# Beam Heating in the Accumulator

#### 

#### Valeri Lebedev

> FNAL, May 13, 2002.

# [Contents]{.underline}

-   Heating mechanisms

-   Review of gas scattering

-   Review of IBS theory

-   Comparison of the theory with experimental data

-   Conclusions

######## Heating Mechanisms 

1.  Scattering on residual gas

    -   Dominates at small beam current

2.  Intrabeam scattering (IBS)

    -   Presently is major heating mechanism at high current (\~100 mA)

3.  Non-linear resonances

    -   Tunes should not be to close to resonances. Harmful effects of
        resonances (up to 18-th order) have been observed

4.  Instabilities

    -   An instability driven by ions originated from the residual gas
        heats the beam at currents above \~20 mA

5.  Noise on electrodes

    -   We do not have any evidence of beam heating due to noise on
        electrodes

# [Scattering on the residual gas]{.underline}

####### Beam lifetime 

> where
> $\overline{\beta_{x,y}} = \frac{1}{C}\int_{}^{}{\beta_{x,y}\text{ds}}$

-   Total beam lime time measured with low intensity protons \~1400 hour

-   Coulomb scattering \~2800 hour

-   Nuclear absorption \~2700 hour

[Emittance growth rate is closely related to the beam life
time]{.underline}

$\frac{{d\varepsilon}_{x,y}}{\text{dt}} = \frac{2\pi\text{cr}_{p^{2}}}{\gamma^{2}\beta^{3}}\left( \sum_{i}^{}{n_{i}Z_{i^{2}}} \right)\overline{\beta_{x,y}}$

[Presuming]{.underline} that the relative gas composition is
proportional to a single point measurement performed 2 years ago we
obtain the vacuum and the emittance growth rates

> $\begin{array}{r}
> \frac{{d\varepsilon}_{x}}{\text{dt}} = 0\text{.}\text{088}\ \text{mm}\ \text{mrad/hour} \\
> \frac{{d\varepsilon}_{y}}{\text{dt}} = 0\text{.}\text{052}\ \text{mm}\ \text{mrad/hour}
> \end{array}$

-   Measured minimum vertical emittance growth time coincides well with
    the calculations

-   The horizontal one is usually above predictions and grows fast with
    beam current

Twiss parameters for the old and new accumulator lattices

![](media/image5.wmf)

![](media/image6.wmf)

# [Intrabeam scattering theory]{.underline}

![](media/image7.wmf)

If in the beam frame the longitudinal momentum spread is much less than
the transverse one the IBS formulas can be significantly simplified

#####  

##### The IBS growth rate for longitudinal degree of freedom

-   IBS transfers the energy from the transverse degrees of freedom to
    the longitudinal one and the growth rate can be approximated by the
    following formula

$\frac{d}{\text{dt}}\left( \theta_{{||}^{2}} \right) \equiv \frac{d}{\text{dt}}\left( \frac{\overline{p_{{||}^{2}}}}{p} \right) = \sqrt{\frac{\pi}{2}}\frac{e^{4}\text{NL}_{C}}{m_{p^{2}}c^{3}\gamma^{3}\beta^{3}C}\left\langle \frac{\Xi_{||}\left( \theta_{x},\theta_{y} \right)}{\sigma_{x}\sigma_{y}\sqrt{\theta_{x^{2}} + \theta_{y^{2}}}} \right\rangle_{s}$
,

where averaging is performed along the beam orbit,

> $\Xi_{||}(x,y) \approx 1 + \frac{\sqrt{2}}{\pi}\text{ln}\left( \frac{x^{2} + y^{2}}{2\text{xy}} \right) - 0\text{.}\text{055}\left( \frac{x^{2} - y^{2}}{x^{2} + y^{2}} \right)^{2}$,
>
> *L~c~* - is the Coulomb logarithm,
>
> *C* -- is the ring circumference,
>
> the beam sizes and local angular spreads along the ring are determined
> by
>
> $\sigma_{x} = \sqrt{\varepsilon_{x}\beta_{x} + D^{2}\theta_{{||}^{2}}}$
>
> $\sigma_{y} = \sqrt{\varepsilon_{y}\beta_{y}}$
>
> $\theta_{x} = \sqrt{\frac{\varepsilon_{x}}{\beta_{x}}\left( 1 + \frac{\left( D^{'}\beta_{x} + \alpha_{x}D_{x} \right)^{2}\theta_{{||}^{2}}}{\varepsilon_{x}\beta_{x} + D^{2}\theta_{{||}^{2}}} \right)}$
>
> $\theta_{y} = \sqrt{\varepsilon_{y}/\beta_{y}}$

and

$\gamma = \frac{1}{\sqrt{1 - \beta^{2}}}$

##### The IBS growth rate for transverse degree of freedom 

-   The heating of the longitudinal degree of freedom causes cooling for
    both transverse degrees of freedom;

-   Additional mechanism heats the horizontal degree of freedom

    -   At regions with non-zero dispersion, changes in the longitudinal
        momentum change the particles reference orbits, which
        additionally excites the horizontal betatron motion,

$\frac{{d\varepsilon}_{x}}{\text{dt}} = \frac{1}{2}\left\langle A_{x}\frac{{d\theta}_{{||}^{2}}}{\text{dt}} \right\rangle_{s}$
,

where
$A_{x} = \frac{D^{2} + \left( D^{'}\beta_{x} + \alpha_{x}D \right)^{2}}{\beta_{x}}$

