u03b3 B<\/strong>, the effective field B\u2019<\/strong>has no role to\u00a0play. The magnetic moment is without torque in the rotating frame and remains constant with respect to it.<\/span><\/p>\r\n \r\n

ESR experiments employ time dependent components and do not use a constant <\/span>B<\/strong> compared to the constant component. Let total magnetic field have a z component which is of constant magnitude <\/span>B<\/strong> and it also have an oscillating component along x axis with a peak to peak amplitude of 4B1 such that B1 << B The time dependent field is<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

The first term is a field of constant amplitude 1 that rotates about z-axis at frequency \u03c9 in a similar way as electronic precession. The second term is similar but with opposite of electronic precession. The component rotating opposite to the sense of electronic precession does not cause resonance, so neglecting this, total field Be<\/sub><\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

fig. fixed laboratory coordinates x,y,z and rotating coordinates x\u2019,y\u2019,z\u2019 for the angular velocity \u03c9 along Z-axis. The primed and unprimed axes being the same at t=0<\/p>\r\n \r\n

B1 is constant in a coordinate system that rotates about the z direction. In this rotating frame it is shown by<\/p>\r\n\"\"\r\n\r\nB\u2019 <\/strong>= B \u2013\u03c9\/\u03b3. B<\/strong>1<\/strong> is along x\u2019 axis of rotating coordinate system.\r\n\r\n \r\n

B\u2019 <\/strong>is along z or z\u2019 axis. When B<\/strong>=\u03c9\/\u03b3 <\/strong>the condition for magnetic resonance occurs and Be = B1.<\/p>\r\n \r\n

B\u2019 is either nearly parallel to B or antiparallel to B as B1<<B apart from magnetic resonance. The motion of magnetic moment is of a precession about B1 with\u00a0angular velocity \u03b3B1 and every half cycle of this motion it changes from being parallel to antiparallel and back again. So, in the rotating system this motion take place in the plane normal to B.<\/span><\/p>\r\n \r\n

Now, as <\/span>B<\/strong>1<\/strong><<<\/span>B<\/strong>, the precession about B1 occurs at a much lower velocity than that at which <\/span>B<\/strong>1<\/strong> rotates in the laboratory system. Thus In the latter system the motion of the angular momentum and magnetic moment consists of a rapid motion about <\/span>B <\/strong>at an angle varying from 0 to \u03c0and back again. Thus when<\/span> B <\/strong>=\u03c9\/\u03b3 the magnetic moment assume to be initially parallel to <\/span>B<\/strong> can be completely reversed by an application of rotating field <\/span>B<\/strong>1<\/strong>. The value of <\/span>B<\/strong>1<\/strong> is very small.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nRelaxation Mechanisms<\/strong>\r\n\r\n \r\n

The populations of the two levels with M = -1\/2 and M = \u00bd are determined by the Boltzmann distribution, under thermal equilibrium conditions and accordingly,<\/p>\r\n\"\"\r\n

Consider N1 and N2 are the population of M= -1\/2 and M = \u00bd levels, respectively. Though N1-N2 is very small, but the phenomenon of ESR absorption depends on this difference. There is an equal probability of transition from lower level to upper level to that from upper level to lower level. However, there is an excess of upward transitions as N1>N2.and there is a net absorption of energy from microwave field. This leads to an increase of N2 that will continue until N1 = N2 and net absorption of energy will tend to zero.<\/p>\r\n \r\n

In case this situation does not occur, there must therefore be other mechanisms through which energy absorbed and stored in the upper level is dissipated in such a\u00a0way as to allow return to lower level to maintain the population difference. These mechanisms are called relaxation processes.<\/span><\/p>\r\n \r\n

(a) Spin- lattice relaxation<\/strong><\/p>\r\n \r\n

The electron spins are randomly oriented in space and their resultant moment is zeroin the absence of magnetic field. The spins become aligned parallel or antiparallel to <\/span>B<\/strong> when a static magnetic field <\/span>B<\/strong> is applied. Because of the slight less spins having the parallel alignment to give the population difference, magnetization Mz is observed. A certain time interval is required to obtain equilibrium valuethat is same as that needed for the spins to become again randomly oriented when the magnetic field is suddenly switched off. This reorientation time is refereed as spin- lattice relaxation time T1 that measures the characteristic time for recovery of the magnetization of the paramagnetic system along the static field direction after the equilibrium is disturbed.<\/span><\/p>\r\n \r\n

