Measuring magnetic susceptibility by NMR—application The Evans method Measuring magnetic susceptibility by NMR—application Some of the chi’s in this PowerPoint turned into c’s for some users, so I also include a pdf version of the slides. Created by Adam R. Johnson, Harvey Mudd College (adam_johnson@hmc.edu) and posted on VIPEr on June 9, 2016. Copyright Adam R. Johnson, 2016. This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license visit http://creativecommons.org/licenses/by-nc-sa/4.0/

Magnetic Susceptibility Magnetic susceptibility is the ratio between magnetization M of the material in a magnetic field and the field intensity H: M = c·H. Diamagnetic materials have c < 0 Paramagnetic and ferromagnetic materials have c > or >> 0 Paramagnetic materials are attracted to a magnetic field; diamagnetic materials are repelled from a magnetic field, but the effect is orders of magnitude weaker. The 2 unpaired electrons in O2 allow it to be held by a strong magnet even though there are 7 electron pairs in O2

Magnetic Field in an NMR H” is the component of applied field due to given concentration of paramagnetic ions in solution H” = H1 + H2 H1 = Lorentz or cavity field of induced magnetization in a sphere around the paramagnet = (4p/3)M H2 is demagnetizing field, opposing applied field; H2 = -aM (a depends on sample geometry and relative orientation of sample tube and external field) H” = [(4p/3)-a]M But M = cH So H”/H = [(4p/3)-a] c There are other terms in the H” definition but for all practical purposes, they can be neglected.

Given: H”/H = [(4p/3)-a]c But ∆H”/H = ∆f/f ∆f/f = [(4p/3)-a]c Finally, add a correction term for the magnetic susceptibility of the solvent and for the change in density of solution vs. solvent. This was all derived using mass susceptibilities (cm) ∆f/f is the same value as ∆H”/H; magnetic field strength is measured in frequency of resonance of the proton and the two are related by h(nu) = 2(mu)H where nu is the frequency and mu is the magnetic moment of the nucleus

History Originally developed for permanent magnet application Applied field perpendicular to sample Permanent magnet NMR spectrometers are oriented with a transverse cylindrical sample tube and the alpha value is 2pi. This is the original definition by Evans in 1959. Note: by convention, paramagnetic materials have a positive magnetic susceptibility, while diamagnetic materials have a negative magnetic susceptibility. Tabulated diamagnetic susceptibilities are negative, so as long as the absolute value of ∆f is used, this equation will give the correct value. Evans did not include the minus sign in his original derivation, though other sources include it, since by convention, positive shifts in f are movement to weak field. ∆f = peak separation (Hz) f = NMR frequency (Hz) m = mass per cm3 cm = mass susceptibility

History Modern instrumentation: coaxial superconducting magnet Applied field parallel to sample Superconducting magnet NMR spectrometers are oriented with a paralell cylindrical sample tube and the alpha value is 0. The net effect is a negative two-fold change in the ∆f, which can be readily demonstrated experimentally. Note: by convention, paramagnetic materials have a positive magnetic susceptibility, while diamagnetic materials have a negative magnetic susceptibility. Tabulated diamagnetic susceptibilities are negative, so as long as the absolute value of ∆f is used, this equation will give the correct value. ∆f = peak separation (Hz) f = NMR frequency (Hz) m = mass per cm3 cm = mass susceptibility

Relationship to number of unpaired electrons cpT = constant for simple paramagnets (Curie Law) cP = paramagnetic susceptibility T = absolute temperature (K) NA = Avogadro’s number g = magnetogyric constant for electron b = Bohr magneton kB = Boltzmann constant S = Total spin Quantum number n = number of unpaired electrons The constants combine to give a value of approximately ½. Unless you are doing very precise work, ½ should suffice. Since S = n/2, ½[S(S+1)] = ½[n/2(n/2+1)] = ½[n^2/4+n/2] = (1/8)[n(n+2)]. See IUPAC guide and J. Chem. Educ. articles for more discussion of units. Must include Russell-Saunders coupling for 2nd or 3rd row And also spin-orbit coupling for f block (CGS units)

Effective magnetic moment† The constants combine to give g^2(S(S+1); since g ~ 2, this is approximately 4S(S+1) meff is independent of temperature and unitless but is usually reported in units of Bohr magneton: mB cP can be measured at any T and related to n or S (CGS units) †The Chemist’s magnetic unit

Evans’ method Using 1H NMR to determine magnetic susceptibility NMR tube Sample solution Capillary with pure solvent NMR spectrum collected NMR solvent in capillary (shifted peak) NMR solvent in tube (reference peak) Use peak shift to calculate unpaired electrons The sample solution is the paramagnetic material at known concentration in a solvent that has a residual solvent peak (CHCl3 in CDCl3, for example). Often, the NMR solvent will also have 1% TMS. This gives two different signals to calculate the frequency shift from.

Commonly published equation Use c for concentration (mol·L-1 = M) Ignore solvent diamagnetism Ignore density correction Combine constants Ignore sign of ∆f; take positive value ∆f = peak separation (Hz) f = NMR frequency (Hz) m = mass concentration (g/mL) m’ = mass n = moles c = molar concentration cm = molar susceptibility

Diamagnetic correction One unpaired electron can be detected even in a large molecule because the magnitude is ~1000 times greater For careful work with large molecules, correct observed susceptibility for presence of paired e- cD tabulated cD opposite sign to cP!