Dark Matter Detection - Revision history

Cosmo All: /* Problem 11 */

2012-10-04T10:43:08Z

‎Problem 11

Cosmo All: /* Problem 10 */

2012-10-04T10:42:40Z

‎Problem 10

Cosmo All: /* Problem 7 */

2012-10-04T10:42:05Z

‎Problem 7

Cosmo All: /* Problem 7 */

2012-10-04T10:41:09Z

‎Problem 7

Cosmo All: /* Problem 5 */

2012-10-04T10:39:24Z

‎Problem 5

Cosmo All: /* Problem 4 */

2012-10-04T10:39:01Z

‎Problem 4

Cosmo All: /* Problem 2 */

2012-10-04T10:38:25Z

‎Problem 2

Cosmo All at 10:37, 4 October 2012

2012-10-04T10:37:55Z

Cosmo All: /* Problem 9 */

2012-10-04T10:37:09Z

‎Problem 9

Cosmo All at 10:36, 4 October 2012

2012-10-04T10:36:55Z

Show changes

@@ Line 273: / Line 273: @@
 f_{p,n}  = \sum\limits_{q = u,d,s} {f_{T_q }^{(p,n)} a_q
 \frac{m_{p,n}}{m_q} + \frac{2}{27}f_{TG}^{(p,n)}
-\sum\limits_{q = c,b,t} {a_q \frac{{m_{p,n} }}{{m_q }}} },
+\sum\limits_{q = c,b,t} {a_q \frac{m_{p,n}}{m_q}} },
 \]
 where $a_q $ are the WIMP-quark coupling constants and the quantity $f_{T_q
@@ Line 284: / Line 284: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 12 ===
 How will the number of counts for the elastic events detector be affected by transition to heavier target nuclei at fixed detector mass?

@@ Line 243: / Line 243: @@
 The wavelength of the WIMP is
 \[
-\lambda  = \frac{h}{p} = 2\pi \frac{\hbar }{{m_p c}}\frac{m_p}{m_\chi}\frac{c}{v} \simeq 2\pi  \cdot 2 \cdot 10^{ - 14}
+\lambda  = \frac{h}{p} = 2\pi \frac{\hbar }{m_p c}\frac{m_p}{m_\chi}\frac{c}{v} \simeq 2\pi  \cdot 2 \cdot 10^{ - 14}
 \mbox{cm} \cdot 10^{ - 2}  \cdot 10^3  \simeq 1.3 \cdot 10^{ -12} \mbox{ cm}
 \]
@@ Line 254: / Line 254: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 11 ===
 Find the total WIMP-nucleus cross-section determining the elastic events' count rate.

@@ Line 168: / Line 168: @@
 \[
 \begin{array}{l}
-  \left\langle v \right\rangle _{(1)}  = \int_0^\infty  {vf_1 (v)dv = {\frac{{\sqrt 2 }}{\pi }} } v_0 ; \\
+  \left\langle v \right\rangle _{(1)}  = \int_0^\infty  {vf_1 (v)dv = \frac{\sqrt 2}{\pi } } v_0 ; \\
   \left\langle {v^2 } \right\rangle _{(1)}^{1/2}  = \left( {\int_0^\infty  {v^2 f_1 (v)dv} } \right)^{1/2}  = \sqrt {\frac{3}{2}}
   v_0.
@@ Line 185: / Line 185: @@
 Finally one obtains for total events number of the elastic scattering registered by the detector unit mass per unit time the following
 \[
-R = \int_{Q_{th} }^\infty  {\left( {\frac{{dR}}{{dQ}}} \right)dQ},
+R = \int_{Q_{th} }^\infty  {\left( \frac{dR}{dQ} \right)dQ},
 \]
 where $Q_{th} $ is the threshold value of the detector energy.

@@ Line 142: / Line 142: @@
 $F(q)$
 \[
-d\sigma  = \frac{1}{{v^2 }}\frac{{\sigma _0 }}{{4m_r^2 }}F^2
+d\sigma  = \frac{1}{v^2}\frac{\sigma _0}{4m_r^2}F^2
 (q)dq^2,
 \]
-where $\sigma _0 $ is the total cross-section, \( m_r  = m_\chi  m_N /(m_\chi
+where $\sigma _0 $ is the total cross-section, \( m_r  = m_\chi  m_N /(m_\chi+ m_N ) \) is the reduced mass, $q = \sqrt {2m_N Q}$ is the transmitted momentum. Thus the differential counting rate per unit mass of the detector can be presented in the following form:
 + m_N ) \) is the reduced mass, $q = \sqrt {2m_N Q}$ is the transmitted momentum. Thus the differential counting rate per unit mass of the detector can be presented in the following form:
 $$
 \frac{dR}{dQ} = \frac{\rho _0 }{m_\chi  m_N }\int vf_1 (v)d\sigma dv
@@ Line 199: / Line 198: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 8 ===
 Reconstruct the one-dimensional WIMPs' velocities distribution $f_1(v)$  (see the previous problem) basing on the given count rate of the elastic events.

