Leonidas Petrakis Gulf Research & Development
Company Pittsburgh. Pennsylvania
Spectral h e Shapes Gaussian a n d Lorentzian functions in magnetic resonance
Spectroncopic experiments may vary considerably in their details yet they all involve the same fundamental phenomenon: transitions are induced between discrete energy levels in which the syst,em under consideration exists, by irradiating the system with electromagnetic radiation of the appropriate frequency so as to satisfy t,he Bohr frequency condit,ion, AE = hw ( A E is the energy difference between any two energy levels, fi is Planck's contant, and w is the frequency of the transition-inducing electromagnetic field). Selection rules deternii~~e \vhich id t:he many possible tmnsitions are likely to he observed, and transitiotr probabilities determine the relative intensities of the t,ransitions. Analysis of the result,ing spectrum t.hen, i.c., extraction of the appropriate spectroscopic parameters such as transition frequencies, intensities, line shapes, splitting of bands, etc., allows the spectroscopist or chemist to deduce information about the system that is giving rise to the particular spectrum studied. The analytical spectroscopists and chemists have hitherto paid attention mostly to the band frequencies. The widespread use of ir group frequencies and nmr chemical shifts in struct,ure determination problems attest to this fact,. Yet the band shape itselt and the various statistical parameters that can he derived from t,he band shape (e.g., areas, moments, half-widths, etc.) may contain much significant information. For example, vaporphase measurements of ir band intensities make possible in principle the determination of the dipole moment change associated with a given normal vibration as well as the effect of vibrational motion on the polarizat i m of molecules. Condensed-phase measurement,^ of ir band intensities may provide information on the n:lt,nre of intermolecular forces (1). Analysis of line sl~:~pes in magnetic resonance of solids can yield informat,ionon the distribution of the species undergoing resonance (random vs. ordered distribution) and on the strengt,h of their interaction with neighboring species (2-5). I t is natural then that spectral line shapes would receive increasing attention from the practicing spectroscopists. Of course, oft,en band overlap complicates the problem and makes extremely difficult the :~nalysisof spectral line shapes. But for nonoverlap ping or resolvable bands the first step is the determination of the line shape function itself. This not only may be significant informat,ion in itself, but it also may dlow the calculation of the various statistical paramrters used to characterize the band itself from the simple measurements of line width and amplitude. 432
/
Journal of Chemical Education
' I v e y con~monly encountwed line shapes in various branches of molecular spectroscopy are the Gaussian and the Lorentzian. Where collision hroadrning is the main factor that det,ermines the band half-width (as for example in the vapor-phase ir spectra and high resolution nmr) the resulting bands are lrentzian ( I 6 On the ot,her hand, rigid systems i n which the relative orientations and positions of randomly distributed and interacting species do not change with time give rise t,o Gaussian lines (2, 6). 1 1 is not our intention to examine here t,he factors that, determine the line shapes for the various spectroscopic 1.echniques. Rather, we intend to examine the proparties of these two very commonly encountered line shapes, the Gaussian and the Lorent,zian. We will specialize our discussion somewhat, in terms of nmr and vsr spectra, yet what is written here is generally applicable to all spectroscopy where these. two 11andshapes are encountered.
