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Date Created: 10/15/15
The Intensity of Absorption Derivation of an Expression for the Intensity as Function of the Transition Probability per Unit Time The time dependent Schrodinger equation is Hwm at The formal solution is t e thh 5y e iEth 5y where the following definitions apply it a Planck39s constant H a hamiltonian an operator E 2 energy eigenvalue a constant 912 wave function Derivation of an Expression for the Intensity as Function of the Transition Probability per Unit Time We can treat an applied electromagnetic field as a perturbation to the hamiltonian The zeroth order hamiltonian that of system without the field is H0 The field hamiltonian is H1 uE The total hamiltonian is H H0 H1 H0 uoE We consider two states an initial state i and a final state f The transition probability between the two states is Wf 71quot ltfuEi gt28o 0 8w 0 2i2 This expression is an approximate solution known as the Fermi Golden Rule The Golden Rule gives the transition probability per unit time for transitions from i 9 f The above expression is also valid for transitions from f 9 i if we interchange the indices The same inherent transition rate applies for the absorption probability Derivation of an Expression for the Intensity as Function of the Transition Probability per Unit Time We wish to relate the transition rate to the intensity of absorbed radiation It is the intensity that we will use to define the timecorrelator for spectroscopy First we determine the net energy change in the radiation field upivvifnm piT The energy density is u There are N molecules in volume V The pi is the probability that the ith initial state is occupied Q is the partition function Note that McHale calls the partition function z wif is the transition probability per unit time from above and hefi is the energy of an absorbed photon We now substitute in the explicit Golden Rule transition probability per unit time for W u i z pm dp Eigt2503 0 p03 ltfu Eigt2503 03 e hooikT 2h V if Derivation of an Expression for the Intensity as Function of the Transition Probability per Unit Time Since i and fare dummy indices we can interchange them in the second sum Further we use the fact that oa 0 U p pf0 ltfii Eigt25o 0 if Note that pf pie hm kT The time derivative of the energy density becomes u emfW po ltfpi Eigt25co 0 Derivation of an Expression for the Intensity as Function of the Transition Probability per Unit Time Our goal is to relate the transition probability to the intensity of absorbed radiation Recall that we have shown that the absorption coefficient is obtained from the intensity reduction as electromagnetic radiation travels through a medium I lOexp yx IOexp 475T From the definition of the imaginary part of the dielectric constant u a co 2no1lto we have Derivation of an Expression for the Intensity as Function of the Transition Probability per Unit Time We also saw that by comparison with Beer39s law I IOexp 2303ECX IO1O ECX we can express the familiar extinction coefficient in terms of the imaginary part of the dielectric constant relative 39tt39 39t permI V y 8 NAV E 2303an By reestablishing the connection between 8r 03 and 03 the relative intensity we can see that it is necessary to determine the imaginary part of the relative permittivity before we proceed We can express 8r 03 in terms of the energy density u 1 H U 2 8 mu whereu 280E0 Derivation of an Expression for the Intensity as Function of the Transition Probability per Unit Time Using the relation axed 1 e m kTpltfue1gt28oo 03 we can express the imaginary part of the permittivity in terms of the transition probability Note that the amplitude of the radiation E0 has been factored out and instead there remains a unit vector pointed in the direction of the polarization of the radiation We can choose any direction for the incident radiation in the absence of applied static fields Suppose we choose xpolarization The square of the transition moment uif2 can be replaced by 13 H xltif For a given polarization arm 350h1e hww pltfluligtltilulfgt5w ml Derivation of an Expression for the Intensity as Function of the Transition Probability per Unit Time The relative absorption intensity is l 3h808coV D TE1 e hmkTN It is the expression on the right that is used to derive a timecorrelator piltfp igtltiILl 550 D Spectral Broadening Mechanisms Lorentzian broadening Homogeneous Gaussian broadening Inhomogeneous Inertial Doppler broadening special case for gas phase The Fourier Transform NC State University Broadening mechanisms The lines observed in spectra are not infinitely narrow even in the gas phase Broadening can be observed due to Lifetime broadening Pure dephasing Doppler effect gas phase Solvation effects condensed phase Inhomogeneous broadening Lineshape functions DeHa Gaus an Lorentzian I 7600 7800 8000 8200 8400 Absorption wavenumber cm Normalized functions are shown Spectral lifetime broadening The uncertainty