@@ Line 94: / Line 94: @@
 and thus
 \[
-v_{\min }  = \frac{{\sqrt {2m_N Q} }}{{2m_r }}.
+v_{\min }  = \frac{\sqrt {2m_N Q}}{2m_r}.
 \]</p>
    </div>
@@ Line 103: / Line 103: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 6 ===
 Estimate the count rate for the detector registering elastic events of WIMPs.

@@ Line 71: / Line 71: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">Consider the recoil energy as a function of the nucleus mass at given WIMP mass (see problem \ref{11_20}), and show that it has maximum at $m_N  = m_\chi.$</p>
+     <p style="text-align: left;">Consider the recoil energy as a function of the nucleus mass at given WIMP mass (see [[#11_20|problem]] ), and show that it has maximum at $m_N  = m_\chi.$</p>
    </div>
 </div></div>
@@ Line 79: / Line 79: @@
 <div id="DM25"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 5 ===
 Determine the minimum velocity of WIMPs that can transmit energy $Q$ to a nucleus with mass $m_N$.

@@ Line 32: / Line 32: @@
 ) Motion of the recoil nucleus in the lattice can induce creation of the vibrational phonons.<br/>
 ) The energy converted into the phonons can eventually thermalize and lead to minor increase in the detector temperature.
-\end{description}
+<br/><br/>
 Therefore the recoil nucleus can induce three kinds of signals:<br/>
 ) ionization (charge);<br/>
@@ Line 44: / Line 44: @@
 <div id="11_20"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 3 ===
 Show that WIMPs of mass $\sim100GeV$ being elastically scattered on Xenon nuclei with mass $\sim130GeV$ lead to energy recoil $\le40keV$.

@@ Line 13: / Line 13: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">Using \( \rho _0 \approx 0.3\mbox{\it GeV/cm}^3 \) under assumption $m_\chi\approx 100\mbox{\it
+     <p style="text-align: left;">Using \( \rho _0 \approx 0.3\mbox{GeV/cm}^3 \) under assumption $m_\chi\approx 100\mbox{GeV}$, one obtains for the WIMP number density the following expression \( n = \rho _0 /m_\chi
-GeV}$, one obtains for the WIMP number density the following expression \( n = \rho _0 /m_\chi
+\approx 3 \times 10^{ - 3} \mbox{cm}^{ - 3} \). For \(
-\approx 3 \times 10^{ - 3} \mbox{\it cm}^{ - 3} \). For \(
+\left\langle v \right\rangle  \simeq 250\mbox{km/s} \) the flow equals \(J = n\left\langle v \right\rangle  \sim 10^5 \mbox{cm}^{ - 2} \mbox{s}^{ - 1}\)</p>
-\left\langle v \right\rangle  \simeq 250\mbox{\it km/s} \) the flow equals \(J = n\left\langle v \right\rangle  \sim 10^5 \mbox{\it cm}^{ - 2} \mbox{\it s}^{ - 1}\)</p>
    </div>
 </div></div>
@@ Line 111: / Line 110: @@
 \simeq {\rm O}\left( {10^{ - 2} } \right)$ is the weak interaction constant. Therefore $\sigma \approx 10^{ - 9} \mbox{GeV}^{ -
 } \approx 1\mbox{pb}$. (Recall that $1\mbox{GeV}^{ - 1}
-\approx 2 \times 10^{ - 14} \mbox{\it cm}$.)
+\approx 2 \times 10^{ - 14} \mbox{cm}$.)
 <br/>
 Note that the experimenters prefer deal with the
 WIMP-nucleon cross-section rather than WIMP-nucleus one. The former is of order of
-$\sigma_{\chi n} \approx 10^{ - 8} \mbox{pb}.$  The very rough estimate of the detector counting rate gives $R \sim J\sigma /m_N  \approx 10 \mbox{events}\,\mbox{\it kg}^{ - 1} \mbox{year}^{ - 1} $
+$\sigma_{\chi n} \approx 10^{ - 8} \mbox{pb}.$  The very rough estimate of the detector counting rate gives $R \sim J\sigma /m_N  \approx 10 \mbox{events}\,\mbox{kg}^{ - 1} \mbox{year}^{ - 1} $
 (for a target with nuclei containing about $100$ nucleons each, i.e. $m_N
 \sim 100\mbox{GeV} = 1.77 \times 10^{ - 27} \mbox{kg}$ ). It means that every day several WIMP (precise number depends on the detector material) will hit the atomic nuclei which the detector is composed of. Of course the above given speculations are nothing ore than an estimate. Accurate calculations should take into account the interaction of WIMP with quarks and gluons, then its consequent translation into the interaction with nucleons and at last to the interaction with the nuclei.</p>

@@ Line 211: / Line 211: @@
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 9 ===
- Construct a model-independent scheme for WIMPs' mass determination using the results of WIMPs' elastic scattering events detection for two or more sets of experimental data with detectors of different composition.
+Construct a model-independent scheme for WIMPs' mass determination using the results of WIMPs' elastic scattering events detection for two or more sets of experimental data with detectors of different composition.
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
@@ Line 224: / Line 224: @@
 <div id="DM27"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 10 ===
 Show that if a WIMP has the mass of the order $100GeV$ and velocity of the order of $300km/s$, then it coherently interacts with nucleons of a detector nuclei.