There is an interaction between the spins and lattice vibrations that generate phonons. however, the spin- magnetic moment is not influenced directly by vibrations of the lattice. The coupling of the lattice vibrations with magnetic spin states occurs indirectly through residual spin-orbit coupling. The crystalline electric field in ionic compounds or chemical bonds in molecular free radical raise the degeneracy of the orbital states, usually leaving an orbital singlet as ground state. The orbital singlet behaves like an atomic S state. However, orbital magnetic field is not completely quenched because of a second- order admixture of the singlet orbital ground state with higher orbital states. Therefore, there is slight orbital magnetic field acting on the spin moment. The orbital moments are\u00a0\u00a0<\/span>strongly coupled to the lattice through strong crystalline electric fields. If the lattice is vibrating, interatomic distances vary and therefore crystalline field also vary. As a result, the residual orbital magnetic field acting on the electron spin is effectively modulated by all vibrational modes. Practically,the concern is with the phonons having frequency 3-30GHZ corresponding to the residual orbital field component that is transverse to <\/span>B<\/strong>. In general one expects large value of T1 with decrease of temperature because of the freezing of lattice vibrations. Three processes are proposed for spin-lattice relaxation.<\/span><\/p>\r\n \r\n

(i)\u00a0<\/strong>Direct Process<\/strong><\/p>\r\n \r\n

The direct process involves phonons of the same frequency as the ESR resonance quantum hv below<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

The figure illustrates that a spin makes a transition to the lower state emitting a phonon at the resonant frequency v<\/em>. Only a small fraction of the normal distribution of the thermal energy is concentrated in vibrational frequencies as low as the ESR frequencies. Therefore, phonons at the resonance frequencies are normally scarce. The process is prominent at low temperatures.<\/p>\r\n \r\n

(ii)\u00a0<\/strong>Raman Process<\/strong><\/p>\r\n \r\n

In this process lattice phonons are scattered by the spins and (Fig.9.5), with the ESR frequency adds to or subtracted from the frequency of scattered phonons. This is a two-phonon process.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

The figure shows the relaxation by the Raman process where phonon hv\u2019<\/em> is absorbed and phonon hv\u2019\u2019<\/em> is emitted accompanied by down transition of the spin<\/p>\r\n \r\n

There must be lattice modes having difference frequencies equal to the ESR frequency.Such vibrations, however, can be in the higher frequency region, more densely populated at room temperature. In this process, many phonons pair can participate since the only requirement is that their frequency difference be equal to ESR frequency. Effectiveness of this process decreases with the decrease in temperatureas the vibration at higher frequencies gets frozen at low temperatures.<\/p>\r\n\r\n<\/div>\r\n \r\n\r\n(iii)\u00a0\u00a0<\/strong>Orbach Process<\/strong>\r\n\r\n \r\n

This process involve absorption of a phonon by direct process to excite the spin system from the upper level to a much higher level at an energy \u03b4 above the ground doublet, then dropping back into lower Zeeman level of the ground state by emitting another phonon of slightly different energy.<\/span><\/p>\r\n\r\n

\r\n\r\n\"\"\r\n

The figure shows the energy level diagram that explains the Orbach process. This mentions level \u2018b\u2019 and a relax by way of two direct transitions involving a third level \u2018c\u2019.In this process, the paramagnetic ion is indirectly transferred from upper Zeeman level to the lower Zeeman level of the ground doublet. It is more restricted than the Raman process because two specific phonons are involved.Only the Direct Process is significant at liquid- Helium temperature whereas Raman and Orbach processes dominate at higher temperatures.<\/p>\r\n \r\n\r\n(b) Spin-spin relaxation\r\n\r\n \r\n

It contains all those mechanisms in which the spins can exchange energy amongst themselves, instead of giving it back to the lattice, or molecular system.<\/p>\r\n \r\n

