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## Microsoft word - mdr2002.doc

**APL, a powerful research tool in **
**Magnetic Resonance Spectroscopy **
**Claude Chachaty **
**e-mail : **claude.chachaty@wanadoo.fr

**Keywords** : NMR, ESR, spin, hyperfine coupling, spectra, simulations, automated fitting.
**I-Introduction **

In spite of its outstanding scientific potential,

**APL** is up to now ignored or scarcely

exploited by research workers. During 15 years as the head of the Magnetic Resonance

Laboratory of the Nuclear Research Center at Saclay, the author has extensively used

**APL** in

his works [1-3] and continues to promote its scientific applications.

The Magnetic Resonance Spectroscopy (

**MRS**) includes two main branches, the Nuclear

Magnetic Resonance (

**NMR**) and the Electron Spin Resonance (

**ESR**) also called Electron

Paramagnetic Resonance (

**EPR**). The

**NMR** is a priviledged method for the identification and

conformational analysis of organic and biological molecules and is well known for its medical

application, the Magnetic Resonance Imaging. The

**ESR/EPR** which is the main subject of

this topics, is the specific method for studying paramagnetic molecules i.e. molecules

possessing at least one unpaired electron, namely the free radicals resulting from the breaking

of a chemical bond, triplet fundamental (e.g. the oxygen of air) or lowest excited states and

some metal coordination complexes. Most of these species are very reactive and are initiators

or intermediates in a large number of chemical and biological processes : oxidation,

combustion, polymerization, radiation damaging, photosynthesis etc… An important

application common to the

**NMR **and

**ESR **is the molecular dynamics which provides

thorough information on some physical properties of condensed matter.

**II**-

**Principles **
Electrons and most of the nuclei possess a spin angular momentum, denoted S and I ,
respectively, as well as magnetic moments µ = g β S = γ η
ge and gI are the spectroscopic factors, the latter being specific of each nucleus, γ
the relevant gyromagnetic ratios, β
e and n the Bohr and nuclear magnetons and η the
Planck's constant divided by 2π . In a magnetic field B0 , the spins and magnetic moments
undergo a precession of angular frequency ω = γ
B about B0 . For a spin quantum
number S or I, multiple of ½, the spins and magnetic moments take 2S+1 or 2I+1 orientations
defined by the projections M = S
,S -1,.1- S,-S or M
To each magnetic quantum number MS or MI corresponds an energy level. A magnetic resonance experiment consists in submitting a small sample (0.1-1 ml) placed
in a very strong and homogeneous magnetic field B0 to a rotating radiofrequency (

**NMR**) or

microwave (

**ESR**) magnetic field B

1 perpendicular to B0 ( B
0 ). The resonance
phenomenon corresponds to a transition between adjacent energy levels which occurs when
the angular frequency of B
B and involves the absorption of a photon of

*Claude Chachaty *
energy hν = gβB
0 , ν0 being the spectrometer frequency and h the Planck's constant. For
technical reasons, the resonance is obtained by varying ν0 at constant field (

**NMR**) or B0 at

constant frequency (

**ESR**) and the

**ESR** spectra are usually recorded as the first derivative.

The nuclear and electron spins are seldom isolated and generally experience local magnetic

fields due to other spins. The 2S+1 fundamental energy levels of a spin S interacting with a

spin I are splitted into 2I+1 sublevels and the resulting (2S+1)(2I+1) levels are :

E←(B0×(ge×be×MS)°.-gn×bn×MI)+a×MS°.×MI

where

**a** is the hyperfine coupling constant expressed in energy units. The allowed

**ESR** transitions between these levels follow the selection rule ∆M = ±1, ∆M =

expression is easily extended to any number of spins of any quantum number and is an usual approximation when its first term is much larger than the second one. Figure 1 shows a simple application of these principles to a S=1/2, I=1/2 system, the H• atom, the smallest and one of the most reactive free radical.
MS MI

**I ** -1/2 1/2

**II **-1/2 -1/2

**III** 1/2 -1/2

**IV **1/2 1/2

Magnetic field (gauss)

Figure 1 : Energy levels and ESR resonance lines of the hydrogen atom. Hyperfine coupling

constant a = 1.42 GHz or 508 gauss (1 gauss = 0.1 mT), spectrometer frequency ν0 = 9.24

GHz. Allowed transitions : I⇔IV and II⇔III.

** **

III-Applications to the ESR spectroscopy.

The interpretation of the

**ESR **spectra in terms of identification of the paramagnetic species

we are dealing with, of the hyperfine coupling parameters and sometimes of dynamical

behaviour is generally not feasible without the help of computer simulations. A visual

comparison of the experimental spectum with the simulated one tell us if the starting