principle gives us an estimate of the extent of broadening due to the lifetime or pure dephasing time Spectral broadening has a mathematical form called the lineshape function Lo Delta function Lo 50 00 Lorentzian function Lltcogt NW 2 co 02 Gaussian function Lo 1M7 exp 0 DOVr2 The delta function Level 2 T Energy E2 hv l Level 1 Energy E1 The delta function is applied theoretically to the case where there is no broadening The two energy levels involved in a transition are infinitely narrow The delta function A delta function is an infinitely narrow infinitely high function whose area is normalized to one This is a little difficult to understand unless we give a model for a delta function One model is the square function Imagine a function whose value is 1a over the range a2 lt X lt a2 and 0 outside these values This function is plotted below for three values of a Properties of the delta function 1 The delta function is the eigenfunction of the position operator For a free particle we can operate with the position operator x hat The eigenvalue equa onis Q8X X0 X08X X0 The eigenvalue is x0 the actual position of the particle The delta function specifies that of all the possible x values only x0 is nonzero Properties of the delta function 2 The integral properties of a delta function are as follows A The integral over 5x x0 is equal to the function evaluated at x0 0 fX8X X0dX fXO 00 B The area under the delta function is one r 8X X0dX 1 Properties of the delta function 3 The value of a delta function is zero everywhere except where the argument is zero 5X XOOf0I X XO 5XOf0I X O 4 A change of argument by a factor results in multiplication by the inverse of the factor 8kx i8X lkl To see this consider the above rectangular function The delta function is 1a over the limits a2 lt x lt a2 Thus the height is 1a and the base is a If we multiply the height by k then it becomes ka This means that we should multiply the base by 1k In other words since 0 6Xdx 1 we have qr 5kxdx 1 thus k6kx 5x Lifetime broadening T1 Pure dephasing T The uncertainty principle gives a relation ship between the natural linewidth and the lifetime T1 of a state For example a state with a 1 ps lifetime will be 5 cm1 in width Pure dephasing of ground and excited state vibrational wave functions also contributes to the energy width The time T2 indicates the time required to lose coherence between ground and excited state during absorption NMR spectroscopy The Nuclear Magnetic Resonance Phenomenon The Magnetization Vector Spin Relaxation Linewidths and Rate Processes The Nuclear Overhauser Effect The Nuclear Magnetic Resonance Phenomenon Nuclei may possess a spin angular momentum of magnitude II1h The component around an arbitrary axis is mlh where mI 1 11 The nucleus behaves like a magnet in that it tends to align in a magnetic field The nuclear magnetic moment u has a component along the zaxis uz y mI n The magnetogyric ratio and nuclear magneton The magnetogyric ratio is y where yh quN The nuclear gfactor ranges from ca 10 to 10 Typical gfactors 1H g5585 13C g1405 14N g0404 The nuclear magneton is uN where MN e 505 gtlt1O27J 7 1 2mp Nuclear magnetic moments are about 2000 times smaller than the electron spin magnetic moment because uN is 2000 times smaller than the Bohr magneton The Larmor frequency Application of magnetic field B to a spin1I2 system splits the energy levels Em1 23 39Yth1 B m1 12 The Larmor frequency vL is the antiparallel precession frequency of the spins Em1 m1th AE m1 12 parallel V L a 27 For spin 12 nuclei the resonance condition is B Eml th Increasing magnetic eld The classical vs quantum view According to a classical picture the nuclei precess around the axis of the applied magnetic field BZ or BO In the quantum view a sample is composed of many nuclei of spin 1 12 The angular momentum is a vector of length 11 1 2 and a component of length mI along the zaxis The uncertainty principle does not allow us to specify the x and y components In either case the energy difference between the two states is very small and therefore the population difference is also small N This small population difference 5 e hVLkT gives rise to the measured N a magnetization in a NMR experiment The bulk magnetization vector B M0 Precessing The bulk nuclear spins magnetization The applied magnetic eld B causes spins to precess at the Larmor frequency resulting in a bulk magnetization M0 The Bloch Equations The magnetization vector M obeys a classical torque equation iMMxB dt where B is the magnetic field vector M precesses about the direction of an applied field B with an angular frequency yB radianssecond The Vector Components of the Bloch Equations d2 leVBZ MZBy dM dt y vMZBx M z dMZ dt yM ByMVBX If no radiofrequency fields are present then dMXdt O and dMydt O and we simply have rotation about the static field 82 We will also call this BO The static field causes precession of nuclear spins B0 or BZ MOZ M The static eld