The Resonance Phenomenon
I n magnetic resonance experimwts O I I C d~serves transitions between the Zeeman levels I elect,rons (em) or nuclear Zeeman levels of nuclei (nmr). A st.ationary magnetic field H removes the degeneracy associated with electron or nuclear spins by causing them to assume definite orientations with respect to the field direction. Since electrons and nuclei can be viewed as microscopic magnet,ic dipoles, they interact with the applied external field, the energy of intermtion l? being. E
=
-#.H
=
-*II
eos8
(1)
where fi is the magnetic dipole moment, 11 is the exlernally applied field, and 0 is the angle between @ m d H. For situations where the spin is '/% R (electrons, protons, ' g F nuclei, nuclei, etc.) only two angles itre admissible according to the laws of quantum mechanics, 0' and 180". Therefore, the energy difIrrence between the two levels is and as a result transitions between these two levels will occur when the system is irradiated with energy of frequency w = 2 p H / 6 = ?H (3) This is the famous Larmor equation. It states that the frequency at which resonance will occur is proportional to the applied magnetic field responsible for the removal of the spin degeneracy, the proportionality
constant being ihe nlaglletogyric ratio of the species undergoing resonance. Normally, in magnetic resonance experiments the frequency is kept constai~t, and the field is swept.. The resulting spectra then arc traces of the ahsorpt,ion versus the field H, or by using the approprixt,e experimental setup the first or even thc, secoud derivative of the absorption may he recorded. In fact, this is stmdnrd practice in ear and \r.irlr-lilic nmr. The Gaussian and Lorentzian Functions Absorpt,i~moccurs uot a t one single frequency, hul, rather over a range of frequenciee resulting in broadened lines or hauds. This, of course, is charnrteristic 01' all spectroscopic: t,echniques although the spreific factors that cause the broadening may be different. The moat commonly encountered line shapes arr 1111: Gaussiau a11d t . 1 1 ~ l,nre~ltzia~~, eqns. (4) a11d ( 5 ) , rrspert,ively. Figure 1. Zero derivative curves for Goursion [solid line1 and Lorenlrian function; lo) curve, h a v e t h e same half-line width ot (broken line1 holf-maximum intensity; (b) curves have t h e some peak-to-peak line
width.
where P(H - Ho)is the amplitude of the functio~l,6 is one-half of t.he width of the function a t half-maximun~ intensity, and (I1 - Ho) is the value of the field mensured from the ceuter of the resonance (Table 1). A plot of these two functions is shown in Figure lo. It is assumed that both functions have the same halfline width. Two significant differences between thew two line shapes are evident: the Lorentzian is :L sharper, narrower funct,ion toward the center; also, it. falls off iu the wings not as rapidly as the Gaussian. At a distance from the center equal to three half-line widths, the amplitude of the Gaussian is essentially zero, while a t t,he same point the amplitude nf thr Lorentaian is approximately 10% of its n~wzi~nun~ amplitude (Tahle 2 ) . -- -
Table 1. Analytical Expressions for Gaussian and Lorentzian Functions Gaussian
F"(H - Ho),., F"(H - H C , ) ~ : , , K* K" Sewnd momelil Fourth moment
It is also clear f n ~ mFigure la that although tllc half-line widt,hs at ha,lf-maximum intensity are the same for the two functions, t,he point a t which thc slope is maximum for earh curve is very dierent.. Close examination of the curves indicates that for the Gaussian the maximum slope of the absorption is w r y close to the half-inteniit,y mark, while for a Lorentzinu the maximum slope is at about t,he three-quarter m:rk of the maximum int,ensity. If each of expressions (1) and ( 5 ) is differentiated twice, set equal to zero and solved for (H - Ho), thc values of the field (H - Hp) are obtained at which thc slope is maximum. These values are, respectively, for t,he Gaussiau and the Lorentaian, 6 / 4 2 and 6 . These arr thr so-wllrd peak-to-peak liue widths
-
Lorentzim
- I J ~ % A ~
Z/&ZA~ 1.%2 0.517 A2 3A4
See eqn. (5). Wee eqn. (6).
Volume 44, Number 8, Augurt 1967 / 433
Table 2. A
6
Numerical Values for Absorption and Derivatives of Gaussian and Lorentzian Functions
FlH - Hn)
Gaussian F ' ( H - H.1
or line widths between slope extremes, and they are indicated here by A. They allow one to write readily the analytical expressions associated with these two line shape functions in terms of either of t,he two half-widths. This is done in Tahle 1, and since in magnetic resonance the peal-to-peak line width is more useful, all the expressions, except for thezero-derivative curve, are given in terms of A. Figure l a indicates the ~eak-to-~eak half-line widths A as well as the halfI I I I ~ : widt11 ti[ I ~ i t l f - n l : ~ x i ~ilirensity n ~ ~ ~ n 6 for the two line 41:1pcfutict i