(i)Dipole interaction: It arises from the influence of the magnetic field of one of the paramagnetic ion on the dipole moments of neighboring ions. The actual local field at any given site will depend on the arrangement of the neighbors and the direction of their dipole moments. If the external magnetic field acts on the paramagnetic compound, the local field at each ion must be added vectorially to it. If the local field is small compared with the external field (which might be - tesla), only the component of the former parallel to the latter is important. The size of this component varies from site to site, giving a random displacement to the resonance frequency of each ion. If the paramagnetic ions are identical, so that they precess at the same frequency in the external magnetic field, there is an additional resonance interaction. The precessing components of one magnetic dipole set up an oscillatory field at another dipole which is just at right frequency to cause\u00a0<\/span>magnetic\u00a0 resonance transitions and\u00a0 vice\u00a0 \u00a0 versa.\u00a0 \u00a0 The\u00a0<\/span>mutual interaction produces resonance transitions that are equivalent to the exchange of quanta between neighboring ions. Thus the quanta are exchanged among neighboring ions by mutual spin flip and spins are in thermal equilibrium is disturbed, it is re-established exponentially with time constant T2 which is called spin-spin relaxation time. TI - 10-6 see and T2 - 10 -10 sec. T2 is independent of temperature.<\/span><\/p>\r\n \r\n

(ii)\u00a0 Exchange Coupling<\/span><\/p>\r\n \r\n

This is important only in undiluted crystals where the paramagnetic sites are so close together that the orbital of the unpaired electrons overlap. Here the spins interact electrostatically through a short-range interactionthat is called exchange interaction. This results in a change in width of absorption lines in the ESR spectrum.<\/span><\/p>\r\n \r\n

c) Cross Relaxation<\/span><\/p>\r\n \r\n

Considering a sample having two spin systems with different resonance frequencies. If the tails of the two absorption curves corresponding to the spin systems overlap, aflip flop spin exchange can take place in the overlapping region. This process is refereed to Cross- relaxation.As spin \u2013spin relaxation times are much shorter than spin lattice relaxation time, this process is effective for dissipating energy than direct transfer to the lattice.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

The figure depicts two overlapping lines centered on fields B and B\u2019<\/p>\r\n \r\n

Let us assume for example T1x<\/sup> (for species X) is very long and that T1Y<\/sup> (for species Y) is very short. If a quantum of energy absorbed by a spin of type X is given to a spin of type Y through a spin exchange, the quantum will have a high probability of being transferred to the lattice by the Y spin before it is transferred back to the X spin through a reverse exchange (because of short T1Y<\/sup>). The spin-lattice relaxation time of the X spin system is thus effectively reduced.<\/p>\r\n \r\n

Examples<\/span><\/strong><\/p>\r\n \r\n

    \r\n \t
  1. Calculate the ESR resonance frequency of an unpaired electron (g = 2) in a magnetic field of 0.335T\u00a0<\/span>From Eq.(9.8)<\/span><\/li>\r\n<\/ol>\r\n \r\n

    V\u00a0 =<\/span><\/p>\r\n \r\n

    2. Calculate the g value if the methy1 radical show ESR at 0.329 T in an ESR spectrometer operating at 9.230 GHz.<\/span><\/p>\r\n \r\n

    From Eq. (9.8)<\/span><\/p>\r\n \r\n

    g\u00a0 =<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n

    3. The ESR spectrum of an unknown sample shows a single line at 0.335 T at a frequency. The DPPH with g = 2.0036 show a line at 0.3375 T at same frequency. Determine the g value of ESR line of unknown sample.<\/p>\r\n \r\n\r\nThe resonance condition (Eq.9.8) for unknown sample becomes\r\n\r\n \r\n

    Hv = gsample \u03b2 e<\/sub> x 0.335T<\/p>\r\n \r\n

    For DPPH the resonance condition is<\/p>\r\n \r\n

    Hv = 2.0036x \u03b2e<\/sub> x 0.3375T<\/p>\r\n

    Equating above two equations<\/p>\r\n \r\n

    gsample<\/sub>\u00a0 =<\/p>\r\n \r\n

    4. Computer the population difference of two states of an electron spin in a magnetic field of 0.3 T at 300 K.<\/p>\r\n \r\n\r\nFrom Eqs. (9.8) and (9.46)\r\n\r\n \r\n\r\nFor g = 2, the above equation is\r\n\r\n \r\n\r\n2. Nuclear Magnetic Resonance (NMR)<\/strong>\r\n\r\n \r\n

