Corrections to equations, resolving issues 18, 19

Notebooks/MR Physics - Bloch Equation.ipynb

@@ -408,7 +408,7 @@
     "\\begin{bmatrix}\n",
     "e^{-t/T_2} & 0 & 0 \\\\\n",
     "0 & e^{-t/T_2} & 0 \\\\\n",
-    "0 & 0 & 1\n",
+    "0 & 0 & e^{-t/T_1}\n",
     "\\end{bmatrix}\n",
     "\\vec{M}(0)\n",
     "+\n",

Notebooks/Spin Physics.ipynb

@@ -79,11 +79,11 @@
     "\n",
     "And has an amplitude of that is defined as the Equilibrium Magnetization:\n",
     "\n",
-    "$$M_0(\\vec{r}) = \\frac{N(\\vec{r}) \\bar{\\gamma}^2 \\hbar^2 I_Z (I_Z +1) B_0}{3 k T}$$\n",
+    "$$M_0(\\vec{r}) = \\frac{N(\\vec{r}) \\bar{\\gamma}^2 h^2 I_Z (I_Z +1) B_0}{3 k T}$$\n",
     "\n",
     "where $N(\\vec{r})$ is the spin density,\n",
-    "$\\hbar$ is Planck's constant, \n",
-    "$\\bar{\\gamma}$ is the gyromagnetic ratio, \n",
+    "$h$ is Planck's constant (6.63e-34 J/Hz), \n",
+    "$\\bar{\\gamma}$ is the gyromagnetic ratio (42.58e6 Hz/T for $^1 H$), \n",
     "$B_0$ is the magnetic field,\n",
     "$I_Z$ is the spin number (1/2 for hydrogen),\n",
     "$k$ is Boltzmann’s constant (8.62e-5 eV/K or 1.38e-23 J/K), and \n",
@@ -92,19 +92,19 @@
     "\n",
     "### Fractional Polarization\n",
     "\n",
-    "Another helpful formulation to describe the preference of spins to align with the magnetic field is to look at the fraction of spins that, if observed, would be aligned with the magnetic field.  This can be done using the Boltzman distribution that describes the ratio of spins in the spin-up, $N_\\uparrow$, to the spin-down, $N_-$, states.  \n",
+    "Another helpful formulation to describe the preference of spins to align with the magnetic field is to look at the fraction of spins that, if observed, would be aligned with the magnetic field.  This can be done using the Boltzman distribution that describes the ratio of spins in the spin-up, $N_\\uparrow$, to the spin-down, $N_-$, states, shown here for spin 1/2 nuclei($I_Z = 1/2$):\n",
     "\n",
-    "$$ \\frac{N_\\uparrow}{N_\\downarrow} = e^{\\hbar \\bar{\\gamma} B_0 / kT}$$ \n",
+    "$$ \\frac{N_\\uparrow}{N_\\downarrow} = e^{h \\bar{\\gamma} B_0 / kT}$$ \n",
     "\n",
-    "(This can also be used to derive the equilibrium magnetization above.)\n",
+    "This uses the separate between the two energy states, $\\Delta E = h \\bar{\\gamma} B_0$.  It can also be used to derive the equilibrium magnetization above.\n",
     "\n",
     "From this distribution, a fractional polarization can be defined as the difference between spin up and spin down states, divided by the total number of spins, which describes the fraction of spins that contribute to the MR signal:\n",
     "\n",
-    "$$ \\frac{N_\\uparrow - N_\\downarrow}{N_\\uparrow + N_\\downarrow} = \\frac{ 1 - e^{-\\hbar \\bar{\\gamma} B_0 / kT} }{ 1 + e^{-\\hbar \\bar{\\gamma} B_0 / kT} }$$ \n",
+    "$$ \\frac{N_\\uparrow - N_\\downarrow}{N_\\uparrow + N_\\downarrow} = \\frac{ 1 - e^{-h \\bar{\\gamma} B_0 / kT} }{ 1 + e^{-h \\bar{\\gamma} B_0 / kT} }$$ \n",
     "\n",
     "If the exponent is small (or, when the polarization is low), this simplifies to\n",
     "\n",
-    "$$ \\frac{N_\\uparrow - N_\\downarrow}{N_\\uparrow + N_\\downarrow} \\approx \\frac{\\hbar \\bar{\\gamma} B_0}{2 kT}$$ \n"
+    "$$ \\frac{N_\\uparrow - N_\\downarrow}{N_\\uparrow + N_\\downarrow} \\approx \\frac{h \\bar{\\gamma} B_0}{2 kT}$$ \n"
    ]
   },
   {