The bulk magnetization The magnetic field vector M precesses about B0 The spins precess at the Larmor frequency 0 yB0 The effect of a radiofrequency field A Equilibn um Effect of a nZ pulse is to Precession in the Xy plane rotate M into the Xy plane leads to an oscillating magnetic eld called a free induction decay The static magnetic eld B0 The magnetic eld due to an applied rf pulse is B1 The magnetization along the zaXis is zero after a saturating TE2 pulse and precesses in the Xy plane Relaxation times T1 and T2 B0 A Longitudinal B ll Transverse Relaxation 0 Relaxation Time T1 Time T2 Spins that precess at different Precessmg rates due to spinspin coupling vector M and they dephase due to spin ips The longitudinal relaxation time governs relaxation back to the equilibrium magnetization along the zaxis The transverse relaxation time is the time required for spin dephasing in the xy plane as the spins precess Quadrature detection and the FID In order to obtain phase information detection along both x and y directions is required Instead of using two coils to detect the radiofrequency signals one uses two detectors in which one has the phase of the reference frequency shifted by 90 These correspond to the real and imaginary components of the free induction decay FID The observed spectrum is the Fourier transform of the FID FT Experimental aspects of quadrature detection Rotating M vector Z Receiver coil Detector Y Direction of Precession l Receiver coil Detector X Illustration of receiver coils at 90 to one another The free induction decay Magnetization au I I I I I I O 100 200 300 400 500 600 Time ms Real art p FIDt exp tT2cosmt Imaginary part FlDt exp tT2sinoat Measuring relaxation Relaxation is the rate of return to the ground state In magnetic resonance this means restoration of the M vector to its initial position There is longitudinal relaxation T1 and tranverse relaxation T2 LL1 T2 2T1 T T2 is also called the pure dephasing T1 is also called the spinlattice relaxation time T2 is also called the spinspin relaxation time Fourier Transform A Fourier series represents any periodic function as a sum of sine and cosine functions with appropriate coefficients Since the sinusoids each have a representative frequency a periodic function in time can be analyzed in terms of its frequency For non periodic functions we use a Fourier transform to decompose a function of time arbitrary into its frequency components We call time and frequency conjugate variables Likewise position and momentum are conjugate variables Conjugate variables For any conjugate variables x and k we can write a Fourier transform as gk if e XfXdX fX e ngkdk If we consider the timefrequency pair we have L 00 4031 g03 moo e ftdt ft 0 e goadoa Lorentzian Fourier Transform The decay of the coherence in NMR and optical spectroscopy can be measured as the T2 time Using the definition F 1T2 we can write the Fourier transform as LD f0w e iwte Ftiootdt 2 00 i03 03 2 1 LmZ 1 F ioo ooo 75Fi03 030F i03 030 LD V F Do TEF2oo ooo2 F2oo ooo2 Doppler broadening Broadening arises in the gas phase due to the frequency shift of molecules moving towards or away from the radiation source A source receding from or approaching the observer at velocity v has a frequency shift V Z V receding I 1 i We approaching This is known as a Doppler shift and the result for an ensemble leads to broadening Determination of the Doppler linewidth Use the kinetic theory of gases approach The distribution of velocities of gas phase molecules is a Gaussian 2 mv ex p 2kT The FWHM of the velocity distribution is 5V 2 21nnkT Determination of the Doppler linewidth The frequency s ift is V 1 WQV l vc 1 vc 1 vc C The spread in frequency is given by v 2v 212kT 5V E5V 7 HT Therefore 5vv 2 x 106 for N2 at 300 K For a typical rotational line of 1 cm1 30 GHz the Doppler linewidth is 70 kHz vu Avv The role of the Gaussian in Condensed Phases A Gaussian line shape is often used to represent an inhomogeneous distribution However very rapid inertial motions may also contribute to a Gaussian lineshape The Gaussian is a very convenient function for two reasons 1 Gaussian integrals are analytic 2 The Fourier transform of a Gaussian is a Gaussian Gaussian Fourier Transform Inertial motion is described using glttgt The Fourier Transform is t GD l e imte t2F2dt 0 t 30 t i f exp ioat t2F2dt 0 t G tieA2m2f ex t2F2 i t A2 2dt 0 2n 0 p 03 03 To solve this we must complete the square 2 1 21 2 i03t A2032 A03 Completing the square The cross terms determines that value of A e E I A 2 r 2 f 2 Gco eWZMJ exp m dt w t w 03 G03t21nerz 2 4 exp F 22 dt 60 e rz 2 4 Relationship of Conjugate Variables The relationship between the two functions is 9 et2F2 30 e FZMM Note that t and o are inversely proportional These functions can be inserted into the commutation relation for energy and time They have a reciprocal relationship As the time t required for a process gets shorter the bandwidth 0 gets larger