    Nuclear magnetic resonance (NMR) was discovered in 1945 independently by Bloch and co-workers and by Purcell and co workers. NMR is the form of spectroscopy concerned with radio frequency induced transitions between magnetic energy levels of the nucleus having a nuclear angular momentum.<\/span><\/p>\r\n \r\n

    Difference between ESR and NMR<\/strong>:<\/span><\/p>\r\n \r\n

    There is the similarity of magnetic resonance phenomenon that occurs in electron spin resonance (ESR), except that it is of same type that of magnetic dipole moment of the nucleus that is involved. The nuclear magnetic moment is smaller than that of electron,as a result of mass difference. The resonance is correspondingly lower in frequency and fall in the radio frequency region.<\/span><\/p>\r\n \r\n

    Another important difference from ESR is that there is no longer a restriction to molecules with unpaired electrons. Any molecule with a magnetic nucleus will produce NMR spectrum. The NMR and ESR are similar in the sense that NMR deals with nuclear ground state while ESR deals with electronic ground state.<\/span><\/p>\r\n \r\n

    Principle<\/strong><\/p>\r\n \r\n

    Consider a nucleus having a magnetic moment \u03bcN.<\/span><\/p>\r\n \r\n

    The energy of interaction berween the nuclear magnetic moment and magnetic field <\/span>B<\/strong> is<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n

    E = - \u03bcN<\/sub>.B<\/p>\r\n \r\n

    The magnetic moment is a vector, which is collinear with angular momentum J<\/strong>. The operator for the vectors are related by<\/p>\r\n

    \u03bcN<\/sub> =\u03b3 J<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    where\u03b3=gN<\/sub>e<\/sub>\/2Mp<\/sub> is called gyromagnetic ratio, gN is nuclear g factor and Mp is mass of the proton. \u03b3 varies with the state it can have positive or negative value and is characteristic of the nuclei.<\/span><\/p>\r\n \r\n\r\nA dimensionless angular momentum operatorI<\/strong>is definedby\r\n\r\n \r\n

    J = (h\/2\u03c0) I<\/p>\r\n \r\n

    I2<\/sup> has eigenvalues I (I+1) where I is an integer or half integer. The component Iz<\/sub> has eigenvalue m where m can take values I, I-1, I-2,\u2026\u2026,-I. From above<\/p>\r\nequations\r\n\r\n \r\n

    \u03bcN = \u03b3 (h\/2\u03c0)I<\/p>\r\n \r\n\r\nSo the quantum mechanical Hamiltonian from\r\n\r\n \r\n

    H = - \u03b3 (h\/2\u03c0) I.B<\/p>\r\n \r\n

    If the magnetic field is assumed to be in the z direction, i.e. Bx = By= 0 and Bz = Bo, the Hamiltonian is<\/p>\r\n \r\n

    H = - \u03b3 (h\/2\u03c0) Iz<\/sub>Bo<\/sub><\/p>\r\n \r\n\r\nThe eigenvalues of Hamiltonian are just the eigenvalues of Iz, that is\r\n\r\n \r\n

    Em = -\u03b3 (h\/2\u03c0) Bo<\/sub>m<\/p>\r\n \r\n

    For I = 3\/2, m = 3\/2, \u00bd, -1\/2, -3\/2 .The figure depicts the energy levels for I = 3\/2 (constant magnetic field)<\/p>\r\n\"\"\r\n\r\n \r\n\r\nThe separation between levels are\r\n\r\n \r\n

    \u2206 E = \u03b3 (h\/2\u03c0)Bo<\/sub><\/p>\r\n\r\n<\/div>\r\nThe separation between the levels increases linearly with Bo<\/sub>. As the energy is related to frequency,<\/span>\r\n

    \r\n\r\n \r\n\r\n\u2206 E = hv = \u03b3 (h\/2\u03c0)Bo<\/sub>\r\n\r\n \r\n\r\n\u03bd= (\u03b3 Bo<\/sub>)\/2\u03c0\r\n\r\n \r\n\r\n\u03c9\u00a0 = \u03b3 Bo<\/sub>\r\n\r\n \r\n

    \u03c9\u00a0 is called Larmor\u2019s frequency. For H1 in normal magnetic field (2.35 \u2013 18.6T) the frequency is in the range 100 \u2013 800 MHz . Putting the value of \u03b3 in above equation<\/p>\r\n \r\n\r\n\u03bd= gN(e Bo)\/4\u03c0Mp<\/sub>\r\n\r\n \r\n\r\nhv<\/sub>= gN<\/sub> Bo (eh\/4\u03c0Mp)=gN<\/sub>\u03b2N<\/sub>B0<\/sub>\r\n\r\n \r\n\r\nwhere\r\n\r\n \r\n