assumptions made about this species are likely or not. The

**hresol** function listed below is a

simplified version for the simulation of high resolution

**ESR **spectra of radicals in solution.

*APL and Magnetic Resonance *
hresol;ŒIO;A;Y;i ŒIO„1 ª 'Central field (mT) :' ª Bc„Œ 'Nuclear spin quantum numbers :' ª NS„½SN„,Œ L0:'Hyperfine coupling constants (mT)' ª …(NS¬½HFC„,Œ)/L0 SW„Bc+˜0.5+¯1.2 1.2×SN+.× HFC Spectral window centered on Bc 'Number of points:' ª dH„--/SW÷NPTS„Œ C0„SW[1],SW[1]+dH×¼NPTS ª DIM1„1+2×SN DIM2„½V„(DIM1½¨HFC)×DIM1½¨MI„SN,¨SN-¼¨2×SN ª i„0 ª HR„Bc L1:HR„,HR°.+¹V[i„i+1] ª …(i<DIM2)/L1 ª INT„+/HR°.=HR„HR["HR] MASK„MASK,1†MASK„(¯1‡HR)¬1‡HR ª HR„MASK/HR ª INT„MASK/INT 'Linewidth at half-height : ' Y2„Y×Y„HR°.-C0 ª SP„INT+.×¯2×Y÷A×A„Y2+RHL×RHL„0.5×Œ PLOT CY„C0,[1.5]SP„SP÷+/+ SP Spectrum first derivative
The figure 2 shows the spectrum of the benzyl ( C H −
2 ) radical generated by this
Figure 2. Simulated ESR spectrum (first derivative) of the C H −
2 radical in fluid
solution. The electron spin is coupled to 3 pairs of equivalent protons and a single proton. The interactions between the spins and the magnetic field and between the spins are of the
0

**. g. ** and S.

**A**. I , respectively, where

**g** and

**A** are symmetric second rank tensors

whose components are the sum of an isotropic term (g factor and hyperfine coupling constant)

and an anisotropic one. In fluids, the latter is averaged to zero by fast molecular motions but

is partially or not averaged in anisotropic systems as solids and liquid crystals.

The functions for fitting the spectra of spin S=1/2 species (free radicals, copper and

vanadyl ions) in anisotropic media proceed by the following steps :

**1** - Parameters

Invariant : spectrometer frequency, spectral width and nuclear spin quantum numbers.

*Claude Chachaty *
Adjustable : principal values of

**A, g **and σ (linewidth) tensors, width of Gaussian line

broadening and rate of rotational motion if any.

** **

2 – Angular dependence of the transitions probabilities

*P *(θ , ϕ ,

*M*
σ (θ,ϕ,

*M *, the angles θ and φ defining the orientation of B
0 in the frame of the

**g** tensor.

** **

3 – Calculation of resonance fields

*B *θ ,ϕ,

*M*
**4** – For each transition, summation of spectra over all orientations :

*S *(

*B M I *)= 1

*P *(θ ,φ ,

*M*
(

*B *−

*B *(θ ,φ ,

*M *);σ (θ ,φ ,

*M *)
)sin θ

*d *θ

*d *φ

where B is the scanning magnetic field, F the form function and N the normalization factor.

For a Lorentzian form function F(x) = 1/(1+x2), the relevant APL expression is :

(d1 d2 d3)„½¨B th phi

U„B°.-,Br ª F„÷RU×1+U×U„U÷RU„(d1½1)°.×,sigma

S„S÷+/S„F+.×,P×(1±th)°.×d3½1

** **

5 – The overall spectrum obtained by summing S(B, MI ) over MI is convoluted by a

Gaussian and derived numerically.

**6** – If the agreement with the experimental spectrum is not satisfactory, return to

**1** to reajust

the parameters. This step may be automated by means of an optimization function based on

the Levenberg-Marquardt's algorithm [4] to minimize the variance between the experimental

and computed spectra.

Figure 3 shows an example of an automated fitting using the method outlined above.

.Figure 3. Experimental (solid line) and computed (•••••) spectra of an ESR spin-probe, a

nitroxide radical, in a model phospholipid membrane before (a) and after (b) addition of

cholesterol. This membrane is constituded by phospholipid bilayers separated by water and

behaves as a liquid crystal.The broadening and increased asymmetry of the lines from (a) to

(b) are significant of an increase of the membrane rigidity and molecular ordering upon

cholesterol addition, which may be quantitavely estimated [5].

** **

IV-Conclusion
*APL and Magnetic Resonance *
The theory of magnetic resonance is for a large part founded on matrix algebra, one of the

strong points of

**APL**, making quite easy the programming of spectral simulations and fitting

of experimental data. For this reason the author has chosen

**APL** rather than other

programming languages currently used by the scientific community (Fortran, Basic, C,

Pascal) in spite of its small diffusion and of some problems of portability.

The Magnetic Resonance software is written in APL2 (IBM) and APL+WIN (APL2000).

Descriptions of the workspaces are given in the sites www.garpe.org and

ftp://ierc.scs.uiuc.edu/pub/SoftwareDatabase.

**References **
[1 ] C. Chachaty et G. Langlet

*Logiciels d'étude conformationnelle par RMN, de molécules flexibles. *

Journal de Chimie-Physique et de Physico-Chimie Biologique,

**82**, 613 (1985).

[2] C. Chachaty

* Simulations de spectres de résonance magnétique appliquées à la dynamique de *

molécules en milieux anisotropes.

ibid.,

**82**, 621 (1985).

[3] C. Chachaty and E. Soulié

*Determination of electron spin resonance parameters by automated fitting of the spectra.* Journal de Physique III France,

**5**, 1927 (1995).

[4] D.W. Marquardt

*An algorithm for least-squares estimation of nonlinear parameters. *

Journal of the Society of Industrial Applied Mathematics,

**11**, 431 (1963).

[5] C. Wolf and C. Chachaty

* Compared effects of cholesterol and 7-dehydrocholesterol on sphingomyelin- *

glycerophospholipid bilayers studied by ESR.

Biophysical Chemistry,

**84**, 269 (2000).

*Claude Chachaty *
Source: http://www.afapl.asso.fr/MDR2002.pdf

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