Nuclear magnetic resonance studies—1 on polymers

The NMR of Polymers

“…Thus, these matrices must have the form Iy = (Yil Y12) Y21 Y22 (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) The numbers forming the matrix Iz are readily found. On the basis of the third postulate, the matrix I z and the functions 1fJ 1 and 1fJ 2 must satisfy the relations By multiplying out the matrix products and equating the corresponding elements of the matrices in the right-and left-hand parts of the equations, we find…”

“…Iz =~ ~ 1, (1 0) 2 0-1 (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) To find the numbers forming the matrices Ix and I y , let us examine the projection I n of the spin on any direction in space, defined by the angles a, f3, and 'Y with respect to the axes x, y, and z.…”

“…== ( ~~) and must satisfy the equation By substituting in the expression for In the matrices Ix and Iy from (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) and the matrix Iz from (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20), multiplying out the matrix product in the left-hand part of , and equating the corresponding elements in the right-and left-hand parts, we obtain a system of homogeneous linear equations for finding the numbers c1 and c2:…”

“…If A is the operator of the variable examined, while l/J is its function of state, then by expanding this function into a series with respect to the eigenfunctions of the operator it is readily demonstrated that the mean value will equal Taking into account the fact that the spin and the magnetic moment of the nucleus are related by the equation and that the operators of projections of the spin are represented by the matrices (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) and , it is easy to calculate the mean values of the projections of the magnetic moment of the nucleus if its spin is in a state described by the function By using these relations from equation (1-81) we obtain…”

“…(II- 24) The correlation function of this component may be written in the form If we regard a molecule as a rigid sphere, which executes random rotations in a medium with a viscosity T} at a temperature T; it may be shown [5] that cpz depends on T in accordance with an exponential law: (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26) where The value T C' which is called the correlation time of the stochastic process, is the time in which the internal statistical bonds which characterize the process are weakened by a factor of e.…”