Spectral Broadening echanisms 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 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 L(ω). - Delta function L(ω) = δ(ω - ω ) - Lorentzian function L(ω) = Γ/{π(Γ 2 + (ω - ω ) 2 )} - Gaussian function L(ω) = 1/Γ π exp{- (ω - ω ) 2 /Γ 2 } The delta function hν Level 2 Energy E 2 Level 1 Energy E 1 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 1/a over the range -a/2 < x < a/2 and outside these values. This function is plotted below for three values of a. 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.
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 equation is: xδ x x = x δ x x The eigenvalue is x the actual position of the particle. The delta function specifies that of all the possible x values only x is non-zero. Properties of the delta function 2. The integral properties of a delta function are as follows. A. The integral over δ(x - x ) is equal to the function evaluated at x. fxδx x dx = fx B. The area under the delta function is one. δ x x dx =1 Properties of the delta function 3. The value of a delta function is zero everywhere except where the argument is zero. δ x x =for x x δ x =for x 4. A change of argument by a factor results in multiplication by the inverse of the factor. δ kx = k 1 δ x To see this consider the above rectangular function. The delta function is 1/a over the limits -a/2 < x < a/2. Thus, the height is 1/a and the base is a. If we multiply the height by k then it becomes k/a. This means that we should multiply the base by 1/k. In other words since Lifetime broadening =T 1 Pure dephasing = T 2 * The uncertainty principle gives a relationship between the natural linewih and the lifetime (T 1 ) of a state. For example, a state with a 1 ps lifetime will be 5 cm -1 in wih. Pure dephasing of ground and excited state vibrational wave functions also contributes to the energy wih. The time T 2 * indicates the time required to lose coherence between ground and excited state during absorption. δ x dx =1 we have k δ kx dx =1 thus k δ kx = δ x NR spectroscopy The Nuclear agnetic Resonance Phenomenon The agnetization Vector Spin Relaxation Linewihs and Rate Processes The Nuclear Overhauser Effect The Nuclear agnetic Resonance Phenomenon Nuclei may possess a spin angular momentum of magnitude I(I+1)h. The component around an arbitrary axis is m I h where m I = I, I-1,..,-I. The nucleus behaves like a magnet in that it tends to align in a magnetic field. The nuclear magnetic moment µ has a component along the z-axis µ Z = γ m I h.
The magnetogyric ratio and nuclear magneton The magnetogyric ratio is γ where γh =g I µ N. The nuclear g-factor ranges from ca. -1 to 1. Typical g-factors 1 H g=5.585, 13 C g=1.45 14 N g=.44 The nuclear magneton is µ N where µ N = eh 2m p =5.5 1 27 JT 1 Nuclear magnetic moments are about 2 times smaller than the electron spin magnetic moment because µ N is 2 times smaller than the Bohr magneton. The Larmor frequency Application of magnetic field, B to a spin-1/2 system splits the energy levels. E mi = -µ z B= -γhbm I The Larmor frequency ν L is the precession frequency of the spins E mi = -m I hν L ν L = γb For spin 1/2 nuclei the resonance condition is E mi =hν L β α m I = -1/2 anti-parallel E m I = 1/2 parallel Increasing magnetic field B The classical vs. quantum view The bulk magnetization vector B According to a classical picture the nuclei precess around the axis of the applied magnetic field B z or B. In the quantum view a sample is composed of many nuclei of spin I = 1/2. The angular momentum is a vector of length {I(I +1)} 1/2 and a component of length m I along the z-axis. 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 β N α = e hν L / kt This small population difference gives rise to the measured magnetization in a NR experiment. Precessing nuclear spins The bulk magnetization The applied magnetic field B causes spins to precess at the Larmor frequency resulting in a bulk magnetization. The Bloch Equations The magnetization vector obeys a classical torque equation: d = B where B is the magnetic field vector. precesses about the direction of an applied field B with an angular frequency γb radians/second. The Vector Components of the Bloch Equations d x = γ y B z z B y d y = γ z B x x B z d z = γ x B y y B x If no radiofrequency fields are present then d x / = and d y / = and we simply have rotation about the static field B z. We will also call this B.
