- #1
Athenian
- 143
- 33
- Homework Statement
- Short Question Explanation:
I have a semi-logarithmic graph of free induction decay (FID) amplitude as a function of time. While the primary purpose of this graph was to find the value of the apparent spin-spin relaxation time (##{T_2}^*##) and its uncertainty for each corresponding data point, I am also tasked with finding the value of the magnetic field (##B_0##) and its uncertainty for the sample due to a pair of external rare-earth magnets.
Note that I do know the process for finding ##{T_2}^*## which will be outlined below as it may help with understanding how to obtain the value of ##B_0## (the problem I am having trouble with).
Additional details to the problem are shown below.
- Relevant Equations
- ##f_0## (MHz) ##= 4.258 B_0## (kilogauss)
##\frac{1}{{T_2}^*} = \frac{1}{T_2} + \gamma \Delta B##
##2\Delta B_0 = \frac{2}{{T_2}^* \gamma}##
##M= M_0 e^{\frac{-t}{{T_2}^*}}##
Background Information (Not Strictly Necessary):
As a quick recap, the graph I am dealing with is a semi-logarithmic graph of free induction decay (FID) amplitude as a function of time. To acquire the value for ##{T_2}^*## (and its uncertainty) in the graph, I used the below equation to do so.
$$M= M_0 \; e^{\frac{-t}{{T_2}^*}}$$
However, the graph needs to be a semi-logarithmic one as opposed to an exponential one. Thus, after the "conversion", I found that:
$${T_2}^* = -\frac{\log{(e)}}{m}$$
Note that ##m## above signifies the slope (##m##) in the equation of a linear line on a graph.
For more details how I came up with the above equation, please refer to the below thread (the solution process is in the latter posts).
https://www.physicsforums.com/threads/calculating-numerical-uncertainties-in-an-equation.1001620/
Example Graph Data (Related to the Next Section):
When plotted in a semi-log graph ...
##m=-0.008344 \pm 0.0004948## mV/##\mu##s
##b=2.931 \pm 0.07160## mV
Linear Equation for above ...
##\log{(V_1)} = y = mt+b##
My Attempt at Finding ##B_0##:
As shown in "Relevant Equations", I know that:
$$f_0 = 4.258 B_0$$
where ##f_0## is measured in MHz and ##B_0## measured in kilogauss.
However, my graph is in the form of amplitude vs. time. As the graph does not possesses any values of frequency, I am unable to use the above equation to obtain my ##B_0## for each of the data points.
I have seen an online PowerPoint (link shown below) that states I must translate my amplitude vs. time graph to one that is amplitude vs. frequency.
Link: https://chemistry.mit.edu/wp-content/uploads/2018/08/DCIF-IntroNMRpart1-theory-o07.pdf (Page 10)
If this is possible, I should be able to take my frequency value for each corresponding data point and find ##B_0## for each of the given data points via the equation ##f_0 = 4.258 B_0##. However, I have absolutely no idea how to approach the problem with this potential method.
In another online article (though I do not remember the link), it argues that as
##\frac{1}{{T_2}^*} = \frac{1}{T_2} + \frac{1}{T_1} + \gamma \Delta B_0## (this article adds ##\frac{1}{T_1}## for some reason), I would be able to obtain the below equation.
$$\Delta B_0 = \frac{1}{{T_2}^* \gamma}$$
While an intriguing discovery, knowing the value of ##\Delta B_0## doesn't exactly bring me any closer to finding the respective ##B_0## values for each of my data points (in the above table).
While perhaps an obvious hint, I was instructed to find the values of ##B_0## and its uncertanties for each corresponding data point on the graph. As the uncertainty values are required for this exercise, I know that the calculation process should involve the numerical data obtained on my linear (or potentially exponential) equation (e.g. from ##m##) as those are the only data set I have that possesses any uncertainty values.
Once again, the above question deals with NMR (Nuclear Magnetic Resonance). There are a lot of helpful information on the web, but none has led me to understand how to find for ##B_0##. In theory, though, the process of finding ##B_0## shouldn't be ridiculously difficult and ought to be relatively straightforward. That said, any assistance on the necessary steps I should take to find ##B_0## would be greatly appreciated.
If additional clarification or information is needed that could help streamline the assistance process, please do let me know! Thank you for reading through my question!
As a quick recap, the graph I am dealing with is a semi-logarithmic graph of free induction decay (FID) amplitude as a function of time. To acquire the value for ##{T_2}^*## (and its uncertainty) in the graph, I used the below equation to do so.
$$M= M_0 \; e^{\frac{-t}{{T_2}^*}}$$
However, the graph needs to be a semi-logarithmic one as opposed to an exponential one. Thus, after the "conversion", I found that:
$${T_2}^* = -\frac{\log{(e)}}{m}$$
Note that ##m## above signifies the slope (##m##) in the equation of a linear line on a graph.
For more details how I came up with the above equation, please refer to the below thread (the solution process is in the latter posts).
https://www.physicsforums.com/threads/calculating-numerical-uncertainties-in-an-equation.1001620/
Example Graph Data (Related to the Next Section):
| t (##\mu##s) | V_1(mV) |
90 | 156.3 |
100 | 131.3 |
120 | 75 |
150 | 45.3 |
175 | 34.4 |
200 | 17.2 |
When plotted in a semi-log graph ...
##m=-0.008344 \pm 0.0004948## mV/##\mu##s
##b=2.931 \pm 0.07160## mV
Linear Equation for above ...
##\log{(V_1)} = y = mt+b##
My Attempt at Finding ##B_0##:
As shown in "Relevant Equations", I know that:
$$f_0 = 4.258 B_0$$
where ##f_0## is measured in MHz and ##B_0## measured in kilogauss.
However, my graph is in the form of amplitude vs. time. As the graph does not possesses any values of frequency, I am unable to use the above equation to obtain my ##B_0## for each of the data points.
I have seen an online PowerPoint (link shown below) that states I must translate my amplitude vs. time graph to one that is amplitude vs. frequency.
Link: https://chemistry.mit.edu/wp-content/uploads/2018/08/DCIF-IntroNMRpart1-theory-o07.pdf (Page 10)
If this is possible, I should be able to take my frequency value for each corresponding data point and find ##B_0## for each of the given data points via the equation ##f_0 = 4.258 B_0##. However, I have absolutely no idea how to approach the problem with this potential method.
In another online article (though I do not remember the link), it argues that as
##\frac{1}{{T_2}^*} = \frac{1}{T_2} + \frac{1}{T_1} + \gamma \Delta B_0## (this article adds ##\frac{1}{T_1}## for some reason), I would be able to obtain the below equation.
$$\Delta B_0 = \frac{1}{{T_2}^* \gamma}$$
While an intriguing discovery, knowing the value of ##\Delta B_0## doesn't exactly bring me any closer to finding the respective ##B_0## values for each of my data points (in the above table).
While perhaps an obvious hint, I was instructed to find the values of ##B_0## and its uncertanties for each corresponding data point on the graph. As the uncertainty values are required for this exercise, I know that the calculation process should involve the numerical data obtained on my linear (or potentially exponential) equation (e.g. from ##m##) as those are the only data set I have that possesses any uncertainty values.
Once again, the above question deals with NMR (Nuclear Magnetic Resonance). There are a lot of helpful information on the web, but none has led me to understand how to find for ##B_0##. In theory, though, the process of finding ##B_0## shouldn't be ridiculously difficult and ought to be relatively straightforward. That said, any assistance on the necessary steps I should take to find ##B_0## would be greatly appreciated.
If additional clarification or information is needed that could help streamline the assistance process, please do let me know! Thank you for reading through my question!