|
|
| Line 1: |
Line 1: |
| {{For|the wavefunction of a particle in a periodic potential|Bloch wave}}
| | Lots of people still thinks that Stock market is very risky market and should not invest money in this market. But actually it is not true. [http://www.Sharkbayte.com/keyword/Stock+Market Stock Market] or Share Market is the best field to [http://news.goldgrey.org/gold-futures/ gold invest] money. This is the Market that can make you rich in very short time period.<br><br>Let's face it, creating cash in the stock marketplace is really an objective of many people. They say money cannot buy you happiness, however it sure can make life easier! And for numerous individuals, besides their fulltime job, they don't have an additional supply of income. stock trading offers an chance to earn more cash.<br><br>You aren't alone. This is a familiar feeling to many traders. It's called "hindsight bias." Hindsight bias is the experience many traders have after a trade in which they realize they should have seen the obvious mistakes they made and should have steered clear of the trade all together.<br><br><br><br>3) You can do stocks trading online. This is the most popular online money making idea. You need to have a DEMAT account online also an account with the bank that operates online. investing in stocks and trading fetches you handsome income online. You keep some amount in your online bank account; you invest it in stocks and trade. You get free tips online which help you to invest.<br><br>Here is what I'm advocating. Let's say you have a $50,000 trading account. You never risk over 5% or $2500 on any one trade (and most of the time you risk less than that). Let's assume you have a margin requirement of $200 per contract (if it is much more than that, consider using another broker). You probably never go over 10 contracts on any one trade. So, actually you would only need to have $2000 in your account to make the same trades you normally take.<br><br>The number one thing to keep in mind if you are new to trading stocks is to start small and work your way up. The last thing you want is to jump into a shark tank unprepared and lose thousands of dollars. Start with low lots of shares such as 100 as this is much easier to take in if the shares go against you.<br><br>The number one tool you have in order to evaluate an advisor is to ask them a lot of questions. Some might not like that and will try to hurry you into a commitment, but don't be afraid of that. Those are usually the ones who don't provide you with serious advice, so the first sorting is very easy. The questions you should ask them about cover their experience, their education, their expertise and their philosophy in their specialty. Pay attention on whether they answer the questions or try to talk about something else. If they try to talk about something else, it could be because they know they have a [https://www.vocabulary.com/dictionary/weak+spot weak spot].<br><br>You as an investor must think of ways on how to deal with worst scenario. You have to think out of the box, or keeping your heads up. These things can be avoided, if you have precautionary measures in tail; like you are able to investigate further, the person or family that wants to rent your house for example. It is not enough that they are able to provide you details of their income, doing a back ground check will help (history from where they last rented). In that manner you will have idea how to deal them. Remember, your house is an investing real estate and not a charity institution. |
| | |
| In physics and chemistry, specifically in [[nuclear magnetic resonance]] (NMR), [[magnetic resonance imaging]] (MRI), and [[electron spin resonance]] (ESR), the '''Bloch equations''' are a set of macroscopic equations that are used to calculate the nuclear magnetization '''M''' = (''M''<sub>''x''</sub>, ''M''<sub>''y''</sub>, ''M''<sub>''z''</sub>) as a function of time when [[relaxation (NMR)|relaxation times]] ''T''<sub>1</sub> and ''T''<sub>2</sub> are present. These are [[Phenomenology (science)|phenomenological]] equations that were introduced by [[Felix Bloch]] in 1946.<ref>[[Felix Bloch|F Bloch]], ''Nuclear Induction'', Physical Review '''70''', 460-473 (1946)</ref> Sometimes they are called the [[equations of motion]] of nuclear magnetization.
| |
| | |
| ==Bloch equations in laboratory (stationary) frame of reference==
| |
| Let '''M'''(''t'') = (''M<sub>x</sub>''(''t''), ''M<sub>y</sub>''(''t''), ''M<sub>z</sub>''(''t'')) be the nuclear magnetization. Then the Bloch equations read:
| |
| | |
| :<math>\frac {d M_x(t)} {d t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _x - \frac {M_x(t)} {T_2}</math>
| |
| :<math>\frac {d M_y(t)} {d t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _y - \frac {M_y(t)} {T_2}</math>
| |
| :<math>\frac {d M_z(t)} {d t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _z - \frac {M_z(t) - M_0} {T_1}</math>
| |
| | |
| where γ is the [[gyromagnetic ratio]] and '''B'''(''t'') = (''B''<sub>''x''</sub>(''t''), ''B''<sub>''y''</sub>(''t''), ''B''<sub>0</sub> + Δ''B''<sub>''z''</sub>(t)) is the [[magnetic field]] experienced by the nuclei.
