Efficient polynomial analysis of magic-angle spinning sidebands and application to order parameter determination in anisotropic samples

Abstract Chemical shift tensors in 13 C solid-state NMR provide valuable localized information on the chemical bonding environment in organic matter, and deviations from isotropic static-limit powder line shapes sensitively encode dynamic-averaging or orientation effects. Studies in 13 C natural abundance require magic-angle spinning (MAS), where the analysis must thus focus on spinning sidebands. We propose an alternative fitting procedure for spinning sidebands based upon a polynomial expansion that is more efficient than the common numerical solution of the powder average. The approach plays out its advantages in the determination of CST (chemical-shift tensor) principal values from spinning-sideband intensities and order parameters in non-isotropic samples, which is here illustrated with the example of stretched glassy polycarbonate.

Meaning of the averaging symbol: For sake of shortness of the expressions, the index γ is omitted in the following.
2. Proof for odd exponents: If the sine power m is odd: We set a = -π, split the integration range in [-π , 0] and [0 , π] and substitute in the first part γ -γ: If the cosine power n is odd: We substitute γ π/2 − γ and get again an integral with odd sine power which therefore will be zero likewise.

Proof for even exponents
This will be done by mathematical induction from n to n + 2 with base case n = 0.The latter has to be proven with an induction from m to m + 2.
Base case n = 0: The proposition eqn.
(1) has here the form: To prove this by a further induction we check first that the base case is obviously valid for m = 0.The inductive step m → m + 2 is done by Inserting the integration limits: first term at the right-hand side is cancelled, and we get That means, the validity of the sub-proposition (4) for m implies the validity of that also for m + 2. This proves sub-proposition (4) for all even m.Therefore the base case for the following induction is valid.
Inductive step n → n + 2: Suppose proposition (1) is valid for a particular n.We investigate this expression for n n + 2 by replacing cos n+2 γ = cos n γ 1 − sin 2 γ : which is exactly the proposition for n + 2. The validity of proposition (1) for n implies the validity of the proposition also for n + 2. This proves that proposition for all even m and n.

B. Polar average
Meaning of the averaging symbol: For sake of shortness of the expressions, the index cos α is omitted in the following.
For even n: Prove by mathematical induction m → m + 2; therefore two base cases (here m = 0 and m = 1) are needed.
Base case 2 (m = 1): To be shown here: This is done by the following steps: Partial integration of a cosine power: Inserting this rule into eqn.( 13): It can be shown easily that eqn.( 12) is fulfilled for n = 0.This together with eqn.( 16) as induction step proves base case 2.
Inductive step m → m + 2: Suppose proposition (8) would be valid for a particular m and all even n.We calculate the average for m + 2 by replacing sin m+2 γ = sin m γ 1 − cos 2 γ : (f n := 1 for even n and π/2 for odd n) which is exactly proposition (8) for m → m + 2.
This inductive step together with the proven base cases proves proposition (8) ∀{m, n} ⊂ N.

S2. POWDER-AVERAGED PHASE POWERS
Listed up to eigth power: The following passages can be copied and pasted into Mathematica, possibly with an in-between step of pasting the text into a simple text editor to remove formatting information.Note that on some systems it seems that the curly brackets { and } are pasted as letters f and g, respectively.Since f and g do not appear in the formulae, search-and-replace can be used.
First, we define an auxiliary function with the independent variables #1, #2 and #3 being identified with n, k, and η, respectively: The general function for SSB generation, with #1, #2, #3 and #4 identified with the SSB order m, δω 0 /ω r , η and the maximum order n of the polynomial, respectively, is then: With this, one can readily generate the SSB polynomials, e.g. for the centerband (m = 0) up to order n = 6: rsb[0, delta, eta, 6] // Simplify