The static field causes precession of nuclear spins B or B z z The effect of a radiofrequency field z X B 1 Y B B The static field The bulk magnetization The magnetic field vector precesses about B. The spins precess at the Larmor frequency ω = -γb. Equilibrium Effect of a π/2 pulse is to rotate into the x,y plane Precession in the x,y plane leads to an oscillating magnetic field called a free induction decay. The static magnetic field B. The magnetic field due to an applied rf pulse is B 1. The magnetization along the z-axis is zero after a saturating π/2 pulse and precesses in the x,y plane. B Relaxation times T 1 and T 2 Longitudinal Relaxation Time T 1 B Spins that precess at different rates due to spin-spin coupling and they dephase due to spin flips. Transverse Relaxation Time T 2 Precessing vector The longitudinal relaxation time governs relaxation back to the equilibrium magnetization along the z-axis The transverse relaxation time is the time required for spin dephasing in the x,y 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 9 o. 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 vector z X Receiver coil Detector (Y) The free induction decay Y Receiver coil Detector (X) Direction of Precession Illustration of receiver coils at 9 o to one another. Real part FID(t) = exp( t/t 2 )cos ωt Imaginary part FID(t) = exp( t/t 2 )sin ωt
easuring relaxation Relaxation is the rate of return to the ground state. In magnetic resonance this means restoration of the vector to its initial position. There is longitudinal relaxation (T 1 ) and tranverse relaxation (T 2 ) 1 = 1 + 1 * T 2 2T 1 T 2 T 2 * is also called the pure dephasing T 1 is also called the spin-lattice relaxation time T 2 is also called the spin-spin 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 nonperiodic 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: g(k) = f(x) = 1 1 e ikx f(x)dx If we consider the time/frequency pair we have: g(ω) = f(t) = e ikx g(k)dk 1 e iωt f(t) 1 e iωt g(ω)dω Lorentzian Fourier Transform The decay of the coherence in NR and optical spectroscopy can be measured as the T 2 time. Using the definition Γ = 1/T 2 we can write the Fourier transform as: L(ω) = 2 π e iωt e Γt + iω o t L(ω) = 2 π e {i(ω ω o ) Γ}t = 2 1 π Γ + i(ω ω o ) L(ω) = 2 1 Γ i(ω ω o ) π Γ + i(ω ω o ) Γ i(ω ω o ) L(ω) = 2 Γ π Γ 2 +(ω ω o ) i(ω ω o ) 2 Γ 2 +(ω ω o ) 2 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. ν = ν - receding 1 ± v/c + approaching This is known as a Doppler shift and the result for an ensemble leads to broadening. Determination of the Doppler linewih Use the kinetic theory of gases approach. The distribution of velocities of gas phase molecules is a Gaussian. exp mv 2 2kT The FWH of the velocity distribution is: δv =2 2ln2kT m
Determination of the Doppler linewih The frequency shift is: ν = ν ν 1 v/c = 1 v/c ν 1 v/c ν 1 v/c = vν c The spread in frequency is given by: δν = ν c δv = 2ν 2ln 2kT c m Therefore, δν/ν = 2 x 1-6 for N 2 at 3 K. For a typical rotational line of 1 cm -1 (3 GHz) the Doppler linewih is 7 khz. 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 g(t) = e t 2 /Γ 2 The Fourier Transform is: G(ω) = 1 t e iωt e t 2 /Γ 2 t G(ω) = 1 exp iωt t 2 /Γ 2 t G(ω) = 1 e A 2 ω 2 exp t 2 /Γ 2 iωt A 2 ω 2 To solve this we must complete the square: t 2 /Γ 2 iωt A 2 ω 2 = t Γ + Aω 2 Completing the square The cross terms determines that value of A. i = 2A Γ, A = iγ 2 G(ω) = 1 e Γ 2 ω 2 /4 exp Γ t + iγ 2 2 ω G(ω) = 1 e Γ2 ω2 /4 exp G(ω) = Γ 2 e Γ2 ω 2 /4 t + iγ2 2 ω 2 Γ 2 Relationship of Conjugate Variables The relationship between the two functions is: g(t) = 1 πγ e t 2 /Γ 2 G(ω) = Γ 2 π e Γ2 ω2 /4 Note that t and ω 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 bandwih, ω, gets larger.