| |
| The ''z'' component of the magnetic field '''B''' is sometimes composed of two terms:
| |
| *one, ''B''<sub>0</sub>, is constant in time,
| |
| *the other one, Δ''B''<sub>''z''</sub>(t), may be time dependent. It is present in [[MRI|magnetic resonance imaging]] and helps with the spatial decoding of the NMR signal.
| |
| '''M'''(''t'') × '''B'''(''t'') is the [[cross product]] of these two vectors.
| |
| ''M''<sub>0</sub> is the steady state nuclear magnetization (that is, for example, when t → ∞); it is in the ''z'' direction.
| |
| | |
| ===Physical background===
| |
| With no relaxation (that is both ''T''<sub>1</sub> and ''T''<sub>2</sub> → ∞) the above equations simplify to:
| |
| | |
| :<math>\frac {d M_x(t)} {d t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _x</math>
| |
| :<math>\frac {d M_y(t)} {d t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _y</math>
| |
| :<math>\frac {d M_z(t)} {d t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _z</math>
| |
| | |
| or, in vector notation:
| |
| | |
| :<math>\frac {d \bold {M}(t)} {d t} = \gamma \bold {M} (t) \times \bold {B} (t) </math>
| |
| | |
| This is the equation for [[Larmor precession]] of the nuclear magnetization ''M'' in an external magnetic field '''B'''. | |
| | |
| The relaxation terms,
| |
| | |
| :<math>\left ( -\frac {M_x} {T_2}, -\frac {M_y} {T_2}, -\frac {M_z - M_0} {T_1} \right ) </math>
| |
| | |
| represent an established physical process of transverse and longitudinal relaxation of nuclear magnetization '''M'''.
| |
| | |
| ===Bloch equations are macroscopic equations===
| |
| These equations are not ''microscopic'': they do not describe the equation of motion of individual nuclear magnetic moments. These are governed and described by laws of [[quantum mechanics]].
| |
| | |
| Bloch equations are ''macroscopic'': they describe the equations of motion of macroscopic nuclear magnetization that can be obtained by summing up all nuclear magnetic moment in the sample.
| |
| | |
| ===Alternative forms of Bloch equations===
| |
| | |
| Opening the vector product brackets in the Bloch equations leads to:
| |
| | |
| :<math>\frac {d M_x(t)} {d t} = \gamma \left ( M_y (t) B_z (t) - M_z (t) B_y (t) \right ) - \frac {M_x(t)} {T_2}</math>
| |
| :<math>\frac {d M_y(t)} {d t} = \gamma \left ( M_z (t) B_x (t) - M_x (t) B_z (t) \right ) - \frac {M_y(t)} {T_2}</math>
| |
| :<math>\frac {d M_z(t)} {d t} = \gamma \left ( M_x (t) B_y (t) - M_y (t) B_x (t) \right ) - \frac {M_z(t) - M_0} {T_1}</math>
| |
| | |
| The above form is further simplified assuming
| |
| | |
| :<math>M_{xy} = M_x + iM_y \text{ and } B_{xy} = B_x + iB_y\, </math> <!--- do not delete "\,": It improved display of the formula in certain browsers.--->
| |
| | |
| where ''i'' = √(-1). After some algebra one obtains:
| |
| | |
| :<math>\frac {d M_{xy}(t)} {d t} = -i \gamma \left ( M_{xy} (t) B_z (t) - M_z (t) B_{xy} (t) \right ) -
| |
| \frac {M_{xy}} {T_2} </math>.
| |
| :<math>\frac {d M_z(t)} {d t} = i \frac{\gamma}{2} \left ( M_{xy} (t) \overline{B_{xy} (t)} -
| |
| \overline {M_{xy}} (t) B_{xy} (t) \right )
| |
| - \frac {M_z - M_0} {T_1}</math>
| |
| | |
| where
| |
| | |
| :<math>\overline {M_{xy}} = M_x - i M_y </math>.
| |
| | |
| is the complex conjugate of ''M<sub>xy</sub>''. The real and imaginary parts of ''M<sub>xy</sub>'' correspond to ''M<sub>x</sub>'' and ''M<sub>y</sub>'' respectively. | |
| ''M<sub>xy</sub>'' is sometimes called '''transverse nuclear magnetization'''.