Coefficient ½ is related to the fact that the beam is unbunched

-   Finally, one can write for the emittance growth rates

$$\frac{{d\varepsilon}_{x,y}}{\text{dt}} = \frac{\sqrt{2\pi}e^{4}\text{NL}_{C}}{8m_{p^{2}}c^{3}\gamma^{3}\beta^{3}C}\left\langle \frac{1}{\sigma_{x}\sigma_{y}\sqrt{\theta_{x^{2}} + \theta_{y^{2}}}}\begin{bmatrix}
2A_{x}\Xi_{||}\left( \theta_{x},\theta_{y} \right) - \frac{\beta_{x}}{\gamma^{2}}\Xi_{}\left( \theta_{x},\theta_{y} \right) \\
 - \frac{\beta_{y}}{\gamma^{2}}\Xi_{}\left( \theta_{y},\theta_{x} \right)
\end{bmatrix} \right\rangle_{s}\ \text{.}$$

> where

$\ \Xi_{}(x,y) \approx 1 + \frac{2\sqrt{2}}{\pi}\text{ln}\left( \frac{\sqrt{3x^{2} + y^{2}}}{2y^{2}}x \right) + \frac{0\text{.}\text{5429}\text{ln}(\frac{y}{x})}{\sqrt{1 + \text{ln}^{2}\left( \frac{y}{x} \right)}}$
.

**[Comparison of the old and new accumulator lattices from the
horizontal emittance growth rate due to IBS]{.underline}**

![](media/image8.wmf)

> Old lattice

![](media/image9.wmf)

> New lattice

# Comparison with experiment 

[Protons]{.underline}, I.~beam~=13.5 mA, Lifetime = 1400 hour, beam is
scraped from 100 mA, Apr.30.02

![](media/image10.wmf)

[Antiprotons]{.underline}, I.~beam~=12.1 mA, Lifetime=900 hour, beam is
scraped from 24 mA,Apr.30.02

![](media/image11.wmf)

-   Green line shows the emittance growth from the gas scattering only

-   X-Y coupling causes additional growth for vertical emittance, κ =
    0.04

-   Beam scraping makes distribution non-Gaussian. It affects the growth
    rates but we do not have a clear answer how much

[Protons]{.underline}, I.~beam~=48.2 mA, Lifetime = 800 hour, Mar.26.02

![](media/image12.wmf)

[Antiprotons]{.underline}, I.~beam~=50.8 mA, Lifetime = 650 hour,
Mar.27.02

![](media/image13.wmf)

-   Green line shows the emittance growth from the gas scattering only

-   X-Y coupling causes additional growth for vertical emittance, κ =
    0.04

-   Presence of ions in the antiproton beam causes decrease of beam
    lifetime due to worsening of the effective vacuum and drives
    additional emittance growth due to ion instability

[Antiprotons]{.underline}, I.~beam~=100 mA, Lifetime = 550 hour,
Mar.27.02

![](media/image14.wmf)

**[Remarks about measurements and theory]{.underline}**

-   Both IBS and residual scattering are taken into account in the
    theoretical model

    -   Both models have logarithmic accuracy with expected accuracy of
        about 10%

-   IBS theoretical model is build for the Gaussian distribution
    function

    -   It is not quite accurate for the proton beam and not always true
        for the antiproton beam

    -   That brings additional uncertainty but it is difficult to say
        how much

-   Longitudinal momentum spread for all presented date sets was fudged
    by factor of 1.15 to get coincidence between theory and measurements
    for the momentum spread growth rate.

    -   Momentum spreads reported by A:SIGMAP and A:FRWDTH is different
        by \~40% and the are not always proportional to each other in
        the both cases of proton and antiproton beams

    -   We need to investigate the reason of this discrepancies

**[Conclusions]{.underline}**

-   Vacuum lifetime in the Accumulator is about 1400 hours

    -   Single intrabeam scattering (Touschek effect) affects the
        lifetime for both proton and antiproton beams

    -   Ions stored in the antiproton beam cause additional drop of the
        antiproton lifetime

-   In the absence of ion instability summed effects of the residual gas
    scattering and the IBS describe well the observed emittance growth
    rates

-   In the case of antiproton beam the ions cause the two stream
    (ion-antiproton) instability above \~20 mA beam current.

    -   The instability can be monitored at lowest (1-Q) betatron
        sideband.

    -   This instability causes additional emittance growth comparable
        to the growth rate due to IBS

-   We do not have any indications of the external noise causing the
    emittance growth

-   We plan to change operation of the accumulator.

    -   After stacking is completed the present optics (optimized for
        stacking) will be dynamically changed to the "low IBS optics".

    -   After about 30 min cooling the beam will be ready for extraction

-   This optics change will bring

A decrease of the horizontal heating by more than two times, which,
consequently, yields smaller beam emittances. Together with the core
cooling upgrade we expect the emittance decrease by

> $$\left( \frac{G_{\text{new}}}{G_{\text{old}}}\frac{H_{\text{old}}}{H_{\text{new}}} \right)^{\frac{2}{5}} \approx (3)^{\frac{2}{5}} \cdot \left( 0\text{.}4 + \frac{0\text{.}6}{2\text{.}5} \right)^{\frac{- 2}{5}} \approx 1\text{.}8$$

-   Decreasing of beta-functions in the extraction aria additionally
    decreases beam size of the extracted beam and will prevent scraping
    at the beam extraction. Presently, we lose 5-10% at the beginning of
    the accumulator-to-MI beam line

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-   Ion instability is expected to be the major heating mechanism after
    new optics is introduced