We consider the powder spectra (lineshapes) due to the anisotropic nuclear Zeeman interaction of the ^{31}P-nucleus.

The corresponding coupling tensor ó of the interaction between the external field and the nuclear magnetic moment is called the CSA (Chemical Shift Anisotropy Tensor).

Also the effects of fast, restricted rotation about an arbitrary axis, cross polarization by neighboring protons, andalignment due to the external magnetic field are considered during the simulation.

The input file is called inputCP^{31}P.txt &myNML
! POWDER AVERAGE PARAMETERS
NthPNT = 90
thBEG = 0 ! Degrees
thEND = 90 ! Degrees
! LINESHAPE PARAMETERS
brd = 4.5 ! Intrinsic broadening of spin packets
sHspn = 150 ! Half Spectral extend
! CHEMICAL SHIFT ANISOTROPY (CSA) TENSOR (Principal Components)
CSAtD = -65.0, -5.0, 126.0
! ORIENTATION PARAMETERS
! Direction of the fast Rotation axis with respect to the CSA
! principal frame
! Azimuthal (Rphi=alfa) Polar (Rthe=beta) and angles
alfa=16, beta=37
!NOE parameters
eNOE=4
! Constant angle between the Director and the Rotation axis:
Rangl=42
! RESTORING POTENTIAL
nVtyp= 1 ! (1= harmonic / parabolic)
Vdep= 1.5 ! (The depth of the potential well in Kelvin)
temp= 322 ! (The absolute temperature)
! this input file must end with slash, all the arrays
! have to be separated by comma?
/

It is important to keep the initial & myNML as well as the final slash / as well as the names of the parameters before the equality signs in order for the input file to work.

All the cases that are implemented in the program and will be studied further separately.

The following cases will be studied further and implemented in programming.
The exe code is called CP^{31}Plip.exe

1) Pure axial CSA tensor

The parameters of the input for these simple cases are the following:
! POWDER AVERAGE PARAMETERS NthPNT = 90
is the number of the theta points to be used in the spatial averaging procedure.

thBEG = 0 ! Degrees thEND = 90 ! Degrees
Here are the limits for the polar angle J theta

The "scanning" of the field orientation through all angles phi, j , between 0 to 360 degrees which is necessary in the rhombic case is generated automatically by the program.

! LINESHAPE PARAMETERS brd = 4.5 ! Intrinsic broadening of spin packets
This is the broadening imposed to the peak for each orientation.
The above value of brd = 4.5 and al the following parameters are only reasonable examples.

sHspn = 150 ! Half Spectral extend
The half of the total span of frequencies for which we run the spectrum

This is a typical rhombic tensor with three different components.

The program recognizes automatically the symmetry (axial or rhombic) of the CSA tensor and produces a file which consists of two columns, after the computation of the spectrum. The first column contains the chemical shift values in ppm and the second column the intensities. This file, called fort.17 , can be plotted by any appropriate software, e.g. ORIGIN.

The fast rotation of the CSA tensor around one arbitrary axis creates an overall axial tensor with axis of highest symmetry the axis of rotation.

The effectively axial CSA tensor is first ø computed and further used for the construction of the powder lineshape, as if it was the axial case, number 1) above.

! Direction of the fast Rotation axis with respect to the CSA
! principal frame
! Azimuthal (Rphi=alfa) Polar (Rthe=beta) and angles alfa=16, beta=37
In the general case of a rhombic CSA tensor, we need two angles to specify the direction of the rotation axis with respect to the principal CSA frame.
These angles are as usual a polar angle, here â (beta), and an azimuthal angle, here á (alpha).

Figure 2.The structure of the phosphate fragment around the ^{31}P atom and
the relative orientation of the principal ó -tensor frame and the rotation axis of the phosphate fragment

This routine considers the field-orientation dependent differential enhancement of the lineshape due to cross polarization CP from protons coupled to the 31 P nucleus by the Dipolar interaction .

!NOE parameters eNOE=4
An empirical parameter determining the overall efficiency of the cross polarization is needed by the program. The value eNOE = 0 means no cross polarization. A value eNOE =10 gives max effect in the present version.
(NOE = Nuclear Overhauser Effect)

It is assumed that the external magnetic field exerts a force on the membrane surface to align along with it. We consider a potential well with a certain depth and minimum at the angle of equilibrium. The program asks about this angle on the run , by the name Dangl , see Fig. 3.

The rest of the parameters are seen in the following.
! RESTORING POTENTIAL nVtyp= 1 ! (1= harmonic / parabolic) Vdep= 1.5 ! (The depth of the potential well in Kelvin) temp= 322 ! (The absolute temperature)

In order to measure the orientation of the bilayer / membrane surface a unit vector perpendicular to it is defined called the Director.
By this way the program allows an arbitrary angle between the field and the Director

! Constant angle between the Director and the Rotation axis: Rangl=42 Rangl is the angle between the Director and the Rotation axis of the fast rotation which leads to an effectively axial CSA tensor. By this angle it is allowed for the rotation axis to obtain any angle with respect to the lamellar surface.

Figure 3.An overall view of the entire model system and basic geometrical parameters, such as the angles Rangl = angle between the Director and Rotation axis,
Dangl = angle between Director and Magnetic Field H_{0} , â = polar angle of the axis of rotation and the principal axis of maximum symmetry of the tensor ó