| |
| | |
| ==Bloch equations in rotating frame of reference==
| |
| In a rotating frame of reference, it is easier to understand the behaviour of the nuclear magnetization '''M'''. This is the motivation:
| |
| | |
| ===Solution of Bloch equations with ''T''<sub>1</sub>, ''T''<sub>2</sub> → ∞===
| |
| Assume that:
| |
| *at ''t'' = 0 the transverse nuclear magnetization ''M''<sub>xy</sub>(0) experiences a constant magnetic field '''B'''(''t'') = (0, 0, ''B''<sub>0</sub>);
| |
| *''B''<sub>0</sub> is positive;
| |
| *there are no longitudinal and transverse relaxations (that is ''T''<sub>1</sub> and ''T''<sub>2</sub> → ∞).
| |
| | |
| Then the Bloch equations are simplified to:
| |
| | |
| :<math>\frac{d M_{xy}(t)} {d t} = -i \gamma M_{xy} (t) B_{0}</math>,
| |
| :<math>\frac{d M_z(t)} {d t} = 0 </math>.
| |
| | |
| These are two (not coupled) [[linear differential equations]]. Their solution is:
| |
| | |
| :<math>M_{xy}(t) = M_{xy} (0) e^{-i \gamma B_{0} t}</math>,
| |
| :<math>M_z(t) = M_0 = \text{const} \,</math>. <!-- do not delete "\,", it improved display of the formula on certain browsers. --->
| |
| | |
| Thus the transverse magnetization, ''M''<sub>xy</sub>, rotates around the ''z'' axis with [[angular frequency]] ω<sub>0</sub> = γ''B''<sub>0</sub> in clockwise direction (this is due to the negative sign in the exponent).
| |
| The longitudinal magnetization, ''M''<sub>z</sub> remains constant in time. This is also how the transverse magnetization appears to an observer in the '''laboratory frame of reference''' (that is to a '''stationary observer''').
| |
| | |
| ''M''<sub>xy</sub>(''t'') is translated in the following way into observable quantities of ''M''<sub>x</sub>(''t'') and ''M''<sub>y</sub>(''t''): Since
| |
| | |
| :<math>M_{xy}(t) = M_{xy} (0) e^{-i \gamma B_{z0} t} = M_{xy} (0) \left [ \cos (\omega _0 t) - i \sin (\omega_0 t) \right ]</math>
| |
| | |
| then
| |
| | |
| :<math>M_{x}(t) = \text {Re} \left (M_{xy} (t) \right ) = M_{xy} (0) \cos (\omega _0 t)</math>,
| |
| :<math>M_{y}(t) = \text {Im} \left (M_{xy} (t) \right ) = -M_{xy} (0) \sin (\omega _0 t)</math>,
| |
| | |
| where Re(''z'') and Im(''z'') are functions that return the real and imaginary part of complex number ''z''. In this calculation it was assumed that ''M''<sub>xy</sub>(0) is a real number.
| |
| | |
| ===Transformation to rotating frame of reference===
| |
| | |
| This is the conclusion of the previous section: in a constant magnetic field ''B''<sub>0</sub> along ''z'' axis the transverse magnetization ''M''<sub>xy</sub> rotates around this axis in clockwise direction with angular frequency ω<sub>0</sub>. If the observer were rotating around the same axis in clockwise direction with angular frequency Ω, ''M''<sub>xy</sub> it would appear to him rotating with angular frequency ω<sub>0</sub> - Ω. Specifically, if the observer were rotating around the same axis in
| |
| clockwise direction with angular frequency ω<sub>0</sub>, the transverse magnetization ''M''<sub>xy</sub> would appear to him stationary.
| |
| | |
| This can be expressed mathematically in the following way:
| |
| * Let (''x'', ''y'', ''z'') the Cartesian coordinate system of the '''laboratory''' (or '''stationary''') '''frame of reference''', and
| |
| * (''x''′, ''y''′, ''z''′) = (''x''′, ''y''′, ''z'') be a Cartesian coordinate system that is rotating around the ''z'' axis of the laboratory frame of reference with angular frequency Ω. This is called the '''rotating frame of reference'''. Physical variables in this frame of reference will be denoted by a prime.
| |
| | |
| Obviously:
| |
| | |
| :<math>M_z' (t) = M_z(t)\,</math>.<!--- do not delete "\,": it improves display of the formula in certain browsers. --->
| |
| | |
| What is ''M<sub>xy</sub>''′(''t'')? Expressing the argument at the beginning of this section in a mathematical way:
| |
| | |
| :<math>M_{xy}'(t) = e^{+i \Omega t} M_{xy}(t)\,</math>.<!--- do not delete "\,": it improves display of the formula in certain browsers. --->
| |
| | |
| ===Equation of motion of transverse magnetization in rotating frame of reference===
| |
| | |
| What is the equation of motion of ''M<sub>xy</sub>''′(''t'')?