    \u03b2N<\/sub>= eh\/4\u03c0Mp<\/sub> is called nuclear magneton. The above vequations lead to resonance condition. The resonance for NMR may be achieved by varying either Bo<\/sub> or by varying driving frequency. The figure shows the energy levels for a proton (I = \u00bd)in a magnetic field.Radio- frequency of energy hv causes resonance at a fieldgiven by the above equation.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

    \r\n\r\n \r\n

    The separation between the two levels increases linearly with the magnetic field and transition between them can be induced by magnetic field component B1<\/sub>of radio-frequency, which is at right angle to Bo<\/sub>. The transitions, which give rise to NMR spectra, are magnetic dipole in origin. The selection rule for the magnetic quantum number m is \u2206 m = \u00b11.<\/p>\r\n \r\n\r\nType of Nuclei Viewed<\/strong>\r\n\r\n \r\n

    NMR requires the presence of nuclear magnetic moment associated with non- zero nuclear spin. The occurrence of non-zero nuclear spin is common in the periodic table, and thus NMR can be observed in isotopes of most elements. The nuclei can be divided into the following categories:<\/p>\r\n \r\n\r\n(a)\u00a0 All the nuclei with I = 0 with magnetic moment and quadrupole moment are zero.\r\n\r\n(b) All the nuclei with I = \u00bd have magnetic moment but no quadrupole moment.\r\n\r\n(c)\u00a0\u00a0\u00a0 All the nuclei with I > 1, posses both magnetic and quadrupole moment.\r\n\r\n \r\n\r\n \r\n\r\n3.\u00a0\u00a0\u00a0\u00a0 <\/strong>M\u00d6ssbauer Spectroscopy Isomer Nuclear Transitions<\/strong>\r\n\r\n \r\n

    A nucleus has discrete energy levels. When a transition from upper level to lower energy level takes place, gamma rays are emitted. The time during which the nucleus remains in a state determine its mean lifetime for that state. Two nuclei with equal charge and mass number but in different excited states are called isomer\u00a0nuclei. The lifetime of the excited state is ~ 10-6 to 10-8 s. The isomer transitions that are used in Mossbauer spectroscopy are shown in the figure for nuclei of iron and tin.<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n

    The gamma emission of Fe57<\/sup> nucleus with energy of 14.4 KeV occurs as a result of isomer transition from the excited state to ground state of a Fe57<\/sup> nucleus. As in the case of a Fe57<\/sup> an isomer with energy of 14.4 keV has a mean life time of ~ 1.4 x 10-7 s, in practice cobalt isotope Co57<\/sup> with a life of 270 days is taken as a source of gamma ray irradiation; then through electron capture transforms into an excited Fe57<\/sup> isomer. In the case of tin, its isomer with a long lifetime ~250 days is used.<\/span><\/p>\r\n \r\n\r\nResonance Fluorescence<\/strong>\r\n\r\n \r\n\r\n(a) Atomic resonance fluorescence<\/strong>\r\n\r\n \r\n

    When light beam from sodium flame is focused on a bulb containing vapours of sodium, a faintYellow glow is observed. The sodium atoms in the bulb are absorbing energy from the incident beam of yellow light (sodium D line) due to transition from lower 2S1\/2 energy level to upper 2P\u00bd,3\/2 energy levels. Because of finite lifetime of upper levels, the absorbed energy is reemitted in all directions as a result of reverse transitions from 2P \u00bd,3\/2 to 2S1\/2. This lies at the base of resonance fluorescence (equal frequencies of primary and secondary emission). A comparison of the light which has passed through the bulb with that coming directly from the source shows that the result of passage through the sodium vapour is not simply weaken the sodium D lines, but to reduce the intensity of their wings. This is because of difference in temperature of the atoms in the flame and those in the bulb. Atoms in the flame, which are at higher temperature moves more rapidly, and therefore, the Doppler Effect broadens the light, which they emit. The cooler atoms in the bulb absorb only the central portion of the broadened line.<\/span><\/p>\r\n \r\n

