The slowly relaxing local structure (SRLS) approach, developed for NMR spin relaxation analysis in proteins, is applied herein to amide N-15 relaxation in deoxy and carbonmonoxy hemoglobin. Experimental data including N-15 T-1, T-2 and N-15-{H-1} NOE, acquired at 11.7 and 14.1 T, and 29 and 34 degrees C, are analyzed. The restricted local motion of the N-H bond is described in terms of the principal value (S-0(2)) and orientation (beta(D)) of an axial local ordering tensor, S, and the principal values (R-parallel to(L) and R-perpendicular to(L)) and orientation (beta(O)) of an axial local diffusion tensor, R-L. The parameters c(0)(2) (the potential coefficient in terms of which S-0(2) is defined), R-parallel to(L), beta(D), and beta(O) are determined by data fitting; R-perpendicular to(L) is set equal to the global motional rate, R-C, found previously to be (5.2-5.8) x 10(6) l/s in the temperature range investigated. The principal axis of S is (nearly) parallel to the C-i-1(alpha)-C-i(alpha) axis; when the two axes are parallel, beta(D) = -101.3 degrees (in the frame used). The principal axis of R-L is (nearly) parallel to the N-H bond; when the two axes are parallel, beta(O) = -101.3 degrees. For "rigid" N-H bonds located in secondary structure elements the best-fit parameters are S-0(2) = 0.88-0.95 (corresponding to local potentials of 8.6-19.9 k(B)T), R-parallel to(L) = 10(9)-10(10) l/s, beta(D) = 101.3 degrees +/- 2.0 degrees, and beta(O) = 101.3 degrees +/- 4 degrees. For flexible N-H bonds located in loops the best-fit values are S-0(2) = 0.75-0.80 (corresponding to local potentials of 4.5-5.5 k(B)T), R-parallel to(L) = (1.0-6.3) x 10(8) l/s, beta(D) = -101.3 degrees +/- 4.0 degrees, and beta(O) = -101.3 degrees +/- 10 degrees. These results are important in view of their physical clarity, inherent potential for further interpretation, consistency, and new qualitative insights provided (vide infra).