| |
| | |
| :<math>\frac {d M_{xy}'(t)} {d t} = \frac {d \left ( M_{xy}(t) e^{+i \Omega t} \right )} {d t} =
| |
| e^{+i \Omega t} \frac {d M_{xy}(t) } {d t} + i \Omega e^{+i \Omega t} M_{xy} =
| |
| e^{+i \Omega t} \frac {d M_{xy}(t) } {d t} + i \Omega M_{xy}'
| |
| </math>
| |
| | |
| Substitute from the Bloch equation in laboratory frame of reference:
| |
| | |
| :<math>\begin{align} \frac {d M_{xy}'(t)} {d t} & = e^{+i \Omega t} \left [-i \gamma \left ( M_{xy} (t) B_z (t) - M_z (t) B_{xy} (t) \right ) -
| |
| \frac {M_{xy}} {T_2} \right ] + i \Omega M_{xy}' \\
| |
| | |
| & = \left [-i \gamma \left ( M_{xy} (t) e^{+i \Omega t} B_z (t) - M_z (t) B_{xy} (t) e^{+i \Omega t}\right ) -
| |
| \frac {M_{xy} e^{+i \Omega t} } {T_2} \right ] + i \Omega M_{xy}' \\
| |
| | |
| & = -i \gamma \left ( M_{xy}' (t) B_z' (t) - M_z' (t) B_{xy}' (t) \right ) + i \Omega M_{xy}' -
| |
| \frac {M_{xy}'} {T_2} \\
| |
| | |
| \end{align}
| |
| </math>
| |
| | |
| But by assumption in the previous section: ''B''<sub>z</sub>′(''t'') = ''B''<sub>z</sub>(''t'') = ''B''<sub>0</sub> + Δ''B''<sub>z</sub>(''t''). Substituting into the equation above:
| |
| | |
| :<math>\begin{align} \frac {d M_{xy}'(t)} {d t} & = -i \gamma \left ( M_{xy}' (t) (B_0 + \Delta B_z(t)) - M_z (t) B_{xy}' (t) \right ) + i \Omega M_{xy}' -
| |
| \frac {M_{xy}'} {T_2} \\
| |
| | |
| & = -i \gamma B_0 M_{xy}'(t) - i \gamma \Delta B_z(t) M_{xy}'(t) + i \gamma B_{xy}' (t) M_z (t)+ i \Omega M_{xy}' -
| |
| \frac {M_{xy}'} {T_2} \\
| |
| | |
| & = i (\Omega - \omega_0) M_{xy}'(t) - i \gamma \Delta B_z(t) M_{xy}'(t) + i \gamma B_{xy}' (t) M_z (t) -
| |
| \frac {M_{xy}'} {T_2} \\
| |
| | |
| \end{align}
| |
| </math> | |
| | |
| This is the meaning of terms on the right hand side of this equation:
| |
| * ''i'' (Ω - ω) ''M''<sub>xy</sub>′(''t'') is the Larmor term in the frame of reference rotating with angular frequency Ω. Note that it becomes zero when Ω = ω<sub>0</sub>.
| |
| * The -''i'' γ Δ''B''<sub>z</sub>(''t'') ''M''<sub>xy</sub>′(''t'') term describes the effect of magnetic field inhomogeneity (as expressed by Δ''B''<sub>z</sub>(''t'')) on the transverse nuclear magnetization; it is used to explain ''T''<sub>2</sub><sup>*</sup>. It is also the term that is behind [[MRI]]: it is generated by the gradient coil system.
| |
| * The ''i'' γ Δ''B''<sub>xy</sub>′(''t'') ''M''<sub>z</sub>(''t'') describes the effect of RF field (the Δ''B''<sub>xy</sub>′(''t'') factor) on nuclear magnetization. For an example see below.
| |
| * - ''M''<sub>xy</sub>′(''t'') / ''T''<sub>2</sub> describes the loss of coherency of transverse magnetization.