    (b) Nuclear gamma resonance fluorescence<\/span><\/p>\r\n \r\n

    Consider a nucleus in its excited state whose energy is E. Nucleus emits gamma rays as a result of transition from excited state to the ground state. The energy of this gamma ray photon is E\u02e0 = hv. If this gamma ray photon is directed on another identical nucleus in its ground state, the photon may be absorbed, resulting in lifting of the nucleus from ground state to its excited state. The process which is possible only because the energy of photon is exactly equal to the resonance absorption does not take place in the nucleus, because when the nucleus emits gamma rays, the nucleus recoils and the energy of the emitted gamma rays is reduced by the amount of the recoil energy. The energy of the emitted gamma rays is<\/span><\/p>\r\n\r\n<\/div>\r\n \r\n

    Ee<\/sub> = E \u2013 ER<\/sub><\/span><\/p>\r\n\r\n

    \r\n\r\n \r\n\r\nEe = energy of the emitted photon\r\n\r\n \r\n\r\nER = recoil energy of the emitter\r\n\r\n \r\n

    Similarly, the absorbing nucleus recoils forward as it absorbs the photon, acquiring some translational kinetic energy, and consequently, if the absorption is to take place, the photon energy must be slightly greater than E, that is<\/p>\r\n \r\n

    Ea<\/sub> = E + ER<\/sub><\/p>\r\n \r\n\r\nwhere Ea<\/sub> is the energy of the absorbing nucleus.\r\n\r\n \r\n

    shows the position of Ee<\/sub> and Ea relative to the hypothetical recoil free situation. Since Ee<\/sub><Ea<\/sub>, the emitted photon does not appear to have enough energy to excite the second nucleus. Therefore, resonance absorption is not usually observed in nuclear case.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

    \r\n

    Considering an isolated atom in the gas phase and the energy difference between the ground state Eg and excited state Ee is given by<\/p>\r\n \r\n

    E = Ee \u2013 Eg<\/p>\r\n \r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nIt is to be noted that the energy and momentum are conserved in the gamma emission process<\/span>\r\n\r\n \r\n

    Let the emitting atom of mass M is moving with a velocity Vx<\/sub> in the x-direction.The linear momentum of the atom before emission of gamma rays is\u00a0<\/span>MVx<\/sub>. After emission of gamma ray, assumed in x-direction, the linear momentum of the system (gamma ray plus de-excited nucleus) must still equal to MVx<\/sub> that is<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n

    MVx<\/sub>=(h\u03bd\/c)+M(Vx<\/sub>+\u03bd)<\/p>\r\n \r\n

    where (Vx<\/sub> +v) is the velocity of the atom after emission of the gamma ray, v is vector and therefore, it can be negative.<\/p>\r\n \r\n

    v\u00a0\u00a0 =(-h\u03bd \/Mc)<\/p>\r\n \r\n

    recoil energy ER<\/sub> is therefore<\/p>\r\n \r\n

    ER = Mv2<\/sup>\/2= E2<\/sup>\/2Mc2<\/sup><\/p>\r\n \r\n

    before emission of gamma ray, the total energy above the ground state nucleus at rest is (E + MVx)2<\/sup>\/2). After emitting the gamma ray of energy E\u02e0, the nucleus will have a new velocity ( Vx + v) due to recoil. The total energy of the system is<\/p>\r\n \r\n\r\nE\u02e0 + M (Vx + v)2<\/sup>\/2.\r\n\r\n \r\n

    From conservation of energy<\/p>\r\n \r\n\r\nE+(MVx2<\/sup>)\/2= E\u02e0+ M (Vx + v)2<\/sup>\/2.\r\n\r\n \r\n\r\n\u03b4E=E- E\u02e0=(MV2<\/sup>)\/2+ M Vx<\/sub>\r\n\r\n \r\n\r\nOr\r\n\r\n<\/div>\r\n

    \r\n\r\nE = ER + ED\r\n\r\n \r\n

    Where \u03b4E is difference between energy of the nuclear transition E and energy of the emitted gamma ray photon E\u02e0 . This difference depends<\/p>\r\n \r\n\r\n(i)\u00a0 On the recoil kinetic energy which is independent of the velocity Vx.\r\n\r\n \r\n

    (ii) On the term ED = MvVx which is proportional to the atom velocity Vx and is the Dopper effect energy.<\/p>\r\n \r\n

