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\title{Effect of RF Fields During Pulse on Rotational Diffusion: Influence on Spectral Density}
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             \author[1]{Dennis  Sorce}

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\date{\small \em Received: 12 December 2019 Accepted: 1 January 2020 Published: 15 January 2020}

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\begin{abstract}
        


The effect of the applied RF field in an NMR experiment on the magnitude of the Spectral Density for a Dipolar Relaxation Mechanism is demonstrated theoretically. The effect was shown with Sin Cos Pulse as a concrete example. The order of magnitude of the magnetic moment where these effects will be significant for typical Rf amplitude values was derived. The effect may be of utility in providing an alternate method of control for MRI Tissue Contrast applications with further development.

\end{abstract}


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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
n contemporary NMR methodologies, it is common to find experimental scenarios where the relaxation of the magnetization during a pulse train is important to be able to model and quantify. \hyperref[b0]{(1,}\hyperref[b1]{2,}\hyperref[b2]{3)} In this note, we suggest that for some molecular species the Rotational Diffusion can be affected and modified by the Magnetic Field Torque of the applied radio-frequency pulse. During the course of working on this concept, it has come to our attention that the Russian investigator Sitnitsky \hyperref[b3]{(4)} has investigated this phenomenon.\par
This proposed influence may be used in some models for explaining experimental data, such as for Liquid Crystals \hyperref[b12]{(14)}. We demonstrate the derivation of this effect on the spectral densities following the classic treatment of Abragam \hyperref[b4]{(5)} and gives some ranges of where this effect may be of importance.\par
We note that the proposed effects may be useful as another avenue to control the spin-dynamics of an experimental system while the pulse is on. Also, the proposed effects have been dealt with rigorously in the Physics Literature \hyperref[b13]{(15)}. 
\section[{II.}]{II.} 
\section[{Theory}]{Theory}\par
The "Toy Model" we propose to explicate this effect is the following.\par
We envision a spin system, transformed to the Tilted Doubly Rotating Frame (TDRF,6). In this frame there will be defined a so-called "effective field." We can write down an effective Hamiltonian for the applied RF of the following form:1 [ ] [ ] [ ] RF x z H t I t I t ? ? = + ? [1]\par
For the exposition here we consider the Sin/Cos pulse, defined as:1 1 1 [ ] Sin[ ] M M t t ? ? ? = [2a] 1 1 [ ] Cos[ ] M M t t ? ? ? ? = [2b]\par
Where 1 M ? is a constant (See for example the relevant papers of the Garwood Group \hyperref[b6]{(7,}\hyperref[b7]{8,}\hyperref[b8]{9)}.\par
In the TDRF the effective field can be seen from geometric arguments to be defined as:1 2 2 [ ] [ ] [ ] eff t t t ? ? ? = + ? [3]\par
Substituting Eq [2 a, b] into Eq \hyperref[b2]{[3]}, one easily appreciates that . So that as required for our treatment the effective field defined as:1 [ ] M eff particle B t ? ? = [4]\par
Where particle ? is the particle gyromagnetic-ratio. Now we consider a Molecular Species in solution with a defined dipole moment particle µ . In a constant field eff B , there is a potential energy of interaction (10) between the moment and the field defined as:\par
Here the angle ? is defined as the angle between the vectorial directions of the dipole moment and the constant field. We change to the convenient notation:0 particle eff K B µ = [6a]\par
So that:\par
Knowing the geometry between the effective field and the magnetic moment in the TDRF, it is Seen that the angle ? is defined as:1 [ ] [ ] Tan[ ] [ ] t t Arc t ? ? ? = ? [7] I © 2020 Global Journals 1\par
Year 2020\par
Author: Retired: CMRR, Mn USA 6 Stonegate CtCockeysville. e-mail: dennissorce1@comcast.net 
\section[{Global Journal of}]{Global Journal of} 
\section[{Medical Research}]{Medical Research}\par
Volume XX Issue I Version I( D ) 0 [ ] [ ] U K ? ? = ? [6b] Cos [ ] [ ] particle eff U B ? µ ? = ? [5] 
\section[{Cos}]{Cos}\par
Using Eq[2a,b] in Eq \hyperref[b6]{[7]}we see that: [ ] M t t ? ? = [8]\par