| |
| | |
| Similarly, the equation of motion of ''M<sub>z</sub>'' in the rotating frame of reference is:
| |
| :<math>\frac {d M_z(t)} {d t} = i \frac{\gamma}{2} \left ( M'_{xy} (t) \overline{B'_{xy} (t)} -
| |
| \overline {M'_{xy}} (t) B'_{xy} (t) \right )
| |
| - \frac {M_z - M_0} {T_1}</math>
| |
| | |
| ==Simple solutions of Bloch equations==
| |
| | |
| ===Relaxation of transverse nuclear magnetization ''M<sub>xy</sub>''===
| |
| Assume that:
| |
| * The nuclear magnetization is exposed to constant external magnetic field in the ''z'' direction ''B''<sub>z</sub>′(''t'') = ''B''<sub>z</sub>(''t'') = ''B''<sub>0</sub>. Thus ω<sub>0</sub> = γ''B''<sub>0</sub> and Δ''B''<sub>z</sub>(''t'') = 0.
| |
| * There is no RF, that is ''B''<sub>xy</sub>' = 0.
| |
| * The rotating frame of reference rotates with an angular frequency Ω = ω<sub>0</sub>.
| |
| | |
| Then in the rotating frame of reference, the equation of motion for the transverse nuclear magnetization, ''M''<sub>xy</sub>'(''t'') simplifies to:
| |
| | |
| :<math>\frac {d M_{xy}'(t)} {d t} = - \frac {M_{xy}'} {T_2}
| |
| </math>
| |
| | |
| This is a linear ordinary differential equation and its solution is
| |
| | |
| :<math> M_{xy}'(t) = M_{xy}'(0) e^{-t / T_2}</math>.
| |
| | |
| where ''M''<sub>xy</sub>'(0) is the transverse nuclear magnetization in the rotating frame at time ''t'' = 0. This is the initial condition for the differential equation. | |
| | |
| Note that when the rotating frame of reference rotates ''exactly'' at the Larmor frequency (this is the physical meaning of the above assumption Ω = ω<sub>0</sub>), the vector of transverse nuclear magnetization, ''M''<sub>xy</sub>(''t'') appears to be stationary.
| |
| | |
| ===90 and 180° RF pulses===
| |
| Assume that:
| |
| * Nuclear magnetization is exposed to constant external magnetic field in ''z'' direction ''B''<sub>z</sub>′(''t'') = ''B''<sub>z</sub>(''t'') = ''B''<sub>0</sub>. Thus ω<sub>0</sub> = γ''B''<sub>0</sub> and Δ''B''<sub>z</sub>(''t'') = 0.
| |
| * At ''t'' = 0 an RF pulse of constant amplitude and frequency ω<sub>0</sub> is applied. That is ''B'<sub>xy</sub>''(''t'') = ''B'<sub>xy</sub>'' is constant. Duration of this pulse is τ.
| |
| * The rotating frame of reference rotates with an angular frequency Ω = ω<sub>0</sub>.
| |
| * ''T''<sub>1</sub> and ''T''<sub>2</sub> → ∞. Practically this means that τ ≪ ''T''<sub>1</sub> and ''T''<sub>2</sub>.
| |
| | |
| Then for 0 ≤ ''t'' ≤ τ:
| |
| | |
| :<math>\begin{align} \frac {d M_{xy}'(t)} {d t} = i \gamma B_{xy}' M_z (t)
| |
| \end{align}
| |
| </math>
| |
| :<math>\frac {d M_z(t)} {d t} = i \frac{\gamma}{2} \left ( M'_{xy} (t) \overline{B'_{xy}} -
| |
| \overline {M'_{xy}} (t) B'_{xy} \right )
| |
| </math>
| |
| | |
| ===Relaxation of longitudinal nuclear magnetization ''M<sub>z</sub>''===
| |
| {{Empty section|date=February 2011}}
| |
| | |
| ==See also==
| |
| *The [[Bloch–Torrey equation]] is a generalization of the Bloch equations, which includes added terms due to the transfer of magnetization by diffusion.<ref>
| |
| {{cite journal
| |
| | doi=10.1103/PhysRev.104.563
| |
| | last=Torrey | first=H C
| |
| | title=Bloch Equations with Diffusion Terms
| |
| |year=1956
| |
| |journal=Physical Review
| |
| |volume=104
| |
| |issue=3
| |
| | pages=563–565
| |
| | bibcode=1956PhRv..104..563T}} (1956)</ref>
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==Further reading==
| |
| * [[Charles Kittel]], ''Introduction to Solid State Physics'', John Wiley & Sons, 8th edition (2004), ISBN 978-0-471-41526-8. Chapter 13 is on Magnetic Resonance.
| |
| | |
| {{DEFAULTSORT:Bloch Equations}}
| |
| [[Category:Nuclear magnetic resonance]]
| |
| [[Category:Magnetic resonance imaging]]
| |