    The mean kinetic energy per translational degree of freedom of a free atom in a gas with random thermal motion is given by<\/p>\r\n\"\"\r\n\r\n \r\n

    <Vx2<\/sup>> is mean square velocity of the atom, kB is the Boltzmann constant and T is the absolute temperature. Now<\/p>\r\n\"\"\r\n\r\n \r\n\r\nUsing ED = Mv<\/sub>Vx<\/sub> and Eq.(11.8) we have mean broadening\r\n\r\n \r\n

    <ED>=Mv(< V2<\/sup>x>)1\/2<\/sup> = 2(ER<Ek>)1\/2<\/sup><\/p>\r\n \r\n

    The gamma ray distribution is displaced by ER and broadened by twice the geometric mean of the recoil energy and the average thermal energy. The distribution is Gaussian.<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n

    For gamma rays of energy 104eV, and a mass M = 100 amu, it is found that ER = 5.4 x 10-4<\/sup>eV and <ED> ~ 5 x 10-3<\/sup>eV at 300 K. Thus resonance overlap for free atom resonance is small.<\/p>\r\n \r\n\r\nMossbauer Effect<\/strong>\r\n\r\n \r\n

    In 1958, Mossbauer used an Ir191 gamma rays source of 129 ke V, for which the Doppler broadening at room temperature is about twice the value of ER, therefore, the lines overlapped a little and resonance fluorescence could be observed. He expected that on cooling, the emitter and absorber, the absorption should decrease in Doppler width and thus in the overlap. However, he observed an increase in absorption. The absorption was as much as would be expected if there had been no recoil at all. The qualitative explanation of this fact is that at sufficiently low temperature an atom in a solid cannot recoil individually. The recoil momentum is absorbed by the crystal as a whole. The effective mass in ER = Mv2\/2= E2\/2Mc2 is therefore the mass of the crystal, which is so much larger than that of atom that the recoil energy is completely negligible. From Eq.\u00a0\"\"<\/p>\r\n

    will also be negligible. The nucleus is not bound rigidly in crystal, but is free to vibrate. The recoil energy of a single nucleus can be taken up either by the whole crystal as discussed above or it can be transferred to the lattice by increasing the vibrational energy of the crystal. The vibrational energy levels of the crystal are quantized. Therefore, it can be excited only if the recoils, leading to negligible recoil energy. Thus the necessary condition for the Mossbauer effect to occur is that the nucleus emitting the gamma ray photon should be in a atom, which has established vibrational integrity with the solid matrix. The vibrational energy of the\u00a0lattice as a whole can change by discrete amounts 0, \u00b1h \u03c9, \u00b12h \u03c9, \u2026\u2026\u2026.. . If ER<\u00a0<\/span>h \u03c9, no transfer of energy will take place as either zero of h \u03c9 units of vibrational but nothing intermediate can be transferred. When an average is taken over many emission processes, the energy transfer per event is exactly the free atom recoil energy. Let f be fraction of events which takes place without transferring energy to the lattice (ER< h \u03c9), then a fraction (1-f) will transfer one photon energy h\u03c9, neglecting two, three etc. quantum transitions to a first approximation and therefore<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n

    ER=(1-f)\u03c9h\/2\u03c0<\/p>\r\n

    f=1-(2\u03c0ER\/h\u03c9)<\/p>\r\n

    f is often called Mossbauer Co-efficient<\/p>\r\n \r\n\r\n \r\n\r\nMossbauer Nuclei:<\/strong>\r\n\r\n \r\n\r\nTo obey the Mossbauer spectroscopy one nuclei must obey\r\n\r\n \r\n

    1.\u00a0\u00a0\u00a0 The element which is to be investigated must have a source of gamma ray emission<\/p>\r\n \r\n\r\n2.\u00a0 The parent nuclei must have larger half-life time.\r\n\r\n \r\n\r\n3.\u00a0 The emission energy must lies in few keV to lower value of hundreds keV.\r\n\r\n \r\n

    4.\u00a0\u00a0 The life time of isomer excited level must ranges in 10-3 to 10-6s. If larger then it will decrease the width to energy ratio resulting to diminishing resonance selectivity.<\/p>\r\n\r\n<\/div>\r\n \r\n\r\nSummary\r\n