Suppose we take the Nuclear Species of Interest to be in a molecule that we model and approximate as a sphere. We assume that the Rotational Brownian Motion can be represented as a series of small incremental rotations. We seek to find the Correlation Function, which characterizes the rotational diffusion. As treated, in for example Abragam Chapter VIII or other places in the literature \hyperref[b9]{(11,}\hyperref[b10]{12)} we can define the Correlation Function in terms of the spatial part of the Dipolar Interaction Hamiltonian. If we adopt the notation of Abragam, we can define the Correlation Function as: Where we set and consider the case where p m is zero. So, to carry out this program we need to an expression for the Probability Density Function .\par
This PDF will be a solution of the so-called Smoluchowski Equation (SE), where the effects of the applied RF Torque will be included. As one can infer there are numerous assumptions one can apply to the formulation of the SE. The solution in general, (see for example, the classic papers of Coffey's group (10) are not trivial, usually the derivation of series solutions which involve the solution of iterative expressions for the expansion coefficients, or continued fraction solutions.\par
We have chosen to present and use the solution of Sitnitsky (4) which is the most easily implemented solution we have found to program for demonstration of our methods.\par
Please see Appendix I for a detailed definition of the terms in the series expression for the PDF.\par
The PDF can be taken to be the approximate solution of the following partial differential Equation using our expression for the Potential Energy Term. (  {\ref 4})2 , ,\textbf{2 2 [ , , ] [ , , ] [ , , , ] (2 [ ]) [ , , , ]( [ ]}\par
)i u i u i u R u R i u u R u w x x K t w x x K t w x x K t K C x w x x K t K C x K t x x ? ? ? ? ? ? = ? + + + ? ? ? [10a] 1 1 1 1 [ ] Cos[ ]( ) [ | |] Cos[ ] b C x d x k x x x Exp x x k x ? ? ? ? ? ? = ? ? ? ? [10b]\par
Here b is a constant defined in (4).\par
In Eq  {\ref [10]} x is defined to be as Cos[ ] ? where u K measure as defined previously the interaction between the Moment and the RF field with the definition:0 u B K K k T = [11]\par
We can use the definition of the PDF to calculate the Correlation Function as In Eq \hyperref[b8]{[9]} and then compute the corresponding Spectral Density as:0 [ , , , ] [ , , ] [ ] t u p p u p p p J K t m G t K m Exp i t dt ? ? = ? [12]\par
In Figure \hyperref[fig_1]{1} we show the dependence of the Spectral Density as given in Eq \hyperref[b10]{[12]}, for the case mp=0.\par
As can be seen, there is found to be an appreciable dependence of the Spectral Density on the parameter u K this is taken to indicate that the RF Field, with a range of values which will be discussed below, can affect the Spectral Density which is used to compute relaxation functions. \hyperref[b4]{(5,}\hyperref[b9]{11,}\hyperref[b10]{12)} So that the RF field, through interaction on the Rotational Brownian Motion, can influence the values of the calculated relaxation functions during a pulse sequence.\par
To the knowledge of the author, this possibility has not been fully appreciated in the NMR literature. 
\section[{III.}]{III.} 
\section[{Discussion}]{Discussion}\par
The reader may wonder what is a lower bound on the magnetic moment of the particle of Interest for a typical value of the pulse amplitude.\par
In the Garwood papers \hyperref[b6]{(7,}\hyperref[b7]{8,}\hyperref[b8]{9)}, the pulse amplitude is typically on the order of 3 3.610 Hz. Then we reason that the interaction energy of the magnetic moment with the field in the TDRF should be greater than the thermal energy of the surrounding liquid medium. So, we propose:1 u K particle eff B u B k T Or particle B eff k T u B\par
We note that the units of a magnetic moment can be seen in CGS units to be ergs Gauss Practical lower bound on the magnetic moment of the particle for an effect of the RF field on the Rotational Brownian Motion of the particle and consequently on the Spectral Density for a dipolar relaxation mechanism. 
\section[{Appendix i}]{Appendix i}\par
The following is the series definition of the Probability Density Function used in the text. ( see,4 )  2 Cos[ ]Sin[ Cos[ ]] 2 1 [1 Sin[ 2 ]] 2 i n i n n n q A q q ? ? = ? 2 Cos[ ]Cos[ Cos[ ]] 2 1 [1 Sin[ 2 ]] 2 i n i n n n k B k k ? ? = + With 1 Tan[ ] [ ] 1, 2,3,...... n n n n q q k Cot k n ? ? ? = ? = = 2 2 1 2 2 2 2 1 [ ] 2 [ ][ ] n n n k k b k k ? ? ? ? ? = + +\begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_1}1}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-3.png}
\caption{\label{fig_4}Figure 1 :}\end{figure}
  			\footnote{© 2020 Global Journals} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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\end{document}
